<%BANNER%>

Wave Loading on Bridge Superstructures

Permanent Link: http://ufdc.ufl.edu/UFE0042461/00001

Material Information

Title: Wave Loading on Bridge Superstructures
Physical Description: 1 online resource (385 p.)
Language: english
Creator: Marin, Justin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: bridges,forces,loading,waves
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Coastal and Oceanographic Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A theoretical, numerical, and experimental study of vertical and horizontal forces due to non-breaking, monochromatic water waves propagating past select bridge superstructure spans is presented. A theoretical model is developed as the sum of the individual forcing components, drag, inertia, buoyancy, and slamming for a finite thickness slab span and for slab spans with beams. Forces were categorized by forcing frequency into lower frequency quasi-static forces and higher frequency slamming forces. A computer model is developed to evaluate the theoretical equations and assess the sensitivities of certain parameters. Stream function theory is used to calculate wave particle kinematics for the numerical model. Physical model tests were conducted for a range of fluid and structure parameters based on prototype cases of Florida coastal bridge structures and hurricane-generated waves. The experimental data was used to evaluate the drag and inertia coefficients in the theoretical model. Once calibrated the computer model was run for a wide range of structure, and wave conditions and the results used to develop parametric equations for use in design. The accuracy of the model was checked using data from two field sites where major bridges were exposed to significant storm surge and wave loading during a hurricane. The model performed well for both test cases.
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.
Statement of Responsibility: by Justin Marin.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Sheppard, DonaldM.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042461:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042461/00001

Material Information

Title: Wave Loading on Bridge Superstructures
Physical Description: 1 online resource (385 p.)
Language: english
Creator: Marin, Justin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: bridges,forces,loading,waves
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Coastal and Oceanographic Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A theoretical, numerical, and experimental study of vertical and horizontal forces due to non-breaking, monochromatic water waves propagating past select bridge superstructure spans is presented. A theoretical model is developed as the sum of the individual forcing components, drag, inertia, buoyancy, and slamming for a finite thickness slab span and for slab spans with beams. Forces were categorized by forcing frequency into lower frequency quasi-static forces and higher frequency slamming forces. A computer model is developed to evaluate the theoretical equations and assess the sensitivities of certain parameters. Stream function theory is used to calculate wave particle kinematics for the numerical model. Physical model tests were conducted for a range of fluid and structure parameters based on prototype cases of Florida coastal bridge structures and hurricane-generated waves. The experimental data was used to evaluate the drag and inertia coefficients in the theoretical model. Once calibrated the computer model was run for a wide range of structure, and wave conditions and the results used to develop parametric equations for use in design. The accuracy of the model was checked using data from two field sites where major bridges were exposed to significant storm surge and wave loading during a hurricane. The model performed well for both test cases.
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.
Statement of Responsibility: by Justin Marin.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Sheppard, DonaldM.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042461:00001


This item has the following downloads:


Full Text

PAGE 1

1 WAVE LOADING ON BRIDGE SUPERSTRUCTURES By JUSTIN M. MARIN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010

PAGE 2

2 2010 Justin M. Marin

PAGE 3

3 For Mom, Dad and the Bean Cheers

PAGE 4

4 ACKNOWLEDGMENTS The following is a work that could not have been completed without the efforts and support of many. The extensive lab work completed on the topic would not have been possible without the (often unpaid) efforts of Jim Joiner, Vic Adams, Sidney Schofield, and Richard Booze of the University of Florida Coastal Engineering Lab, my home away from home. I thank them for all the time, tools, jokes, patience, and advice on how not to be a professor they have lent. I would also like to thank the entire staff at OEA Inc., especially Phil Dompe, for suffering silently through 1 22 versions of the model and counting. For the funding and backing, I thank Ric Renna and the Florida DOT. Thanks also go to my committee members, Dr. Robert J. Thieke, Dr. Arnoldo Valle-Levinson, and Dr. Bruce Carroll. For the work, experience, and the understanding, I thank my advisor, Dr. D. Max Sheppard, who put up with the odd hours and the army of one attitude that drove him insane. The man gives laid back a new name. Lastly, I thank my parents, Michelle and Juice, and my better (looking) half, Tina, who have loved and backed me n o matter how long I postpone the real world. To everyone involved, Im buying.

PAGE 5

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...............................................................................................................4 LIST OF TABLES ...........................................................................................................................9 LIST OF FIGURES .......................................................................................................................10 ABSTRACT ...................................................................................................................................20 CHAPTER 1 INTRODUCTION ..................................................................................................................21 Bridge Failures Attributed to Wave Loading .........................................................................21 Hurricane Camille ...........................................................................................................22 Hurricane Frederic ...........................................................................................................22 Hurricane Ivan .................................................................................................................22 Hurricane Katrina ............................................................................................................23 Motivation ...............................................................................................................................24 Economics of Bridge Failure and Design ........................................................................24 Frequency of Occurrence ................................................................................................26 Storm surge, wind setup, tides, and wave height .....................................................27 Effects of phasing and geography ............................................................................28 Outside trends contributing to future incidents ........................................................29 Existing Methods for Design ...........................................................................................30 Introduction to the Problem ....................................................................................................31 The Structure ...................................................................................................................31 Bridge superstructure types ......................................................................................32 Bridge support structure types ..................................................................................33 The Wave .........................................................................................................................33 Wave/structure Interaction ..............................................................................................35 The generalized force curve and its components .....................................................35 Clearance heights .....................................................................................................36 Air entrapment ..........................................................................................................37 Relative size ratios ...................................................................................................38 Added mass and change in added mass ...................................................................38 Organization ...........................................................................................................................39 2 LITERATURE REVIEW .......................................................................................................54 Previous Work on Bridge Superstructures .............................................................................54 Denson (1978) .................................................................................................................55 Denson (1980) .................................................................................................................56 U.S. Army Corp of Engineers (2008) ..............................................................................57 Morison et al. (1950) .......................................................................................................58

PAGE 6

6 Previous Work on Offshore Platforms and Decks ..................................................................58 Kaplan (1992) ..................................................................................................................59 Kaplan et al. (1995) .........................................................................................................60 Suchithra and Koola (1995) .............................................................................................61 Bea et al. (1999) ..............................................................................................................62 Baarholm and Faltinsen (2004) .......................................................................................64 Previous Work on Offshore Jetties .........................................................................................65 Overbeek and Klabbers (2001) ........................................................................................65 Tirindelli et al. (2002) ......................................................................................................66 McConnell et al. (2003) ...................................................................................................67 Cuomo et al. (2003) .........................................................................................................68 Tirindelli et al. (2003) ......................................................................................................69 Cuomo et al. (2007) .........................................................................................................70 Da Costa and Scott (1988) ...............................................................................................71 Sulisz et al. (2005) ...........................................................................................................71 Previous Work on Flat Plates and Docks ...............................................................................72 El Ghamry (1963) ............................................................................................................73 Wang (1970) ....................................................................................................................73 French (1970) ..................................................................................................................74 French (1979) ..................................................................................................................75 Isaacson and Bhat (1996) ................................................................................................77 Current Work on Bridge Superstructures ...............................................................................77 Schumacher et al. (2008) .................................................................................................78 Douglas et al. (2006) .......................................................................................................78 Numerical Methods ................................................................................................................79 Foundation for This Work ......................................................................................................79 Marin (2009) ....................................................................................................................79 3 THE THEORETICAL QUASI STATIC FORCE ..................................................................83 Model Considerations .............................................................................................................83 Model Assumptions .........................................................................................................83 Dimensional Analysis ......................................................................................................85 Dimensionless buoyancy force .................................................................................86 Dime nsionless inertia and drag forces .....................................................................87 Dimensionless slamming force ................................................................................88 The Buoyancy Force ...............................................................................................................88 The Drag and Inertia Forces ...................................................................................................89 The Drag Force ................................................................................................................90 Partially inundated horizontal drag scenarios ..........................................................91 Partially inundated vertical drag scenarios ...............................................................92 The Inertia Force .............................................................................................................93 Time dependent mass ...............................................................................................94 The inertial and mass rate force equations ...............................................................96 The effective mass and mass rate equations .............................................................98 Adjustments to Theory for Beam and Slab Spans ................................................................104 Updated Model Assumptions ........................................................................................104

PAGE 7

7 Updated Dimensional Analysis .....................................................................................105 Updated Force Equations ..............................................................................................106 4 THEORETICAL SLAMMING FORCE ..............................................................................116 Slamming Observations ........................................................................................................116 Horizontal Slamming .....................................................................................................116 Vertical Slamming .........................................................................................................118 On the effect of clearance .......................................................................................119 On t he effect of wave shape ...................................................................................119 On the effect of fluid buildup .................................................................................120 A Model for Slamming .........................................................................................................121 A Note on FluidStructure Interaction Rate ..................................................................124 Beam and Slab Slamming .....................................................................................................125 5 THE NUMERICAL MODEL ...............................................................................................139 Overview ...............................................................................................................................139 The Elemental Grids .............................................................................................................139 The Structure Grid .........................................................................................................140 The Fluid Grid ...............................................................................................................140 The Required Inputs .............................................................................................................141 Structural Parameters .....................................................................................................141 Fluid Parameters ............................................................................................................142 Evaluative Parameters ...................................................................................................142 Considerations for Evaluating the Theoretical Equations ....................................................143 Trapped Air ...................................................................................................................143 Added Mass Distribut ion ...............................................................................................146 Change in Effective Mass ..............................................................................................147 Stages of inundation and mass growth ...................................................................147 Negative change in effective mass .........................................................................148 Periods of cutoff .....................................................................................................149 6 PHYSIC AL MODEL TESTS ...............................................................................................159 Model Considerations ...........................................................................................................159 Frequency Effects ..........................................................................................................159 Scale Effects ..................................................................................................................163 Physical Models ....................................................................................................................163 The Support Structure ....................................................................................................164 Instrumentation ..............................................................................................................165 Model Setups .................................................................................................................166 Setups for investigation of the general wave loading ............................................166 Setups for investigation of slamming .....................................................................168 Physical Model Tests ............................................................................................................169 Data Processing ....................................................................................................................169 Spectral Analysis ...........................................................................................................170

PAGE 8

8 Signal Filtering ..............................................................................................................171 Assumption Verification ...............................................................................................171 Significant Parameters Extracted from the Data ...........................................................172 7 RESULTS AND ANALYSIS ...............................................................................................186 Coefficients ...........................................................................................................................188 Slab Span Structures ......................................................................................................188 Vertical inertia coefficients ....................................................................................189 Horizontal inertia coefficients ................................................................................190 Drag coefficients ....................................................................................................191 Buoyancy coefficient ..............................................................................................192 Beam and S lab Span Structures .....................................................................................193 Vertical inertia coefficients ....................................................................................194 Horizontal inertia coefficients ................................................................................195 Drag coefficients ....................................................................................................195 Buoyancy coeff icient ..............................................................................................195 Additional Adjustments .................................................................................................196 Overhang effects on vertical loading .....................................................................197 Overhang effects on horizontal loading .................................................................198 Comparisons .........................................................................................................................198 Physical Model Data ......................................................................................................199 Prototype Bridges ..........................................................................................................199 Parametric Equations ............................................................................................................200 Limitations ............................................................................................................................201 8 CONCLUSIONS AND RECOMMENDATIONS ...............................................................207 Conclusions ...........................................................................................................................207 Recommendations .................................................................................................................209 APPENDIX A COMPARISONS BETWEEN PHYSICAL AND NUMERICAL MODELS .....................211 B PHYSICAL MODEL DATA QUASI STATIC FORCES ................................................332 C PHYSICAL MODEL DATA SLAMMING FORCES ......................................................369 LIST OF REFERENCES .............................................................................................................382 BIOGRAPHICAL SKETCH .......................................................................................................385

PAGE 9

9 LIST OF TABLES Table page 6-1 Test breakdown by target and count. ...............................................................................185 6-2 Range of fluid variable values covered in the physical model testing. ............................185 B-1 Structure and fluid parameters for all physical model tests. ............................................333 B-2 Significant force values for all physical model tests. ......................................................351 C-1 Structure and fluid parameters for all physical model tests. ............................................370 C-2 Significant force values for all physical model tests. ......................................................376

PAGE 10

10 LIST OF FIGURES Figure page 1-1 Dauphin Island Causeway spans removed by Hurricane Frederic (longshot). ..................40 1-2 Dauphin Island Causeway spans removed by Hurricane Frederic (closeup). ...................40 1-3 I-10 Bridge Escambia Bay damage from Hurricane Ivan. .................................................41 1-4 I-10 Bridge Escambia Bay spans removed by Hurricane Ivan. .........................................41 1-5 I-10 Bridge Escambia Bay spans displaced by Hurricane Ivan. ........................................42 1-6 I-10 Bridge Escambia Bay pile bents damaged by Hurricane Ivan. ..................................42 1-7 I-10 Bridge Escambia Bay tie-downs damaged by Hurricane Ivan. ..................................43 1-8 I-10 Mobile Bay onramp spans displaced by Hurricane Katrina. ......................................43 1-9 I10 Bridge Lake Pontchartrain damage from Hurricane Katrina. ....................................44 1-10 I-10 Bridge Lake Pontchar train spans removed by Hurricane Katrina. .............................44 1-11 I-10 Bridge Lake Pontchartrain support structures damaged by Hurricane Katrina. ........45 1-12 I10 Bridge Lake Pontchartrain spans displaced by Hurricane Katrina. ...........................45 1-13 US 90 Bridge Biloxi Bay damaged by Hurricane Katrina. ................................................46 1-14 US 90 Bridge St. Louis Bay damaged by Hurricane Katrina. ...........................................46 1-15 Effect of bridge location and wind direction on wind setup. .............................................47 1-16 Effect of storm path and wind alignment on wind setup. ..................................................47 1-17 Common bridge superstructure types in Florida.. ..............................................................48 1-18 Commo n pile-support setups for bridge superstructures. ..................................................49 1-19 Typical vertical force curve on a sub aerial, flat bottomed structure. ...............................50 1-20 Typical vertical force curve on a submerged, flat -bottomed structure. .............................50 1-21 Typical vertical force curve on a sub aerial girdered structure. .........................................51 1-22 Water particle kinematics in a progressive wave. ..............................................................52 1-23 Water particle velocities distribution over different structures. .........................................53

PAGE 11

11 3-1 Slab span overview. .........................................................................................................108 3-2 Partially inundated span height. .......................................................................................108 3-3 Inundation of leading edge only in horizontal drag flow. ................................................109 3-4 Inundation of midsection only in horizontal drag flow. ..................................................109 3-5 Inundation of trailing edge only in horizontal drag flow. ................................................109 3-6 Partial deflection by leading edge in vertical drag flow. .................................................110 3-7 Complete deflection by midesction in vertical drag flow. ...............................................110 3-8 Partial deflection by trailing edge in vertical drag flow. .................................................110 3-9 Springmass-dashpot system. ...........................................................................................111 3-10 Spheres in an unsteady flow. ...........................................................................................111 3-11 Existing added mass equations predict two separate masses for the same object rotated 90 in the flow. .....................................................................................................112 3-12 Existing added mass equations predict two separate masses for a object and the same object made up of three separate objects. ........................................................................112 3-13 Orientation of rectangle in flow for added mass equation from Sarpkaya (1981). ..........112 3-14 Comparison of predictive Equation 3Sarpkaya (1981) to predictive Equation 3-28. .................................................................113 3-15 Comparison of added mass predictive Equation 3-30 to experimental data from Yu (1945). ..............................................................................................................................113 3-16 Periods of changing added mass during inundation. .......................................................114 3-17 Beam an d slab span overview. .........................................................................................115 4-1 Progressive nonbreaking wave at a vertical wall. ...........................................................128 4-2 Progressive near breaking wave at a vertical wall. ..........................................................128 4-3 Progressive breaking wave at a vertical wall. ..................................................................128 4-4 Representative force profiles on a vertical wall of a progressive wave and a near breaking wave. .................................................................................................................129 4-5 Representative occurrence of slamming for a smooth -bottomed structure. ....................129

PAGE 12

12 4-6 No occurrence of slamming in early waves in train followed by occurrence of slamming in later waves in train. .....................................................................................130 4-7 Representative occurrence of slamming for a beam and slab span with slamming appearing in each air chamber. ........................................................................................130 4-8 Effect of clearance height on slamming occurrence. .......................................................131 4-9 Slightly dispersive wave with flattened wave face shape. ...............................................131 4-10 Wav e sealing off chamber and trapping air as it propagates through span. ....................132 4-11 Fluid deflected by structure occupying empty trough in front of steepsloped wave. .....132 4-12 Barrel wave forms beneath structure due to buildup of deflected fluid from steep sloped wave. .....................................................................................................................133 4-13 Wedge forms beneath structure due to buildup of deflected fluid from mild-sloped wave. ................................................................................................................................133 4-14 Variation of volume between wave and structure based on steepness of wave face ........................................................................................134 4-15 Four stages of forcing on a rectangular object in a vertically accelerating water column..............................................................................................................................135 4-16 Presence of added mass for a flat plate in an accelerating water column. .......................136 4-17 Orientation of the plate lengthens the time rate of change of mass and added mass buildup. ............................................................................................................................136 4-18 Large volume of fluid mass affected over short period due to steep faced near breaking wave on a vertical wall. ....................................................................................137 4-19 Small volume of fluid mass affected over same period due to shallow -faced nonbreaking wave reflecting on a vertical wall. ....................................................................137 4-20 Significant periods of effective mass variation in a chambered structure. ......................138 5-1 Structure/air grid example (low resolution) for a beam and slab span. ...........................151 5-2 Structure/air grid example (low resolution) for a beam and slab span. ...........................151 5-3 Fluid grid example (low resolution) for a two wave train. ..............................................152 5-4 Example of fluid grid stepping through structure/air grid. ..............................................152 5-5 Air activity in chambers between girders.. ......................................................................153

PAGE 13

13 5-6 Air activity beneath the overhang ledge.. ........................................................................153 5-7 Added mass distributions used in program.. ....................................................................154 5-8 Periods of changing added mass during inundation.. ......................................................155 5-9 Significant periods of effective mass variation in a chambered structure.. .....................156 5-10 Time rate of change of effective mass and resultant forcing. ..........................................157 5-11 Representative time series over the course of inundation of a slab span. ........................158 6-1 Air/sea wave tank wave height limits by period and depth. ............................................173 6-2 Springmassdash pot system under forcing. ....................................................................173 6-3 Amplification effect vs. damping and frequency ratios. ..................................................174 6-4 Typical power spectral density of total vertical forcing. .................................................174 6-5 Three structure components form one continuous structure. ...........................................175 6-6 Structure support carriage (profile side of tank). ..........................................................175 6-7 Structure support carriage (profile down the tank). ......................................................176 6-8 Working model with wave gauges (profile side of tank). .............................................177 6-9 Slab structure model. .......................................................................................................178 6-10 Beam and slab structure model. .......................................................................................178 6-11 Beam and slab with overhangs structure model. .............................................................179 6-12 Beam and slab with overhangs and rails structure model. ...............................................179 6-13 Alternate beam and slab with overhangs and rails structure model. ................................180 6-14 Flat plate structure model for slamming. .........................................................................180 6-15 Beam and slab with overhangs and rails structure model for slamming. ........................181 6-16 Typical power spectral density of l oad cell pair vertical forcing. ....................................181 6-17 Total filtered (noise removed) vertical force examples for a typical subaerial slab case. .................................................................................................................................182 6-18 Total quasisteady vertical force examples for a typical subaerial slab case. ..................183 6-19 Total slamming force examples for a typical subaerial slab case.. ..................................184

PAGE 14

14 7-1 Representative surface curve for the vertical mass rate coefficient for slab spans for a constant value of the wave steepness. ..............................................................................202 7-2 Representative surface curve for the horizontal inertial coefficient for slab spans. ........202 7-3 Re presentative surface curve for the horizontal drag coefficient for slab spans. ............203 7-4 Representative surface curve for the buoyanc y coefficient for slab spans. .....................203 7-5 Representative surface curve for the vertical mass rate coefficient for beam and slab spans for a constant value of the wave steepness. ...........................................................204 7-6 Representative surface curve for the buoyancy coefficient for beam and slab spans. .....204 7-7 Comparison of the numerical model against prototype case of the I-10 bridge over Escambia Bay, Florida. ....................................................................................................205 7-8 Comparison of the numerical model against prototype case of the SR687 Big Island Gap over Old Tampa Bay, Florida. ..................................................................................206 A-1 Measured vs. predicted forces for SLAB -022. ................................................................212 A-2 Measured vs. predicted forces for SLAB -023. ................................................................213 A-3 Measured vs. predicted forces for SLAB -024. ................................................................214 A-4 Measured vs. predicted forces for SLAB -025. ................................................................215 A-5 Measured vs. predicted forces for SLAB -026. ................................................................216 A-6 Measured vs. predicted forces for SLAB -027. ................................................................217 A-7 Measured vs. predicted forces for SLAB -028. ................................................................218 A-8 Measured vs. predicted forces for SLAB -029. ................................................................219 A-9 Measured vs. predicted forces for SLAB -030. ................................................................220 A-10 Measured vs. predicted forces for SLAB -041. ................................................................221 A-11 Measured vs. predicted forces for SLAB -042. ................................................................222 A-12 Measured vs. predicted forces for SLAB -043. ................................................................223 A-13 Measured vs. predicted forces for SLAB -044. ................................................................224 A-14 Measured vs. predicted forces for SLAB -045. ................................................................225 A-15 Measured vs. predicted forces for SLAB -046. ................................................................226

PAGE 15

15 A-16 Measured vs. predicted forces for SLAB -047. ................................................................227 A-17 Measured vs. predicted forces for SLAB -048. ................................................................228 A-18 Measured vs. predicted forces for SLAB -049. ................................................................229 A-19 Measured vs. predicted forces for SLAB -050. ................................................................230 A-20 Measured vs. predicted forces for SLAB -052. ................................................................231 A-21 Measured vs. predicted forces for SLAB -053. ................................................................232 A-22 Measured vs. predicted forces for SLAB -055. ................................................................233 A-23 Measured vs. predicted forces for SLAB -056. ................................................................234 A-24 Measured vs. predicted forces for SLAB -057. ................................................................235 A-25 Measured vs. predicted forces for SLAB -058. ................................................................236 A-26 Measured vs. predicted forces for SLAB -059. ................................................................237 A-27 Measured vs. predicted forces for SLAB -060. ................................................................238 A-28 Measured vs. predicted forces for SLAB -062. ................................................................239 A-29 Measured vs. predicted forces for SLAB -063. ................................................................240 A-30 Measured vs. predicted forces for SLAB -064. ................................................................241 A-31 Measured vs. predicted forces for SLAB -065. ................................................................242 A-32 Measured vs. predicted forces for SLAB -066. ................................................................243 A-33 Measured vs. predicted forces for SLAB -069. ................................................................244 A-34 Measured vs. predicted forces for SLAB -070. ................................................................245 A-35 Measured vs. predicted forces for SLAB -081. ................................................................246 A-36 Measured vs. predicted forces for SLAB -083. ................................................................247 A-37 Measured vs. predicted forces for SLAB -087. ................................................................248 A-38 Measured vs. predicted forces for SLAB -091. ................................................................249 A-39 Measured vs. predicted forces for SLAB -092. ................................................................250 A-40 Measured vs. predicted forces for SLAB -093. ................................................................251

PAGE 16

16 A-41 Measured vs. predicted forces for SLAB -094. ................................................................252 A-42 Measured vs. predicted forces for SLAB -095. ................................................................253 A-43 Measured vs. predicted forces for SLAB -096. ................................................................254 A-44 Measured vs. predicted forces for SLAB -097. ................................................................255 A-45 Measured vs. predicted forces for SLAB -099. ................................................................256 A-46 Measured vs. predicted forces for SLAB -102. ................................................................257 A-47 Measured vs. predicted forces for SLAB -104. ................................................................258 A-48 Measured vs. predicted forces for SLAB -105. ................................................................259 A-49 Measured vs. predicted forces for SLAB -106. ................................................................260 A-50 Measured vs. predicted forces for SLAB -109. ................................................................261 A-51 Measured vs. predicted forces for SLAB -131. ................................................................262 A-52 Measured vs. predicted forces for SLAB -132. ................................................................263 A-53 Measured vs. predicted forces for SLAB -133. ................................................................264 A-54 Measured vs. predicted forces for SLAB -134. ................................................................265 A-55 Measured vs. predicted forces for SLAB -135. ................................................................266 A-56 Measured vs. predicted forces for SLAB -137. ................................................................267 A-57 Measured vs. predicted forces for SLAB -142. ................................................................268 A-58 Measured vs. predicted forces for SLAB -144. ................................................................269 A-59 Measured vs. predicted forces for SLAB -145. ................................................................270 A-60 Measured vs. predicted forces for SLAB -148. ................................................................271 A-61 Measured vs. predicted forces for BSXX -002. ................................................................272 A-62 Measured vs. predicted forces for BSXX -003. ................................................................273 A-63 Measured vs. predicted forces for BSXX -004. ................................................................274 A-64 Measured vs. predicted forces for BSXX -006. ................................................................275 A-65 Measured vs. predicted forces for BSXX -008. ................................................................276

PAGE 17

17 A-66 Measured vs. predicted forces for BSXX -010. ................................................................277 A-67 Measured vs. predicted forces for BSXX -011. ................................................................278 A-68 Measured vs. predicted forces for BSXX -012. ................................................................279 A-69 Measured vs. predicted forces for BSXX -013. ................................................................280 A-70 Measured vs. predicted forces for BSXX -014. ................................................................281 A-71 Measured vs. predicted forces for BSXX -015. ................................................................282 A-72 Measured vs. predicted forces for BSXX -016. ................................................................283 A-73 Measured vs. predicted forces for BSXX -017. ................................................................284 A-74 Measured vs. predicted forces for BSXX -019. ................................................................285 A-75 Measured vs. predicted forces for BSXX -021. ................................................................286 A-76 Measured vs. predicted forces for BSXX -022. ................................................................287 A-77 Measured vs. predicted forces for BSXX -023. ................................................................288 A-78 Measured vs. predicted forces for BSXX -024. ................................................................289 A-79 Measured vs. predicted forces for BSXX -025. ................................................................290 A-80 Measured vs. predicted forces for BSXX -026. ................................................................291 A-81 Measured vs. predicted forces for BSXX -030. ................................................................292 A-82 Measured vs. predicted forces for BSXX -041. ................................................................293 A-83 Measured vs. predicted forces for BSXX -042. ................................................................294 A-84 Measured vs. predicted forces for BSXX -043. ................................................................295 A-85 Measured vs. predicted forces for BSXX -044. ................................................................296 A-86 Measured vs. predicted forces for BSXX -051. ................................................................297 A-87 Measured vs. predicted forces for BSXX -052. ................................................................298 A-88 Measured vs. predicted forces for BSXX -053. ................................................................299 A-89 Measured vs. predicted forces for BSXX -054. ................................................................300 A-90 Measured vs. predicted forces for BSXX -055. ................................................................301

PAGE 18

18 A-91 Measured vs. predicted forces for BSXX -056. ................................................................302 A-92 Measured vs. predicted forces for BSXX -057. ................................................................303 A-93 Measured vs. predicted forces for BSXX -059. ................................................................304 A-94 Measured vs. predicted forces for BSXX -061. ................................................................305 A-95 Measured vs. predicted forces for BSXX -062. ................................................................306 A-96 Measured vs. predicted forces for BSXX -063. ................................................................307 A-97 Measured vs. predicted forces for BSXX -065. ................................................................308 A-98 Measured vs. predicted forces for BSXX -066. ................................................................309 A-99 Measured vs. predicted forces for BSXX -069. ................................................................310 A-100 Measured vs. predicted forces for BSXX -082. ................................................................311 A-101 Measured vs. predicted forces for BSXX -083. ................................................................312 A-102 Measured vs. predicted forces for BSXX -086. ................................................................313 A-103 Measured vs. predicted forces for BSXX -091. ................................................................314 A-104 Measured vs. predicted forces for BSXX -093. ................................................................315 A-105 Measured vs. predicted forces for BSXX -096. ................................................................316 A-106 Measured vs. predicted forces for BSXX -101. ................................................................317 A-107 Measured vs. predicted forces for BSXX -102. ................................................................318 A-108 Measured vs. predicted forces for BSXX -104. ................................................................319 A-109 Measured vs. predicted forces for BSXX -107. ................................................................320 A-110 Measured vs. predicted forces for BSXX -121. ................................................................321 A-111 Measured vs. predicted forces for BSXX -122. ................................................................322 A-112 Measured vs. predicted forces for BSXX -124. ................................................................323 A-113 Measured vs. predicted forces for BSXX -128. ................................................................324 A-114 Measured vs. predicted forces for BSXX -131. ................................................................325 A-115 Measured vs. predicted forces for BSXX -132. ................................................................326

PAGE 19

19 A-116 Measured vs. predicted forces for BSXX -134. ................................................................327 A-117 Measured vs. predicted forces for BSXX -137. ................................................................328 A-118 Measured vs. predicted forces for BSXX -141. ................................................................329 A-119 Measured vs. predicted forces for BSXX -144. ................................................................330 A-120 Measured vs. predicted forces for BSXX -148. ................................................................331

PAGE 20

20 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy WAVE LOADING ON BRIDGE SUPERSTRUCTURES By Justin M. Marin December 20 10 Chair: D. Max Sheppard Major: Coastal and Oceanographic Engineering A theoretical, numerical, and experimental study of vertical and horizontal forces due to nonbreaking, monochromatic water waves propagating past select bridge superstructure spans is presented. A theoretical model is devel oped as the sum of the individual forcing components, drag, inertia, buoyancy, and slamming for a finite thickness slab span and for slab spans with beams. Forces were categorized by forcing frequency into lower frequency quasistatic forces and higher frequency slamming forces. A computer model is developed to evaluate the theoretical equations and assess the sensitivities of certain parameters. Stream function theory is used to calculate wave particle kinematics for the numerical model. Physical model tests were conducted for a range of fluid and structure parameters based on prototype cases of Florida coastal bridge structures and hurricane-generated waves The experimental data was used to evaluate the drag and inertia coefficients in the theoretica l model. Once calibrated the computer model was run for a wide range of structure, and wave conditions and the results used to develop parametric equations for use in design. The accuracy of the model was checked using data from two field site s where major bridges were exposed to significant storm surge and wave loading during a hurricane. The model performed well for both test cases.

PAGE 21

21 CHAPTER 1 INTRODUCTION Over the course of 2004 and 2005, several bridges sustained critical damage during major storm events. The cause of these failures was attributed to the combined effects of storm surge and water wave loading, a forcing mechanism previously unaccounted for in the design of coastal bridges. The purpose of this study is to provide a single coherent investigation of wave loading on select bridge superstructures and a methodology for predicting these loads. To this end a theoretical, physics based model was developed and a corresponding numerical model was created to evaluate it. Extensive physical model testing was carried out for the purposes of providing information needed to evaluate empirical coefficients in the theoretical model. The model was then used to generate data for the development of parametric wave force equations for design. Bridge Failures Attributed to Wave Loading B ridge failure is de fined as any damage that render s a bridge unsafe for continued use. This damage includes the failure of span tie -downs, the displacement of spans from their initial position on their pile caps, the complete removal of spans fr om their pile caps, the shift or collapse of bents and piles from the above movements of the spans, the removal or wash out of railings, and the excessive cracking or spal ling of concrete. The severity of these incidents ranges from visible stress and fatigue of bridge components to the complete destruction and collapse of entire bridges. The recent bridge failures of 2004 and 2005 are not the first ones attributed to waves. Similar incidents during hurricanes were recorded as early as 1969 with the failures displaying similar levels of damage. That it took 40 years and multiple failures to spur the development of

PAGE 22

22 bridge/wave loading guidelines has proven to be an expensive mistake a mistake highlighted by the fact that two of the bridges that failed in recent storms previously failed in the sa me manner some 40 years ago A brief review of incidents involving wave damage to bridges during hurricanes follows. Hurricane Camille In 1969 Hurricane Camille severely damaged the US 90 bridges over St. Louis Bay, Mississippi and Biloxi Bay, Mississip pi ( U.S. Army Corp of Engineers 1970). At landfall Camille was a Category 5 storm on the Saffir-Simpson scale with unknown maximum wind speeds ( though Army Corp estima tes r un as high as 200mph) and a storm surge over 24 feet. About one-third of the spans of the St. Louis Bay bridge and onehalf of the spans of the Biloxi Bay b ridge were either completely removed or displaced significantly. These same two bridges were again destroyed by Hurricane Katrina in 2005. Hurricane Frederic In1979 Hurricane Frederic caused similar damage to the Dauphin Island C auseway Alabama, and an I-10 onramp near Mobile, Alabama ( U.S. Army Corp of Engineers 1981). A Category 3 storm on the Saffir-Simpson scale at landfall, wind gusts of 145mph and a storm surge in the range of 8ft to 13ft were recorded at the bridge site. From the causeway, 135 spans were completely removed from their pile caps (Figures 1 1, 1-2). At the I -10 onramp, s everal spans were displaced varying distances. This same I -10 onramp suffered similar damage in Hurricane Katrina. Hurricane Ivan In 2004 Hurricane Ivan made landfall in the Florida panhandle, causing massive damage (Figure 1 3) to the I -10 bridge over Escambia Bay (Maxey 2006). This Category 3 storm on the Saffir Simpso n scale made landfall with sustained winds of 130mph and an estimated surge of

PAGE 23

23 11ft at the bridge site. 51 spans from the eastbound bridge and 12 spans from the westbound bridge were completely removed (Figure 1-4). 33 spans from the eastbound bridge and 19 spans from the westbound bridge were displaced varying distances (Figure 1-5). Support structures were also affected as 25 pile bents from the eastbound bridge and 7 pile bents from the westbound bridge suffered irreparable damage (Figure 1 -6) by the displacement and collapse of the above superstructures. Failed tie downs between bents and the decks showed both shear and tension failure in the bolts. In some caps, the tie down systems had been completely ripped out, as seen by the cracking and spalli ng in the concrete where the bolts and bearing pads had previously been (Figure 1-7). Hurricane Katrina In 2005 Hurricane Katrina decimated the gulf coasts of Louisiana and Mississippi. Three major bridges (I -10 over Lake Pontchartrain, Louisiana; US 90 over St. Louis Bay, Mississippi; US 90 over Biloxi Bay, Mississippi) were brought down by the resulting storm surge and waves. An I10 onramp near Mobile Bay, Alabama also suffered the displacement of several spans (Figure 1 8). At landfall Katrina was a Category 3 storm on the SaffirSimpson scale with sustained maximum winds of 125mph and a storm surge that varied along the coast with ranges of 10ft to 19ft in Louisiana, 17ft to 28ft in Mississippi, and 10ft to15ft in Alabama. Damage to the I-10 bridge over Lake Pontchartrain (Figure 19) was extensive (Chen et al. 2005). 38 spans from the eastbound bridge and 20 spans from the westbound bridge were completely removed (Figure 110) from their support structures. Supporting pile structures were also dam aged from the collapsed spans (Figure 1-11) while 379 more spans were laterally displaced varying distances (Figure 1-12). The US -90 bridge over Biloxi Bay was completely obliterated (Schumacher et al 2008). With the exception of the approach spans and higher elevated drawbridge spans, all of its 124

PAGE 24

24 spans were removed by the storm (Figure 1-13). The US -90 bridge over St Louis Bay experienced the same level of destruction (Figure 1 -14). In Chapter 7 the incident specifics of the I-10 bridge over Escambia Bay are examined in more depth. By combining the structural characteristics of the bridge with hindcast met -ocean conditions, a series of test cases were created for checking the validity of the developed predictive models. This was also done for a bridge which sustained wave loading but did not fail (SR687 Big Island Gap over Old Tampa Bay, Florida during Hurricane Gladys in 1961). Motivation The previously described failures were all major bridges The loss of commerce fro m the destruction of the traffic routes, the cost of replacement bridges, the risk of repeat failures, and the lack of standardized predictive and design methods are the driving motivations behind this work. Economics of Bridge Failure and Design While the reconstruction of four major bridges in two years is a significant monetary consideration, it is not the only financial burden. Du e to the extensiveness of the damage to the US 90 bridge over St. Louis Bay, traffic flow across the bay was detoured almost 60 miles during reconstruction. Besides the commercial losses incurred due to the absence of an expedient alternative transportation route, e mergency services also suffer from the loss of access ( as was seen during rescue efforts following Katrina ). From a pure capital standpoint, a new bridge by itself is an expensive project, but replacement bridges are more expensive than the original s. Not only must the new bridge account for the lost value from the unfulfilled design life of the original, but replac ement structures are often over-designed in order to compensate for the failures of the previous one. These tighter design specifications are reflected in additional material and cost. In the case of

PAGE 25

25 the recent wave loading failures, the beefing up of the design is compounded by the lack of a method with which to predict any wave loading, so it easy to imagine the extent of the safety factors added to the process. These extreme designs tend towards what appears to be a simple and immediate fix for the problem build bridges higher. If the low chord of a bridge superstructure is elevated above the maximum crest height of a wave, then the threat of wave loading is removed. While undeniably true, accommodations must be made for the structure to attain these ( sometimes ridiculous) clearance heights. For suitable elevations over the entire body of water, the approaches must be lengthened and set back thereby increasing construction cost. In some cases, the necessary headway for the approaches to reach the se design heights may not be available, resulting in the need for land acquisition or eminent domain expenditures to satisfy the increased land coverage. If dealing with a replacement bridge rather than a new bridge, debris and remnants from the previous structure may limit location and routing options Finally and most importantly, raising the elevation of the superstructure itself results in more material and additional cost. As an example, the replacements for the recently failed bridges were examined Each of the se recent replacement bridges was constructed to significantly higher elevations and t he cost of this extreme conservatism can be seen in their price tags The eleva tion of the new 5.5 mile I10 bridge over Lake Pontchartrain, Louisiana was raised 21ft over the old one to a clearance of 30ft, which produces a structure practically unreachable by storm waves. The twin spans will have three lanes and two shoulders and over 3200 individual concrete piles. The project came in at a total cost of $803 million ( U. S. Department of Transportation 2006), the most expensive publics work contract in the history of Louisiana.

PAGE 26

26 In Florida, the new 2.6 mile long bridge over Escambia Bay was raised 13ft over the old one to a clearance of 25ft. Similar to the Lake Pontchartrain bridge, each of the twin spans has three lanes and two shoulders. The project came in at a total cost of $243 million. The new 2.1 mile US 90 bridge over St. Louis Bay in Mississippi was raised considerably with approaches set to an elevation of 30ft and rising to a maximum height of 85ft at the highest point, well out of the range of possible wave action. The single span bridge contains four lanes, two shoulders, and a pedestrian walkway and was contracted at a cost of $267 million. Also in Mississippi, the new 1.6 mile US 90 bridge over Biloxi Bay runs in a full sloped grade to a maximum clearance of 95ft, completely removing the structure from the threat of future wave attack. The side by side twin span bridge contains six traffic lanes, two shoulders, and a shared use walkway. The project was contracted at a cost of $338 million. Even if raising future bridge elevations was not fiscally demanding, somethi ng would still need to be done for existing bridges. Existing bridges cannot be raised, but their risk must be evaluated and a retrofitting solution devised if they are found vulnerable. Frequency of Occurrence The four hurricanes responsible for the aforementioned failures were all significant storms, and in the case of Camille and Katrina, two of the more devastating ones on record. That all of these failures occurred during highly destructive events of a somewhat rare frequency suggests that an except ionally strong storm is required to induce failure conditions but this is untrue A weaker hurricane can produce nearly the same forcing situations as one of these major storms given the right set of meteorological and environmental conditions. While d etermining design storm criteria and analyzing risk potential are outside the scope of this work, an understanding of the factors leading to possible failure conditions provides a broader view of the problem that may make it easier to understand

PAGE 27

27 At its co re, the two determining factors of whether or not a bridge deck will experience wave loading are the rise in water level and the wave heights in the vicinity of the structure. These two parameters are governed by a combination of four components1) storm surge, 2) local wind setup, 3) astronomical tides, and 4) windgenerated wave heights. How a weaker storm can bring about similar conditions to a strong storm involves a combination of the phasing of these four components and the geographic environment. Storm surge, wind setup, tides, and wave height Storm surge. Storm surge is the rise in water level associated with the storm system itself. It is the product of four storm -related components the barometric tide, the wind stress tide, the Coriolis tide and the wave setup (Dean and Dalrymple 2002). The barometric tide is caused by the hydrostatic imbalance between the low pressure at the center of the storm and the higher pressure outside the storm, and the wind stress tide is produced by the frictional drag of the wind over the water. If a storm produces strong shoreline currents, a setup is also induced (the Coriolis tide) to balance the hydrostatic variation caused by the Coriolis force. Finally, the wave setup is caused by the transfer of momentum from breaking waves to the water column, but since this occurs only within the breaking zone, it is set aside for the current problem Local wind setup. The local wind setup is similar to the wind stress tide of the storm surge, but is confined to the body of water upon which the structure of interest is located (bays, sounds, lakes, inlets, estuaries, etc.). Frictional drag on the water surface from a sustained wind creates a hydrostatic imbalance that produces a sloped water surface with the higher water level near the downwind shore. Astronomical t ides. The astronomical tides are governed by the gravitational pull of planetary objects and the rotational characteristics of the earth. While there are hundreds of individual components that can be taken into account, the main components are the moon, the

PAGE 28

28 sun, and the rotation of the earth. Between the earths rotation and the orbit of the moon about the earth, an area can experience diurnal tides, semidiurnal tides, or a mixture of both depending on la titude. As such, a bridge location can see a variety of tides based on the cycles of these components including extreme cases such as spring and neap tides. Wave h eights. Waves are produced near the structure by a sustained wind blowing in line over an expanse of water in front of the structure (the fetch). The height of the wave at the structure is determined by the water depth, the sustained wind speed and its duration, the wind direction, and the fetch length in the direction of the wind. If the structure is near the mouth of an inlet or open to the ocean, contributions from sea swell may come into play. Effects of phasing and geography Knowing the components that make up the rise in water level and the wave heights, the vulnerability of a structure c an be examined by considering when and how these components act. If each component is at a critical condition at the location of the structure at the same time (i.e. in phase), the structure is at its greatest risk. If there is any lag between the compone nts, though, conditions will be less severe. Considered in addition to this timing is the location of the bridge, the layout of the body of water (or the general geography of the area), and the direction of the storms path. Looking at the rise in water l evel first, the tides are the most straightforward. The difference between high and low tides (or spring and neap tides in the extreme case) could produce a difference in several feet of water, positive or negative. The effects of this are subject to tid e magnitudes at the site and the whims of when a storm makes landfall, as a storm could break during high or low tide. The local wind setup can also contribute either positive or negative water elevation. Depending on the direction of the wind and the location of the bridge, the bridge may not

PAGE 29

29 experience set -up at all, but instead set-down (Figure 1-15). Because of the importance in location and wind direction, the approach angle and position of the storm itself becomes crucial. If the storms eye were t o pass on either side of the structure and the body of water, two radically different wind setups would occur (Figure 116). The se effects from the storm path and wind direction are compounded in the case of the storm surge. The surge may be significant, but from one side of a storm to the other, there could be a large drop off. Similar to the local wind setup, a storm passing on either of the structure could avoid any serious effects. In some cases the geography itself can increase the surge. During Hurricane Katrina, the hooklike shape of the coastline between Louisiana and Mississippi acted as funnel for the surge being swept in by the storm. Consider now the phasing of these three elements. In a worst case scenario, the storm surge would arrive a t the bridge location with the storm following such a path that the wind direction falls directly in line with the maximum fetch length in front of the structure. If this all occurs during high tide, the effect is substantial. This of course can play out the opposite as well. At the time the maximum surge is present at the bridge site, the winds may actually be counter -productive in terms of wind setup and wave heights. Wave heights themselves fall under the same restraints of the wind setup. Unless fetch orientation and wind direction align, waves may be nonexistent at the structure location at the time of maximum surge, in which case, the bridge may not be at risk at all. However, all it takes is a slow moving storm with the right phasing, direction, and geography. Outside trends contributing to future incidents Outside of the direct para meters of the problem itself, two other factors may contribute to possible bridge susceptibility. The first concern is water temperature. Warm waters are a driving fa ctor in both hurricane formation and intensity. With the possible effects of global warming

PAGE 30

30 phenomenon, an increase in storm frequency and magnitude is plausible. This may also lead to a rise in ocean water levels, lessening the required surge magnitude necessary to bring waves within reach of existing structure s. Even if someone were to debate the validity of global warming as a product of natures cyclic tendencies, hurricane frequency and strength would still increase, because whether the trend of warm er waters is a product of greenhouse effects or a naturally occurring warming period, the general effect on hurricanes is the same. The second concern is that the population of citizens living in coastal areas has increased every year for the last twenty years and risen by 75% in Florida alone (Crossett et al. 2004). The growth in population along the coast requires a growth in coastal infrastructure. This results in larger bridges or a greater number of bridges, which equals more potential targets. The growth in general infrastructure also limits available space for future bridges and upgrades, meaning some bridges may be limited to what heights they may economically or spatially achieve. Existing M ethods for D esign The most perplexing motivating factor for this work is the lack of research in the area. Despite two major failures in 1969 and 1979, little research effort has been devoted to this topic. Until the recent bridge failures due to storm surge and wave loading, only two studies (Denson 1978, 1980) were directed specifically at wave loading on bridge superstructures. However, if we expand the search beyond bridges to work done on similar structures, numerous studies regarding wave loading on offshore platforms, decks, jetties, and docks can be fo und and even more regarding horizontally suspended cylinders and cylindrical elements. Unfortunately, the cross-application of these methods to the case of a bridge is problematic due to the inherent differences between the offshore/nearshore environment and their structures. Major differences include structure to wave size ratios, finite thickness of the structures, air

PAGE 31

31 trapping concerns, and the variability between shallow water, intermediate, and deep water waves. Because of these differences, the bri dge/wave problem is unique. However, some principles from previous work (especially in the offshore industry) provide an excellent foundation from which to begin. A complete literature search of relevant previous work is given in Chapter 2. Introduction to the Problem In the broadest of strokes, the problem being examined is one of a suspended structure of arbitrary shape being struck by a progressive wave. The first element is the structure. Its shape can be as simple as a flat plate or as complex as a beam and slab bridge with overhangs and rails. Its location can be either subaerial or submerged relative to the still water level (SWL) with its orientation relative to an approaching wave variable in three dimensions. The second element is the wav e. It can be of varying height, period, and length, and the portion of the wave in contact with the structure is dependent upon the location of the structure and the depth of the water in the area of the structure. The goal is to determine how the interaction of these two elements governs the forcing. The S tructure In its most basic form a bridge is little more than a slab supported at both ends. At its most complex, a bridge is an amalgam of beams, diaphragms, decks, rails, overhangs, pile caps, pilings, bents, tie downs, box girders and in some cases, suspensions or arches. These individual components themselves vary in size and shape from bridge to bridge. To categorically define a bridge by a single setup would exclude any number of bridge types that also inhabit coastal waters. However, it would be neither practical nor cost effective to investigate every conceivable setup and conduct physical model tests covering the full variety of those parameters.

PAGE 32

32 The most common bridge superstructure elements used in bridges in coastal environments in Florida are simple slab spans, beam and slab spans, and segmental box girder spans. Bridge superstructure types For comparison, cross sectional diagrams of the three aforementioned span types are shown in Figure 1-17. Each of these structures interact differently with a wave propagating past it, but they each contain similarities in shape that can be exploited when a more broad analysis tool is required. Simple s lab spans. Simple s lab spans are basic in shape and complexity, normally rectangular, and of constant thickness (Figure 1-17). The thickness of these spans, which is essentially all deck, is considerably thicker than the decks on girder spans or segmental spans. They are primarily used for bridge approaches near the shoreline and give way to other span types for the major portion of the bridge. Beam and slab spans. Beam and slab spans are more complex with a deck of nominal thickness atop a series of girders running the full longitudinal len gth of the span (Figure 1-17). Girders vary greatly in size and application from one type to the next. Diaphragms usually run laterally between girders, creating pockets or chambers with the capability of entrapping air between them. Decks usually exten d past the outer girders creating overhangs. It is interesting to note that this is the span type of all previous wave loading failures. Segmental box girde r spans. Segmental box girder spans consist of singlepiece fully enclosed trapezoidal/rectangul ar segments that join together end to end in a series to create a full span (Figure 1-17). The box girder itself is a sealed air space of considerable volume. Similar to the beam and slab spans, the box girder spans can have significant overhangs depending on the style of box girder used.

PAGE 33

33 Each of the three span types described may also contain various types of guard rails (solid or open) These rails may create increased projected area normal to wave propagation thereby increasing the effects of both inertia and drag loading. Bridge support structure types The system used for the linking and supporting of adjacent spans also plays a role in the analysis of the forcing. A span may be simply supported and rest on a pile cap or bent with no additional supports (Figure 1-18). Longer spans may extend over several bents or pile caps, creating an indeterminate system of forcing with variable structural characteristics (especially in terms of natural frequency). These are often referred to as a continuous span (Figure 1-18). In rare cases, some spans may be non pile -support connected, a situation where a span is completely supported by the two adjacent spans without its own bents or caps (Figure 1-18). The variability of these setups can make computing resi stive forces and moments complex. While less likely to experience a severe moment, an indeterminate structure with multiple supports would be more likely to experience frequency or vibrational effects due to high frequency wave slamming components. A vibrational effect could include a situation where a structure that is excessively tied down to pile supports (creating an extremely stiff structure with high natural frequency) may not fail, but could cause damage and/or failure to the supporting structure (damage to the substructure or liquefaction of the sediment around the driven piles). The dynamic response properties of the structure can be important as well, especially for loads due to slamming. An analysis of structural response falls outside the scope of this work. For prediction and design purposes, all structures are treated as rigid. The W ave One of the key differences between offshore structures and bridges is the characteristics surrounding the local wave field. Offshore structures are genera lly located in deep water while

PAGE 34

34 bridges are located in areas of s hallow or intermediate water depth. The limits of these depth designations are based on the product of the water depth (ZD) and the wave number (k), where the wave number is a function of the wave length ( ) as defined in Equation 1-1. The limits of deep, intermediate, and shallow water de pth are defined as in Equation 1-2. 2 k = (1-1) DkZ > (Deep Water) (1-2a) D < kZ < (Intermediate Water) 10 (1-2b) D kZ < (Shallow Water) 10 (1-2c) These boundaries as applied to waves affect certa in properties of the wa ve field. In deep water, wave particles travel in circular motions while these motions flatten into ellipses as the wave propagates into shallow wate r. Wave lengths also shorten during the change from deep to shallow water and the wave shoals, altering th e wave height and steepness which leads to breaking. Lastly, the wave celerity (c), while vari able in deep water, is only a function of water depth in shallow water. The effects of individual particle kinematic s, wave length, and wave celerity play important roles in the loading experienced by a structure during wave inundation. So the differences inherent to the variation between o ffshore and nearshore wave climates also become important. For the majority of cases, bridges that are susceptible to wave attack are located in intermediate or shallow water depths. Another wave concept that is important in th e study of wave loading is dispersion. In reality, a wave field consists of many waves of varying heights and fre quencies propagating at

PAGE 35

35 assorted speeds and directions. The distribution of wave heights and therefore wave energy can be quantified with directional wave energy density spectra. This ene rgy content of a wave spectrum is critically different from offshore to nearshore. In the deepwater realm, ocean swell (very large period waves) can make up a significant portion of the energy content. In the bays and coastal waterways, swell is usually negligible. For bridges, the wave spectrum is composed almost entirely of shorter, locally wind generated, waves. Wave/structure Interaction It is important to understand some of the basic ideas that play a role in wave loading. A significant amount of variation in forcing magnitude and distribution occurs according to the ideas and limits of these concepts. Each are looked at in more detail in subsequent chapters. The generalized force curve and its components A single wave propagates into a suspended structure, inundates it, and propagates away from it. In the review of literature where physical model testing was done, the same generalized forcing time series over this interaction was consistently found by the researchers. An example of this forcing is presented in Figure 1-19. What is obvious at first glance is the presence of two separate forcing mechanisms working at radically different frequencies As the wave first strikes the structure a sharp large magnitude spike occurs, followed by a slowly building positive force. As the wave propagates away from the structure, the force becomes negative, though not as large in magnitude (due to the countering of buoyancy and the dissipation of wave energy). The large short -duration (high frequency) spike is often referred to as the slamming force or the impact force. It is occurs during the initial contact between the structure and the air/water interface where a large exchange of momentum takes place quickly. The slowly varying force is sometimes referred to as the quasi static force. It occurs over the full inundation cycle of the structure by the wave and has a forcing frequency nominally

PAGE 36

36 equivalent to the wave frequency. The loading over this inundation cycle is not from single source but rather is composed of several independent forcing contributions present during the interaction For the remainder of this document, the components of forcing are r eferred to by separate terms. The short-duration (high frequency) forcing is referred to as the slamming force, a force that is highly variab le and possibl y susceptible to the response characteristics of the structure, wave shape, and structure shape. The slowly -varying force is referred to as the quasistatic force, but is subdivided into three separate components. These three components are the buoyancy force (vertical direction only ), the drag force, and the inertia force. While the slamming and quasistatic forces are vastly different in terms of frequency, they both originate from the same mechanisms. Factors surrounding these mechanisms are what determine whether or not the slamming force will exist Clearance heights The clearance height (ZC) is the distance between the still water level (SWL) and the lowest chord of the structure (when SWL is referred to in this document it includes all contributions from storm surge, wind setup, and tides at the location of the structure ). For a beam and slab b ridge, the base of the girder would constitute the lowest chord. In general, all previous works on the topic examined structures that were subaerial, ranging from the low chord being directly at water level or above. However, with the recent storms and fa ilures having produced situations of submerged and partially submerged structures, it is necessary to extend the investigation further into the possibility of negative clearance heights. This has a significant dynamic effect when dealing with interaction between the wave, the structure, and possible trapped air.

PAGE 37

37 Instances of negative clearance are also important because they produce a different typical force curve (Figure 1-20). The slamming force drops out of the forcing and the negative quasistatic f orces are, in general, larger in magnitude for structures that have negligible thickness It should be note d that the wave induced forces on the superstructure may be smaller for the submerged or partially submerged case than for the subaerial case due to the absence of slamming For a given situation, then, a more severe storm surge could be beneficial to the survival of a bridge span. The combination of bridge size and the variation in clearance between a structure and SWL also calls into play an important factor regarding the inertia component of the quasistatic force the concepts of added mass and time rate of change in added mass. These concepts are discussed in Chapte r 3 Air entrapment The natural voids created by the girders and diaphragms of a bridge deck are adept at trapping air as a wave propagates past the structure. The immediate concern regarding these air cavities is the impact they have on the buoyancy force. In fact, for many new bridges the girder types are such that the volume of potentially trapped air is greater than that displaced by the structural components. What is less evident, however, are the effects these pockets have on the slamming force. The series of chambers of air between the girders can actually act as spring mass systems when confronted with the periodic forcing of the wave. They provide a cushioning system for a slamming force to act in every chamber (for a structure with positive clearance). Of greater concern, though, is that the spring-like compression of the air lowers the forcing frequency of the slamming force, creating a forcing frequency that is potentially more aligned with the fundamental modes of vibration of the bridge supe rstructure

PAGE 38

38 Q uasi static forces will also be present in these trapped air chambers. The combination of these forces produce another unique forcing time series (Figure 1 -21) where t he number of peaks in the vertical force will be equal to the number of chambers created by the girders (number of girders minus one). Relative size ratios When dealing with the nearshore environment, both wave periods and wave lengths are generally smaller than their offshore counterparts. Unlike the offshore environment, this creates a forcing system where the bridge width and wave length are of comparable size. In contrast, for a bridge width that is much smaller than the length of the approaching wave, there is little variation in wave kinematics over the width of the bridge. This allows for the use of uniform velocities and accelerations over the width of the bridge at any point in time. Since bridge widths and wave lengths in coastal bays and waterways are comparable, there are significant variations in the water velocitie s and accelerations over the width of the bridge at any instant in time. To visualize this, Figure 1 22 shows the water particle velocity and acceleration magnitudes and directions throughout a progressive wave. Flow exists in all directions within the wave. If we place bridge superstructures and offshore platforms into these wave fields (Figure 1 23), it is clear that the offshore platform experiences a more uniform flow field. Further complicating the calculation of wave forces on bridge superstructur es is the fact that the closer the bridge width is to the wave length the greater the impact of the structure on the wave. This results in possible variations in wave properties as the wave propagates past the structure due to the larger reduction in energ y. Added mass and change in added mass The discrepancy in (bridge width to wave length) size ratio presents another problematic change in the form of a wave/s tructure interaction system where the structure experiences

PAGE 39

39 periodic inundation. Because the structure is rarely (if ever) fully submerged by the wave, the added mass (or virtual mass) of the structure is in constant flux. The time-varying nature of the mass displaced by the structure and the added mass introduces an additional inertia term in the wa ve force equations. This adds to the complexity of computing storm surge and wave loads on coastal bridge superstructures. Further adding to the complexity of the computations is the finite thickness of the superstructures since most of the empirically determined added mass equations are for thin structures. This issue is discussed in Chapter 3 Organization The results of a literature review on the topic of wave loading on horizontal structures are presented in Chapter 2. This is followed by the development of a theoretical model for horizontal and vertical quasistatic forces on horizontal, bridge superstructure type structures. An introduction to vertical slamming forces is then given with a theoretical model introduced for beam and slab spans. Next is a description of physical model experiments that were performed with both flat plates and model bridge superstructures in a wave tank in the Coastal Engineering Laboratory at the University of Florida. This is followed by a description of a computer mo del ( referred to as the Physics Based Model, or PBM) that was used to evaluate the theoretical equations developed in Chapter 3. The PBM is then validated against the physical model data and prototype field cases for the developed empirical coefficients. From this, a set of parametric equations are developed using data generated by the PBM. Finally, the summary and conclusions chapter includes recommendations for future work on this topic.

PAGE 40

40 Figure 1-1. Dauphin Island Causeway spans removed by Hurricane Frederic (longshot). Figure 1-2. Dauphin Island Causeway spans removed by Hurricane Frederic (closeup).

PAGE 41

41 Figure 13. I -10 Bridge Escambia Bay damage from Hurricane Ivan. Figure 1-4. I -10 Bridge Escambia Bay spans removed by Hurricane Ivan.

PAGE 42

42 Figure 15. I -10 Bridge Escambia Bay spans displaced by Hurricane Ivan. Figure 16. I -10 Bridge Escambia Bay pile bents damaged by Hurricane Ivan.

PAGE 43

43 Figure 17. I -10 Bridge Escambia Bay tie-downs damaged by Hurricane Ivan. Figure 1-8. I -10 Mobile Bay onramp spans displaced by Hurricane Katrina.

PAGE 44

44 Figure 19. I 10 Bridge Lake Pontchartrain damage from Hurricane Katrina. Figure 1-10. I-10 B ridge Lake Pontchartrain span s remov ed by Hurricane Katrina.

PAGE 45

45 Figure 1-11. I-10 Bridge Lake Pontchartrain support structures damaged by Hurricane Katrina. Figure 1-12. I-10 Bridge Lake Pontchartrain spans displaced by Hurricane Katrina.

PAGE 46

46 Figure 1-13. US 90 Bridge Biloxi Bay damaged by Hurricane Katrina. Figure 1-14. US 90 Bridge St. Louis Bay damaged by Hurricane Katrina.

PAGE 47

47 Figure 1-15. Effect of bridge location and wind direction on wind setup. Figure 1-16. Effect of storm path and wind alignment on wind setup.

PAGE 48

48 A B C Figure 1-17. Common bridge superstructure types in Florida. A) S lab span. B) Beam and slab span. C) S egmental box girder span

PAGE 49

49 A B C Figure 1-18. Common pile -support setups for bridge superstructures A) S imple span support. B) Continuous span support. C) L inked cantilever span support.

PAGE 50

50 Vertical Force Time Figure 1-19. Typical vertical force curve on a su b aerial flat-bottomed structure. Vertical Force Time Figure 1-20. Typical vertical force curve on a submerged, flat-bottomed structure.

PAGE 51

51 Vertical Force Time Figure 1-21. Typical vertical force curve on a sub aerial girdered structure.

PAGE 52

52 A B Figure 1-22. Water particle kinematics in a progressive wave. A) V elocity magnitudes and directions B) Acceleration magnitudes and directions.

PAGE 53

53 A B Figure 123. Water particle velocities distribution over different structure s A) O ffshore structure small in size relative to wave B) N earshore structure nominal in size relative to wave

PAGE 54

54 CHAPTER 2 LITERATURE REVIEW If one were to consider only the existing work specifically dealing with bridge superstructures, the compendium developed would be quite small. Prior to the recent resurgence of interest in the topic, only two studies (Denson 1978, 1980) were found that i nvestigate the effects of wave forcing on bridge decks. In general, there are very few studies that consider vertical forcing on any structure that is of comparable size to the wave length. Looked at broadly, the number of works done on topic of wave l oading is overwhelming. However, the breadth of that work is fairly narrow While one could spend years reviewing the works done on horizontal wave loading and its variants, the vertical loading library is quite small. The majority of work published on this topic is dedicated to offshore platform structures which has limited applicability to bridges due to the size ration between structure and wave. That this work is often proprietary and not in the public domain further restricts its usefulness. Avail able w ork from the areas of bridges, standardized methods, the offshore industries, and simpler structures are considered and reviewed. Previous Work on Bridge Superstructures The only studies located that dealt with wave forces on bridge superstructures are those by Denson (1978, 1980). Both papers were purely empirical physical model studies. The deficiencies of both papers are docum ented by Douglass el al. (2006), but in brief, Denson (1978, 1980) contains several si gnificant flaws : The importance of wave period (only a single value was used in testing) and wavelength on the forcing was dismissed as insignificant due to the shallowwater environment. While the works of El Ghamry (1963, 1971), French (1970, 1979), and Wang (1970) were alr eady published, Denson (1980) stated that no useful papers could be found as these papers referred to offshore structures.

PAGE 55

55 The differentiation of the forcing mechanisms into a slowly-varying force and a shortduration slamming force was never made, despite the consensus found in the rest of the literature. That the separation between the low and high frequency components of the loading was never found in the raw data suggests possible errors in instrumentation or sampling frequency. The lone graph of raw data in their report shows no high frequency (slamming) contributions No model similitude or scaling is discussed. If scaling is carried out, then the test wave period of 3 sec would correspond to almost a 15 sec period in the prototype, an unrealistic wa ve frequency for bays or coastal waterways For dimensional considerations, Denson (1980) states that the choice of a significant length parameter does not affect the results, only the shape of the plot i.e. the width, length, and thickness of the structu re are interchangeable quantities. There are large discrepancies between the measured forces of Denson (1978) and Denson (1980), which is explained away as the inclusion of diaphragms in the later models. These issues severely limit the usefulness and question the validity of the presented results As such, this study provides no useful information towards the development of a new model. Denson (1978) Denson (1978) experimentally examined forces on a slab and beam type bridge. The work was motivated by the damage to the Bay St. Louis bridge in Mississippi caused by storm surge and wave action from Hurricane Camille. Denson (1978) states that no useful previous papers related to the topic could be found. Physical model testing was carried out in a 44 ft long, 2 ft wide wave basin at Mississippi State University. A scale model (1:24) of the Bay St. Louis b ridge was subjected to monochromatic waves (all normal to the structure) with a range of wave heights, water depths, and clearance heights relative to SWL ( some tests were done with a partially submerged structure ). Only a single wave period (T = 3.0 s) was used. Strain gauges were used to record vertical and horizontal forces. No mention of filtering or description of the features of the physical model data is given. Maximums in the vertical and horizontal direction are referred to as a lift force (FL) and a drag

PAGE 56

56 force (FD), respectively. The se measured forces were non 2 and maximu m moments (M) were non3 to produce forces per unit width. The nondimensionalized forces and moments were then plotted against three parameters (Equation 2-1), where D is the water depth, ZC is the clearance height of the bridge relative to the still water level ( SWL ), W is the width of the bridge parallel to wave propagation direction and H is the wave height. DCDC LD 223 DDZZZZ FF MH ,,f,, (2 -1) The series of plots created were then presented as design curves. Denson (1978) concluded that the failures were mainly caused by the rolling moment and could have been avoided by using slightly larger anchoring systems. Denson (1980) Denson (1980) continued the experimental work of Denson (1978). Al ong with the beam and slab model (1:24) of the Bay St. Louis bridge, a box-girder model (1:24) of an I-110 connector was also tested. The physical model tests were performed in 40 ft. long, 16 ft. wide wave basin at Mississippi State University. The models were subjected to monochromatic waves with a range of wave heights, water depths, clearance heights relative to SWL (some tests were done with a partially submerged structure), and wave incidence angles. Only a single wave period (T = 3.0 sec.) was used Denson (1980) stated that the effect of wave length and period on the loading were insignificant for shallow water waves. Vertical and horizontal forces were measured using a single 6-axis strain gauge and moments calculated from the data. Maximum pos itive and negative forces in the vertical and horizontal direction are referred to as a lift force (FZ) and drag forces (FX, FY), respectively. These measured forces were non -

PAGE 57

57 dimensionalized by W3 and maximum positive and negative moments (M) were nondimensionalized by W4. The non-dimensionalized forces and moments were then plotted versus three parameters (Equation 2-2), where is the unit weight of water, ZD is the water depth, ZC is the clearance height of the bridge relative to SWL, W is the width of the bridge parallel to wave propagation, H is the wave height, and is the angle of incidence of the wave. XYZ DC 34 DDF ZZ MH W WZZ,,,f,, (2-2) The series of plots created were presented as design curves. The forces were found to decrease with increasing angle of incidence. (i.e. waves propagating no rmal to the bridge resulted in the greatest forces). Denson (1980) concluded that inexpensive anchorage systems would prevent failure. U.S. Army Corp of Engineers (2008) The Coastal Engineering Manual (CEM) by the U.S. Army Corps of Engineers (2008) breaks up the case of a horizontally oriented stru cture into two separate groups submerged (or partially submerged) structures and emergent (subaerial) structur es. For a submerged structure, forces (FL) are calculated using a rudimentary lift-ba sed flow relationship (Equation 2-3), where u is the horizontal water velocity, CL is an empirically determined lift coefficient for each structure of interest, is the unit weight of water, g is gr avity, and A is the projected area of the structure on a vertical plane. 2 LLu FCA 2g (2-3) For an emergent structure, vertical forces (FS) are calculated using a single slamming-type relationship (Equation 2-4), where w is the vertical water velocity (w), CS is an experimentally

PAGE 58

58 determined coefficient for each structure, g is gravity, is the unit weight of water, and A is the projected inundation area of the stru cture on a horizontal plane. 2 SSw FCA 2g (2-4) Due to the limited work done on the topic, the Coastal Engineeri ng Manual recommends the use of problem-specific numerical and physical modeling for the purposes of design. This limits its usefulness in that the majority of proj ects involving bridge construction will not include detailed numerical modeling. Morison et al. (1950) The Morison Equation was developed by Morison et al. (1950) for computing wave forces on a single vertical pile. This equation separates the forces into the two major components, drag and inertia. For the case of the ve rtical cylinder, drag is propor tional to the approach velocity squared and the inertia proportional to the acceleration of the water particles. Versions of this equation have been used by the offshore industry fo r a number of years. Direct application is limited to cases where the structures are slender compared to the lengths of the waves encountered. Previous Work on Offshore Platforms and Decks The amount of work done on wave loading and offshore platforms is extensive, although the majority of this work is for cylindrical components. The works investigated here were limited to those dealing specifically with deck or platform like struct ures with a horizontal orientation. As noted earlier in this document, the value of this work to the wave forces on bridge superstructures is limited due to open water storm wave parameters and the wave length/structure width ratios. Also, the work done on horizontal platform structures is limited to objects of negligible thickness.

PAGE 59

59 Kaplan (1992) Kaplan (1992) further con tinued the work done by Kaplan (1979) on wave forcing on cylindrical elements and extended the work to horizontal platform decks (of negligible thickness). However, the work on flat decks was limited to theoretical model development. The proposed model for vertical wave forces on horizontal decks was briefly described as the sum of the momentum and drag forces. The time derivative of the momentum resulted in two separate components due to the intermittent submergence of the structure by the wave. The variation in inundation with time produced a ti me dependent added mass. To capture the variation in inundation levels, an expression for the added ma ss (Equation 2-5) was proposed, where L is the length of the structure (perpendicul ar to the direction of wave propagation), W* is the wetted width of the structure (parallel to the direction of wave propagation), and is the mass density of water. 2 AddedMass LW 8* (2-5) The total vertical force in Kaplans model is the sum of the drag and inertia forces. The expression is as given in Equation 2-6, where CD is an empirical drag coefficient, w is the vertical water velocity, w/ t is the vertical acceleration, and W*/ t is the time rate of change of the wetted width. 2 D w W1 TotalVerticalForcing LW LwC LWww 8t4t2* ** (2-6) In order to evaluate the equation, several assumptions were made. The W*/ t term was only included as long as the forcing was positive. Since the model structure was treated as a plate of negligible thickness, neither buoyancy nor the mass of the water displaced by the deck was included. Also, the wave was considered to be significantly larger than the deck so that

PAGE 60

60 there was no variation in wave kinematics over the width of the plate. Finally, the deck length, L, was treated as infinitely long. Kaplan et al. (1995) Kaplan et al. (1995) expande d on the methods described by Kaplan (1992) for predicting the wave forcing on suspended hor izontal platform decks of neg ligible thickness. The expanded theoretical model was presented and compared against physical model test data to assess predictive capability. Kaplan (1995) extended the 1992 theoretical model to situations of plated decks with finite aspect ratios. The resulting vertical force equation (FV) is given in Equation 2-7, where the time rate of change of the wetted width term, W*/ t, has been replaced with the wave celerity, c. 2 2 2 V D 32 2 21W 1 2L LWw 1 F WLcwC Aw 8t42 W W 1 1 L L* / (2-7) For the rate of change in momentum, a modification to the added mass equation was needed for use with structures of finite meas urements. An expression derived by Payne (1981) for thin plates was used (Equation 2-8). 2 2 LW AddedMass 8 W 1 L* (2-8) The expression for total horizontal force (FH) is given in Equation 2-9 where D is the wetted vertical height of the structure, D* is the time rate of change of the vertical wetted height, u the horizontal water particle velocity, and u/ t is the horizontal water particle acceleration. The added mass for this orientati on was adapted to Equation 2-10.

PAGE 61

61 22 HD2u4D1 F DL DLuC DLu t t2* (2-9) 22 AddedMass DL (2-10) In evaluating these equations, linear theory was used since this was suitable application was for deep water. The theoretical model was compared with e xperimental data obtained from physical model studies with offshore platform m odels (Murray et al. 1995). Agreement between measured and predicted forces was good except in cases where additional structures were in front of the test platform causing significant wave diffraction. Suchithra and Koola (1995) Suchithra and Koola (1995) examined the for ces acting on a horizontal slab for regular and freak waves. Physical model tests were done at the Ocean Engineering Centre at the Indian Institute of Technology, Madras, India in a 10m long, 0.3m wide wave tank. The model slab was 0.25m long, 0.25m wide, 0. 08m thick and made of Perspex. A series of tests were also run with girder-like stiffene rs beneath the slab runni ng in the longitudinal, lateral, and both longitudinal and lateral directions. Measurements were taken with a single load cell sampled at 1000Hz. Tests were done for a range of wave periods and clearance heights relative to SWL (no tests were done with submerge d or partially submerged slabs). In the data and the discussion of the data all forc ing is treated as slamming forcing. The predictive relationship developed for the maximum forcing (FS) is given in Equation 2-11 where is the fluid density, A is the area of contact, w is the vertical water particle velocity, and CS is an empirical coefficient. This equa tion is referred to as a slamming equation. 2 SS1 FC Aw 2 (2-11)

PAGE 62

62 A formula for the coefficient, CS, is given in Equation 2-12 where ZC is the clearance 0 is the deep water wave length, and CNS is a modified coefficient of singular value based on the clearance height. Neither a method for obtaining CNS nor the origin of the ter m is given. 0 SNS C CC Z (2 -12) Suchithra and Koola (1995) state that the wave period s and the clearance heights are the only variables that have an effect on the forcing magnitude. For tests done with the girderlike stiffeners, it was noted that the slamming force was noticeably reduced due to the presence of trapped air and it is suggested that adding air pockets to the structures would reduce the vertical forces on horizontal slabs. Bea et al. (1999) Bea et al. (1999) concentrated on offshore platform decks that were suspended beneath the structure, specifically dealing with failed decks in the field. Data was compared from laboratory tests for wave forces on offshore platforms (Finnigan and Petrauskas 1997; Dean et al. 1985; Faltinsen et al. 1977; Jue 1993; Kjeldsen and Myrhaug 1979; Kjeldsen and Hasle 1985; Kjeldsen et al. 1986; Weggel 1997) with the guidelines of the American Petroleum Institute. The current methods were found excessively conservative and more extensive work and modifi cations to the guidelines were recommended. An analytical model was presented as the sum of forcing components (Equation 2-13) similar to those derived from the Morison equation by Kaplan et al. (1995) While vertical components are discussed, the decks in question were porous or grated, so the concentration of the work was aimed at the horizontal component of the forcing. 5th order Stokes wave theory was used to compute the wave kinematics.

PAGE 63

63 BuoyancySlammingDragLiftInertiaFFFFFF (2-13) Each component of the total forcing was then developed. The drag term (Equation 2-14) was identical to that given by Kaplan et al. (1995). The inertia term (Equation 2-15), though, contained neither a time dependent mass term nor the added mass itself in the calculation, instead basing the calculation on the volume of the structure, V the water particle acceleration, u/ t, and a mass coefficient, CM. A lift term was also included (Equation 2-16) where CL is a lift coefficient and A is the projected area of the structure in the vertical plane. 2 DragD1 F CAu 2 (2-14) Inertia M1 F CV 2 u t (2-15) 2 LiftL1 F CAu 2 (2-16) A slamming force equation with the force proport ional to the square of the water velocity, u, was also developed (Equa tion 2-17) using an empirical slamming coefficient, CS. 2 SlammingS1 F CAu 2 (2-17) While the equation itself was simple, the importance of the dynamic characteristics of the structure and the wave were also included by using a multiplication factor to give an effective slamming force. This required knowledge of the structures primary modes of vibration as well as the frequency of the slamming force among other variables. No method was provided for estimating these variables for the purposes of design. The analytical models were evaluated numeri cally and compared with results from field data (Stear and Bea 1997; Imm et al. 1994; Ca rdone and Cox 1992; Vannan et al. 1994) for existing structures that had been subjected to wa ve loads. The horizontal force predictions were

PAGE 64

64 found to be extremely conservative. Further comparisons were made in Bea et al. (2001), though no changes to the analytical model were presented Baarholm and Faltinsen (2004) Baarholm and Faltinsen (2004) performed a numerical and experimental study on the vertical wave force on an offshore platform. Physical model tests were carried out at the Department of Marine Hydrodynamics, NTNU, Trondheim, Norway in a 13.5m long, 0.6m wide wave tank. The model was a box-shaped deck that spanned the full width of the tank with two vertical plates attached to the front and back of the deck to prevent any overtopping. Regular waves were used in the experiments. In the measured data, a difference in the impact process was found between a slowly varying force and a short duration force of significant magnitude. The short duration force was left out of the scope of the study. The theoretical model devised was the 2D Laplace equation with dynamic (Bernoulli) and kinematic free surface boundary conditions. No bottom boundary condition was used as the water depth was assumed to be deep enough relative to the wave length to ignore. An additional boundary condition of a nonpermeable structure was used. The boundary value problem is solved numerically three separate ways a Wagner -based method (Wagner 1932) that solves the perturbation velocity potential and two variations of a boundaryelement method that uses Greens second identity ( Faltinsen 1977) that solves the total velocity potential. The method described by Kaplan et al. (1995) is recommended for the propagation of the wave away from the structure when using the Wagner-based method. The boundary-element methods showed good agreement to the experimental data for the single plot showed. A nearly identical study was done by Lai and Lee (1989).

PAGE 65

65 Previous Work on Offshore Jetties Offshore jetties (not to be confused with jetti es used in coastal shore management) are used for the berthing, loading and offloading of tankers and other sizab le craft. These structures consist of deck or dock-like platforms suspe nded over supportive piles and on occasion beam or girder-like support elements. In ge neral, these structures have the same limitations as those with offshore platforms, though many jetties extend fro m shore and the shallow water environment. Overbeek and Klabbers (2001) Overbeek and Klabbers (2001) were involved in the analysis and desi gn of two jetty-type structures in the Caribbean. 100 years of statistical data was used to determine the design event for each structure. The prediction of the vertic al forcing for the given conditions was divided into a slowly-varying pressure and an impact pressure. For the slowly-varying case, a simplistic hydrostatic pressure expression (Equation 2-18) similar to Wang (1970) and French (1970) was used, where PVE is the maximum pressure, is the fluid mass de nsity, g is gravity, max is the elevation of the wave crest above the SWL, and ZC is the clearance of the deck above SWL. ve maxCP g Z (2-18) For the impact pressure, another simplisti c expression (Equation 2-19) was used, where Hmax is the maximum wave height, and C is an em pirical coefficient. This equation was taken from Goda (2000). ve maxPC gH (2-19) The two jetties summarily experienced signifi cant wave action due to Hurricanes Iris (1995) and Lenny (1999). For Iris, design conditions were not reached. For Lenny, design conditions were believed to be exceeded. Both jetties sustained significant damage including missing spans, excessive concrete spalling and tension cracks. Pressu re vents and blowout

PAGE 66

66 panels were thought to have reduced the loading experienced even though failure occurred. The authors concluded that further work on the topic with practical guidelines was needed. Tirindelli et al. (2002) Tirindelli et al. (2002) investigated the current design methods for wave forcing on shipping jetties and performed experimental testing with which to compare the predictive capabilities of those methods. The three methods considered were those of Kaplan et al. (1995), Shih and Anastasiou (1992), and Bea et al. (1999). Physical model tests (1:25) were conducted at a wave tank at HR Wallingford in the United Kingdom. Random waves with JONSWAP spectra were used and vertical and horizontal measurements taken via four load cells attached to standalone elements within the model. Two separate water dep ths (0.60 m, 0.75 m) and four separate clearance heights from four still water levels (0.01 m, 0.06 m, 0.11 m, 0.16 m) were used. No submerged or partially submerged tests were reported. Three separate model setups, a flat deck, a deck with beams, and a d eck with beams and vertical side panels were used. Only the slowly -varying component of the forces was analyzed and compared with theoretical predictions. Despite considerable scatter, it was concluded that both the vertical and horizontal maximum forces (positive and negative) varied linearly with wave height. Period and wave length were considered to be non important parameters Comparisons with the data were made with the models of Kaplan et al. (1995) and Shih and Anastasiou (1992). Both models under predicted versus the experimental data. It should be noted that the authors of the paper erroneously applied the equations of Kaplan et al. (1995) by dropping the leading inertia term of the model, mistaking it for a slamming term. Further clouding the validity of the comparison was the selection of input parameters. The significant wave height (HS) was used as input for the models, but the models were being

PAGE 67

67 compared against the average of th e 4 highest forces in 1000 waves (F1/250). The use of HS was clearly not the proper wave he ight for these predictions. McConnell et al. (2003) McConnell et al. (2003) builds on the work presented by Tiri ndelli et al. (2002), breaking down and analyzing the experimental data collected in Tirindelli et al. (2002). That experimental data consisted of forces collected from a mode l structure where the measurements were taken through a single instrumented element in a larger structural model. In the analysis of the force data, the paper concentrated on the slow ly-varying (referred to as the quasi-static) component while the shor t-duration slamming force data was analyzed by Cuomo (2003). The significant forces derived were taken as the average of the 4 largest magnitude loads in 1000 waves (F1/250). The horizontal and vert ical forces were then nondimensionalized by a basic wave force (Equa tion 2-20, Equation 2-21a, and Equation 2-21b) derived from hydrostatic pr inciples in the vein of Wang (1970) and French (1970). In the basic wave force equations, max is the maximum wave crest elevation, W is an element width, L is an element length, D is an element depth, ZC is clearance above SWL, p1 is the hydrostatic pressure at the top of the element, and p2 is the hydrostatic pr essure at the botto m of the element. V2FWLp* (2-20) 2 Hmaxmaxp FW Lfor LD 2* (2-21a) 12 Hmaxpp FWDfor LD 2* (2-21b) Once the forces were non-dimensionalized they were plotted against a single nondimensional parameter, ( max ZC)/HS, and best fit regression lines were fitted to each data set (positive vertical quasi-static force, negative vertical quasi-static force, positive horizontal quasi-

PAGE 68

68 static force, ne gative horizontal quasistatic force). From these fits, a general empirical relationship was developed (Equation 2-22) where a and b are constants. qs b maxC SF a F H* (2 -22) It was noted that the data exhibited a significant degree of scatter though no further comment was given. Tables for empirical coefficients a and b are given in McConnell et al. (2003) for the setups tested. Several case studies were then worked with the developed equations. In the case studies where failures occurred, the quasi static forces obtained from Equation 2-22 did not always predict those failures, a fact explained away as the contribution of the slamming force, stating that slamming forces may be up to 4 times great than quasistatic vertical forces, the application and determination of which is not provided. This study did not include the effects of the wave period or the wave length on the forcing. A ll predictive forces were based on the experimental extrapolation of the forcing experienced by two instrumented elements of the model structure rather than the structure itself. By assuming the loading on the structure is uniform over the entire structure, application is limited to scenarios where the structure is much smaller than the wave length. Cuomo et al. (2003) Cuomo et al. (2003) builds on the work of Ti rindelli et al. (2002) by re -examining the shortduration vertical slamming force data. Again, that experimental data consisted of forces collected from a model structure where the measurements were taken through a single instrumented element in a larger structural model. Wavelet analysis was described and used to filter out possible dynamic effects in the instrumentation.

PAGE 69

69 With oscillations filtered out, the maximum vertical slamming forces (Fmax) were nondimensionalized by the positive vertical quasi st atic force (Fqs+) of the corresponding test and plotted against the rise time (tr) of the slamming force non-dimensionalized by the wave period (T). Envelope fits were done producing Equation 2-23, an empirical relationship for the slamming force. Cuomo et al. (2003) provided c oefficients a and b for the testing setups examined in Tirindelli et al (2002). b max r qsF t a FT (2 -23) No method of obtaining the rise time of the slamming force was provided. O scillations found in the force time series data were attributed to the instrumentation rather than the structure itself. Possible amplification of forcing magnitudes due to the similarity of forcing and structure natural frequencies was not discussed Tirindelli et al. (2003) Tirindelli et al. (2003) essentially reproduces the content of McConnell et al. (2003), with the exception of the means of non-dimensionalizing the significant forces. The positive quasistatic force is now nonSA (where A is the p rojected maxZC)/HS. The negative quasistatic force is non -dimensionalized by what appears to be a shear oA. The negative force data is plotted against the wave steepness, HS0 0 is the deep water wave length). Brief discussions of the slamming force and the horizontal forcing are also presented with the importance of the wave period on forcing finally recognized. No equations for the h orizontal direction and slamming are given.

PAGE 70

70 Cuomo et al. (2007) Cuomo et al. (2007) further expands on the work of McConnell et al. (2003), Tirindelli et al. (2003), and Cuomo et al. (2003). A brief description of the problem of offshore loading on jettie s is given, followed by a description of the current predictive methods scattered throughout the literature. Particular attention is given to the methods described in Kaplan (1992) and Kaplan et al. (1995). Working from the same experimental dataset of Tirindelli et al. (2002), the data was filtered using wavelet analysis (as described in Cuomo et al. 2003) to account for dynamic effects in the instrumentation. The forcing consisted of a slam ming force and a quasistatic force. From this filtered data set, the values of the maximum positive and negative quasistatic force and the maximum slam ming force were identified Each was calculated as the average of the 4 highest loadings experienced in 1000 waves (F1/250). The wave field was composed of a JONSWAP spectr um. max-ZC)/ZD, a newly introduced parameter where ZD is he water depth Foregoing the power curve fits and the basic wave force parameter used in McConnell et al. (2003), the vertical an d horizontal slowly -varying forces are now nonsA (a buoyancy force based parameter) and linear fits are used for the predictive relationships (Equation 2 -24) where a and b are coefficients. qs-1/250 maxC SDF ab (2 -24) No rela tionships for the coefficients were provided, though a table of values was given for the model setup used. For the slamming force, a straight multiplying factor relationship was developed based on the corresponding positive quasistatic force (Equation 2-25) where a is an

PAGE 71

71 empirical coefficient. No relationship for the coefficient was derived though a table of values for the model setup was. slam 1/250qs+ 1/250FaF (2 -25) Comparisons between Equation 2-25 and select existing methods were calculated for the quasistatic forces of the experimental data. Equation 2-25 showed better fit to the data than the other methods tested. Da Costa and Scott ( 1988) Da Costa and Scott (1988) examined the failure of the Jones Island East Dock in Milwaukee, Wisconsin, which failed under wave action from a moderate storm on Lake Michigan. The dock was a cantilevered concrete structure extending out from a vertical wall over the lake A hindcast was done to determine the wave characteristics and sea state. From this data it was found that the methods developed by El Ghamry (1971) and French (1979) failed to predict the failure of the dock for the given meteorological conditions. In analyzing the failure, it was determine d that the shortduration slamming force was responsible for the failure as the existing dock design was able to withstand the slowly -varying wave forces. At the time, the conclusion was that no existing method was sufficient to predict slamming -type forces. The backing vertical wall was thought to have produced standing wave effects and also contributed to the failure. Sulisz et al. (2005) Sulisz et al. (2005) conducted laboratory experiments and used theoretical methods to study the vibration of deckli ke structures subjected to progressive waves, building off the work done with standing waves by Wilde et al. (1998). Physical model experiments were performed

PAGE 72

72 in 64 m long, 0.6 m wide wave tank at the Institute of Hydroengineering in Gdansk, Poland. The model was rectangular box composed of Plexiglas and stiffened internally by ribs. The support system was combination of strings and springs, while measurements were taken using accelerometers and wave gauges. The structure was tested for free vibrations in air and submerged in water Large variations were found in the vertical direction while the horizontal showed little variation between air and water. Forced vibrations were then measured from running waves past the structure. In the decomposition of the pressure transducer signals, significant forcing was found in the frequency of the wave period and in the structural frequency associated with the submerged structure. Four separate vibration frequency stages were found during the wave inundation cycle with the frequency range in each stage being higher than the wave frequency. In the first stage as first contact between structure and wave is made wave slamming induces very high frequency vibration in the structure as well as producing residual waves in the tank (some with acoustic frequencies). In the second stage the structure is fully submerged and the structure vibrates at the free vibration frequency of the structure submerged in water, lower than in the first range. I n the third stage as water is shed from the structure, high frequency vibrations are produced again in the structure. In the fourth stage where the wave has left the structure, the structure vibrates at the free vibration frequency of the structure in ai r. The slamming force appeared to be what initiated all vibrations throughout the inundation. Previous Work on Flat Plates and Docks In the realm of nearshore structures, some work on dock structures has been done. In terms of experimental testing, the representative model used is almost always a flat plate. As a simple structure, a flat plate provides an excellent basis from which to develop and test initial

PAGE 73

73 theories and their viability. Expansion from the thin plate model to realistic bridge superstructure models, however, requires considerable work El Ghamry (1963) El Ghamry (1963) studied vertical wave forces on docks and conducted physical model experiments in a 105 ft long, 1 ft wide wave tank at the University of California, Berkeley. A simple setup was used with the docks represented by pile supported rectangular horizontal decks. A more complex model setup was also used with beamlike structures added to the underside of the deck to create airtight cavities. In the tests, monochromatic waves were used and quantities of wave height, wave period, and the clearance of the deck above the waterline were varied. No tests were conducted with submerged or partially submerged decks. Tests were conducted with both flat and sloping channel beds. Testin g showed a slowly-varying force-time curve that changed greatly with the aforementioned wave characteristics, especially the wave period. Also shown in the data was an extremely large magnitude, short duration slamming force. The short duration slamming forces were attributed to compression of trapped air between the structure and the wave. Data showed that the slamming forces for cases with air entrapment were an order of magnitude (in some cases two orders of magnitude) larger than those without air entrapment. This large discrepancy in the forces is suspected to be a vibration al problem with the model or the instrumentation (since these ratios do not appear in other experimental results). No equations or predictive methods were presented Wang (1970) Wang (1970) studied vertical wave forces on horizontal plates and conducted physical model experiments in a wave basin at the Naval Civil Engineering Laboratory in Port Hueneme, California. In the tests, random waves generated from a central plunger were used with the

PAGE 74

74 model situated at four different locations spaced in an increasing radial line from the plunger. The waves used were meant to simulate explosio n-generated waves. Wave heights ranged from 0.16 ft to 0.50 ft with the cleara nce height varied from 0 ft to 0.125 ft above SWL (no tests were done for submerged or partially submerged plates). Pressure transducers were used to measure pressures at various lo cations in the plate. Testing showed a slowly-varying force along wi th a short-duration impact force. Wang (1970) produced Equation 2-26 for the slowly-var ying force on the underside of the plate, where P is the maximum pressure, is the unit weight of water, max is the maximum wave crest elevation, ZC is the clearance of the pl ate relative to SWL, and C is an empirical coefficient between 1 and 2. maxCPC Z (2-26) The equation essentially stated that the fo rcing is proportional to the difference in hydrostatic pressure between the plate and the wave crest. For the short-duration impact force (Pi), Equations 2-27 was produced by taking the instantaneous ch ange in momentum at the incipient moment of interaction between the wa ve and the plate for progressive waves, where is the unit weight of water, ZD is the water depth, H is the wave height, and is the wave length. The mass used was that derived by Von Ka rman (1929) for landing seaplane floats. 2 C iD 24Z P2 Z Htanh1 2 H (2-27) French (1970) French (1970) studied vertical wave forces on a flat horizontal plate. Physical model tests were conducted in 1.3 ft wide, 98 ft long wave ta nk at the California Institute of Technology. The model setup included a plate that spanned th e full width of the tank. Monochromatic waves

PAGE 75

75 were used and tests were done with the properties of wave height, water depth, and clearance height from still water level varied. Wave period was not varied. Pressure transducers were used to measure pressures at various locations in the plate. The data showed measurements similar to those of Wang (1970) with both a slowlyvarying pressure and a short-duration, high magnitude slamming pressure. For the slowlyvarying pressure, French (1970) concluded the maximum was related t o the hydrostatic difference of the wave crest and clearance height from still water level, similar to Wang (1970) French (1979) French (1979) expanded on the work of French (1970) and conducted a theoretical and experimental investigation of vertical wav e forces on a horizontal plate. Physical model tests were carried out in a 30m long, 0.4m wide wave tank. The physical model was an aluminum plate 13mm thick, 1.5m long, and 0.4m wide. The plate spanned the full width of the tank and was supported from above. To avoid any overtopping effects, a flat vertical plate was added. Solitary waves were run past the plate for a range of wave heights, wave periods, water depths, and clearance heights. No tests were conducted with a submerged or partially submerged plate. Pressures were measured with two pressure transducers mounted flush to the bottom of the plate. In the measured data there appeared two separat e frequencies of forcing, a slowlyvarying component, and a short-duration slamming component. The predictive equations presented by French (1979) were based on conservation of mass and momentum principles. The peak pressure (Ppeak) was equated to the stagnation pressure (Equation 228) with the wave celerity, c, taking the place of the velocity component in the Bernoulli equation and the clearance height, ZC, used as the elevation peak 2 CP 1 cgZ (2 -28)

PAGE 76

76 The slowly varying force was divided into two individual components, the positive uplift pressure, Pu+, and the negative uplift pressure, Pu-. French (1979) used conservation of momentum, with the wave celerity as the characteristic velocity to produce Equation 2 -29, where ZD relative location of the wa ve front along the plate (measured from the leading edge of the plate). CCC 2 uC DDD 2 CC DD D D C DD DZZZ 2 Wx PZ ZZZ c 1c ZZ Z 221 1 ZZ Z (2 -29) The wave celerity was thought to be highly variable and follow conservation of mass once inundation occurred. Therefore the wave celerity was a funct ion of the clearance height and water depth and two separate celerities were used, cp for positive pressures (Equation 2-30a) and cn for negative pressures (Equation 2max is the water surface elevation of the wave if the structure were not present. When compared to the experimental data, the equations produced excessively conservative predictions of the pressures. C 22 p max Cmax C D C D DDDDD D DZ 1 c Z H1H 37 111 Z Z Z2ZZZ4Z4 gZ Z (2 -30a) C CC nD DDD CC DDZ 2 ZZ cZ 1 gZZZ ZZ 112 ZZ (2 -30b)

PAGE 77

77 Isaacson and Bhat (1996) Isaacson and Bhat (1996) conducted a theoretical/experimental study of vertical forces on a rigid, suspended flat plate of negligible thickness. Physical model tests were done in a 20m long, 0.62m wide wave tank at the Hydraulics Laboratory at the U niversity of British Columbia. The rectangular plate model did not extend the full width of the tank and was supported from above. Wave height, wave period, and clearance height relative to still water level were varied among the tests. Water depth was constant. Two load cells, one near the leading edge of the plate and one near the middle of the plate, were used to record the resulting forces. In total, 69 tests were performed. No tests were conducted for a submerged or partially submerged plate. Low p ass filters were used to filter out the slamming force components from the load cell signals, so only the quasistatic force was analyzed. The theoretical expression developed was mathematically similar to that developed by Kaplan et al. (1995). The ver tical force was assumed to be the sum of the time-varying components (time -rate of change of momentum, drag, and buoyancy). Included were the effects of added mass in the calculations. The added mass equation was for a nonspecific structure shape with a single empirical coefficient. The equations were evaluated numerically. Due to the uncertainties associated with the change in added mass as the latter portion of the wave strikes the structure no attempt was made to predict forces beyond the first half of the wave. Selected results wer e presented in table form along with a single plot of predicted versus measured vertical force Current Work on Bridge Superstructures T he recent bridge failures of 2004 and 2005 have renewed interest in the topic. The initial works of the following studi es are ongoing. These studies should be re-evaluated upon completion.

PAGE 78

78 Schumacher et al. (2008) Schumacher et al. (2008) conducted large scale (1:5) physical model experiments in a 104m long, 3.66m wide wave tank at the O.H. Hinsdale Wave Research Laboratory at Oregon State University. The model was an actual reinforced concrete slab with scaled AASHTO Type III girders inc luding diaphragms. Tests were done with the structure braced as a rigid structure and also as a flexible structure in the horizontal direction with springs. Forces in the vertical and horizontal direction were measured with a series of load cells. 12 pres sure transducers were located through the model deck and girders. Tests were done for a variety of ranges of water depth, wave heights, wave periods, and clearance height relative to SWL (some tests were done with partially submerged structures). Testing was ongoing at the time of presentation and no measurements were produced. It was noted though, that peak horizontal and vertical forces do not occur simultaneously, an observation not unreasonable given the absence of buoyancy force in the vertical dire ction Douglas et al. (2006) Douglas et al. (2006) studied wave forces on bridge superstructure decks using the failure of the US 90 Bridge over Biloxi Bay, Mississippi as a case study. Physical model tests were conducted at the Haynes Coastal Engineering Laboratory at Texas A&M University. A scale model (1:15) of the LA1 bridge in Louisiana was used. Lead weights were used to hold the model decks down and the decks did not extend the full width of the tank. A single six component load cell was used to measure forcing. The structure was subjected to a variety of wave heights and two wave periods (1.3s and 1.8s). A theoretical force prediction model, similar to that of French (1970), Wang (1970), and McConnell et al. (2004) was developed. In the model the forcing was divided into a quasi static

PAGE 79

79 component and a slamming component. For the quasistatic component the force was a function of hydrostatic head and an empirical coefficient. For the slamming force, the relation developed by McConnell et al. (2004) was used. Neither method takes into account the effects of wave period or the inherent kinematics of the wave. Numerical Methods Numerical modeling and simulation as well as 3D flow solvers have not been reviewed in depth for the purposes of this study. While these methods can be extremely useful and accurate in specific cases, the application of single models to complex structures can be problematic. The rigidity of the models limits their use. With the future advancement of computing technology an d processing speeds, these models may yet reach full design viability. For works dealing with this topic in conjunction with vertical wave forcing on suspended decklike structures, the reader is referred to Lai and Lee (1989), Ren and Wang (2003), Kleefsman et al. (2005), and Colleter (2008). Foundation for This Work As discussed in Chapter 1, the initial step in the road to analyzing a complex bridge structure is the examination of a flat plate. T he work done in Marin (2009) lays the foundation for the work to follow and acts as both the basis and the initial validation of the method chosen. Marin (2009) Marin (2009) completed a theoretical and experimental study of wave loading on flat plates. The forcing was divided into two main components the slamming force and the quasi static force. For the quasistatic force, a theoretical model was developed based on a Morison type equation system first described in Kaplan et al. (1995). The model was expanded to situations where the wave length and the plate length were of comparable size (similar to that for coastal bridges) and to include partially and fully submerged structures.

PAGE 80

80 The theoretical model for vertical wave for ce was divided into three components. The buoyancy force is given in Equation 2-31, where V S is the time dependent submerged volume of the structure. The drag and inertia components are given in Equa tions 2-32 and 2-33 where w is the time and location dependent water particle velocity, w/t is the time and location dependent water particle acceleration, me is the effective mass (a combination of added and structure displaced mass), me/ t is the time rate of change of effec tive mass, A is the projected area of the structure in the plane of interest, CD is an empirical drag coefficient, CI is an empirical inertial coefficient, and CM is an empirical mass rate coefficient. BuoyancySF gV (2-31) DragD1 FC Aww 2 (2-32) e InertiaMeMm w FCmCw tt (2-33) Due to the variation in struct ure inundation caused by the sim ilar size ratio of wave length to structure width, the inertia force has two terms. The first term is known as the inertial component of the inertia and th e second term as the mass rate component of the inertia. To accurately evaluate the tw o inertia components, a method for determining the effective mass (me) and the time rate of change of effective mass ( me/ t) is required. A method described by Payne (1981) was modified and used to develop the expression for added mass for this problem. The effective mass, which is the sum of the mass of water displaced by the wetted portion of the structure and the added mass, is gi ven in Equations 2-34 and 2-35, where L is the plate length, W* is the wetted wi dth of the plate, and D* is th e wetted vertical thickness of the plate (for a plate with finite thickness).

PAGE 81

81 22 1 4 eS 22 LW m V LW* (2-34) 3 1 2 4 e 22 22W L m WDW t LDW1 tttLW LW* ** ** (2-35) A computer program was written to evaluate the theoretical equations using an overlapping grid system of wave and structure. Stream function theory was used to determine the wave kinematics at each grid intersection. The wave was time stepped through the grid and past the fixed structure and forces calcula ted at every grid element. The elements were then summed for the total forcing at that time step. A total of 292 physical model tests were conduc ted in a 6ft wide, 120ft long wave tank at the Coastal Engineering Laboratory at the University of Florida. The scaled plate model (1:8) was based on the deck dimensions of the failed I-10 bridge over Escambia Bay, Florida. Tests were run for a continuous plate setup (plate extendi ng the full width of the tank) and a finite plate setup (single plate centered in the tank). Tests were run for a wide range of periods, wave heights, water depths, and clea rance heights (including subaeria l, submerged, and partially submerged). Forces were measured using four thr ee-axis load cells located at the corners of the plate. Water surface elevations upstream and dow nstream of the structure were measured using three capacitance-type wave ga uges. All instrument sampling frequencies were 480Hz and monochromatic waves were used. Filters were created based on power spectral density calculations and implemented to remove ambient noise and separate the slammi ng and quasi-static forces. Drag and inertia coefficients in the theoretical model were determ ined using the measured data. Expressions for the coefficients were develope d based on significant parameters. These parameters were the

PAGE 82

82 wave steepness, the relative clearance, ZCmax, and the relative width, Forcing was treated as a per unit length quantity. From the physical model data, empirical relationships for the slamming force were developed with the recognition that further work on the topic was required. Two separate equations were developed, one for a continuous plate and one for a plate of finite length. As a check on the validity of the model and the coefficient expressions, the program was run against the experimental data of Isaacson and Bhat (1996). The model showed good agreement.

PAGE 83

83 CHAPTER 3 THE THEORETICAL QUASI STATIC FORCE The work on wave loading on horizontal flat plates (Marin 2009) outlined in Chapter 2 is the fo undation for this work. The next step is to evolve the theoretical model to horizontal structures with finite thickness specifically simple slab spans and beam and slab spans. The addition of finite thickness to the structure results in an increase in added mass and introduces horizontal forcing to the problem. The problem is examined first through a simple slab span and the theoretical model for quasistatic forcing developed. Exceptions and adaptations for the beam and slab span are then discussed at the end of the chapter. A definition sketch is presented in Figure 31 showing the relevant parameters for a simple slab span. Model Considerations Before developing the theoretical model, it is essential to address the simplifying assumptions and var iables associated with the problem. Some variables may have only a minor impact on the loading or can be reduced to a constant value and therefore neglected in the development of practical design equations. Through the model assumptions and dimensional analysis, the relevant variables are identified Model Assumptions The flow and fluidstructure interaction processes associated with wave impact on structures with comparable widths and wave lengths are complex. For this reason certain simplifying assumptions are made in order to develop a manageable theoretical model: The structure is rigid and horizontal (no slope to the structure). The waves are monochromatic, nonbreaking and two-dimensional. The waves approach the structure normal to the length of the structure (i.e. the waves propagate normal to the roadway).

PAGE 84

84 The effects of the structure on the waves can be accounted for through the experimentally determined drag and inertia coefficients. The total forcing can be treated on a per unit length basis for a given wave and structure. The total forcing is the sum of individual, independent forcing components. These assumptions are discussed below. Rigid structure. From the standpoint of loading, the rigid structure assumption is important with re gards to harmonic frequency effects of the span. This concern, though, is more likely to play an influential role in the slamming force, whose higher frequency is closer to the fundamental modes of vibration for a bridge span. The impact on the quasistat ic force should be minimal (up until the point of structure failure when free motion under forcing is no longer restricted). Monochromatic waves. Monochromatic waves are thought to produce conservative forces compared to random waves with the same overall energy content. Complexities arise, though, when considering the slamming force. Because the slamming force is dependent upon the shape of the wave and the air/surface interaction between the bridge and the wave, smooth bottomed slab spans are subject to large variations in forcing dependent upon the waves dispersive characteristics. It is not clear, then, whether monochromatic waves will generate conservative loading magnitudes versus dispersive wave fields of the same energy content. To address this would require a dedicated multi-frequency wave slamming study for slab spans, and for that reason. Conversely, slamming in beam and slab systems is addressed in detail in Chapter 4 due to the ability of trapped air in the chambers to limit the effect of the initial wave profile by deforming it through dampening. Wave propagation. The waves approaching normal to the leading edge of the structure produce the most conservative forces and moments. For a 2D wave, normal propagation will

PAGE 85

85 create a situation with the least possibility of 3D flow effects and air escape. The angle of incidence ( ) is therefore set as a constant, normal to the structure and neglected from further consideration as a variable. Wave symmetry. While the problem is treated tw o-dimensionally and the forces and moments computed on a per unit length basis, it shoul d be noted that the eff ect of finite structure length on the calculation of added mass must be taken into cons ideration as the variation of the added mass quantity is not linear w ith structure length. However th e impact of this non-linearity only affects the 2D legitimacy for extremely narrow spans (i.e. span length < 5 times span width), a situation that is rare in coastal bridges. Dimensional Analysis For the total wave-induced vertical (FZ) and horizontal (FX) forces on a slab span, 12 independent wave, fluid, and stru ctural parameters were identified as possibly important. These variables are given in Equation 3-1, where W is the slab width (par allel to the direction of wave propagation), L is the slab length (perpendicular to the direction of wave propagation), D is the slab thickness, HR is the railing height, ZC is the clearance height, ZD is the water depth, H is the wave height, T is the wave period, is the fluid density, g is gravity, is wave angle of incidence, and is the dynamic fluid viscosity. ZX RCDFWLDHZZHT g f ,,,,,,,,,,, (3-1) There are, however, a number of other variable s that are important to the problem that are dependent upon these independent variables. Including these de pendent variables and applying the previous model assumptions (i .e., neglecting the wave angle of incidence), the total forcing becomes a function (Equation 3-2) of 14 total variables, where TIN is the inundation time (the

PAGE 86

86 amount of time the wave acts on the structure), is the wave length, and max is the maximum wave crest elevation. ZX RCD INmaxFWLDHZZHT g T ,f,,,,,,,,,,,,, (3-2) The contribution of these variables to the total forcing also varies depending on the situation of loading considered. The variables that are important for the vertical forces are not necessarily the ones important for the horizontal fo rces and vice versa. Taken a step further, variables within the forcing orientation will behave in a similar manner a variable that is important to the vertical drag ma y not be important to the vertical inertia. Dimensionless groups needed to be identified for each forcing mechanism. To this end, a dimensional analysis was perf ormed with the fluid density, gravity, and the wave length taken as the three independent variables. A non-di mensionalized force (Equation 33) was developed from this analysis as a f unction of 11 initial dimensionless groups. These 11 non-dimensional parameters were then further separated and combined to obtain the most meaningful quantities (some of which are established groups in the study of fluid and wave dynamics) for each forcing mechanism isolated. 2 2 CmaxIN RD 3Z gT HZ FWLDHgT g g f,,,,,,,,,, (3-3) Dimensionless buoyancy force Beginning with the simplest component, the buoyancy force (FB), the non-dimensionalized general function becomes a simp ler equation (Equation 3-4), where each of the given groups is either directly from Equation 3-3 or obtained through a combination of them. CminIN BZ T F WH gWLD DTf,,, (3-4)

PAGE 87

87 In this function the force is non-dimensi onalized by the maximum possible buoyancy for a slab span, gWLD. The individual nondimensionalized groups of note are a relative width parameter, W/ a wave steepness parameter, H/ a relative clearance parameter, ZCmin/D, and a submergence parameter, TIN/T. The wave trough elevation, min, is a previously unmentioned variable. It is arrived at by subtracting the wave height, H, from the crest elevation, max. It should be noted that each gr oup in this equation relates to a geometric aspect of the wave or the structure. The absence of dynamic groups in the equation is logical in that the buoyancy is solely a product of the hydros tatic forcing on the structure. Dimensionless inertia and drag forces The non-dimensionalized inertia (FI) and drag (FD) forces (Equation 3-5) are more complex than the buoyancy. Each of the given groups is either directly from Equation 3-3 or obtained through a combination of them. DI CminIN 2 R DF Z T DHW gHLH DHT DgZ,f,,,,, (3-5) In this function the force is non-dimensiona lized by the wave energy across the span, gHL. The individual non-dimens ionalized groups of note are a re lative height parameter, D/H, a wave steepness parameter, H/ a relative width parameter, W/ a relative clearance parameter, ZCmin/D+HR, a submergence parameter, TIN/T, and a unique term, /D gZD. This last term can be evaluated two separate ways. The first way, is to recognize that the term gZD is the shallow water celerity, c, and consider this then to be a representative velocity in the flow. That would make this last term the Reynolds number with the form shown in Equation 3-6a. The second way to look at this term is as a ratio of the Reynolds number and the Froude number with the form shown in Equation 3-6b.

PAGE 88

88 Dd Re Dc DgZDgZ (3-6a) DD DRe (Velocity) Fr D(Velocity) DgZDgZ gZ (3-6b) The Reynolds number quantifies the importance of the inertia forces versus the viscous forces. The Reynolds to Froude ratio quantifies the importance of the viscous versus gravity forces. Because of the size ratio of structure a nd wave, inertia forces dominate the forcing and the Reynolds number is suitably large enough to disregard viscous effects. Dimensionless slamming force The non-dimensionalized slamming force is a similar equation to the inertia and drag forces equation (Equation 3-7). Each of the give n groups is either directly from Equation 3-3 or obtained through a combination of them. SC 2 max DFZ HW gHL WgZ f,,, (3-7) In this function the force is non-dimensiona lized by the wave energy across the span, gHL. The individual groups of note are a wave steepness parameter, H/ a relative width parameter, W/ a relative clearance parameter, ZC/ max, and the Reynolds number. While these groups are similarly defined for the inertia and drag, the magnitude of the slamming force is governed to a large extent by th e shape of the wave. For this work, the analysis of slamming forces on slab structures with physical model testing is limited to empirical relationships. The Buoyancy Force The buoyancy force (FB), which is simply the net hydr ostatic force, is the most straightforward of the force components and acts only in the vertical dire ction. Any submerged portion of the slab will experien ce a buoyancy force equivalent to the weight of the volume of

PAGE 89

89 water displaced by the structure. Equation 38 shows the hydrostatic relation for the buoyancy force, where V S is the submerged volume of the structure, is the fluid density, and g is gravity. BSF gV (3-8) The submerged volume is a time dependent vari able a function of the wetted dimensions of the structure. Because of the similarity in size ratio between the wave and the structure, there will be large portions of time where the struct ure is not fully inundated. The buoyancy is therefore redefined for V S*, the wetted volume. With the assumption propagation normal to the orientation of the slab, the final expression for the buoyancy at point during wave/structure interaction is rewritten (Equation 3-9), where AS* is time dependent wetted cross sectional area. BSF gLA* (3-9) Even though the slab is a rectangular object, the surface area quantity is used in place of the wetted width and wetted thickness. The reas on for this is that the size ratio between the structure and the wave creates non rectangular submerged cross-sec tions (Figure 3-2) due to the slope of the wave and the potential thickness of the structure. A purely rectangular cross section would not be able to catch this. The Drag and Inertia Forces The forces exerted on a completely submerge d body in an accelerating fluid are due to drag and inertia. At first glance, these are both lower frequency (in the range of the wave period) forces derived from introducing the stru cture into the kinematic wave field. The sum of the drag and inertia forces as the total forcing was first introduced in the Morison equation (Morison et al. 195 0). Dealing with the forces on piles and elements that are small in comparison to wave, the Morison equati on does not account for the effects of localized drag components, time-varying mass quantities, or the distribution of variable kinematics over

PAGE 90

90 the entirety of the structure. This was update d to a degree by Kaplan et al. (1995) and Isaacson and Bhat (1996) for flat plat es with small size ratios. In the case of the drag and inertia forces, while both are taken into account for the total forcing, one is predominant over the other depending on the orient ation of forcing considered. For slab spans, the inertia forces are the domin ant for vertical forcing regardless of external parameters. For horizontal forcing the dominan ce of inertia/drag is a function of certain parameters a combination of sl ab thickness (including possible ra ilings) and clearance heights. The Drag Force The drag force (FD) is composed of shear (skin frict ion) and normal (pressure drag) components with the normal com ponent resulting from flow separation around the body. Most analytical treatments of drag combine the two co mponents together with an expression that is proportional to the projected area of the structure, the mass density of the fluid and the square of the approach velocity. The experimentally dete rmined constant of proportionality is a drag coefficient, which in general is a function of the structure shape, the Reynolds Number (based on the structure width and approach ve locity), surface roughness, etc. The general expressions for both horizontal (FDX) and vertical (FDZ) drag force are given (Equation 3-10a and 3-10b), where CDX is the horizontal drag coefficient, CDZ is the vertical drag coefficient, AX is the projected area of the submer ged slab in the vertical plane, AZ is the projected area of the submerged slab in the hori zontal plane, u is the time and position dependent horizontal water particle velocity, w is the time and position dependent vertical water particle velocity, and is the fluid density. DXDXX1 FC Auu 2 (3-10a)

PAGE 91

91 DZDZZ1 FC Aww 2 (3-10b) For a rectangular slab with waves propagating normal to it, the wetted dimensions of the structure can be substitu ted into the projected areas variab les (Equations 3-11a and 3-11b), where W* is the wetted width and D* is the wetted thickness. DXDX1 FC LDuu 2* (3-11a) DZDZ1 FC LWww 2* (3-11b) Wetted dimensions are time dependent variab les that represent only the submerged portion of a specific dimension. For structures and waves of similar size, these parameters are more physically meaningful than the fu ll structure dimensions. In some instances, these parameters represent combinations of multiple variables fo r example, the wetted thickness may include the slab thickness plus the railing height. This concept of wetted dimensions is important due to the fact that at a given point in time during wave/structure interaction, portions of the structure may not be inundated at all due to the relative sizes of each. This play s out in three interesting scen arios regarding the drag force (horizontal and verti cal) and a partially inundated structure. Partially inundated horizontal drag scenarios In the first scenario, as the wave inundates th e slab, flow separation occurs at the head of the structure, but there is no region of low pre ssure flow separation at the opposite end of the structure (Figure 3-3). As such, the pressure drag and the drag coefficient at this point are based on the difference in pressure between the pressu re along the portion of th e structure experiencing flow and the atmospheric pressure.

PAGE 92

92 In the second scenario, if the ends of the structure are protruding from either end of the wave, the structure is only experiencing a horizontal shear drag force (Figure 3-4) over the area of the structure that the wave is propagating. In essence there is no separation at this point because the size of the structure may restrict the wave from flowing around it. The drag coefficient would fluctuate throughout. In the th ird scenario, a s the wave propagates away from the structure, the tail end of the structure may be experiencing kinematic flow opposite of the propagat ion direction (Figure 3 -5). This is similar to the first scenario where there is separation, but no spec ific region of lower pressure flow on the opposite end of the structure. As before, the drag force and coefficient must be based on the pressure difference between the portion of the structure experiencing flow and the atmospheric pressure. Partially inun dated vertical drag scenarios In the first scenario, as the wave inundates the bridge, flow separation occurs at the head of the structure, but only around one end of the structure. As flow separates around the front face, part of the flow is also deflect ed upstream along the unders ide of the structure (Figure 3 -6). As such, the pressure drag and the drag coefficient at this point must be tempered by the fact that flow separation occurs as if a structure is being slowly introduced into a flow. In the second scenario, if the ends of the structure are protruding from either end of the wave, the structure is experiencing no flow separa tion at all, but simply flow deflection (mainly upstream but also downstream) along the underside of the structure (Figure 3 -7). In essence, the structure is acting as a massive deflector plate. A drag force may not even be considered viable here, as there is no real drag action occurring because there is no actual flow around the structure

PAGE 93

93 In the third scenario, as the wave propagates away from the structur e, the tail end of the structure may be experiencing kinematic flow sim ilar to the first scenario (Figure 3-8). As before, the drag force and coefficient must be mo dified by the fact that the vertical flow is occurring over only one half of se paration and one half deflection. Over the full cycle of loading for all of the scenarios (horizontal a nd vertical), the drag coefficient and force magnitude will vary signific antly, where each drag coefficient needed is a function of the local conditions along the structure and the type of interaction that occurs. The Inertia Force The inertia force is a normal force that re sults from the acceleration-induced pressure gradient field in the vicinity of the structure. The force component is proportional to the time rate of change (the derivative) of the linear mome ntum of the flow. For structures that are fully submerged or very small in comparison to the wave, the common representation of the Morison equation is often used (Equation 3-12), where CM is an inertia coefficient, is the fluid density, V is the volume of the fluid affected by the presence of the structure, and U is the velocity of the flow. InertiaMd FC VU dt (3-12) For the case of simple slab span, complexitie s in the problem arise from the intermittent submergence of the structure due to the comparab le size of the wave and the structure. This intermittent submergence creates a time dependent mass term (stemming from the variable volume term), a forcing component not present in a fully submerged structure and left out of the Morison equation (though, the term is not necessary in the Morison equation because the Morison equation is intended for use with slende r elements and/or structures significantly smaller than the waves of interest).

PAGE 94

94 Kaplan et al. (1995) included the time depe ndent mass component, but used averaged water velocities and accelerations over the width of the structure as well as mass quantities derived for a thin plate only. For structures and waves of comparab le lengths, though, the variation in wave kinematics over the width of the structur e is significant and averaged kinematics cannot be used. Carrying out the de rivation of Equation 3-12 for a variable volume quantity, we see the development of a second term in brackets in Equation 3-13, where U/ t is the acceleration of the fluid field and V / t is the time rate of change of the affected fluid volume. InertiaMUV FC V U tt (3-13) Time dependent mass The next step is to consider how these separa te inertia terms are rela ted to the changes over the inundation a bridge structur e would be likely to see. Consider two separate scenario s (Figure 3-10) of an object submerged in an unsteady flow, U(t) 1) a sphere of constant radius, R, and 2) a sphere of ch anging radius, R(t). If drag is disregarded, this flow field imposes a force upon th e sphere equivalent to the time rate of change in momentum of the flow field as defined by Equation 3-14. Equation 3-14 is the same as Equation 3-12, except the water is now assumed incompressible (constant density) and volume quantities have been replaced by mass quantities, specifically me, the effective mass of the complete system. The effective mass is the co mbination of the mass of water displaced by the structure, ms, and the added mass, ma (sometimes known as the virtual mass). InertiaMe MsaF CmU(t) CmmU(t) tt (3-14)

PAGE 95

95 To evaluate this equation, the mass values are needed. The structure displaced mass (Equation 3-15) and the added mass (Equation 3-16) for a sphere are shown with the formula for the added mass of a sphere taken from Sarpkaya (1981). Interestingly, the added mass for a sphere is half the mass of the sphere itself, but it is important to realize that this is not true for all shapes and in this case just coincidence (as me ntioned earlier, the added mass of an object is a function of the shape, dimensions, and orie ntation of the structure in the flow). 3 ssphere4 m R 3 (3-15) 3 asphere2 m R 3 (3-16) Substituting the mass quantities into the for ce expression (Equation 3-14) we can now evaluate the time derivative of th e momentum (Equation 3-17) for th e first scenario of a constant radius, R. Taking the derivative, it is found that the only time dependent term is the unsteady flow, U(t), since both the radius and the fluid density are cons tant. The equation essentially reduces to force equals a mass times acceleration. 33 3 42 MM 33dU F C R RU(t) C2 R dtt (3-17) If the mass quantities are substituted into the force expression (Equation 3-14 for the second scenario of a sphere with changing radius, then when the derivative is taken a much different result is produc ed (Equation 3-18). 33 32 42 MM 33dU R F C R(t) R(t)U(t) C2 R6 R(t)U dt t t (3-18) With a changing radius, the mass quantities are no longer constant so there are two time dependent terms creating two separate momentum components. The first term in Equation 3-18 is the more common mass times acceleration term, identical to the case of the constant radius

PAGE 96

96 sphere. The second term is a ve locity times a mass rate term, where mass rate is the rate of accumulation of affected fluid mass in the system. Hereafter in this document, the first term in the linear momentum derivation will be referre d to as the inertial component (FI) of the inertia force and the second term will be referred to as the mass rate component (FM) of the inertia force. Defining what the mass rate component of the inertia force means from a physical standpoint can be diff icult. What is important to note though, is that the rate at which this change takes place is extremely important being defined by the time period over which the change in mass takes place (i.e., rate of inundation). An even more diluted intricacy is the concept of negative mass rate. If effective mass in the system is decreasing (as represented by a wave receding from a structure) the time rate of change of the effective mass becomes negative. T here is debate as to whether the reduction in momentum in the flow would correspond to merely a cutoff in loading on the body due to this component or an actual reversal in loading. The inertia l and mass rate force equations The inertia force is the result of the taking the time derivative of the linear momentum of the system in the horizontal (Equation 3-19a) and vertical (Equation 3-19b) directions, where mex is the effective mass of the system in the horizontal direction, mez is the effective mass of the system in the vertical direction u is the time dependent horizontal water particle velocity, w is the time dependent vertical water particle velocity, and CM is the empirical momentum coefficient for the structure. XInertiaMexd FCmu dt (3 -19a) ZInertiaMezd FCmw dt (3 -19b)

PAGE 97

97 In the original M orison Equation, the coefficient was included in the equations to account for the added mass of the system. Since a functional relationship for added mass quantity is developed here later, the coefficient included here exists in order to account for the increased effect the structure has on the wave field during interaction due their similar size ratio Carrying out the derivation of the equations leads to equations for the horizontal (Equation 3-20a ) and vertical (Equation 3 -20b) directions where du/dt is the horizontal water particle acceleration, dw/dt is vertical water particle acceleration, and dmex/dt is the time rate of change of effective mass in the horizontal direction, and dmez/dt is the time rate of change of effective mass in the vertical dir ection ex XInertiaMexdm du FCmu dtdt (3 -20a) ez ZInertiaMezdm dw FCmw dtdt (3 -20b) The forces are now separated into individual components to isolate them and allow them to be dealt with systematically As stated earlier, the first term of the inertia force is referred to as the inertial force (FI) and the second term is referred to as the mass rate force (FM). The momentum coefficient is similarly divided into separate entities. In the inertial component, the coefficient (CI) is to handle discrepancies in the effect of the structure upon the wave field due to increased size, specifically for the added mass quantity. In the mass rate component, the coefficient (CM) is to handle not only the added mass discrepancy, but also the difficulty in as sessing rate of change, specifically with regards to the growth rates of wetted dimensions and the degrading wave field. The individual component equations are given for the horizontal and vertical inertial forces (Equation 3-21a and 3-21b) and mass rate forces (Equation 3-22a and 3-22b).

PAGE 98

98 IXIXexdu FCm dt (3-21a) IZIZezdw FCm dt (3-21b) ex MXMXdm FCu dt (3-22a) ez MZMZdm FCw dt (3-22b) The effective mass and mass rate equations In order to evaluate the previously define d inertia force equations (Equations 3-21 and 322), a means for determining the effective mass and the time rate of change of the effective mass (or mass rate) is needed. This can be done by breaking the effective mass into its two components, the structure displaced mass and the added mass, and deriving expressions for each. The structural displaced mass is fairly st raightforward as it is only based upon the complexity of the structure shape. The displa ced mass will be equivalent to the mass of the volume of water that is directly taken up by the presence of the st ructure within the fluid, or simply the submerged volume, V S. Similar to the buoyancy force, this equation reduces to a straight volume times a density with the wetted volume, V S*, used to account for the intermittent submergence. Also similar to the buoyancy force, because we are dealing with waves approaching normal to the slab, the wetted length will always be a constant value and can be pulled out of the wetted volume. This leaves an expression for the structure displaced mass (Equation 3-23) where AS* is the wetted cross sectional area. It should be noted that the structure displaced mass has no directionality, i.e. it has th e same value in both the vertical and horizontal directions. sSm AL* (3-23)

PAGE 99

99 Deriving t he added mass is a more involved process. For many simple shapes, theoretical equations for added mass have been developed from potential flow theory and can commonly be found in offshore structures texts such as Sarpkaya (1981) These equations are often presented as a two dimensional shape oriented a certain way in the flow. Using coefficients based on the shapes geometric ratios, the added mass is then calculated per unit length orthogonal to the direction of the flow However, because these relationships were developed from shapes that were infinitely long (which allows the subsequent use of a per unit length application at the cost of specific orientations ), two separate proble ms arise. The first problem is that the quantities predicted by the equations change with orientation of the object regardless of whether the orientation has an effect. Turning the object in the flow does not alter how the flow reacts to the pr esence of the object (Figure 311 ) because it will still see the exact same dimensions of the object with the same face projection. Different values would be calculated for a 4ft wide plate taken per unit length out to 6ft and a 6ft wide plate taken per unit length out to 4ft, despite the fact that the flow recognizes no difference between the two. The second problem is that the per unit length designation makes the equation linear in one direction. While this would suggest that added mass has an additive property, experimental tests show that it is no t (Yu 1945). If you had two separate objects where the second object was composed of a quantity of the first object laid end to end, then by virtue of the equations in the literature, the second object would just be that many times the add e d mass of the first (Figure 3 12). This also has been shown to be false in experimental tests (Payne 1981). So f or a rectangular slab of finite thickness, an expression must be developed that calculates the added mass but satisfie s the condition that the orientation of the flow is arbitrary and the length of the slab is not linearly additive, but consistent with established experimental

PAGE 100

100 data. It would also be useful to substitute in a relationship in order to remove the need for a coefficient table. Payne (1981) recognized the discrepancy in orientations between existing added mass equations and developed a modifier to be us ed in conjunction with those equations that eliminated these inconsistencies. This expression (Equation 3-24) eliminated the linear relationship of a per unit length sy stem and removed the orientati on restriction of the normal face in the flow produced the same result. 22L PaynesMultiplier LW' (3-24) Before applying Paynes factor, an expression for the added mass of finite thickness slab must be developed. From Sarpkaya (1981) a pe r unit length expression for a rectangle is given in Equation 3-25 with the orientation as shown in Figure 3-13, where A is a dimension of the face the perpendicular to flow, B is the dime nsion of the face para llel to the flow, and (A,B) is the mass coefficient based on geometric ratio. 2 aA m 2 (3-25) The inputs and dimensions for this expression ar e now adjusted to reflect the variables of interest for a slab span, with the vertical direction chosen as the initial flow direction of interest. The dimension A is then replaced by the slab wi dth and the dimension B is then replaced by the slab thickness. The length of the slab L is i nput to remove per unit length, giving Equation 3-26 for added mass in the vertical direction. Adding Paynes multiplier gives Equation 3-27. 2 1 a 4m WL (3-26) 22 1 a 4 22WL m LW (3-27)

PAGE 101

101 The next step is to replace the coefficient with a relationship that covers the range of geometric ratios of interest for the problem. For completeness, the range chosen was 1/ to Empirically determined coefficients were given in Sarpkaya (1981) as functions of the ratio A to B (or in the case of the span, D to W). Plotti ng these coefficients ag ainst the ratios of the corresponding widths and thicknesses gives the graph in Figure 3-14. Curve fitting was done on this data to produce a simple expr ession (Equation 3-28) for coefficient as a function of the thickness to width st ructure ratio, D/W. 2 5 1 2D 1 W (3-28) Substituting this expression for the coefficient into Equation 3-27 produces the complete added mass expression (Equation 3-29) for a rectangular slab of finite thickness submerged in a flow in the vertical direction. This expressi on was checked against experimental data from Yu (1945) and showed excelle nt fit (Figure 3-15) 8 2 5522 2 11 a 48 2222WLWDL m LWLW (3-29) Once again the intermittent submergence must be taken into account. Both the width and the thickness of the structure are time dependent parameters and must be replaced by their wetted counterparts W* and D*, respectively. The wette d length is still constant due to the normal propagation of the wave. The final time depende nt expression for the added mass is now given in Equation 3-30. 8 2 5522 2 11 a 48 2222WLWDL m LWLW** ** (3-30) Adding the structure displaced mass (Equation 323) to this gives the complete expression for the time dependent effective mass of the stru cture (Equation 3-31a). For the horizontal flow,

PAGE 102

102 it easy to note that the orientation of our ini tial dimensions A and B and L from Figure 3-13 are simply rotated 90 degrees about the axis that is the slab length. Wette d widths and thicknesses may simply be interchanged with each other to produce the complete time dependent expression (Equation 3-31b) for the effective mass in the horizontal direction. 8 2 5522 2 11 ezS 48 2222WLWDL m AL LWLW** ** (3-31a) 8 2 5522 2 11 exS 48 2222DLDWL m AL LDLD** ** (3-31b) To obtain the time rate of change of effective mass quantities needed for the mass rate force calculation, the time derivatives (Equati on 3-32a and 3-32b) of the effective mass expressions given in Eq uation 3-31 were taken. 3 223 2 55 22 ezS 3 32 5 22WL1W1DW1WD 2t5Wt20Dt WL mA L tt WL1W1DW 4t8Wt WL****** ** **** (3-32a) 3 223 2 55 22 exS 3 32 5 22DL1D1WD1DW 2t5Dt20Wt DL mA L tt DL1D1WD 4t8Dt DL****** ** **** (3-32b) One of the benefits of the normal wave pr opagation assumption can be seen here. For scenarios other than wave propaga tion normal to the structure, the wetted length rather than the full length would be required, adding a third time dependent variable and leading to an extraordinarily complex derivative.

PAGE 103

103 By inserting the effective mass equations (Equation 3-31) and the effective mass rate equations (Equation 3-32) into the inertia force equations (Equation 3-21 and Equation 3-22) the forces can now be calculated. The full expression for the total inertia forces (inertial and mass rate) in the vertical direction is given in Equation 3-33 for levity. 8 2 55 3 222 2 11 IZMZIZS 48 2222 2 5 2 S MZ MZ 3 22 5 3 32 5 MZ 22WLWDLdw FFC AL dt LWLW 1W1DW 2t5Wt A WL C Lw C w t WL 1WD 20Dt WL1W1DW C 4t8Wt WL w (3-33) Evaluating the effective mass and mass rate equa tions, though, can be difficult due to the rather complex nature of the time dependence of the effective mass and its interaction over the full inundation cycle of the structure. For example as the wave initially strikes the slab, the effective mass is changing with time (Figure 3-16). If the trailing edge of the wave reaches the structure before the leading edge exits then there is a period of time when the effectiv e mass is mathematically constant. For a wave wider than the structure at elev ation, a similar period of time occurs when the effective mass is mathematically constant as the slab is fully submerged. The effective mass becomes time dependent once again as the wave exits the structure. If one adheres to the strict mathematics of the equations, forces would cutoff and cutback with unnatural, jagge d characteristics. Application of the equations when dealing with these problems is discussed in the development of the numerical model.

PAGE 104

104 Adjustments to Theory for Beam and Slab Spans For a beam and slab span, some adjustments must be taken into account when applying the previously developed equations to account for the differences in structural characteristics. A definition sketch of the beam and slab span problem is given in Figure 3-17. The major difference between the slab span and the beam and slab span is the presence of girders and overhangs, both of which introduce elements of air trapping. Overhangs also introduce plate like characteristics. To assess these effects, previous assumptions and analysis are re-examined. Updated Model Assumptions The majority of assumptions between the slab span and the beam and slab span cases are identical. However, two assumptions must be revised that the structure is rigid and that the total forcing can be assessed as the sum of the individual and independent forcing components. They are revised as follows: The structure is semi -rigid and horizontal, as areas of sealed air respond dynamically to forcing. The total forcing is the result of th e interaction of the individual forcing components which are dependent upon each other. The purpose and effect of some of these new assumptions are further addressed below. Semi -r igid structure. The presence of trapped air between the structure and the wave creates a dynamic and compressible interface. The air, which has no means of escape between the girders, dampens any forcing between the structure and the wave. While this may reduce the loading, it also creates the possibility that the high frequency slamming forcing which may have been beyond the natural frequency of the structure is being lowered perhaps to the point where the structure has time to respond to it. Dependent forcing Because any trapped air is compressible, the wetted volume of the structure at any given moment in time is now not only a function of the wave location (a function

PAGE 105

105 of time) but also of the forcing. For exam ple, the buoyancy is a function of the submerged volume, but because the volume of trapped air is a function of the pressure in the chamber (i.e., the wave loading in the given chamber), the buoyan cy cannot be calculated separate of the other vertical forces. The most disconcerting aspect of this interaction is that the slamming force becomes integral to the overall forcing. Th e equations must now be solved simultaneously. Updated Dimensional Analysis Adding the new assumptions to the old, 17 important independent and dependent wave, fluid, and structural parameters are now identified for the total vertical (FZ) and horizontal (FX) loading on a beam and slab span. These variab les are given in Equation 3-34, where the new variables are the girder height, HG, the girder number, NG, and the overhang width, WO. ZX RGGOCDINmaxFWLDHHNWZZHTT g f ,,,,,,,,,,,,,,,, (3-34) Another dependent variable that is not include d but plays a critical role is the girder spacing, WG. This variable is a direct function of the span width, girder number, and overhang width (Equation 3-35). On beam and slab spans, overhangs are not separate structures attached to the bridge. They are the continuous extens ion of the deck slab, so when specifying an overhang width, this does not cha nge the overall width of the span. O G GW2W W N1 (3-35) A dimensional analysis was performed with th e fluid density, gravity, and the wavelength taken as the three independent va riables. From the analysis, the non-dimensionalized force was found to be a function (Equation 3-36) of 14 dimensionless groups. 2 2 CmaxOGGIN DR g 3Z WWHgT ZH FWLDHgT N g g f,,,,,,,,,,,,,, (3-36)

PAGE 106

106 Again, these 14 non-dimensional parameters do not represent the most physically meaningful quantities, but by combing some of the groups we obtain more meaningful parameters. Because the trapped air scenario cr eates interaction between the different forcing mechanisms, it is anticipated that all forcing components will share certain common parameters related to this issue. The updated dimensionless forces for a beam and slab span are given for buoyancy (Equation 3-37), inertia and drag (E quation 3-38), and slam ming (Equation 3-39). GCm i n I N B GWZ T FWH gWLDH DHTf,,,, (3-37) DI GORGCminIN 2 RG DF WWDHHZ T HW gHLHWH DHHT DgZ,f,,,,,,, (3-38) SGOCCG 2 maxmax DFWWZZH HW gHLH WgZ f,,,,,, (3-39) Several new parameters have been introduced to the problem. Relative clearance and relative thickness parameters have been altered du e to the addition of the girder height. The presence of the overhang creates a separate relative width, WO/W, and relative clearance, WO/ parameter as well. The most significant new parameter, though, is the WG/H term. This common parameter among the loading components represents a comparison of the trapped air pocket to wave size ratio. These considerations and the direct application of the equations for the beam and slab span case are looked at further in Chapter 5. Next, the equations developed for the individual force components are updated. Updated Force Equations The buoyancy equation is the same as in the case of the slab span however, now the air trapped between the girders will fluctuate be tween contributing to the buoyancy (when the

PAGE 107

107 chamber between the girders is sealed) and not (when the chamber between the girders is open to the atmosphere) So the crosssectional area is now also a function of the air entrapment an d not just the wave profile T he drag equations are also identical to their slab span counterparts but what has changed again is where they are valid F or times when the chambers between the girders are sealed, the surface that drag force is acting upon can be air rather than solid. When the chambers are open, the wave can then also create drag on individual girder elements. Also of note is that the presence of girders, overhangs, and rails create larger projected areas in the wave. Before re-visiting the actual force equations for the inertial and mass rate components, the mass equations are re-examined. Again, the submerged crosssectional ar ea is tempered by the reduction/addition of possible trapped air between the girders and treated the same as in the buoyancy case. Similar eff ects must now be considered for wetted width as well. The addition of alternate structural components (girders, rails, etc.) c alls for a more complex wetted thickness that ma y vary significantly over time depending on these parameters and air entrapment. The key takeaway when examining the beam and slab span is the complexity added to the problem by the sealing and unsealing of trapped air between the girders. The time dependent wetted dimensions and cross section areas necessary to evaluate the previously developed equations is impossible without the use of computer assistance. The development of the numerical model for evaluating the theoretical equations is looked at in Chapter 5.

PAGE 108

108 Figure 3-1. Slab span overview. Figure 32. Partially inundated span height.

PAGE 109

109 Figure 3-3. Inundation of leading edge only in horizontal drag flow Figure 3-4. Inundation of midsection only in horizontal drag flow. Figure 3-5. Inundation of trailing edge only in horizontal drag flow

PAGE 110

110 Figure 3-6. Partial deflection by leading edge in vertical drag flow Figure 3-7. Complete deflection by midesction in vertical drag flow. Figure 3-8. Partial deflection by trailing edge in vertical drag flow

PAGE 111

111 Figure 3-9. Springmass-dashpot system. A B Figure 3-10. Spheres in an unsteady flow. A) C onstant radius B) V ar iable radius.

PAGE 112

112 Figure 3-11. Existing added mass equations predict two separate masses for the same object rotated 90 in the flow. Figure 3-12. Existing added mass equations predict two separate masses for a object and the same object made up of three separate objects. Figure 3-13. Orientation of rectangle in flow for added mass equation from Sarpkaya (1981).

PAGE 113

113 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 0 2 4 6 8 10 Added Mass Coefficient D*/W* Equation 3 28 Sarpkaya (1981) Figure 3-14. Comparison of predictive Equation 3-28 to added mass coefficient Sarpkaya (1981) to predictive E quation 3-28. 130.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0 0 1 2 3 4 5 6 Added Mass (g) Wetted Thickness (cm) Equation 3 30 Yu (1945) Figure 3-15. Comparison of added mass predictive Equation 3 -30 to experimental data from Yu (1945).

PAGE 114

114 A B C D Figure 3-16. Periods of changing added mass during inundation. A) Structure entering wave field. B) Structure wider than wave. C) Structure narrower than wave. D) Structure emerging from wave field.

PAGE 115

115 Figure 3-17. Beam and slab span overview.

PAGE 116

116 CHAPTER 4 THEOR ETICAL SLAMMING FORCE The slamming force is the most difficult component of wave loading to dissect. When encountered previously, studies have tended to reduce the slamming to empirical relationships dismissing it as a stochastic process that cannot be treated by theoretical means (U .S. Army Corp of Engineers 2006). As stated earlier, the development of a slamming model for smooth bottomed slab spans requires a separate detailed investigation of multi frequency wave effects, but as a first step, observ ations on slamming are assessed and a high level predictive slamming concept introduced. This concept is then developed for beam and slab spans, where air pockets remove the importance of wave profile/flatness. Slamming Observations The observations on slamming are broken down directionally into horizontal and vertical occurrence. While the focus of the work here lies in vertical slamming, horizontal slamming in the coastal environment has been studied in depth for structures such as caissons, sea walls, bulkheads, etc. and their conditions for occurrence are well documented (U.S. Army Corp of Engineers 2006). Despite their different orientations, the processes that appear common to horizontal slamming are relatable to vertical slamming Horizontal Slamming Waves propagating into a vertical wall can be categorized (Wood et al 2000) into three separate groups progressive (Figure 4-1), near-breaking/breaking (Figure 4 -2), and broken (Figure 4 -3). Progressive waves and broken waves produce a smooth, slowl y varying force similar to the quasistatic force. Near -breaking/breaking waves, though, produce high frequency slamming forces in addition to the slowly varying force of the progressive and broken waves. Representative force profiles for near -breaking/breaking versus progressive or broken are given

PAGE 117

117 in Figure 4-4. Since broken waves are outside the scope of this work, focus will be directed to the progressive wave when considering the nonslamming occurrence case. From the wave profiles given in Figures 41 through 4 -3, t he most recognizable difference between those that produce slamming and those that do not is the slope of the wave face. The slope of the wave face controls two distinct physical interactions between the vertical wall and the wave the rate at which the momentum of the wave is transferred between the wall and the wave and the rate at which air vacates the space between the wall and wave. Looking at the momentum transfer rate, t he steeper the wave face for a constant wavelength, the shorter the period over which the momentum exchange over the full wave height takes place. For a mil der sloped wave this excha nge is slow as the reflection of the wave off the wall occurs over the gradual rise of the water surface at the wall. Taken to t he next step, when the near-breaking wave (consider a wave face near or even past 90) is exchanging momentum at the wall, the period this happens over is extremely high frequency. The near vertical steepness of the near breaking wave brings air entrapment into play as well When wave b reaking is imminent the wave creates a pocket between the fluid and the structure where air is trapped or entrained in the wave This trapped air then does three things. The first thing is that air acts as a pseudoexten sion of the structure because the air cannot completely escape This expedites the momentum exchange, though at a dampened rate. The energy exchange once again takes place over a high frequency and slamming occurs. The second thing the air does is compress. Behaving like a spring, the air dampens the loading and introduces pulsating characteristics. The final thing the air does is escape. At some point, the compression of the air is released through the back side of the wave crest, a mechanism known

PAGE 118

118 as blowout or bounce back which causes a pulsating force on the wall similar to a spring being released. In observations and laboratory measurements, this situation was found to create the highest slamming forces (U .S. Army Corp of Engineers 2006). These t rapped air cases proved difficult to predict due to large variances in slamming magnitude for nominally identical waves (Wood et al 2000) a situation similar to that of vertical slamming on smooth-bottomed str uctures such as slabs and flat plates. Vertical Slamming The vertical slamming force manifests differently depending on the structure in question, or specifically, the structure surface with which the wave is making contact. Here there are two separate surfaces of interest the smooth-bottomed underside of the flat plate or slab span and the chambered underside of the beam and slab span. For the smooth-bottomed surface, slamming occurs a single time during the initial stage of structure wave contact (F igure 45) In lab tests, though, slamming does not occur for every single wave that strikes the structure. It is not uncommon for slamming to be absent for one wave in a train and then present the very next one (Figure 4 -6). In the past, this randomnes s often led to the conclusion that slamming was indefinable through parameterization. This same randomness, though, was present in horizontal slamming. The corollary to be drawn is that the shape of the wave at the face of the structure (a highly variabl e parameter) governed slamming occurrence. For the vertical orientation, the important wave shape/slope has just been rotated 90 from the horizontal. For the chambered surface of the beam and slab span, though, the slamming force is far more consistent. For all waves in a given train, if one wave generates slamming then all subsequent waves can be observed to generate it. Also, the chambered underside produces

PAGE 119

119 slamming in each pocket over the full course of inundation, creating a series of slamming spikes that coincide with the number of pockets (Figure 4-7). The presence of air removes the uncertainty of the wave profile shape. O n the e ffect of c learance To see the effect of clearance, a structure is moved through the water column for a given wave height (Figure 4-8) S tart ing with a structure that is suspended above the maximum elevation of the wave crest, the elevation of the structure is then lower ed to the point of initial contact. Slamming may now occur The slamming increases with decreasing clearance until a maximum is reached near SWL After this point, the slamming force drops off sharply to zero as clearance continues to decrease. Regarding the clearance height where slamming no longer occurs, it was found in the physical model tests that this elevation never falls below the wave trough, providing an easily identifiable (if not obvious) slamming boundary. Although this seems to be a common sense observation, it highlights a key component of vertical slamming breaks in continuity of the structure/fluid interface If the elevation of the structure is required to be above the minimum trough elevation, then at some point before slamming occurs (even if the structure begins submerged) it must be possible for air to enter the space betwe en the structure and the wave. O n the e ffect of wave shape Another observation taken from the physical model tests is the occurrence/non occurrence of slamming The frequency of occurrence of slamming increased with the degradation of the monochromatic wave field. In a controlled environment, the waves produced in a tank are purely monochromatic and exceedingly well structured as they are usually generated from perfectly still water. If the wave maker is allowed to run continuously, though, wave s will invariably be reflected off of the

PAGE 120

120 structure side walls, and even the wave absorbers back upstream. As the reflected waves interact with the waves off the paddle, dispersive effects are introduced and wave shape deterioration occurs (Figure 4 -9). These slightly dispersive waves have a compound wave shape that results in a flatter wave face slope and increased likelihood of slamming occurrence. The comparison of monochromatic and dispersive wave profiles shapes is the mirror of the wave steepness in hor izontal slamming The well-formed monochromatic wave inundates the suspended structure in a more gradual and continuous manner than the flattened slightly dispersive wave, similar to how the non -breaking wave gradually is reflected by the wall where as the steeply sloped wave face is not. The dispersive waves result in an increased rate of structure inundation and a corresponding increase in the rate of momentum transfer, hence the more likely occurrence of vertical slamming. For a beam and slab span, the wave shape is far less critical. As a propagating wave seals off a chamber (Figure 4 -10) the air becomes the new interface surface between the wave and the structure and will assume a near horizontal shape/slope under loading. Because of this, wave shape will not dictate whether or not slamming occurs in a chamber The limiting case would be a near vertical wave face that was able to sweep out the air in the chamber before sealing it off. O n the e ffect of fluid buildup A furth er contributor to slamming occurrence is the deformation and deflection of the wave b the structure. Due to comparable sizes of the wave and structure, the effect can be significant. When a wave inundates a span the structure interrupts the water partic le orbits in the wave field and deflects the flow B ecause there is still a wave propagating behind the deflected water, the flow is deflected downstream ahead of the wave, falling into the emptiness of the leading trough (Figure 4-11). This deflected flow accumulates in front of the wave and builds as

PAGE 121

121 it is pushed forward by the onward propagation of the wave. Depending on the initial steepness of the wave, the excess water will behave in one of two ways. If the wave is steep ly sloped, then the deflected flow will build on the wave face before break ing forward to cr eate a small barrel wave that propagates steadily in front of the initial wave (Figure 4-12). In this case, slamming is less likely to occur as the rate of change of momentum is not increased by the presence of additional fluid in front of the wave because the fluid is rolling into the trough at a near constant rate and no buildup is sustained. However, if the wave is mildly sloped, then the excess deflected flow will not have as much vertical space within the trough to fill. Instead, the deflected flow will be deflected further in front of the wave, forming an artificial wedge between the wave and structure (Figure 4-13). This extended wedge will create an increase in momentum transfer between the wave and the structure as the wetted surface of structure grows faster than the propagation rate of the wave. This process is best illustrated in Figure 4 -14, where the variation in storage volume for deflected below between the wave and the structu The flatter the wave, the larger the increase in downstream inundation rate due to deflection, the higher the chance for slamming to occur. This is also a function of the wave height relative to the structure size and clearance, but larger slamming is certainly anticipated for larger waves. For the scenario of the chambered underside, this effect is noticeably neutralized by the excess available volume present in the downstream open air chambers that have yet to be sealed by the wave. Water may simply enter these chambers without wetted length buildup. A Model for Slamming A simplified representation of the wave propagating into a suspended structure is looked at first. Consider a rectangular object fixed above an accelerating water column with four separate stages of flow (Figure 4 -15).

PAGE 122

122 In the first stage, the flow has yet to reach obj ect and the initial flow is unaffected. In the second stage, initial contact be tween the flow and object has ta ken place and the effect of structure/fluid interaction throughout the flow begins to spread. In the third stage, inundation of the object by the flow occurs. At this point sla mming, if it has occurred, is already over. In the fourth stage, the flow has passe d the fully inundated object and the flow has fully adapted to the presence of the object. In stage one, there is no loading. In stage two, the flow is at the initia l point of contact. In stage three, the flow is conti nuing to experience changes induced by the presence of the structure resulting in loading from buoyancy, drag, and in ertia (including the mass rate force due to the changing added mass as submergence increases). In stage four, the loading is constant, the sum of buoyancy, drag, and inertia (the mass rate force now zero). For slamming, the area of interest is the s econd stage. To simplify the problem, the structure is now taken to be a flat plate (Figure 4-16a). The effect of finite thickness is negligible as it is expected for slamming to have already o ccurred before the full inundation of the object. If the flow approaches perpendicular to the f ace of the plate, then there is a fully wetted surface affecting the flow at the initial point of contact. In theory, added mass is present in the flow a mere moment after contact. From the flat plate added mass e xpression of Payne (1981), the added mass in the flow the moment after co ntact is given by Equati on 4-1, where W and L are the width and length of the plate, respectively. 2 1 a 4 2WL m W 1 L (4-1) In this case, the effective mass equals the added mass as a flat plate nominally has no mass. From the moment right before c ontact to the moment right after contact, the flow field has to

PAGE 123

123 have adjusted to the presence of the flat plate in the flow (assuming an incompressible fluid). This adjustment is represented by the presence of added mass in the flow (Figure 4 -16b). Disregarding drag, the loading on the plate at this instant is calculated according to the time derivative of linear momentum (Equation 4-3) where ma is the added mass, dma/dt is the time rate of change of added mass, V is the time dependent velocity of the flow field, and dV/dt is the acceleration of the flow field Again there are the inertial and mass r ate components. a adm d dV Force mV m V dt dtdt (4 -3) What is important about this equation in this situation is the dependence of the two derivatives. In the first term, the flow velocity derivative (the acceleration of the flow) is independent of the inundation rate of the structure. However, in the second term, the derivative of the added mass with time is purely a function of the inundation rate of the structure Because of this, the contributio n of the second term (the mass rate force) will fluctuate wildly based solely on how quickly the fluid comes in contact with the structure. This is the high transfer rate of momentum seen in the near 90 wave face of the near -breaking wave in horizontal slamming and the flat wave face of the degraded dispersive wave in vertical slamming. This is best realized by an order of magnitude analysis. The representative magnitudes of the velocities and masses are borrowed from prototype sized waves and spans, with a vertical velocity of 100 f t/s, a vertical acceleration of 100 ft/s2, an added mass of 103 slugs, and a time period of change of 10-1 s ec (theoretically the change time is instantaneous for perpendicular flow and incompressible fluid) Substituting into Equation 4 -3 gives us an order of magnitude force assessment (Equation 4 -4). 3 300 34 1O10 Force O10O10 O10 O10 O10 O10 (4 -4)

PAGE 124

124 Comparing the order of magnitudes between the inertial term and the mass rate term, it is seen that the mass rate term is controlling. This is logical in that the mass rate component of the momentum transfer is a function of the speed at which the mass of the system is affected. For bridge spans and waves, the mass rate will always dominate in the slamming case. This leaves the base mathematical represen tation of slamming ( Equation 4-5), where V is the velocity of mass affected and dma/dt is the rate at which it is affected adm Slamming Force V for dt << T dt (4 -5) The time period over which this momentum exchange takes place becomes the controlling factor. The shorter the duration over which this takes place, t he larger the forces produced. The greater the speed with which the largest portion of the structure is introduced to the flow, the larger the change in mass rate. Because of this, the structure/fluid surface orientation becomes a driving mechanism of slamming magnitude and duration. Imagine the difference in forcing if the flat plate of the previous example was simply turned on an angle to the approaching flow (Figur e 4 -17) and not perpendicular to it. The change in added mass rate would reduce significantly because the added mass grows more slowly (due to the slower growth of the wetted dimensions). Conversely, the same thing would occur if instead of the plate being turned, the surface of the water approached the plate at an angle, as in the case of a wave propagating into a suspended span. A Note on FluidStructure Interaction Rate The importance of a high change in added mass rate and the orientation between flow and structure in producing significant slamming requires the consideration of the initial relationship between the structure and the flow. Now that the rate of mass change is understood to be a driver the observations of slamming in certain situations can be readi ly explained.

PAGE 125

125 For horizontal slamming on a vertical wall, a steeply sloped wave face creates a situation where the mass of fluid affected by the presence of the wall would grow suddenly upon contact with the structure (Figure 4-18) The case of a large trapped air pocket on a near -breaking wave similarly transmits the presence of the wall to the entire wave, again creating a sudden increase in the mass of fluid affected by the wall. Perhaps most telling, though, is the lack of slamming for a non-breaking wave. In this case, the gradual interaction and rise of the water along the structure face corresponds to a gradual rise in the mass of fluid in the wave affected by the wall (Figure 4 19) Beam and Slab Slamming A single chamber is first con sidered Equation 45 rewritten as the slamming force (FS), where w is the vertical water particle velocity, and dme/dt is the time rate of change of effective mass during the occurren ce of slamming (Equation 4-6). e Sdm F w dt (4 -6) The effective mass is used in place of the added mass as a beam and slab span will have some finite thickness that comes into play due to the occurrence of slamming in all of the structures chambers (structure displaced mass will already exist in the flow field while slam ming is occurring in later chambers, unlike the initial occurrence of slamming on a flat plate ). Expressions for the effective mass and the rate of change of effective mass were derived in Chapter 3 for a finite structure. The main inputs for these equations were the wetted width of the span, W*, wetted thickness of the span, D*, the length of the span, L, time rate of change of these wetted dimensions. The chambered beam and slab span is convenient in that it provides set increments of wetted width in crease. While the wetted thickness will vary slightly as the

PAGE 126

126 wave propagates through the structure due to the variation in air levels within the chamber, the wetted width will increase in two distinct and calculable manners (Figure 4 -20). First, when the leading edge of the wave is propagating over a girder flange, the wetted length will increase as the rate of the wave celerity. Second, when the leading edge of the wave is propagating over an open chamber, the wetted width is not increasing at all because air has the ability to escape. A s soon as the leading edge of the wave reaches the next girder flange, though, the chamber is effectively sealed and the wetted width contribution from that entire chamber is added. The addition of this large chunk of ad ditional wetted length at every chamber corresponds to the addition of a large chunk of added mass to the system. T he time period over which these blocks of mass are added to the system is very short, consisting of the time it takes for the air to compress under loading (from all components) once the chamber is sealed. This results in a high time rate of change of added mass. The problem in evaluating this is buried in the dynamic nature of the air chamber. Because the volume of air in the chamber will compress to a minimum volume under max loading and then expand, the magnitudes from all the individual forcing components in addition to the slamming is required to calculate it. This is further complicated by the dependence on the individual components of the forces on structural parameters that include the dynamic air pocket This means all the force components have to be iterated simultaneously This is examined in Chapter 5. Substituting the mass quantity equations from Chapter 3 gives a fina l sla mming expression (Equation 4-7). In this expression W* is the wetted width, D* is the wetted thickness, w is the time dependent vertical water particle velocity, and AS* is the wetted cross sectional area.

PAGE 127

127 3 223 2 55 22 S S 3 32 5 22WL1W1DW1WD 2t5Wt20Dt WL A Fw Lw t WL1W1DW 4t8Wt WL****** ** **** (4-7) Due to the set incremental nature of the air chambers, expressions for the wetted width can be easily derived. The wetted width (Equation 4-8) for a given moment of slamming (for the chosen chamber of interest) is a function of girder spacing, WG, girder flange width, WF, and girder number, NG. The wetted thickness (Equation 4-9) of the structure is now a function of the wave height, H, the clearance, ZC, the girder height, HG, and the deck slab thickness, D. GFG m a xCGW N1WWfor Z < H (4-8a) GFGOm a xCGW N1WWWfor Z H (4-8b) maxCG maxC GD ZHDfor Z < HD (4-9a) Gm a x C GD HD for Z HD (4-9b) Determining the time rate of change of these wetted dimensions is much more complex. The effect that the structure has on the wave field is imposed through the trapped air in the chamber. Because the dynamic compression of the air is a function of all the forcing components, the time that it takes for the air to compress during slamming cannot be calculated directly. This requires th e use of a numerical model as described in Chapter 5.

PAGE 128

128 Figure 41. Progressive nonbreaking wave at a vertical wall. Figure 42. Progressive near breaking wave at a vertical wall. Figure 4-3. Progressive breaking wave at a vertical wall.

PAGE 129

129 0 10 20 30 40 50 60 70 80 90 0 0.5 1 1.5 2 2.5 3 Force Time Progressive Near Breaking Slamming Force Figure 4-4. Representative force profiles on a vertical wall of a progressive wave and a near breaking wave. 150 100 50 0 50 100 150 200 250 300 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 Force Time Figure 45. Representative occurrence of slamming for a smooth -bottomed structure.

PAGE 130

130 40 20 0 20 40 60 80 100 120 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Figure 4-6. No occurrence of slamming in early waves in train followed by occurrence of slamming in later waves in train. 30 20 10 0 10 20 30 40 50 1.0 1.5 2.0 2.5 3.0 3.5 Force (lbs) Time (Sec) Figure 4-7. Representative occurrence of slamming for a beam and slab span with slamming appearing in each air chamber.

PAGE 131

131 Figure 4-8. Effect of clearance height on slamming occurrence. Figure 4-9. Slightly dispersive wave with flattened wave face shape.

PAGE 132

132 Figure 4-10. Wav sealing off chamber and trapping air as it propagates through span. Figure 4-11. Fluid deflected by structure occupying empty trough in front of steep-sloped wave.

PAGE 133

133 Figure 412. Barrel wave forms beneath structure due to buildup of deflected fluid from steepsloped wave. Figure 4-13. Wedge forms beneath structure due to buildup of deflected fluid from mild -sloped wave.

PAGE 134

134 Figure 4-14. Variation of volume between wave and structure based on steepness of wave face

PAGE 135

135 Figure 4-15. Four stages of forcing on a rectangular object in a vertically accelerating water column. A) Stage 1. B) Stage 2 C) Stage 3 D) Stage 4.

PAGE 136

136 A B Figure 4-16. Presence of added mass for a flat plate in a n accelerating water column A) Just before contact. B) J ust after contact. Figure 4-17. Orientation of the plate lengthens the time rate of change of mass and added mass buildup.

PAGE 137

137 Figure 4-18. Large volume of fluid mass affected over short period due to steepfaced near breaking wave on a vertical wall. Figure 4-19. Small volume of fluid mass affected over same period due to shallow-faced nonbreaking wave reflecting on a vertical wall.

PAGE 138

138 A B C Figure 4-20. Significant periods of effective mass variation in a chambered structure. A) Positive along the girder flange. B) Zero along the unsealed air pocket. C) Sudden positive along the sealed air pocket.

PAGE 139

139 CHAPTER 5 THE NUMERICAL MODEL In order to obtain useful results from the theoretical model, a numerical model was needed. The model that was developed is not a flow solver, but a loop calculator that evaluates the theoretical equations at every point of a grid based structure/fluid template. When dealing with actual physical prototypes, certain assumptions must be made in order to process the theoretical equations. These assumptions d eal with both physical and theoretical constructs and are presented here with the thinking used to arrive at their operating assumptions. Overview The numerical model operates on a grid basis. Using an array of user-provided inputs, two separate grids are built, a structure/air grid and a fluid grid. Inside the structure grid a span is constructed. All spans are treated as simply supported spans. Inside the fluid grid the wave train is constructed. The fluid grid is then overlaid onto the structure grid and time stepped through it. At each time step, the equations developed in Chapters 3 and 4 are evaluated in each grid element of the wave and structure overlap The model operates under the 2D assumption that the wave is propagating normal to the st ructure and force is calculated per linear foot. Also calculated is the moment about the lowermost trailing edge of the structure, considered the worst case location for overturning of the structure. The program then moves forward another time step and the process is repeated. The forces of each component are stored in individual time series. The Elemental Grids The two grids, structure/air and fluid, are of different sizes, with the structure/air grid only covering an area in the vicinity of the structure while the fluid grid covers the full distance of the

PAGE 140

140 wave train (the number of waves in the train defined by the user). The grids are a two dimensional plane representing a lateral cross section (cut across the perceived lanes of traffic). The Structure Grid The structure grid covers an area vertically from the seabed to the highest elevation of interest (defined as the higher of the wave crest and structure top). Horizontally, the grid extends upstream and downstream one additional structure width This is necessary as the mass quantities and wave kinematics that create the loading will sometimes extend large distances away from the structure All elements in the grid are square and the resolution is user defined. Computer processing power limits the functional resolution size, but obtainable resolution s are fine enough to resolve the smaller elements of the more detailed span types (girder web thickness, etc.). Based on the structural parameters input, the model creates an accurate representation o f the structure in the grid at the correct elevation (the bottom of the grid corresponding to the seabed). For the more complex spans and span components reference files are used that contains the pre -described dimensions of a large number of existing girder and rail types. User defined components can be easily inserted into the file and given a reference number. Within the grid, each element is either specified as structure or air. E xample structure/air grid s are given for a slab span (Figure 5-1) and a beam and slab span (Figure 5 -2). The figures shown are low resolution and given for visualization only. The Fluid Grid The fluid grid covers an area identical in vertical height to the structure grid but variably different horizontally. The horizon tal length of the fluid grid is determined from the fluid parameters and the number of waves chosen for the model wave train. The element sizes are identical to the structure grid

PAGE 141

141 Based on the input fluid properties parameters, the numerical model calculates the wave shape and particle kinematics using stream function theory with a starting order of 4 and a maximum order of 25. During this process, the wave is broken into gridded elements of the same size as the structure grid. In each of these elements, the particle velocities, accelerations, and water surface elevations are calculated and stored. Like the structure, the wave is assumed to be two dimensional for the full length of the span, taken per unit length along the wave ray. Once the wave ch aracteristics are obtained, a wave train is constructed from an arbitrary number of monochromatic waves placed end to end. Due to the possibility that a structure could be simultaneously struck by two waves in a train if they are of short enough length, a minimum number of waves must be added to the train. On either side of this wave train are placed extra horizontal lengths equivalent to the length of the structure grid. In these areas, water depth begins at SWL and slopes down linearly to the elevation of the first wave trough. Figure 5-3 shows an example grid with a wave train defined within it. An example fluid grid time stepping through a structure /air grid is given in Figure 5-4. The Required Inputs A number of inputs are required in the numerical model. Each input represents either a concrete numerical variable or a choice of an assumption used to realistically apply the theoretical equations. The inputs are therefore separated into three different groupings structural parameters, fluid parame ters, and evaluative parameters. Structural Parameters The structure parameters are used to define the bridge type, dimensions, and location. The structural inputs are as follows: Bridge type. The bridge type specifies whether the bridge is a flat plate, a slab, or a beam and slab span

PAGE 142

142 Clearance height. The height of the lowest chord of the bridge above SWL. Width. The total span width (cross traffic, parallel to the direction of the wave). Length. The total span length (with traffic, perpendicular to the direction of the wave) Deck thickness. The thickness of just the deck in a beam and slab span or t he thickness of the entire slab in a slab span Girder type. T he girder model or design (can be standard or user defined) Girder number. The total number of girders. Rail type. The rail model or design (can be standard or user defined) Overhang. The width of any overhangs Some of these parameters only apply to certain span types. For instance, only the beam and slab type would have girder parameters, the flat plate would not have thickness parameters, etc. Fluid Parameters The fluid parameters are used to define the wave characteristics and water levels in the vicinity of the structure. The fluid inputs are as follows: Water depth. The depth of the water at the location of the structure. Wave period. The period of the design wave. Wave height. The height of the design wave (may be the significant height, the maximum height, or any height of interest). Current. The local current inline with the propagation of the wave at the location of the structure. Water density. The density of the water at the location of the structure. For the water depth it should be noted that this value includes all the contrib utions of storm surge, wind setup, etc. Evaluative Parameters The evaluative parameters are selections made by the user to dictate how the numerical model evaluates certain physical aspects of the fluid/structure interaction process. These are best

PAGE 143

143 case estimates of physical reality of wave/structure interaction based on the individual case parameters and observations from physical model testing. The inputs are as follows: Wave theory. Whether linear or stream function theory is used to determine the wave properties and particle kinematics. Element resolution. The size of the individual elements in the structural and fluid grids. Coefficients. Drag and inertia coefficients for the structure. These coefficients are functions of the structural and wave p arameters. Added Mass Distribution. How the added mass is distributed throughout the wave during the interaction period. Mass Rate Force Distribution. Were the Mass Rate Force is applied on the structure. Air venting. How air pockets and trapping are tr eated. It should be noted that the coefficients parameter can be user defined or based on predictive relationships developed from the physical model testing. These relationships are examined in Chapter 7. The details and assumptions of s ome of these parameters are discussed later in th is chapter. Considerations for Evaluating the Theoretical Equations The next step is to mesh the data in each grid and evaluate the theoretical equations over every element. Each of the theoretical equations (inertia l, mass rate, buoyancy) is calculated for each element in the grid and the results summed. First, though, consideration is given to outside factors that go into evaluating each component as well as aspects that affect the component calculations as a whole. Trapped Air One difficulty in evaluating the equations of Chapters 3 and 4 is tied directly into the span type being modeled. A slab span or flat plate presents little complexity due to their simple

PAGE 144

144 shapes and lack of air trapping members. Application of the equations is straight forward and the individual forcing components can be applied independently of one another. For the case of a beam and slab span, the air pockets between the chambers and the ledges created by the overhangs create sign ificant cross component effects. As the wave propagates through the structure, the air is flushed out of the chamber s between the girders (Figure 5-5) or out from under the overhang ledge (Figure 56) by the wave until the chamber or the ledge is complete ly sealed off by the water. At this point air no longer escapes and it becomes a trapped volume of compressible resistance. From this point, two scenarios are possible closed pocket or open pocket. In a closed pocket system, the air has nowhere to go once the chamber is sealed. It is completely trapped and subject only to the loading of the wave with no escape. In an open pocket system, the air has an alternate vent system. Once the wave seals off the chamber at the girders, compression begins under the wave loading. However, as the air compresses it simultaneously begins to bleed off through openings in the chamber ( either through open diaphragms between the girders or vents in the deck). This creates a complex problem of forcing, compression, and air bleed. This situation is beyond the scope of this work, but is an important aspect of wave loading that needs to be further researched as it may present possible solutions on reducing wave loading magnitudes. In a closed system, the air is trapped an d can be readily examined. F or the assumption of constant temperature, the combined gas law (Equation 5 -1) dictates that this volume of trapped air is a direct function of the p ressure within the trapped area, where P1 and P2 are pressures and V 1 and V 2 are volumes. 1122PV = PV (5 -1)

PAGE 145

145 Applied to trapped air under loading, the pressure, P1, is the atmospheric pressure, Patm, and the volume, V 1, is the initial volume of air in the chamber, V 0, at the time it is sealed On the right hand side of the equation, the pressure, P2, is the combination of all the individual vertical forcing components, FZ, divided by the surface area of the trapped air that is in contact with the wave AW. The volume, V 2, is the maximum compressed volume of air (the volume under the maximum loading), henceforth known as the compressed volume, V C. Substituting these values into Equation 5-1 and rearranging gives an expression (Equation 52) for the compressed volume. atm C0 Z atm WP V = V F P A (5-2) From this equation, a significant problem is found the presence of the vertical forcing in determining the compressed air volume. The force equations that were developed in Chapters 3 and 4 showed dependence upon wetted dimensions and the locations of the kinematics within the wave. With the trapped air acting as an extension of the structure, it in turn becomes a variable in determining the wetted dimensions and structure/wave interaction boundaries. The presence of air means that the vertical forcing is in a roundabout way a function of itself. If the air compression was instantaneous, this problem could be solved by iterating for the correct compressed volume. However this compression will takes place over some finite time period, so the ability to predict the rate of compression is required To solve this, the physical reality of what is happening in the chamber is examined. In order for a volume of air to be compressed, a moving boundary is required. In this problem, that moving boundary is the propagating wave face, which moves at a constant velocity (for a constant water depth) A simplifying conclusion, then, is that the maximum rate at which the air

PAGE 146

146 can be compressed is the rate at which water would fill the chamber, i.e. the rate at which the wave propagates past the structure. This rate is a function of the w ave celerity and the wave slope. The structure will affect this rate to some degree because it will manipulate the wave shape after contact. To solve the problem, though, it is assumed that rate of fill in the chamber would be equivalent to propagation of the unaffected wave shape over that area, or put simply, the wave celerity, c. If the wave profile can only advance as fast as the wave celerity, then there is a l imit to how much compression can occur over a given time step. This is important because the forcing the structure experiences would be limited by this rate. Using these concepts, the time to and volume of the compressed volume can be estimated and subsequently, the forcing over the compression cycle calculated Added Mass Distribution While equations were developed for predicting quantities such as added mass and change in effective mass, there are no simple means for predicting the exact distribution o f the added mass in the flow field. With the large variation in kinematics along the structure at any given point in time, the location of the fluid mass affected can radically alter not only the magnitude of the forces, but also the shape of the force ti me series since the variation of the kinematics applied to these masses will govern the phasing of the individual force components. Two separate methods for selecting added mass location were developed. The first method, called simple spreading, distribut es the mass perpendicular from the wetted surface. If the structure is wetted on both of the faces of a given orientation (vertical or horizontal), the mass is distributed evenly on either side as the available volume of water allows. If the calculated m ass exceeds the available volume of water limited by the simple spread, it is carried

PAGE 147

147 out one element row or column further in parallel until the correct mass is obtained. This method best represents the directionality of the mass effect in the water colu mn. The s econd method, called radial spreading, distributes the mass in a multi -directional diffusion radiating out from the wetted surface of the structure. This represents a more balanced spread of added mass in the wave field and does not consider orientation is distributing the mass. This method best represents an interlinked distribution of the effective mass in the water column. Example distributions of each are shown in Figure 5-7. Change i n Effective Mass In addition to the placement of effective mass are the physics of the time rate of change of the effective mass. The first issue is the presence of negative time rate of change of effective mass and its physical meaning. The second issue is the sudden cutoff from time to time of the predicted values as the wave propagat es through the structure. Both of these issues stem from a vague understanding of the mass rate force and the time rate of change of the effective mass from a physical per spective. Stages of inundation and mass growth The possible stages of inundation for a slab span were briefly touched upon in Chapter 3 and are shown again in Figure 5-8. As the wave initially strikes the slab, the effective mass is increasing with time. If the trailing edge of the wave reaches the structure before the leading edge exits then there is a period of time when the effective mass is mathematically constant because the wetted length and wetted thickness are both constant. The same thing would occur if the wave were much wider than the structure at elevation. For either situation, the effective mass becomes time dependent once again as the wave exits the structure, but now the rate is negative. A beam and slab span differs slightly then the simple slab span, as the presence of air pockets now creates situations where effective mass grows/reduces in a cycle of brief constant

PAGE 148

148 rates as pockets are sealed or unsealed. The periods of these different rates were looked at in Chapter 4 for a single cham ber and are shown in Figure 5-9. Similar to the slab, though, the beam and slab span can experience periods where the effective mass is constant as well as the period of negative change in effective mass as the structure leaves the wave. Negative change i n effective mass For the negative time rate of change in effective mass, the process is examined in terms that are physically meaningful and compared to what is theoretically predicted. The equations that control the mass rate force (Equation 53) predict that loading will result regardless of whether the mass is accumulating or decreasing. e MXMXdm FCu dt (5 -3a) e MZMZdm FCw dt (5 -3b) Looking at a simple case, imagine an object partially submerged in an unsteady horizontal flow with both positive velocity and accelerati on (Figure 5-10). If the object was slowly removed from the flow, the effective mass of the system would decrease. According to the equations, the object would now experience a negative loading (positive velocity, negative change in mass). If removed quickly, the loading would theoretically be large. This situation is analogous to the structure emerging from the wave field as it propagates away. Here, the wave will have negative (vertically downward) water particle velocities and a negative rate of change of effective mass. From the equations, this should produce a positive (vertically upward) force. However, this forcing does not appear in the measured wave loading. From the physical standpoint, the loading contribution of the mass rate force originates from the work done by the structure on the wave field to adjust the flow field to the structures

PAGE 149

149 presence. When the structure is removed, the flow field adjusts to its original state. For the flow field to adjust though, work must be done. T he equations suggest that the structure is the one doing this work as it is removed, but in reality the work is being done the flow itself. If in place of the structure in the flow there was suddenly empty space (creating a ne gative mass rate of there is an infinitely large force as predicted by Equation 5-3. However, the work necessary to direct the flow into the newly empty space has to come from the flow itself since the structure is no longer there to do work. Because of this, it is assumed throughout this study that the mass rate force is only valid for positive rates of change in effective mass. In areas where the rate of change of effective mass is negative, the work necessary to affect the flow is done by the flow itself (or the flow adjusts and this effect is felt upstream). Therefore the mass rate force is turned off for negative time rate of change in effective masses. Periods of cutoff The final concern is how to evaluate the mass rate force for periods where t he time rate of change of effective mass suddenly cut s off. Depending on the confluence of wave length, deck width, wave height, and structure clearance, predicted force curves with on/off transitions or spikes can result These jagged transitions do not appear in the measured force time series of the physical model experiments, so a method of dealing with these erratic forces is needed. Using the case where the slab span is narrower than the wave at elevation, a represen tative time series of the growth in wetted length, time rate of change in wetted length, effective mass and time rate of change in effective mass over the course of inundation of the span is shown in Figure 5-11. From this figure, the areas where forcing continuity would be problematic are evident by the sudden drop in quantities to zero.

PAGE 150

150 In past work on flat plates, different methods of dealing with this problem have been used. Isaacson and Bhat (1996) dropped the mass rate force linearly to zero from th e moment where the wave crest w as over the center of the plate, a method that only provides smooth transition from Kaplan et al. (1995) set the force to zero as soon as the plate reached the point of maximum inundation. With the assumption that there is no mass rate force for negative time rate of change of effective mass values, a natural terminal point for the forcing is given. Using this terminal point, the mass rate force is therefore dropped off linearly from the initial point of cutoff to zero at the location where the time rate of change of effective mass first becomes negative. For a beam and slab span, the jagged cutoffs of the time rate of change of effective mass are not a concern, because they are expected due to the presence of the chambered underside and should appear in the measured forcing.

PAGE 151

151 Figure 5-1. Structure/air grid example (low resolution) for a beam and slab span. Figure 5-2. Structure/air grid example (low resolution) for a beam and slab span.

PAGE 152

152 Figure 53. Fluid grid example (low resolution) for a two wave train. A B C Figure 5-4. Example of fluid grid stepping through structure/air grid. A) T ime step 00 B) T ime step 30. C) T ime step 60 (overlapped fluid/structure elements in black). Note: tail end of fluid grid has been cut off for element clarity.

PAGE 153

153 A B Figure 5-5. Air activity in chambers between girders. A) A ir is still being flushed from the chamber B) A ir is sealed (gray elements) within the chamber by the wave. A B Figure 56. Air activity beneath the overhang ledge A) A ir is still being flushed from underneath the overhang. B) A ir is sealed (gray elements) between the overhang and the wave.

PAGE 154

154 A B Figure 5-7. Added mass distributions used in program. A) S imple distribution. B) R adial distrobution

PAGE 155

155 A B C D Figure 5-8. Periods of changing added mass during inundation. A) Structure entering wave field. B) Structure wider than wave. C) Structure narrower than wave. D) Structure emerging from wave field

PAGE 156

156 A B C Figure 5-9. Significant periods of effective mass variation in a chambered structure. A) Positive along the girder flange. B) Zero along the unsealed air pocket. C) Sudden positive along the sealed air pocket.

PAGE 157

157 A B Figure 5-10. Time rate of change of effective mass and resultant forcing A) Non-moving partially -submerged object with zero change. B) Moving partially-submerged object with negative change.

PAGE 158

158 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.0 1.5 2.0 2.5 3.0 Non dimensionalized Qunatity Time Wetted Length Change in Wetted Length A 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.0 1.5 2.0 2.5 3.0 Non dimensionalized Qunatity Time Effective Mass Change of Effective Mass B Figure 5-11. Representative t ime series over the course of inundation of a slab span. A) W etted length and time rate of change of wetted length. B) E ffective mass and time rate of change of effective mass.

PAGE 159

159 CHAPTER 6 PHYSICAL MODEL TESTS All tests were conducted in the a ir/sea wave tank located at the Coastal Engineering Laboratory at the University of Florida. The wave tank is 6 ft wide by 6 ft deep by 120 ft long. The wave maker has the capability of both paddle and piston modes of operation but was used solely in piston mode. Wave absorbers are located behind the paddle and at the downstream end of the tank in order to minimize wave reflection. A series of glass panels run the length of one side of the tank for viewing. The ranges of wave heights and periods as a function of water depth achievable in the tank are shown in Figure 6-1. The tests were laid out in increasing levels of model complexity with differing arrays of instruments to capture various facets of interest in the forcing mechanisms. The increasing levels of complexity allowed for the contributions of normal bridge additions (i.e. overhangs, rails, girder spacing, etc.) to be determined through comparisons to similar testing conditions in their absence. In all, 1100 individual tests were completed where the ranges of testing conditions were chosen to represent the practical limits encountered in a nearshore environment. Model Considerations Wave force data from previous physical model studies exhibit considerable scatter, a fact that is attributed to the highly dynamic nature of the forces being measured. Care was taken to design a robust system that would avoid these dynamic effects. Therefore, in constructing the model and its support systems, two important factors were considered in the design, frequency effects and scale effects. Frequency Effects Adverse frequency effects were particularly crucial to avoid because they could largely manipulate the recorded measurements of the various forcing components. To record the loading

PAGE 160

160 on a test model, the strain bridge elements in load c ells and pressure sensors respond to the minutest movements induced under the forcing. In order for these measurements to be accurate, the structure must transfer these loads to the instruments at a frequency significantly higher (at least an order of magnitude or more) than the forcing frequency, otherwise distortions or resonance occurs. Unless they are made of rubber, most model structures will have no problem producing an undistorted response to the quasistatic wave forces. But while the se low freq uency forces do not create a problem, the slamming force might, depending on its duration. I f its frequency of forcing is sufficiently short then the structure may not be able respond to it. A springmassdashpot system similar to the one seen in Chapter 3 is given in Figure 6-2. The difference between this system and the earlier system is that now forced motion is considered while before free motion was considered. The forcing of this system is described by a differential equation (Equation 6-1) where z is the displacement, m is the object mass, c is a damping coefficient, k is the spring constant and F(t) is the forcing 2 2dzdz F(t)mcmkz dt dt (6 -1) There are three important frequencies to this single degree of freedom system They are the system n), and the systems d). How these frequencies relate to each other determines the motion of the system under a given forcing situation. Expressions for the natural frequency (E quation 6-2) and the damped frequency (Equation 6-3) are given. The forcing frequency is an arbitrary value determined by the specific forcing being considered. The critical damping for the system occurs n.

PAGE 161

161 nk m (6 -2) 2 dn nc 2m (6 -3) Depending on how close the forcing frequency is to the structure dampened frequency, the magnitude of the displacement can be amplified or attenuated. The structure response as a function of damping magnitude and ratio of forcing to natural frequency is shown in Figure 6-3. Note that as the forcing frequency approaches the natural frequency the response is significantly amplified. As the mass of the structure becomes large (as in the case of a bridge), the frequency of its fundamental mode of vibration becomes very low and thus far removed from the slamming frequencies. From Figure 63, it can be seen that the consequences regarding the effects of slamming is related directly to the structure itself. The structures natu ral frequency and shape will determine whether or not it even experiences any loading from the slamming component. An exceptionally high frequency slamming component on a structure with a low natural frequency may not have time to respond the forcing and therefore not feel any effect from it. On the other hand, if there was significant air trapping between the structure and the wave, the slamming force would be cushioned and the frequency that it acts at would be lowered. This air trapping, then causes a situation where the structure is more likely to experience slamming effects because of the air presence. The dynamic variation of the frequency depending on structure bottom characteristics requires consideration when analyzing any measured data. When considering the possible effects of slamming in a design environment, the issue of partial submergence and added mass would then appear to play an important part. The natural frequency of the structure as well as the response characteristics of the structure varies with the

PAGE 162

162 added mass quantity. A single structure could theoretically respond to the slamming force at one clearance height and not at another, depending on how the natural frequency is altered by the magnitude of the added mass. To avoid these am plification problems, the structure and support system was made as stiff as possible, producing a model that is functionally a rigid body in terms of the expected forcing frequencies. Due to the anticipated damping of the water when the structure is submerged or being inundated, the structure had to be stiffened enough to maintain a suitably high damped frequency that would also avoid possible resonance or amplification effects (since it can be seen in Equation 63 that as damping increases, the system frequency decreases). Ping tests were done on all of the constructed models to determine the system frequencies for the structure suspended in air and also for the structure fully submerged in water. This gave the two extreme limits of the system frequency range under wave loading. Power spectral densities for each model confi guration are given in Figure 6-4. Maintaining these high system frequencies was essential to recording accurate experimental data. In order to resolve the measured data quantities in later analysis, the sampling on all instrumentation needed to be at least twice the highest expected forcing frequency of interest, in this case the slamming frequency (This sampling frequency limitation is referred to as the Nyquist Frequency) For al l tests done, the sampling frequency used was 480Hz, capable of resolving frequencies up to 240Hz. The deviation from a rounded off number (such as 500Hz) lies in the usefulness of filtering multiples. In order to remove any background noise from measured signals, it is much easier to resolve the standard 60 cycle background effects out of a 480HZ sample frequency than it is for a 500Hz sample frequency due to the 480HZ rate being a multiple of the 60Hz background effects.

PAGE 163

163 Scale Effects In all physical model work, the effects of scaling upon the parameters being considered must be handled to insure verisimilitude between prototype and model. Geometric, kinematic, and dynamic similarity must be held throughout the scaling of all parameters. The geometric scaling is handled by maintaining a single length scale between the prototypes and the models. For all models used in the study, the geometric scaling ratio was 1:8. The kinematic and dynamic scaling was handled by maintain a single velocity scale ratio between the prototypes and the models. The similarity criteria chosen was Froude scaling. For the ratio of structure to wave size that is being dealt with, the inertia forces are the dominant term, while viscous forces are minimal. Maintaining Froude s caling, the ratio of inertia to gravity forces, is far more critical to the processes at work than maintaining Reynolds scaling, the ratio of inertia to viscous forces. From the Froude number similarity between prototype and model, the velocity scaling ra tio is the square root of the length scale ratio, 1: forces are fairly insignificant, the large model scale chosen for this study helps to minimize the distortion of any Reynolds dominated effects. Physical Model s The models represent a s trip of a continuous span running the width of the wave tank with waves propagating into the structure normal to the span length (i.e., the waves approach the bridge normal to the flow of traffic). All wave tests were done with the models in the most conservative setup and orientation of zero incline and wave propagation perfectly normal (90) to the span length. To reduce the effects of the wave tank walls on wave generation and propagation, each model was divided into three completely independent structu res of equal size. When placed side by side they formed a complete continuous span of the model running the full width of the tank

PAGE 164

164 (Figure 6 -5). The center section of the three structure continuous model was the only instrumented one with the two, non-instrumented lateral sections serving to buffer the measured data from any end effects. Several tests were also run without the two side sections to determine the impact of cases where there would be free flow around the ends. While this situation is not d irectly applicable to coastal bridges, it helped to clarify any discrepancies in added mass comparisons between continuous spans and those with free flow around the sides. The data may be useful to future studies of isolated structures. The Support Structure In order to avoid any effects induced by the presence of additional structures in the flow field, the models were supported from above by a fixed carriage system (Figure s 66 and 6 -7). The carriage consisted of a rigid steel frame locked onto the steel channel running the length of the concrete walls of the wave tank. From the main steel frame, an H frame composed of steel channels was suspended by four 1 in steel travel screws used to set the elevation of the model decks. From the H frame, 30in long 1in diameter steel pipes were used to connect the models. F langes were welded to the pipe ends and attached to the models by four in bolts for each flange For the center section (the instrumented model), a three -axi s load cells was placed in aluminum housing on the end of the steel pipes in place of the flange. All center sections of the models ran through these load cells and were connected by a locked in bolt. Because the model is fully suspended from the carriage, the same care needed to be taken to avoid any possible distortions of data measurements by creating a sufficiently stiff structure that would have a suitable natural frequency. So a number of stiffeners were added to make the support structure more rigid. Steel bar cross bracing wa s placed between each set of pipes running longitudinally down the length of the tank. Aluminum channel bracing was run laterally

PAGE 165

165 across the pipes as well. Eight in turnbuckles were added as an additional longitudinal connection between the H frame and t he solid steel carriage Instrumentation Four multidirectional (three axis) load cells were placed in aluminum housings and attached near the four corners of the center section. The load cells measured forces in the X (longitudinally down the tank), Y (l aterally across the tank), and Z (vertical) directions. The electronics for the load cells were housed above the structure carriage in a grounded electrical box. The wire leads and cables from the load cells were routed through the support pipes. In the X and Y directions, the load cell working range was 50lbs per cell. In the Z direction, the load cell working range was 200lbs per cell. The frequency response of the load cells was 1000Hz. All sampling was done at 480Hz. For the slamming specific tests, a series of pressure transducers were run along the width of the models. This enabled the measurement of localized forcing maximum and minimums for determining load distribution. The transducers were mounted in such a way that the ports of each sens or were flush with the bottom side of each model. The working range of the pressure transducers was 0 5psig. Depending on the model setup used, strings of six or ten transducers ran the width of the model. The frequency response of the pressure transducers was 1000Hz. All sampling was done at 480Hz. T hree capacitor -type wave gauges developed by engineers at the University of Florida were used to monitor the water surface elevation and wave heights. Two wave gauges were located 32ft and 8ft upstream of the leading edge of the models, while the third gauge was located 8ft downstream from the trailing edge of the models The wave gauges were positioned so that the effects of the structure on the wave field upstream and downstream of the structure could be monitored. The frequency response of the wave gauges was 1000Hz. All

PAGE 166

166 instrumentation was sampled at 480 Hz A ll measurements taken on all instruments were produced using LabView. The model section s were located near the center of the tank, 68ft from the paddle. A working diagram of the model and wave gauge is shown in Figure 6-8. Model Setups The model setups are divided into two separate categories consisting of cases dealing with the complete general loading on a span and cases dealing explicitly with the investigation of slamming forces. Within each of these categories a number of setups were tested. The dimensions of all the physical model setups involving beam and slab models are based on a 1:8 scale twolane bridge representative of the spans of the failed I -10 bridges of Louisiana and Florida. The dimensions of the physical model for the slab tests were also based on the aforementioned I-10 bridges, but this was done for easy comparison to later beam and slab tests. The slab models are not scaled to represent typical approach slab spans as seen in Florida. The various tested models are described below in the order that they were studied. Setups for investigation of the general wave loading The models detailed below were only instrumented with load cells for the main purpose of measuring the total wave loading time series for the study of the quasistatic forces. While the slamming component of the force was also recorded in all data and is beneficial for comparative purposes, it was not the target goal of these tests. Slab structure of finite thickness. The slab structure of finite thickness was constructed of 1in thick sheets of polypropylene formed into a rectangular shell with internal girders also made of polypropylene for stiffening. All joints were sealed with silicone for water tightness. Each individual section of the three structures modeled measured 48in wide by 24in long by 7in thick When installed edge to edge a continuous rectangular slab of 48in wide by 72in long by 7in thick was created (Figure 6 -9).

PAGE 167

167 Beam and slab structure. The beam and slab structure was constructed of a 1in thick polypropylene deck with a series of seven fiberglass girders (at 8in on center spacing) running along the underside of the deck. Polycarbonate glass sheets were used as diaphragms to enclose the girders on each side of the deck. The glass sheets also acted as a viewin g window into air bleed and wave penetration. All joints between the girders, diaphragms, and the deck were sealed with silicone for water tightness. Each individual section of the three structures modeled measured 48in wide by 24in long by 7in thick (in cluding the vertical girder height). When installed edge to edge, a continuous beam and slab structure of 48in wide by 72in long by 7in thick (including the vertical girder height) was created (Figure 6 -10). Beam and slab structure with overhangs. The beam and slab structure with overhangs wa s the same model as the beam and slab structure. Overhangs (1in thick, 4in long) were added to both the upstream and downstream edges of each model section. T he new size of each section measured 56in wide (including overhangs) by 24in long by 7in thick (including the vertical girder height). When installed edge to edge, a continuous beam and slab structure with overhangs of 56in wide (including overhangs) by 72in long by 7in thick (including the vertical girder heig ht) was created (Figure 6 -11). Beam and slab structure with overhangs and rails. The beam and slab structure with overhangs and rails wa s the same model as the beam and slab structure with overhangs Rails (1in thick, 3in tall) were added to both the ups tream and downstream overhangs of each model section. The new size of each section measured 56in wide (including overhangs) by 24in long by 10in thick (including the vertical girder height and rails ). When installed edge to edge, a continuous beam and slab structure with overhangs of 56in wide (including overhangs) by 72in long by 10in thick (including the vertical girder height and rails ) was created (Figure 6 -12).

PAGE 168

168 Beam and slab structure with overhangs and rails. The beam and slab structure with overhangs and rails was the same model as the beam and slab structure with overhangs. Rails ( 1in thick, 3in tall) were added to both the upstream and downstream overhangs of each model section. T he new size of each sect ion measured 56in wide (including overhangs) by 24in long by 10in thick (including the vertical girder height and rails). When installed edge to edge, a continuous beam and slab structure with overhangs of 56in wide (including overhangs) by 72in long by 10in thick (including the vertical girder height an d rails) was created (Figure 6 -13). Alternate beam and slab structure with overhangs and rails The beam and slab structure with overhangs and rails was put through a separate set of tests with an alternate girder spacing. Four girders were used instead of seven to examine the effects of girder spacing on air entrapment and forcing. For these tests, the additional instrumentation of six pressure transducers (at 8in on center spacing) was included in the model for comparison with similar measurements taken in later slamming tests. Two transducers were tapped and mounted for each air cavity aligned along the center of the model (Figure 6-14). Otherwise, the model dimensions and construction remained the sa me as the beam and slab with overhangs and rails model. Setups for investigation of slamming The models detailed below were instrumented with load cells as well as pressure transducers for the purpose of measuring both the total wave loading time series as well as the localized individual slamming force and its distribution along the structure. While the quasistatic component of the force was also recorded in all data and is beneficial for comparative purposes, it was not the target goal of these tests. F lat plate A flat plate structure of negligible thickness was constructed from a 1in thick polypropylene sheet. In addition to the load cell supports at the corners ten pressure transducers were tapped and mounted along the centerline of the plate. The transducers were variably spaced

PAGE 169

169 with a higher concentration of sensors towards the upstream end of the model since the slamming force on flat bottomed structure tends to be isolated to the front of the structure for structure/wavelength ratios of interes t. Each individual section of the three structures modeled measured 48in wide by 24in long by 1in thick. When installed edge to edge, a continuous rectangular slab of 48in wide by 72in long by 1in thick was created. Layout and specific spacing of the pr essure tr ansducers is shown in Figure 615 Beam and slab structure with overhangs and rails. The beam and slab structure with overhangs and rails was the same model used in the tests concentrated on quasistatic forces. However, for these tests, the addi tional instrumentation of six pressure transducers (at 8in on center spacing) was included in the model for monitoring the air pressure within each cavity created by the girders, one for each cavity. Otherwise, the model dimensions and construction remained the same as the beam and slab with overh angs and rails model (Figure 6-16 ). As a note, the girders used in the model studies were based on the AASHTO Type IV Girders use in the bridge decks of the old I10 Bridge over Escambia Bay, Florida. This girder is representative of girders in existing bridges throughout Florida. Physical Model Tests A total of 143 2 tests were performed on the assorted model setups and configurations. The breakdown of tests and the setups are given in Table 6-1. Wa ve heights, wave periods, clearance heights, and water depths were varied over the tests. The range of parameters covered and their prototype equivalents are shown in Table 6-2. For each test, a train of monochromatic waves of the same height and period were run past the structure and measurements taken Data Processing In order to make accurate and meaningful comparisons, the physical model data was first processed to extract the necessary frequency -based forcing time series. This required various

PAGE 170

170 filte ring techniques and signal analysis. Data from the experiments were reduced and analyzed as described below. Spectral Analysis For all tests, p ower spectra were computed for the vertical and horizontal components of each of the load cells as well as for the combined signals of the two upstream (and two downstream) cells. The same was done for all pressure transducers used. In all cases, the frequency with the highest power content was that of the wave frequency. Example power spectrums for sub aerial and subme rged cases for a slab test are shown in Figure 6-17. Example power spectrums for sub aerial and submerged cases for a beam and slab test are shown in Figure 6-18. A few notes can be made from Figures 6 -17 and 6-18. In the submerged cases there is very little energy present in the harmonic multiples of the wave frequency and no energy at all in the range of slamming. For a structure initially submerged or partially submerged, the natural frequency of the structure is lowered by the presence of the added mass of the water. There is also more damping for this situation. In this case, there is less structural response at the higher frequencies. In the sub aerial case the higher frequency slamming force excites the higher harmonics where there is less damping. In the power spectra of the upstream and downstream load cell pairs, similar frequency content divisions were found as in the total forcing. However, from the upstream to the downstream pair a si gnific ant decrease in slamming frequency content is noticed (Figure 6 -19). The presence of detectable action in the higher frequency range of the upstream load cell pairs does not always correspond to action in the downstream load cell pair. This difference suggests that slamming is limited to the upstream end of the structure for wave/structure size ratios of interest.

PAGE 171

171 Power spectra were also calculated for water surface elevations measured (wave heights). These spectra were calculated over the entire wave train series as well as for selected individual waves. As an example a comparison is shown in Figure 6-20 of the spectra for the first wave in a train, the last wave in the train, and the overall wave time series. In all tests a small amount of 60Hz background noise was present. Signal Filtering It was useful to separate the quasi static forces from the slamming forces. A low pass 8th order Butterworth filter was used to filter out the higher frequency components of the force. To determine attenuati on effects of the filter on the lower frequency forcing, maximums and minimums were compared for test cases where higher frequency components were not present. Attenuation of the quasi-static force due to the filter proved to be negligible. The slamming force signal was then determined by subtracting the phasecorrected filtered quasistatic force fr om the original forcing. A subaerial example (so that slamming is present) of the original vertical force time series and its filtered signals are shown for the same test case used in Figure 6-17. Included is the noiseless signa l (Figure 6 -21), the quasistatic signal (Figure 622 ), and the slamming signal (Figure 6 -23). Assumption Verification To simplify both the analysis of the data and create a practical predictive design tool, the wave is assumed to be two dimensional and propagates directly perpendicular to the structure (the orientation established as creating the largest forcing). This allows for the forcing to be treated as a force per unit length. However, wave tank wall shear and drag effects can affect the two -dimensionality of the wave as it propagates. Waves propagating at an angle of incidence other than 90 would create alternate forcing and phase shifts in lateral loading distribution. D ifferences in force measurements in laterally located load cell pairs were compared. The

PAGE 172

172 differences between individual located load cells measured maximum force showed scatter in the range of 0-5%, sufficiently small enough for the two -dimensionality assumption to hold. For further verificat ion, checking lateral axis load measurements in all load cells showed negligible forcing. Significant Parameters Extracted from the Data Once time series of the quasi static slamming force, and water surface elevation were obtained, a number of important parameters could be computed. For tests centered around quasistatic forces (slab tests, beam and slab tests, etc.), m aximum and minimum values of the quasistatic forces (horizontal and vertical) were extracted and th e maximum and minimum values of the moments about the downstream edge of structure calculated In cases where slamming was present, the slamming force and time duration (frequency) were also extracted. Associated forces in the horizontal and vertical directions were taken for the maximum forces of the corresponding opposite. For tests centered around slamming forces (flat plate tests, beam and slab tests), maximum values of the quasistatic and slamming forces (vertical only ) were extracted and the maxim um value of the moment about the downstream edge of structure calculated. The time duration (frequency) of the slamming was also extracted. Associated forces in the vertical direction were taken for the maximum slamming force From the water surface elevation data, wave heights and maximum wave crest elevations corresponding to the forces were obtained. This information is presented in Appendices A, B, and C.

PAGE 173

173 Figure 6-1. Air/sea wave tank wave height limits by period and depth. Figure 6-2. Springmass-dashpot system under forcing.

PAGE 174

174 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 4.0 5.0 Amplification n Damping Ratio (c/c o ) Forcing Frequency n = Natural Frequency c = Damping Coefficient c o = Critical Damping Coefficient Figure 63. Amplification effect v s. damping and frequency ratios. 0 100 200 300 400 500 600 700 800 900 0 2 4 6 8 Power Spectral Density Frequency (Hz) A 0 2000 4000 6000 8000 10000 12000 14000 16000 0 2 4 6 8 Power Spectral Density Frequency (Hz) B Figure 6-4. Typical power spectral density of total vertical forcing. A) Submerged span case. B) Sub aerial span case. Wave period = 2.0 seconds in both cases.

PAGE 175

175 Figure 6-5. Three structure components form one continuous structure. Figure 66. Structure support carriage (profile side of tank).

PAGE 176

176 Figure 67. Structure support carriage (profile down the tank).

PAGE 177

177 Figure 6-8. Working model with wave gauges (profile side of tank).

PAGE 178

178 Figure 6-9. Slab structure model. Figure 6-10. Beam and slab structure model.

PAGE 179

179 Figure 6-11. Beam and slab with overhangs structure model. Figure 6-12. Beam and slab with overhangs and rails structure model.

PAGE 180

180 Figure 6-13. Alternate beam and slab with overhangs and rails structure model. Figure 614. Flat plate structure model for slamming.

PAGE 181

181 Figure 615. Beam and slab with overhangs and rails structure model for slamming. 0 10 20 30 40 50 60 70 80 90 0 2 4 6 8 Power Spectral Density Frequency (Hz) A 0 5 10 15 20 25 30 35 40 45 50 0 2 4 6 8 Power Spectral Density Frequency (Hz) B Figure 6-16. Typical power spectral density of load cell pair vertical forcing. A) Upstream load cell pair. B) Downstream load cell pair.

PAGE 182

182 100 50 0 50 100 150 0.0 5.0 10.0 15.0 Force (lbs) Time (sec) Raw Noise removed A 100 50 0 50 100 150 10.0 10.5 11.0 11.5 12.0 12.5 Force (lbs) Time (sec) Raw Noise removed B Figure 617. Total filtered (noise removed) vertical force examples for a typical subaerial slab case. A) Force time series resulting from the passage of several waves. B) Force time series resulting from a single wave. The unfiltered signal is also presented.

PAGE 183

183 100 50 0 50 100 150 0.0 5.0 10.0 15.0 Force (lbs) Time (sec) Raw Quasi steady A 100 50 0 50 100 150 10.0 10.5 11.0 11.5 12.0 12.5 Force (lbs) Time (sec) Raw Quasi steady B Figure 6-18. Total quasisteady vertical force examples for a typical subaerial slab case. A) Force time series resulting from the passage of several waves. B) Force time series resulting from a single wave. The unfiltered signal is also presented.

PAGE 184

184 100 50 0 50 100 150 0.0 5.0 10.0 15.0 Force (lbs) Time (sec) Raw Slamming A 100 50 0 50 100 150 10.0 10.5 11.0 11.5 12.0 12.5 Force (lbs) Time (sec) Raw Slamming B Figure 6-19. Total slamming force examples for a t ypical subaerial slab case. A) Force time series resulting from the passage of several waves. B) Force time series resulting from a single wave. The unfiltered signal is also presented.

PAGE 185

185 Table 6 -1. Test breakdown by target and count. Test setup Target Number of tests Slab Quasi static 150 Beam and slab Quasi static 150 Beam and slab with overhangs Quasi static 150 Beam and slab with overhangs and rails Quasi static 150 Beam and slab with overhangs and rails (alternate) Quasi static 150 Flat Plate Slamming 180 Beam and slab with overhangs and rails Slamming 150 Beam and slab Repeatability 80 Flat Plate Repeatability 80 Table 6 -2. Range of fluid variable values covered in the physical model testing. Parameter Range minimum Range maximum Water depth (ft) 1.50 3.00 Wave period (s) 1.00 3.50 Wave height (ft) 0.00 1.16 Clearance height (ft) 8.00 4.00

PAGE 186

186 CHAPTER 7 RESULTS AND ANALYSIS The theoretical model for wave loading on bri dge superstructures presented in Chapters 3 and 4 was developed and refined within the conf ines of the numerical model prior to conducting the wave tank tests. The wave tank tests were then performed to not only investigate the physical aspects and characteristics of wave loading (quasi-static and slamming), but also to test the theoretical model and obtain the relevant information needed to establish ranges and relationships for the various coefficients cont ained there within (drag, inertial, mass rate, buoyancy). Also investigated were the effects of certain common structural components to bridge superstructures whose complexity fell outside of the base theoretical model but still influenced the overall forcing of the system. These addi tional components were accounted for through the use of adjustment factors to the overall equations. The flow field for this problem is extrem ely complex and, as w ould be expected, the coefficients are, in general, not constants but functions of the water, wave, and structure parameters. Relationships between these coeffi cients and dimensionless groups (formed with wave and structure variables identified in Chapte r 3) were established based on a portion of the experimental data. The relationshi ps for these coefficients were then tested against the portions of the physical model data that were not used in their development to validate the accuracy of the expressions. Once verified at laboratory scale, the coefficient relationships were integrated into the numerical model and tested against prototype da ta. The bridge prototypes used were two existing structures that both had been subjec ted to significant wave loading during hurricane

PAGE 187

187 events. The prototypes chosen were for one case where a bridge suffe red failures under wave loading and one case wh ere a bridge did not. The prototype case that experienced structural failures was the I-10 bridge over Escambia Bay, Florida during Hurricane Ivan in 2004. The da mage to this structure included the removal of a large number of spans from their founda tion as described in Chapter 1. OEA, Inc. accurately documented the exact location of all failed (and non-failed) spans across the bay and completed a full hindcast of Hurricane Ivan at the behest of the Florida Department of Transportation, the results of whic h provided the water elevations, si gnificant wave heights, peak wave periods, and depth-averaged flow velocities at each span throughout the duration of the storm. The prototype case that did not experience structural failur e was the SR687 Big Island Gap bridge over Old Tampa Bay, Florid a during Hurricane Gladys in 1968. A hindcast for this storm was also done by OEA, Inc., and the same re levant fluid and struct ure parameters were developed at each span location across the bridge. This provided an excellent data set for testing the equations at the prototype scale. The surge/wave loads on these bridge superstructures during these storm events were then computed with the numerical model and compared to th e resistive forces of the structures (and correspondingly whether or not the span failed in reality or the model). Field data of this completeness and accuracy is very difficult to acquire thus it is fortunate for this study that such data exists. Agreement of nonfailed/failed spans between the model and reality was excellent. With the developed relationships validated ag ainst prototype cases, a set of parametric equations was developed for the purposes of design from the final numerical model. Limitations to application of the model are evaluated and ascribed below.

PAGE 188

188 Coefficients E mpirical coefficients for the quasistatic force were determined and expressions developed for both slab span structures and beam and slab span structures. Using the same structural parameters and wave conditions as the physical model tests for input, a horizontal and vertical force time series was calculated using the numerical model for the first three waves in the wave train of each case. Each force time series consisted of the individual time series of the inertia force (separate inertial and mass rate components), the drag force, and the buoyancy (addressed below). This data was output from the model at the same frequency resolution as the experimental data for ease of comparison. From the physical model tests, the filtered quasistatic force was used for comparison with the predictive model. By varying the drag, inertial, mass rate and buoyancy coefficients as well as certain characteristics of the numerical model, a best fit was obtained using the least squares method for the forcing over the full period of a single wave. The collection of estimated coefficients was then plotted against dimensionless groups described in Chapter 3 from the dimensional analysis. Curve-fitting was carried out and expressions developed for the relationship between the coefficients and the relevant dimensionless groups. These expressions were retested against the physical model data. Trends in forcing and their relationships to certain parameters were examined. Slab Span Structures The analysis of vertical and horizontal slamming for the slab span structure is beyond the scope of this work, so mass rate coefficients developed here are strictly for use in predicting quasi-static forcing. However, a previously developed empirical relationship for slamming for smooth-bottomed structures (Marin 2009) is checked against the physical model data to further validate its predictive accuracy and gauged conservatism.

PAGE 189

189 Vertical inertia coefficients The inertia coefficients for inertial force a nd mass rate force in the vertical direction depend most upon a relative size parameter (W*/H), a relative width parameter (W*/ ), a wave steepness (H/ ) parameter, and a relative clearance parameter (ZCmin/D*), where W* is the wetted slab width, H is the wave height, is the wave length, ZC is the clearance height, min is wave trough elevation, and D* is the wetted slab thickness. Best fit values (i.e., values determined from the experimental data) of the inertial coefficient (CIZ) ranged from 0.8 to 1.2 and related well to the relative width parameter given above. Best fit values of the mass rate coefficient (CMZ) ranged from 1.0 to 2.9 and related well to the relative size, relative steepness, and relative cleara nce parameters given above. Expressions for the inertial and mass rate coefficien ts for a slab span in the vertical direction are shown in Equations 7-1 and 7-2 wh ere subscripted K is a constant. The representative curve fit for the mass rate coefficient is given in Figure 7-1. IZ1 1W CKfor K082 *exp (7-1) 11 2 K CMIN MZ 234 5 3 4 5K-02 K20 Z HW CK K KK-f o r K 1 0 HD K-037 K30* *. ln lnexp (7-2) The general trends of these expressions agree with the intuitive physic s of the problem. For the inertial coefficient, as the wetted widt h increases, it follows that the impact upon the wave field would increase. Conversely, as the wave length increa se, the slab would be expected to have less impact on the wave field.

PAGE 190

190 For the mass rate coefficient, as the wave steepness (H ) increases, the rate of inundation decreases and so subsequently would the mass rate force. As the relative size (W*/H) increases, it follows that the impact upon the wave field increases and so subsequently would the magnitude of the mass rate force. As the relative clearance decreases, the mass rate force would decrease since the structure is either losing mass moving towards the point where the mass rate force shuts off (due to the structure always being wetted). Horizontal inertia coefficients The inertia coefficients for inertial force and mass rate force in the horizontal direction IN/T), and a max-ZC/H), where D is the wetted effective thickness (wetted slab thickness plus wetted rail height) C is the clearance max is wave crest elevation, TIN is the time of inundation, and T is the wave period. It s hould be noted that this relative clearance parameter is a variant of the previously used relative clearance parameter. Best fit values (i.e., values determined from the experimental data) of the inertial coefficient (CIX) ranged from 0.05 to 0.3 and related well to the relative size, relative clearance, and submergence parameters. Values of the mass rate coefficient (CMX) varied very little for the quasistatic force and a constant value of 1.0 has been assigned to it (the contribution of this c oefficient to horizontal slamming has not been assessed) Expressions for the inertial and mass rate coefficients for a slab span in the horizontal direction are shown in Equations 73 and 7 -4 where subscripted K is a constant. The representative curve f it for the inertial coefficient is given in Figure 7-2.

PAGE 191

191 1 11 2 maxC IN IX123 4 3 4K015 K375 DD D CKKK K for K069 K277 . (7 -3) MXC10 (7 -4) The general trend of the inertial coefficient expression follows the intuitive physics of the problem. As the relative size increases, it follows that the impact upon the wave field would increase. As the relative clearance increases (in this case, that corresponds to a decrease in actual clearance) the impact upon the wave field would increase again. As the submergence inc reases, the slab is in the wave field for a longer period of time and therefore would affect the wave field to a greater degree as well Drag coefficients The drag coefficients in the vertical and horizontal directions depend most upon a submergence parameter (TIN/T), and a relative clearance parameter (ZCmin/D*), where D* is the wetted effective thickness (wetted slab thickness plus wetted rail height), ZC is the clearance min is wave crest elevation, TIN is the time of inundation, and T is the wave period. Best fit values (i.e., values determined from the experimental data) of the horizontal drag coefficient (CDX) ranged from 1.0 to 3.0 and related well to the submergence and relative clearance parameters. Values of the vertical drag coefficient (CDZ) varied very little (and had very little impact due the inertia dominant nature of the forcing) and a constant value of 2.0 has been assigned to it. The reason that the constant was given a value of 2.0, rather than 1.0, is that previous laboratory testing by the Federal Highways Lab estimated a value of 2.0 for bridge-type structures. Expressions for the horizontal and vertical drag coefficients for a slab span are shown

PAGE 192

192 in Equations 7-5 and 7-6 where subscripted K is a constant. The representative curve fit to the data for the horizontal drag coe fficient is given in Figure 7-3. DZC20 (7-5) 1 2 Cmin IN DX12 3 2 3K300 Z T CKKKfor K020 DT K134*. ln. (7-6) The general trend of the horizontal drag coefficient expression follows the intuitive physics of the problem. As the relative clearance decr eases, the vertically projected area of inundated slab increases, and correspondingly the drag increases. As the submergence increases, the slab is seeing a larger portion of the wa ve and drag increases again. Buoyancy coefficient During comparisons of the numerical model a nd the physical model data, it was found that the theoretical model over predic ted loading on the tail end of wave inundation (i.e., when the wave was leaving the structure). It was realized that due to th e amount of energy pulled from the wave during loading (as well as upstream reflecti on off the structure face), the wave height was being reduced as the wave propaga ted past the structure. This was confirmed in the physical model tests by the wave gauge downstream of the structure, which was showing significant reduction in wave heights upstream and downstream of the structure. The effect of this was that the buoyancy was reducing in physical model due to the reduction in wave height but not in the numerical model where the wave height stayed the same. To account for this, a buoyancy coefficient was a dded to the numerical model that created decay over the inundation cycle of the wave. The deca y function is given in Equation 7-7a, where tIN is the current time of inundation, TIN is the full time of inundation, T is the wave period, and CB is the buoyancy coefficient.

PAGE 193

193 The buoyancy coefficient for slab span de pends most upon a submergence parameter (TIN/T), and a relative clearance parameter (ZCmin/D*), where D* is the wetted effective thickness (in this case, just the wetted slab thickness), ZC is the clearance height, min is wave crest elevation, TIN is the time of inundation, and T is the wave period. Best fit values (i.e., values determined from the experimental data) of the buoyancy coefficient (CB) ranged from 0.0 to 2.0 and related well to the submergence and relative clearance parameters. An expression for the coefficient is shown in Equations 7-7b where subscripted K is a constant. Th e representative curve fit to th e data is given in Figure 7-4. IN B ININt 11T DecayC 22TTtanh (7-7a) 2 1 CMININ B12 2K0953 Z T C1KKfor K2108 DT*. lnexp (7-7b) The general trend of the buoyanc y coefficient expression follows the intuitive physics of the problem. As both the relative clearance and submergence parameters decrease, the slab is lowering further into the wave column to the po int where regardless of the impact upon the wave field, buoyancy will not decrease. Beam and Slab Span Structures In general, simple slab spans and beam and sl ab spans created nearly identical coefficient trends and expressions. For many situations, they are interchangeable structures. Variations occur when the structure has a positive clearanc e and the chambers between girders experience periodic inundation. Because slam ming is inherent to the chambered nature of the beam and slab span, coefficient relationships de veloped here cover both the slam ming and quasi-static forces in the vertical direction (horizontal slamming remains outside the scope of this work). Coefficient expressions and discussi on are given below.

PAGE 194

194 Vertical inertia coefficients The inertia coefficients for inertial force a nd mass rate force in the vertical direction depend most upon a relative size parameter (WS/H), a relative width parameter (W*/ ), a wave steepness (H/ ) parameter, and a relative clearance parameter (ZCmin/D*), where W* is the wetted slab width, WS is the girder spacing, H is the wave height, is the wave length, ZC is the clearance height, min is wave trough elevation, and D* is the wetted slab thickness. Best fit values (i.e., values determined from the experimental data) of the inertial coefficient (CIZ) ranged from 0.8 to 1.4 and related well to the relative width parameter given above. Best fit values of the mass rate coefficient (CMZ) ranged from 1.0 to 2.6 and related well to the wave steepness, relative size, and relative clear ance parameters given above. Expressions for the inertial and mass rate coefficients for a be am and slab span in the vertical direction are shown in Equations 7-8 and 7-9 wh ere subscripted K is a constant. The representative curve fit for the mass rate coefficient is given in Figure 7-5. IZ1 1W CKfor K091 *exp (7-8) 1 2 1 SCmin MZ12 2K-011 WZ H C-1KK-for K143 HD*. lnexplnexp (7-9) The general trends of these expressions agr ee with the intuitive physics of the problem with the inertial coefficient mimicking the case of the simple slab span. For the mass rate coefficient, though, the pres ence of girders and air pocketing creates a notable difference from the simple slab span the impact of the wave steepness. Whereas the increasing wave steepness in the slab span case brought about a reduction in impact due to the lessening of the rate of inundati on, the air chambers of the beam and slab span produce the opposite effect. The rate at which the mass rate force grows is controlled by the rate at which air

PAGE 195

195 is compressed which in turn is controlled by how fast the wave face rises inside the chamber to the point of full compression. A steeper wave would raise vertically faster than a shallow wave, hence the flipped relationship of the wave steepness. The other parameters behave similarly to the simple slab span case. Horizontal inertia coefficients The inertia coefficients for inertial force and mass rate force in the horizontal direction for beam and slab spans mirrored those of the simple slab span. In the case of the inertial coefficient, d eviations in the expressions developed between the two were found to only be a function of the overhang width (WO) but these deviations were small. The effects of the overhang on the total horizontal and vertical forcing are discussed later in this chapter. Therefore the expression s developed for the beam and slab span consist of the expressions previously developed for the simple slab span (Equations 7-3 and 7-4). Drag coefficients The coefficients for the vertical and horizontal drag force fall into the same situation as the inertia coefficients for a beam and slab span. Differences between the coefficient expressions for the slab span case and the beam and slab span case are negligible and all deviations can be attributed to the addition of overhang width (WO). The effects of the overhang on the total horizontal and vertical forcing are discussed later in this chapter. Therefore, t he expressions developed for the beam and slab span consist of the expressions previously developed for the simple slab span (Equations 75 and 7-6). Buoyancy coefficient Similar to the slab span, a decay function was needed to simulate the degradation of the wave as it interacted with the structure. The decay function is given in Equation 7-10a, where tIN

PAGE 196

196 is the current time of inundation, TIN is the full time of inundation, T is the wave period, and CB is the buoyancy coefficient. The buoyancy coefficient for beam and slab span depends most upon a submergence parameter (TIN/T), and a relative clearance parameter (ZCmin/D*), where D* is the wetted effective thickness (the wetted deck thickne ss plus the wetted girder height), ZC is the clearance height, min is wave crest elevation, TIN is the time of inundation, and T is the wave period. Best fit values (i.e., values determined from the experimental data) of the buoyancy coefficient (CB) ranged from 0.0 to 2.2 and an expression for th e coefficient is shown in Equations 7-10b where subscripted K is a constant. Th e representative curve fit to th e data is given in Figure 7-6. IN B ININt 11T DecayC 22TTtanh (7-10a) 2 1 CMININ B12 2K069 Z T C1KKfor K232 DT*. lnexp (7-10b) The variation between the deve loped expressions for the buoya ncy for simple slabs spans and beam and slab spans can be contributed to compressive air chambers. Despite this, the general trend of the para meters and their effects follow those of the simple slab span. As both the relative clearance and submergence parameters decrease, the span is lowering further into the wave column to the point where regardless of the impact upon the wave field, buoyancy will not decrease. Additional Adjustments In addition to the coefficients and the expr essions developed for them, some adjustments are required to account for the effect on the forcin g due to some of the increased complexities of some of the structures, namely overhangs.

PAGE 197

197 Overhangs are only a consideration for beam a nd slab spans, but they can increase the forcing in both the vertical and horizontal directions. In the vertical dire ction, the presence of overhangs adds increased structur e width that both adds air trap ping capabilities and a significant smooth bottomed surface that may or may not gene rate slamming effects. In the horizontal direction, the presence of overhangs disrupts the flow of the wave around the structure, affecting both drag and inertia characteristics. In lieu of applying alterations to the individual force compone nts, adjustment factors and separate expressions that are applied directly to the resultant total forcing have been developed to handle these effects. These adjustment fact ors are applied as shown in Equation 7-11. Adjusted Forcing = Adjustment Factor Total Estimated Forcing (7-11) Overhang effects on vertical loading The overhangs create additional forcing to both quasi-static force and the slamming force. While the slamming force effects are limited to th e overhangs themselves, the quasi-static effects are tied into the rest of the structur e by increasing overa ll wetted widths. For the quasi-static force, overhangs add width to the structure that is offset from the clearance level of the re st of the structure by the girder he ight. This difference in clearance levels is critical to determining the effect the overhangs have. From the physical model testing and the numerical model it was found that the effects of the overha ng are related to an adjusted relative clearance (ZC+HGmin/H) and relative width parameters (WO/W), where W is the span width, WO is the overhang width, H is the wave height, ZC is the clearance height, min is wave trough elevation, and HG is the girder height. The expression for vertical quasi-static force adjustments for overhangs is given in Equation 7-12. OCGm i n QWZH Adj.F1when WW WHtanh (7-12)

PAGE 198

198 From this expression it is seen that as th e water level rises the contribution of the overhangs to the forcing increases. It also impor tant to note the restriction on this adjustment, that this, the adjustment only applies when the wetted width of the structure is equal to the span width. If the wetted width is less than the span width, the overhangs ha ve little or no effect on the quasi-static force, just a slight shift in the phasing. For the slamming force (FS), the empirical relationship deve loped in Marin (2009) for flat plates is adequate for initial assessment pur poses, proving slightly c onservative against the physical model data. For detailed predictive design, further work is needed for slabs. Overhang effects on horizontal loading In the horizontal direction, the overhangs create additional quasi-static forces by limiting the flow of the wave around the structure and by crea ting a larger impact on the wave field. From the physical model tests, it was found that the effects of the overhang are related to a deflection parameter (WO/HG), where WO is the overhang width and HG is the girder height. The expression for horizontal quasi-static force adjust ments for overhangs is given in Equation 7-13. O Q11maxCG GW Adj.F1Kfor K022when ZH H tanh (7-13) From this expression it is seen that as the overh ang increases, its effect increases to a point where it then remains constant. It also is impor tant to note that this adjustment only applies when the crest of the wave is located above the elevation of the overhang. Comparisons To demonstrate the accuracy of the equations for laboratory scale structures, predicted versus measured (vertical and horizontal) force plots were developed for a range of water level and wave conditions. These plots, which are for bot h slab and slab and beam spans, are presented in Appendix A.

PAGE 199

199 Physical Model Data T he numerical model was run against the physical model tests again, this time with the coefficient relationships and adjustments built into the program. These comparisons were done for both slab span and beam and slab spans, for both vertical and horizontal forcing. For the slab spans, only quasi static forces were predicted. Agreement between the model and data was good. A selection of 120 comparisons covering the range of conditions for both slab (60) and beam and slab (60) spans are given in Appendix A. Prototype Bridges After checking the numerical model at the scaled level, it was run against two prototype cases of existing bridges that have undergone wave loading during hurricanes and had accurate hindcasts of the metocean conditions at the time. One bridge was selected that experienced loading and failed and one that experienced loading and didnt fail. The first prototy pe that was tested was the I -10 bridge over Escambia Bay, Florida, which failed during Hurricane Ivan and was discussed in Chapter 1. For this bridge there is a record of which spans running across the length of the bay were either completely removed from the pile caps or displaced from their original positions. The bridge spans were beam and slab type spans with a span width of 35ft, a span length of 60ft, a slab thickness of 8in, overhangs of 2.7ft, two 32 F Shape rails, and six AASHTO Type III girders. The bridge stretched across the bay at a constant elevation, except at the center where it increases in elevation to allow for boat traffic. For the metocean conditions across the bay, wave periods varied from 4.1 to 6.1s and wave heights varied from 2.9 to 9.0ft. The resistive f orce of the individual spans (600 kips) was calculated by combing the weight of the span with the tie down strength (taken as then tension limit of the tie down bolts).

PAGE 200

200 Using these parameters for input to the model, predictions of failed and nonfailed spans were compared to actual results and showed excellent agreement. A plot of the forcing versus failed and nonfailed spans is shown in Figure 7 -7. The second prototype that was tested was the SR687 Big Island Gap over Old Tam pa B ay, Florida, which experienced wave loading during Hurricane Gladys in 1968 and sustained no failed spans. The bridge spans were beam and slab spans that were similar to the I 10 over Escambia Bay spans with a span width of 34ft, a span length of 48ft, a slab thickness of 9in, overhangs of 4.8ft, two 32 F Shape rails, and five AASHTO Type III girders. For the metocean conditions across the bay, wave periods peaked at 2.8s and wave heights at 2.8ft. Notably, clearance levels during the storm were negative. The resistive force of the individual spans (297 kips) was calculated by combing the weight of the span with the tie down strength (taken as then tension limit of the tie down bolts). Using these parameters for inp ut to the model, predictions were compared to actual results and showed excellent agreement. Since there were only 5 spans for this prototype case, the spread of wave conditions was very narrow. Instead of examining each of the five spans, the loading over the full course of the storm wa s examined, where the forcing was compared to the resistive force over the range of storm surge levels and wave heights. A plot of the forcing versus resistive force is shown in Figure 7-8. Parametric Equations With the finalized numerical model validated against prototype cases, parametric equations for slab spans and beam and slab spans were developed A limiting range for each key parameter (ZD, H, T, ZC, W, L, D, WO, WS, HG, NG, and HR) was determined for prototype scale. These parameters, were vari ed over that range individually

PAGE 201

201 while holding all other variables constant and run in the numerical model and forcing output. In all, over 20,000 cases were run to create a database of parameters and forces from which to build the parametric equations. In addition to the key forces of interest (max imum and minimums of vertical and horizontal forces a nd moments), associated forces were also output (e.g., magnitude of the horizontal forcing at the tim e of maximum vertical forcing). With this dataset, forces and parameters were non-dimensionali zed and expressions developed for each force or moment of intere st as functions of the non-dimensionalized parameters. The created equations have been ad opted into the specification Guide Specifications for Bridges Vulnerable to Coastal Storms (AASHTO 2008). Limitations It is recommended that the work done and the equations developed be used within prescribed limits as they have not been verifi ed outside of certain pa rameter ranges. Those ranges are for the parameters as given below: 0 < W/ < 1 0 < H/ < 0.1 min-D < ZC < max (For this limit, the thickness parameter, D, is the effective thickness of the problem in question) L > W/10 Within these limitations, the relationships deve loped here have shown good predictive ability.

PAGE 202

202 Figure 7-1. Representative surface curve for the vertical mass rate coefficient for slab spans for a constant value of the wave steepness. Figure 72. Representative surface curve for the horizontal inertial coefficient for slab spans.

PAGE 203

203 Figure 73. Representative surface curve for the horizontal drag coefficient for slab spans. Figure 74. Representative surface curve for the buoyancy coefficient for slab spans.

PAGE 204

204 Figure 75. Representative surface curve for the vertical mass rate coefficient for beam and slab spans for a constant value of the wave steepness. Figure 76. Representative surface curve for the buoyancy coefficient for beam and slab spans.

PAGE 205

205 0 200 400 600 800 1000 1200 0153045607590105120135150165180195210225Vertical Force (Kips)Span Number Model Span Critical Resistance Damaged Spans Figure 7-7. Comparison of the numerical model against prototype case of the I-10 bridge over Escambia Bay, Florida.

PAGE 206

206 0 50 100 150 200 250 300 350 400 132.5 133.5 134.5 135.5 136.5 137.5 138.5 139.5 Total Vertical Force (kips) Time (Hrs) Model Span Critical Resistance Figure 7-8. Comparison of the numerical model against prototype case of the SR687 Big Island Gap over Old Tampa Bay Florida.

PAGE 207

207 CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS Conclusions A theoretical model for wave loading on bridge structures was developed in this study. It is an extension and modification of the models developed Morison, et al. (1950) and Kaplan et al. (1995), evolved to include bridge superstructure shapes and the meteorological and oceanographic (met/ocean) conditions found in the locations of coastal bridges. The waves in water bodies with limited fetches such as bays, harbors and coastal waterways are shorter in length than those in the open ocean. Wave forces on bridge superstructures are more complex for these conditions where the span widths are similar in magnitude to the wave lengths. This results in large variations in the vertical force over the width of the span at any point in time. By necessity, the predictive equations contain empirical coefficients that must be determined experimentally. To provide the information needed to compute the coefficients wave tanks tests were performed with model structures. Experiments were performed with slab spans and beam and slab (girder) spans. The beam and slab spans were tested with and without overhangs, with and without railings and with two different girder spacings. The tests included a range of wate r depths, span locations relative to the still water level, and wave heights and periods (lengths). Three direction load cells on each corner of the instrumented panel were used to measure the horizontal, transverse and vertical forces. Pressure transducers were used to measure the pressure on the bottom of the spans. The pressure measurements were used primarily for measurement of the high frequency slamming forces. Using the empirical coefficients determined from the laboratory data the predictive equations were used to predict the wave loads on the spans on the I-10 bridge over Escambia

PAGE 208

208 Bay, FL during Hurricane Ivan and SR687 Big Island Gap over Old Tamp Bay, Florida during Hurricane Gladys The equations did a good job of predicting both the cases where the wave loads exceeded and did not exceed the resistance (dead weight and tie-down). The water elevation and wave conditions at the bridge were provided by a detailed hindacasts. The wave force is in general composed of a low frequency component sometime s referred to as the quasi -static force and a high frequency component called the slamming force. Both components were measured and analyzed in this study. The low frequency component has the frequencies of the dominant waves which are typically in the range of 0.1Hz to 0.5 Hz. The slamming frequency is much higher, the magnitude can be equal to or larger but the duration is much shorter. Both the frequency and the duration of the slamming force depend on structure type as well as the wave properties. Th e frequency is lower and the duration longer for beam and slab type spans with air entrapment. Analytically the two forces are treated separately in the slab span with the total force being the superposition of the two and as an integrated force for beam and slab spans. The slamming force can occur before, at or after the peak of the low frequency force. A somewhat conservative but realistic assumption is that the two peaks of the vertical forces occur at the same time. The theoretical model and the expe riments were for waves approaching the spans at right angles. The lengths of the wave crests were assumed to be at least as long as the bridge spans, therefore the wave forces are uniform over the length of the span. Waves approaching at angles other than 90 degrees were not investigated but it is believed that angles other than 90 will produce lesser entrapped air and smaller total forces. In conclusion, this study has advanced the understanding and the ability to predict forces on bridge superstructures due to elevated water levels and waves. Slab and beam type bridge

PAGE 209

209 spans were investigated experimentally and the results used to test the theoretical model and to provide information needed to compute drag and inertia coefficients for the theoretical model. Work was initiated on wave slamming on horizontal structures in this study but this is a complex subject and more work is definitely needed, especially in the area of slamming on flat bottomed structure (specifically dispersive waves and multi -frequency spectra) Recommendations Out of this work, several areas have been identified as needing further dedicated study. The most important aspect of wave loading on bridges that should be investigated is slamming on smooth-bottomed structures. An adequate study of this topic should involve well distributed pressure and force measurements of slamming loads induced by dispersive waves and multifrequency spectra. Care should be taken to insure a sufficiently rigid model structure so simplify post measurement signal work when separating out slamming and quasistatic components. Slab spans and beam and slab spans are not the only bridge span types in use in coastal environments. Box girder (segmental) bridges are also extremely common. Due to its unique shape and sloped sides, box girder bridges present a unique wave face/structure interface. An in depth examination of these span types, including physical modeling is recommended and would greatly benefit a number of coastal areas (these bridges are common withi n the Florida Keys). Although, the work is thorough, the primary proving ground for these equations is based in small scale physical modeling. It is recommended that these equations (and any subsequently developed work) be checked and recalibrated against large scale testing if and when those resources become available. Finally, studies should begin to focus on loading mitigation measures. With an understanding of what contributes to loading and causes the worst case loads, the logical next step is to investigate ways in which existing bridges can lessen their risk of exposure to wave

PAGE 210

210 loading through retrofitting or other means. These methods may consist of air bleeding through vents between girders or the road deck itself or some other means, but once predictive methods are accepted, engineering mitigations from their use should follow.

PAGE 211

211 APPENDIX A COMPARISONS BETWEEN PHYSICAL AND NUMERICAL MODELS The following figures are comparison plots of the numerical model and the physical mode tests performed for simple slab spans and beam and slab spans The tests are differentiated by case number, with SLAB pre fixes for slab span cases and BSXX prefixes for beam and slab cases. Up to the first three waves in the wave train from the tests are included for comparison and both vertical and horizontal forces are given. For the slab span case, the time series that are given are the measured total (FZ-Total and FXTotal) forcing (including slamming) and the quasistatic (FZ-Quasi and FX-Quasi) forcing from the physical model tests and the predicted quasi static forces from the numerical model. For the beam and slab span case, the time series that are given are the measured raw forcing (including slamming) and the quasistatic forcing from the physical model and the predicted total forcing from the numerical model (including slamming effects). The plots included in this appendix are not the full complete battery of comparisons done during this work, but well represent the range of conditions covered in the study for both slab spans and beam and slab spans.

PAGE 212

212 50 0 50 100 150 200 250 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-1. Measured vs. predicted forces for SLAB -022. A) V ertical. B) Horizontal.

PAGE 213

213 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A2. Measured vs. predicted forces for SLAB 023. A) Vertical. B) Horizontal.

PAGE 214

214 0 50 100 150 200 250 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A3. Measured vs. predicted forces for SLAB 024. A) Vertical. B) Horizontal.

PAGE 215

215 100 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A4. Measured vs. predicted forces for SLAB 025. A) Vertical. B) Horizontal.

PAGE 216

216 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A5. Measured vs. predicted forces for SLAB 026. A) Vertical. B) Horizontal.

PAGE 217

217 100 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A6. Measured vs. predicted forces for SLAB 027. A) Vertical. B) Horizontal.

PAGE 218

218 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A7. Measured vs. predicted forces for SLAB 028. A) Vertical. B) Horizontal.

PAGE 219

219 100 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A8. Measured vs. predicted forces for SLAB 029. A) Vertical. B) Horizontal.

PAGE 220

220 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A9. Measured vs. predicted forces for SLAB 030. A) Vertical. B) Horizontal.

PAGE 221

221 50 0 50 100 150 200 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A10. Measured vs. predicted forces for SLAB -041. A) Vertical. B) Horizontal.

PAGE 222

222 30 20 10 0 10 20 30 40 50 60 70 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A11. Measured vs. predicted forces for SLAB 042. A) Vertical. B) Horizontal.

PAGE 223

223 40 20 0 20 40 60 80 100 120 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A12. Measured vs. predicted forces for SLAB 043. A) Vertical. B) Horizontal.

PAGE 224

224 20 10 0 10 20 30 40 50 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-13. Measured vs. predicted forces for SLAB 044. A) Vertical. B) Horizontal.

PAGE 225

225 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A14. Measured vs. predicted forces for SLAB 045. A) Vertical. B) Horizontal.

PAGE 226

226 30 20 10 0 10 20 30 40 50 60 70 80 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A15. Measured vs. predicted forces for SLAB -046. A) Vertical. B) Horizontal.

PAGE 227

227 60 40 20 0 20 40 60 80 100 120 140 160 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A16. Measured vs. predicted forces for SLAB 047. A) Vertical. B) Horizontal.

PAGE 228

228 60 40 20 0 20 40 60 80 100 120 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A17. Measured vs. predicted forces for SLAB 048. A) Vertical. B) Horizontal.

PAGE 229

229 60 40 20 0 20 40 60 80 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-18. Measured vs. predicted forces for SLAB 049. A) Vertical. B) Horizontal.

PAGE 230

230 30 20 10 0 10 20 30 40 50 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.0 0.5 0.0 0.5 1.0 1.5 2.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A19. Measured vs. predicted forces for SLAB 050. A) Vertical. B) Horizontal.

PAGE 231

231 40 20 0 20 40 60 80 100 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.0 0.5 0.0 0.5 1.0 1.5 2.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A20. Measured vs. predicted forces for SLAB -052. A) Vertical. B) Horizontal.

PAGE 232

232 50 0 50 100 150 200 250 300 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A21. Measured vs. predicted forces for SLAB 053. A) Vertical. B) Horizontal.

PAGE 233

233 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A22. Measured vs. predicted forces for SLAB -055. A) Vertical. B) Horizontal.

PAGE 234

234 40 20 0 20 40 60 80 100 120 140 160 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-23. Measured vs. predicted forces for SLAB 056. A) Vertical. B) Horizontal.

PAGE 235

235 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A24. Measured vs. predicted forces for SLAB 057. A) Vertical. B) Horizontal.

PAGE 236

236 40 20 0 20 40 60 80 100 120 140 160 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-25. Measured vs. predicted forces for SLAB -058. A) Vertical. B) Horizontal.

PAGE 237

237 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-26. Measured vs. predicted forces for SLAB -059. A) Vertical. B) Horizontal.

PAGE 238

238 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-27. Measured vs. predicted forces for SLAB -060. A) Vertical. B) Horizontal.

PAGE 239

239 50 0 50 100 150 200 250 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-28. Measured vs. predicted forces for SLAB -062. A) Vertical. B) Horizontal.

PAGE 240

240 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-29. Measured vs. predicted forces for SLAB -063. A) Vertical. B) Horizontal.

PAGE 241

241 50 0 50 100 150 200 250 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-30. Measured vs. predicted forces for SLAB -064. A) Vertical. B) Horizontal.

PAGE 242

242 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-31. Measured vs. predicted forces for SLAB -065. A) Vertical. B) Horizontal.

PAGE 243

243 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-32. Measured vs. predicted forces for SLAB -066. A) Vertical. B) Horizontal.

PAGE 244

244 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-33. Measured vs. predicted forces for SLAB -069. A) Vertical. B) Horizontal.

PAGE 245

245 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-34. Measured vs. predicted forces for SLAB -070. A) Vertical. B) Horizontal.

PAGE 246

246 40 20 0 20 40 60 80 100 120 140 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-35. Measured vs. predicted forces for SLAB -081. A) Vertical. B) Horizontal.

PAGE 247

247 20 0 20 40 60 80 100 120 140 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-36. Measured vs. predicted forces for SLAB -083. A) Vertical. B) Horizontal.

PAGE 248

248 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-37. Measured vs. predicted forces for SLAB -087. A) Vertical. B) Horizontal.

PAGE 249

249 50 0 50 100 150 200 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-38. Measured vs. predicted forces for SLAB -091. A) Vertical. B) Horizontal.

PAGE 250

250 40 20 0 20 40 60 80 100 120 140 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-39. Measured vs. predicted forces for SLAB -092. A) Vertical. B) Horizontal.

PAGE 251

251 50 0 50 100 150 200 250 300 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-40. Measured vs. predicted forces for SLAB -093. A) Vertical. B) Horizontal.

PAGE 252

252 50 0 50 100 150 200 250 300 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-41. Measured vs. predicted forces for SLAB -094. A) Vertical. B) Horizontal.

PAGE 253

253 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-42. Measured vs. predicted forces for SLAB -095. A) Vertical. B) Horizontal.

PAGE 254

254 40 20 0 20 40 60 80 100 120 140 160 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-43. Measured vs. predicted forces for SLAB -096. A) Vertical. B) Horizontal.

PAGE 255

255 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-44. Measured vs. predicted forces for SLAB -097. A) Vertical. B) Horizontal.

PAGE 256

256 100 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-45. Measured vs. predicted forces for SLAB -099. A) Vertical. B) Horizontal.

PAGE 257

257 0 50 100 150 200 250 300 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-46. Measured vs. predicted forces for SLAB -102. A) Vertical. B) Horizontal.

PAGE 258

258 0 50 100 150 200 250 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-47. Measured vs. predicted forces for SLAB -104. A) Vertical. B) Horizontal.

PAGE 259

259 100 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-48. Measured vs. predicted forces for SLAB -105. A) Vertical. B) Horizontal.

PAGE 260

260 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-49. Measured vs. predicted forces for SLAB -106. A) Vertical. B) Horizontal.

PAGE 261

261 100 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 5.0 0.0 5.0 10.0 15.0 20.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-50. Measured vs. predicted forces for SLAB -109. A) Vertical. B) Horizontal.

PAGE 262

262 40 20 0 20 40 60 80 100 120 140 160 180 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-51. Measured vs. predicted forces for SLAB -131. A) Vertical. B) Horizontal.

PAGE 263

263 40 20 0 20 40 60 80 100 120 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.0 0.5 0.0 0.5 1.0 1.5 2.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-52. Measured vs. predicted forces for SLAB -132. A) Vertical. B) Horizontal.

PAGE 264

264 50 0 50 100 150 200 250 300 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-53. Measured vs. predicted forces for SLAB -133. A) Vertical. B) Horizontal.

PAGE 265

265 40 20 0 20 40 60 80 100 120 140 160 180 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-54. Measured vs. predicted forces for SLAB -134. A) Vertical. B) Horizontal.

PAGE 266

266 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-55. Measured vs. predicted forces for SLAB -135. A) Vertical. B) Horizontal.

PAGE 267

267 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-56. Measured vs. predicted forces for SLAB -137. A) Vertical. B) Horizontal.

PAGE 268

268 0 50 100 150 200 250 300 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-57. Measured vs. predicted forces for SLAB -142. A) Vertical. B) Horizontal.

PAGE 269

269 0 50 100 150 200 250 300 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-58. Measured vs. pred icted forces for SLAB -144. A) Vertical. B) Horizontal.

PAGE 270

270 100 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-59. Measured vs. predicted forces for SLAB -145. A) Vertical. B) Horizontal.

PAGE 271

271 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-60. Measured vs. predicted forces for SLAB -148. A) Vertical. B) Horizontal.

PAGE 272

272 60 40 20 0 20 40 60 80 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-61. Measured vs. predicted forces for BSXX-002. A) Vertical. B) Horizontal.

PAGE 273

273 60 40 20 0 20 40 60 80 100 120 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-62. Measured vs. predicted forces for BSXX-003. A) Vertical. B) Horizontal.

PAGE 274

274 80 60 40 20 0 20 40 60 80 100 0.0 1.0 2.0 3.0 4.0 5.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 1.0 2.0 3.0 4.0 5.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-63. Measured vs. predicted forces for BSXX-004. A) Vertical. B) Horizontal.

PAGE 275

275 80 60 40 20 0 20 40 60 80 0.0 1.0 2.0 3.0 4.0 5.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 0.0 1.0 2.0 3.0 4.0 5.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figu re A -64. Measured vs. predicted forces for BSXX-006. A) Vertical. B) Horizontal.

PAGE 276

276 80 60 40 20 0 20 40 60 80 100 0.0 1.0 2.0 3.0 4.0 5.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 5.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-65. Measured vs. predicted forces for BSXX-008. A) Vertical. B) Horizontal.

PAGE 277

277 50 30 10 10 30 50 70 90 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.0 0.5 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-66. Measured vs. predicted forces for BSXX-010. A) Vertical. B) Horizontal.

PAGE 278

278 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-67. Measured vs. predicted forces for BSXX-011. A) Vertical. B) Horizontal.

PAGE 279

279 40 20 0 20 40 60 80 100 120 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-68. Measured vs. predicted forces for BSXX-012. A) Vert ical. B) Horizontal.

PAGE 280

280 100 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-69. Measured vs. predicted forces for BSXX-013. A) Vertical. B) Horizontal.

PAGE 281

281 50 30 10 10 30 50 70 90 110 130 150 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-70. Measured vs. predicted forces for BSXX-014. A) Vertical. B) Horizontal.

PAGE 282

282 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-71. Measured vs. predicted forces for BSXX-015. A) Vertical. B) Horizontal.

PAGE 283

283 40 20 0 20 40 60 80 100 120 140 160 180 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.0 0.0 1.0 2.0 3.0 4.0 5.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-72. Measured vs. predicted forces for BSXX-016. A) Vertical. B) Horizontal.

PAGE 284

284 100 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-73. Measured vs. predicted forces for BSXX-017. A) Vertical. B) Horizontal.

PAGE 285

285 100 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-74. Measured vs. predicted forces for BSXX-019. A) Vertical. B) Horizontal.

PAGE 286

286 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-75. Measured vs. predicted forces for BSXX-021. A) Vertical. B) Horizontal.

PAGE 287

287 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-76. Measured vs. predicted forces for BSXX-022. A) Vertical. B) Horizontal.

PAGE 288

288 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-77. Measured vs. predicted forces for BSXX-023. A) Vertical. B) Horizontal.

PAGE 289

289 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-78. Measured vs. predicted forces for BSXX-024. A) Vertical. B) Horizontal.

PAGE 290

290 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-79. Measured vs. predicted forces for BSXX-025. A) Vertical. B) Horizontal.

PAGE 291

291 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-80. Measured vs. predicted forces for BSXX-026. A) Vertical. B) Horizontal.

PAGE 292

292 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-81. Measured vs. predicted forces for BSXX-030. A) Vertical. B) Horizontal.

PAGE 293

293 60 40 20 0 20 40 60 80 100 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 8.0 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-82. Measured vs. predicted forces for BSXX-041. A) Vertical. B) Horizontal.

PAGE 294

294 60 40 20 0 20 40 60 80 100 120 140 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-83. Measured vs. predicted forces for BSXX-042. A) Vertical. B) Horizontal.

PAGE 295

295 50 0 50 100 150 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 8.0 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-84. Measured vs. predicted forces for BSXX-043. A) Vertical. B) Horizontal.

PAGE 296

296 60 40 20 0 20 40 60 80 100 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-85. Measured vs. predicted forces for BSXX-044. A) Vertical. B) Horizontal.

PAGE 297

297 50 0 50 100 150 200 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-86. Measured vs. predicted forces for BSXX-051. A) Vertical. B) Horizontal.

PAGE 298

298 20 0 20 40 60 80 100 120 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-87. Measured vs. predicted forces for BSXX-052. A) Vertical. B) Horizontal.

PAGE 299

299 50 0 50 100 150 200 250 300 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-88. Measured vs. predicted forces for BSXX-053. A) Vertical. B) Horizontal.

PAGE 300

300 50 0 50 100 150 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-89. Measured vs. predicted forces for BSXX-054. A) Vertical. B) Horizontal.

PAGE 301

301 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 10.0 5.0 0.0 5.0 10.0 15.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-90. Measured vs. predicted forces for BSXX-055. A) Vertical. B) Horizontal.

PAGE 302

302 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-91. Measured vs. predicted forces for BSXX-056. A) Vertical. B) Horizontal.

PAGE 303

303 100 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 10.0 5.0 0.0 5.0 10.0 15.0 20.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-92. Measured vs. predicted forces for BSXX-057. A) Vertical. B) Horizontal.

PAGE 304

304 100 50 0 50 100 150 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 5.0 0.0 5.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-93. Measured vs. predicted forces for BSXX-059. A) Vertical. B) Horizontal.

PAGE 305

305 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-94. Measured vs. predicted forces for BSXX-061. A) Vertical. B) Horizontal.

PAGE 306

306 0 50 100 150 200 250 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-95. Measured vs. predicted forces for BSXX-062. A) Vertical. B) Horizontal.

PAGE 307

307 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-96. Measured vs. predicted forces for BSXX-063. A) Vertical. B) Horizontal.

PAGE 308

308 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-97. Measured vs. predicted forces for BSXX-065. A) Vertical. B) Horizontal.

PAGE 309

309 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 1.0 2.0 3.0 4.0 5.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-98. Measured vs. predicted forces for BSXX-066. A) Vertical. B) Horizontal.

PAGE 310

310 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A-99. Measured vs. predicted forces for BSXX-069. A) Vertical. B) Horizontal.

PAGE 311

311 60 40 20 0 20 40 60 80 100 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A100 Measured vs. predicted forces for BSXX-082. A) Vertical. B) Horizontal.

PAGE 312

312 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 10.0 5.0 0.0 5.0 10.0 15.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A101 Measured vs. predicted forces for BSXX-083. A) Vertical. B) Horizontal.

PAGE 313

313 60 40 20 0 20 40 60 80 100 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 8.0 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A102 Measured vs. predicted forces for BSXX-086. A) Vertical. B) Horizontal.

PAGE 314

314 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A103 Measured vs. predicted forces for BSXX-091. A) Vertical. B) Horizontal.

PAGE 315

315 100 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 10.0 5.0 0.0 5.0 10.0 15.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A104 Measured vs. predicted forces for BSXX-093. A) Vertical. B) Horizontal.

PAGE 316

316 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 10.0 5.0 0.0 5.0 10.0 15.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A105 Measured vs. predicted forces for BSXX-096. A) Vertical. B) Horizontal.

PAGE 317

317 50 0 50 100 150 200 250 300 350 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A106 Measured vs. predicted forces for BSXX-101. A) Vertical. B) Horizontal.

PAGE 318

318 0 50 100 150 200 250 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A107 Measured vs. predicted forces for BSXX-102. A) Vertical. B) Horizontal.

PAGE 319

319 0 50 100 150 200 250 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A108 Measured vs. predicted forces for BSXX-104. A) Vertical. B) Horizontal.

PAGE 320

320 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A109 Measured vs. predicted forces for BSXX-107. A) Vertical. B) Horizontal.

PAGE 321

321 40 20 0 20 40 60 80 100 120 140 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A110 Measured vs. predicted forces for BSXX-121. A) Vertical. B) Horizontal.

PAGE 322

322 40 20 0 20 40 60 80 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 1.0 0.5 0.0 0.5 1.0 1.5 2.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A111 Measured vs. predicted forces for BSXX-122. A) Vertical. B) Horizontal.

PAGE 323

323 100 50 0 50 100 150 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A112 Measured vs. predicted forces for BSXX-124. A) Vertical. B) Horizontal.

PAGE 324

324 100 50 0 50 100 150 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A113 Measured vs. predicted forces for BSXX-128. A) Vertical. B) Horizontal.

PAGE 325

325 50 0 50 100 150 200 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 0.0 2.0 4.0 6.0 8.0 10.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A114 Measured vs. predicted forces for BSXX-131. A) Vertical. B) Horizontal.

PAGE 326

326 40 20 0 20 40 60 80 100 120 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A115 Measured vs. predicted forces for BSXX-132. A) Vertical. B) Horizontal.

PAGE 327

327 50 0 50 100 150 200 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A116 Measured vs. predicted forces for BSXX-134. A) Vertical. B) Horizontal.

PAGE 328

328 100 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 10.0 5.0 0.0 5.0 10.0 15.0 20.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A117 Measured vs. predicted forces for BSXX-137. A) Vertical. B) Horizontal.

PAGE 329

329 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A118 Measured vs. predicted forces for BSXX-141. A) Vertical. B) Horizontal.

PAGE 330

330 50 0 50 100 150 200 250 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A119 Measured vs. predicted forces for BSXX-144. A) Vertical. B) Horizontal.

PAGE 331

331 50 0 50 100 150 200 250 300 0.0 1.0 2.0 3.0 4.0 5.0 Force (lbs) Time (Sec) Fz Total Fz Quasi Model A 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 0.0 1.0 2.0 3.0 4.0 5.0 Force (lbs) Time (sec) Fx Total Fx Quasi Model B Figure A120 Measured vs. predicted forces for BSXX-148. A) Vertical. B) Horizontal.

PAGE 332

332 APPENDIX B PHYSICAL MODEL DATA QUASI STATIC FORCES The following tables are a list of all physical model tests performed and the significant variables and values associated with each test. The tables are divided into variables and forces. The tests can be differentiated by the indi vidual case prefix and reference number Four different setups were used in this section with SLAB signifying tests done with the slab model, BSXX signifying tests done with the beam and slab model with no overhangs or rails, BSOX signifying tests done with the beam and slab model with overhangs but no rails, and BSOR signifying tests done with the beam and slab model with overhangs and rails. Table B -1 contains the relevant fluid and structure parameters for all tests. Table B -2 contains the measured significant forces and moments for all tests. All dimensions are in feet, all forces are in pounds, and all times are in s econds.

PAGE 333

333 Table B 1. Structure and fluid parameters for all physical model tests. Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders SLAB0014.006.000.580.171.420.923.5022.930 SLAB0024.006.000.580.171.420.583.5022.930 SLAB0034.006.000.580.171.420.733.0019.590 SLAB0044.006.000.580.171.420.633.0019.590 SLAB0054.006.000.580.171.420.862.5016.100 SLAB0064.006.000.580.171.420.812.5016.100 SLAB0074.006.000.580.171.420.692.0012.530 SLAB0084.006.000.580.171.420.632.0012.530 SLAB0094.006.000.580.171.420.841.508.820 SLAB0104.006.000.580.171.420.721.508.820 SLAB0114.006.000.580.001.580.743.5024.240 SLAB0124.006.000.580.001.580.593.5024.240 SLAB0134.006.000.580.001.580.683.0020.640 SLAB0144.006.000.580.001.580.463.0020.640 SLAB0154.006.000.580.001.581.112.5016.920 SLAB0164.006.000.580.001.580.792.5016.920 SLAB0174.006.000.580.001.580.692.0013.120 SLAB0184.006.000.580.001.580.582.0013.120 SLAB0194.006.000.580.001.580.801.509.170 SLAB0204.006.000.580.001.580.731.509.170 SLAB0214.006.000.58-0.291.881.223.5026.310 SLAB0224.006.000.58-0.291.880.653.5026.310 SLAB0234.006.000.58-0.291.880.593.0022.310 SLAB0244.006.000.58-0.291.880.453.0022.310 SLAB0254.006.000.58-0.291.881.032.5018.230 SLAB0264.006.000.58-0.291.880.672.5018.230 SLAB0274.006.000.58-0.291.880.782.0014.050 SLAB0284.006.000.58-0.291.880.562.0014.050 SLAB0294.006.000.58-0.291.880.851.509.670 SLAB0304.006.000.58-0.291.880.691.509.670 SLAB0314.006.000.58-0.632.210.963.5028.410 SLAB0324.006.000.58-0.632.210.673.5028.410 SLAB0334.006.000.58-0.632.210.703.0024.020 SLAB0344.006.000.58-0.632.210.453.0024.020 SLAB0354.006.000.58-0.632.211.022.5019.550

PAGE 334

334 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders SLAB0364.006.000.58-0.632.210.822.5019.550 SLAB0374.006.000.58-0.632.210.822.0014.960 SLAB0384.006.000.58-0.632.210.712.0014.960 SLAB0394.006.000.58-0.632.210.901.5010.130 SLAB0404.006.000.58-0.632.210.851.5010.130 SLAB0414.006.000.580.171.750.953.5025.450 SLAB0424.006.000.580.171.750.543.5025.450 SLAB0434.006.000.580.171.750.773.0021.620 SLAB0444.006.000.580.171.750.473.0021.620 SLAB0454.006.000.580.171.750.952.5017.690 SLAB0464.006.000.580.171.750.632.5017.690 SLAB0474.006.000.580.171.750.772.0013.670 SLAB0484.006.000.580.171.750.702.0013.670 SLAB0494.006.000.580.171.750.861.509.470 SLAB0504.006.000.580.171.750.751.509.470 SLAB0514.006.000.580.001.921.023.5026.590 SLAB0524.006.000.580.001.920.503.5026.590 SLAB0534.006.000.580.001.920.603.0022.540 SLAB0544.006.000.580.001.920.393.0022.540 SLAB0554.006.000.580.001.920.862.5018.400 SLAB0564.006.000.580.001.920.662.5018.400 SLAB0574.006.000.580.001.920.632.0014.170 SLAB0584.006.000.580.001.920.462.0014.170 SLAB0594.006.000.580.001.920.771.509.740 SLAB0604.006.000.580.001.920.641.509.740 SLAB0614.006.000.58-0.292.210.793.5028.410 SLAB0624.006.000.58-0.292.210.463.5028.410 SLAB0634.006.000.58-0.292.210.733.0024.020 SLAB0644.006.000.58-0.292.210.443.0024.020 SLAB0654.006.000.58-0.292.210.922.5019.550 SLAB0664.006.000.58-0.292.210.662.5019.550 SLAB0674.006.000.58-0.292.210.952.0014.960 SLAB0684.006.000.58-0.292.210.652.0014.960 SLAB0694.006.000.58-0.292.210.831.5010.130 SLAB0704.006.000.58-0.292.210.651.5010.130

PAGE 335

335 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders SLAB0714.006.000.58-0.632.541.333.5030.310 SLAB0724.006.000.58-0.632.540.723.5030.310 SLAB0734.006.000.58-0.632.540.923.0025.570 SLAB0744.006.000.58-0.632.540.623.0025.570 SLAB0754.006.000.58-0.632.540.962.5020.730 SLAB0764.006.000.58-0.632.540.632.5020.730 SLAB0774.006.000.58-0.632.541.152.0015.740 SLAB0784.006.000.58-0.632.540.782.0015.740 SLAB0794.006.000.58-0.632.541.131.5010.480 SLAB0804.006.000.58-0.632.540.621.5010.480 SLAB0814.006.000.58-0.172.080.603.5027.650 SLAB0824.006.000.58-0.172.080.383.5027.650 SLAB0834.006.000.58-0.172.080.733.0023.400 SLAB0844.006.000.58-0.172.080.483.0023.400 SLAB0854.006.000.58-0.172.080.802.5019.080 SLAB0864.006.000.58-0.172.080.622.5019.080 SLAB0874.006.000.58-0.172.081.002.0014.630 SLAB0884.006.000.58-0.172.080.652.0014.630 SLAB0894.006.000.58-0.172.080.861.509.970 SLAB0904.006.000.58-0.172.080.591.509.970 SLAB0914.006.000.580.002.250.913.5028.660 SLAB0924.006.000.580.002.250.573.5028.660 SLAB0934.006.000.580.002.250.683.0024.230 SLAB0944.006.000.580.002.250.593.0024.230 SLAB0954.006.000.580.002.250.992.5019.710 SLAB0964.006.000.580.002.250.732.5019.710 SLAB0974.006.000.580.002.250.942.0015.060 SLAB0984.006.000.580.002.250.762.0015.060 SLAB0994.006.000.580.002.250.841.5010.180 SLAB1004.006.000.580.002.250.721.5010.180 SLAB1014.006.000.58-0.292.540.983.5030.310 SLAB1024.006.000.58-0.292.540.613.5030.310 SLAB1034.006.000.58-0.292.540.883.0025.570 SLAB1044.006.000.58-0.292.540.603.0025.570 SLAB1054.006.000.58-0.292.540.922.5020.730

PAGE 336

336 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders SLAB1064.006.000.58-0.292.540.632.5020.730 SLAB1074.006.000.58-0.292.541.412.0015.740 SLAB1084.006.000.58-0.292.541.102.0015.740 SLAB1094.006.000.58-0.292.541.081.5010.480 SLAB1104.006.000.58-0.292.540.951.5010.480 SLAB1114.006.000.58-0.632.880.893.5032.050 SLAB1124.006.000.58-0.632.880.523.5032.050 SLAB1134.006.000.58-0.632.880.743.0026.970 SLAB1144.006.000.58-0.632.880.613.0026.970 SLAB1154.006.000.58-0.632.880.752.5021.780 SLAB1164.006.000.58-0.632.880.572.5021.780 SLAB1174.006.000.58-0.632.881.322.0016.410 SLAB1184.006.000.58-0.632.881.212.0016.410 SLAB1194.006.000.58-0.632.881.011.5010.760 SLAB1204.006.000.58-0.632.880.971.5010.760 SLAB1214.006.000.58-0.172.420.973.5029.620 SLAB1224.006.000.58-0.172.420.583.5029.620 SLAB1234.006.000.58-0.172.420.853.0025.010 SLAB1244.006.000.58-0.172.420.603.0025.010 SLAB1254.006.000.58-0.172.420.962.5020.300 SLAB1264.006.000.58-0.172.420.762.5020.300 SLAB1274.006.000.58-0.172.421.142.0015.460 SLAB1284.006.000.58-0.172.420.802.0015.460 SLAB1294.006.000.58-0.172.421.021.5010.360 SLAB1304.006.000.58-0.172.420.781.5010.360 SLAB1314.006.000.580.002.580.973.5030.540 SLAB1324.006.000.580.002.580.553.5030.540 SLAB1334.006.000.580.002.580.793.0025.750 SLAB1344.006.000.580.002.580.523.0025.750 SLAB1354.006.000.580.002.580.832.5020.870 SLAB1364.006.000.580.002.580.742.5020.870 SLAB1374.006.000.580.002.581.152.0015.830 SLAB1384.006.000.580.002.581.262.0015.830 SLAB1394.006.000.580.002.580.991.5010.520 SLAB1404.006.000.580.002.580.851.5010.520

PAGE 337

337 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders SLAB1414.006.000.58-0.292.881.003.5032.050 SLAB1424.006.000.58-0.292.880.723.5032.050 SLAB1434.006.000.58-0.292.881.043.0026.970 SLAB1444.006.000.58-0.292.880.773.0026.970 SLAB1454.006.000.58-0.292.880.922.5021.780 SLAB1464.006.000.58-0.292.880.702.5021.780 SLAB1474.006.000.58-0.292.881.452.0016.410 SLAB1484.006.000.58-0.292.881.322.0016.410 SLAB1494.006.000.58-0.292.881.141.5010.760 SLAB1504.006.000.58-0.292.881.031.5010.760 BSXX001 4.006.000.580.171.420.773.5022.937 BSXX002 4.006.000.580.171.420.453.5022.937 BSXX003 4.006.000.580.171.420.753.0019.597 BSXX004 4.006.000.580.171.420.483.0019.597 BSXX005 4.006.000.580.171.420.862.5016.107 BSXX006 4.006.000.580.171.420.642.5016.107 BSXX007 4.006.000.580.171.420.712.0012.537 BSXX008 4.006.000.580.171.420.592.0012.537 BSXX009 4.006.000.580.171.420.761.508.827 BSXX010 4.006.000.580.171.420.791.508.827 BSXX011 4.006.000.580.001.581.013.5024.247 BSXX012 4.006.000.580.001.580.643.5024.247 BSXX013 4.006.000.580.001.580.753.0020.647 BSXX014 4.006.000.580.001.580.553.0020.647 BSXX015 4.006.000.580.001.581.042.5016.927 BSXX016 4.006.000.580.001.580.712.5016.927 BSXX017 4.006.000.580.001.580.662.0013.127 BSXX018 4.006.000.580.001.580.552.0013.127 BSXX019 4.006.000.580.001.580.761.509.177 BSXX020 4.006.000.580.001.580.841.509.177 BSXX021 4.006.000.58-0.291.880.793.5026.317 BSXX022 4.006.000.58-0.291.880.643.5026.317 BSXX023 4.006.000.58-0.291.880.543.0022.317 BSXX024 4.006.000.58-0.291.880.423.0022.317 BSXX025 4.006.000.58-0.291.880.922.5018.237

PAGE 338

338 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders BSXX026 4.006.000.58-0.291.880.872.5018.237 BSXX027 4.006.000.58-0.291.880.822.0014.057 BSXX028 4.006.000.58-0.291.880.562.0014.057 BSXX029 4.006.000.58-0.291.880.811.509.677 BSXX030 4.006.000.58-0.291.880.771.509.677 BSXX031 4.006.000.58-0.632.211.143.5028.417 BSXX032 4.006.000.58-0.632.210.653.5028.417 BSXX033 4.006.000.58-0.632.210.723.0024.027 BSXX034 4.006.000.58-0.632.210.533.0024.027 BSXX035 4.006.000.58-0.632.210.852.5019.557 BSXX036 4.006.000.58-0.632.210.692.5019.557 BSXX037 4.006.000.58-0.632.210.772.0014.967 BSXX038 4.006.000.58-0.632.210.772.0014.967 BSXX039 4.006.000.58-0.632.210.941.5010.137 BSXX040 4.006.000.58-0.632.210.901.5010.137 BSXX041 4.006.000.580.171.750.653.5025.457 BSXX042 4.006.000.580.171.750.723.5025.457 BSXX043 4.006.000.580.171.750.613.0021.627 BSXX044 4.006.000.580.171.750.583.0021.627 BSXX045 4.006.000.580.171.750.852.5017.697 BSXX046 4.006.000.580.171.750.682.5017.697 BSXX047 4.006.000.580.171.750.782.0013.677 BSXX048 4.006.000.580.171.750.812.0013.677 BSXX049 4.006.000.580.171.750.791.509.477 BSXX050 4.006.000.580.171.750.811.509.477 BSXX051 4.006.000.580.001.920.863.5026.597 BSXX052 4.006.000.580.001.920.583.5026.597 BSXX053 4.006.000.580.001.920.713.0022.547 BSXX054 4.006.000.580.001.920.533.0022.547 BSXX055 4.006.000.580.001.920.832.5018.407 BSXX056 4.006.000.580.001.920.712.5018.407 BSXX057 4.006.000.580.001.920.842.0014.177 BSXX058 4.006.000.580.001.920.912.0014.177 BSXX059 4.006.000.580.001.920.801.509.747 BSXX060 4.006.000.580.001.920.691.509.747

PAGE 339

339 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders BSXX061 4.006.000.58-0.292.210.803.5028.417 BSXX062 4.006.000.58-0.292.210.623.5028.417 BSXX063 4.006.000.58-0.292.210.643.0024.027 BSXX064 4.006.000.58-0.292.210.483.0024.027 BSXX065 4.006.000.58-0.292.210.882.5019.557 BSXX066 4.006.000.58-0.292.210.822.5019.557 BSXX067 4.006.000.58-0.292.211.122.0014.967 BSXX068 4.006.000.58-0.292.210.992.0014.967 BSXX069 4.006.000.58-0.292.210.911.5010.137 BSXX070 4.006.000.58-0.292.210.851.5010.137 BSXX071 4.006.000.58-0.632.541.063.5030.317 BSXX072 4.006.000.58-0.632.540.793.5030.317 BSXX073 4.006.000.58-0.632.540.943.0025.577 BSXX074 4.006.000.58-0.632.540.623.0025.577 BSXX075 4.006.000.58-0.632.540.802.5020.737 BSXX076 4.006.000.58-0.632.540.652.5020.737 BSXX077 4.006.000.58-0.632.541.422.0015.747 BSXX078 4.006.000.58-0.632.540.962.0015.747 BSXX079 4.006.000.58-0.632.541.031.5010.487 BSXX080 4.006.000.58-0.632.541.021.5010.487 BSXX081 4.006.000.58-0.172.080.833.5027.657 BSXX082 4.006.000.58-0.172.080.533.5027.657 BSXX083 4.006.000.58-0.172.080.923.0023.407 BSXX084 4.006.000.58-0.172.080.523.0023.407 BSXX085 4.006.000.58-0.172.080.862.5019.087 BSXX086 4.006.000.58-0.172.080.642.5019.087 BSXX087 4.006.000.58-0.172.080.892.0014.637 BSXX088 4.006.000.58-0.172.080.752.0014.637 BSXX089 4.006.000.58-0.172.080.911.509.977 BSXX090 4.006.000.58-0.172.080.811.509.977 BSXX091 4.006.000.580.002.251.073.5028.667 BSXX092 4.006.000.580.002.250.713.5028.667 BSXX093 4.006.000.580.002.250.873.0024.237 BSXX094 4.006.000.580.002.250.603.0024.237 BSXX095 4.006.000.580.002.251.012.5019.717

PAGE 340

340 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders BSXX096 4.006.000.580.002.250.862.5019.717 BSXX097 4.006.000.580.002.251.042.0015.067 BSXX098 4.006.000.580.002.250.902.0015.067 BSXX099 4.006.000.580.002.251.011.5010.187 BSXX100 4.006.000.580.002.250.961.5010.187 BSXX101 4.006.000.58-0.292.541.143.5030.317 BSXX102 4.006.000.58-0.292.540.693.5030.317 BSXX103 4.006.000.58-0.292.540.923.0025.577 BSXX104 4.006.000.58-0.292.540.603.0025.577 BSXX105 4.006.000.58-0.292.541.032.5020.737 BSXX106 4.006.000.58-0.292.540.722.5020.737 BSXX107 4.006.000.58-0.292.541.282.0015.747 BSXX108 4.006.000.58-0.292.541.022.0015.747 BSXX109 4.006.000.58-0.292.541.081.5010.487 BSXX110 4.006.000.58-0.292.541.071.5010.487 BSXX111 4.006.000.58-0.632.881.113.5032.057 BSXX112 4.006.000.58-0.632.880.753.5032.057 BSXX113 4.006.000.58-0.632.881.053.0026.977 BSXX114 4.006.000.58-0.632.880.783.0026.977 BSXX115 4.006.000.58-0.632.880.942.5021.787 BSXX116 4.006.000.58-0.632.880.612.5021.787 BSXX117 4.006.000.58-0.632.881.562.0016.417 BSXX118 4.006.000.58-0.632.881.352.0016.417 BSXX119 4.006.000.58-0.632.881.281.5010.767 BSXX120 4.006.000.58-0.632.881.091.5010.767 BSXX121 4.006.000.58-0.172.420.853.5029.627 BSXX122 4.006.000.58-0.172.420.573.5029.627 BSXX123 4.006.000.58-0.172.420.823.0025.017 BSXX124 4.006.000.58-0.172.420.633.0025.017 BSXX125 4.006.000.58-0.172.421.052.5020.307 BSXX126 4.006.000.58-0.172.420.722.5020.307 BSXX127 4.006.000.58-0.172.421.192.0015.467 BSXX128 4.006.000.58-0.172.420.982.0015.467 BSXX129 4.006.000.58-0.172.421.101.5010.367 BSXX130 4.006.000.58-0.172.421.051.5010.367

PAGE 341

341 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders BSXX131 4.006.000.580.002.580.823.5030.547 BSXX132 4.006.000.580.002.580.633.5030.547 BSXX133 4.006.000.580.002.580.863.0025.757 BSXX134 4.006.000.580.002.580.553.0025.757 BSXX135 4.006.000.580.002.581.042.5020.877 BSXX136 4.006.000.580.002.580.782.5020.877 BSXX137 4.006.000.580.002.581.232.0015.837 BSXX138 4.006.000.580.002.581.142.0015.837 BSXX139 4.006.000.580.002.581.031.5010.527 BSXX140 4.006.000.580.002.581.011.5010.527 BSXX141 4.006.000.58-0.292.881.203.5032.057 BSXX142 4.006.000.58-0.292.880.783.5032.057 BSXX143 4.006.000.58-0.292.881.043.0026.977 BSXX144 4.006.000.58-0.292.880.813.0026.977 BSXX145 4.006.000.58-0.292.880.972.5021.787 BSXX146 4.006.000.58-0.292.880.632.5021.787 BSXX147 4.006.000.58-0.292.881.642.0016.417 BSXX148 4.006.000.58-0.292.881.392.0016.417 BSXX149 4.006.000.58-0.292.881.201.5010.767 BSXX150 4.006.000.58-0.292.881.111.5010.767 BSOX001 4.006.000.580.171.420.923.5022.937 BSOX002 4.006.000.580.171.420.583.5022.937 BSOX003 4.006.000.580.171.420.733.0019.597 BSOX004 4.006.000.580.171.420.633.0019.597 BSOX005 4.006.000.580.171.420.862.5016.107 BSOX006 4.006.000.580.171.420.812.5016.107 BSOX007 4.006.000.580.171.420.692.0012.537 BSOX008 4.006.000.580.171.420.632.0012.537 BSOX009 4.006.000.580.171.420.841.508.827 BSOX010 4.006.000.580.171.420.721.508.827 BSOX011 4.006.000.580.001.580.743.5024.247 BSOX012 4.006.000.580.001.580.593.5024.247 BSOX013 4.006.000.580.001.580.683.0020.647 BSOX014 4.006.000.580.001.580.463.0020.647 BSOX015 4.006.000.580.001.581.112.5016.927

PAGE 342

342 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders BSOX016 4.006.000.580.001.580.792.5016.927 BSOX017 4.006.000.580.001.580.692.0013.127 BSOX018 4.006.000.580.001.580.582.0013.127 BSOX019 4.006.000.580.001.580.801.509.177 BSOX020 4.006.000.580.001.580.731.509.177 BSOX021 4.006.000.58-0.291.881.223.5026.317 BSOX022 4.006.000.58-0.291.880.653.5026.317 BSOX023 4.006.000.58-0.291.880.593.0022.317 BSOX024 4.006.000.58-0.291.880.453.0022.317 BSOX025 4.006.000.58-0.291.881.032.5018.237 BSOX026 4.006.000.58-0.291.880.672.5018.237 BSOX027 4.006.000.58-0.291.880.782.0014.057 BSOX028 4.006.000.58-0.291.880.562.0014.057 BSOX029 4.006.000.58-0.291.880.851.509.677 BSOX030 4.006.000.58-0.291.880.691.509.677 BSOX031 4.006.000.58-0.632.210.963.5028.417 BSOX032 4.006.000.58-0.632.210.673.5028.417 BSOX033 4.006.000.58-0.632.210.703.0024.027 BSOX034 4.006.000.58-0.632.210.453.0024.027 BSOX035 4.006.000.58-0.632.211.022.5019.557 BSOX036 4.006.000.58-0.632.210.822.5019.557 BSOX037 4.006.000.58-0.632.210.822.0014.967 BSOX038 4.006.000.58-0.632.210.712.0014.967 BSOX039 4.006.000.58-0.632.210.901.5010.137 BSOX040 4.006.000.58-0.632.210.851.5010.137 BSOX041 4.006.000.580.171.750.953.5025.457 BSOX042 4.006.000.580.171.750.543.5025.457 BSOX043 4.006.000.580.171.750.773.0021.627 BSOX044 4.006.000.580.171.750.473.0021.627 BSOX045 4.006.000.580.171.750.952.5017.697 BSOX046 4.006.000.580.171.750.632.5017.697 BSOX047 4.006.000.580.171.750.772.0013.677 BSOX048 4.006.000.580.171.750.702.0013.677 BSOX049 4.006.000.580.171.750.861.509.477 BSOX050 4.006.000.580.171.750.751.509.477

PAGE 343

343 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders BSOX051 4.006.000.580.001.921.023.5026.597 BSOX052 4.006.000.580.001.920.503.5026.597 BSOX053 4.006.000.580.001.920.603.0022.547 BSOX054 4.006.000.580.001.920.393.0022.547 BSOX055 4.006.000.580.001.920.862.5018.407 BSOX056 4.006.000.580.001.920.662.5018.407 BSOX057 4.006.000.580.001.920.632.0014.177 BSOX058 4.006.000.580.001.920.462.0014.177 BSOX059 4.006.000.580.001.920.771.509.747 BSOX060 4.006.000.580.001.920.641.509.747 BSOX061 4.006.000.58-0.292.210.793.5028.417 BSOX062 4.006.000.58-0.292.210.463.5028.417 BSOX063 4.006.000.58-0.292.210.733.0024.027 BSOX064 4.006.000.58-0.292.210.443.0024.027 BSOX065 4.006.000.58-0.292.210.922.5019.557 BSOX066 4.006.000.58-0.292.210.662.5019.557 BSOX067 4.006.000.58-0.292.210.952.0014.967 BSOX068 4.006.000.58-0.292.210.652.0014.967 BSOX069 4.006.000.58-0.292.210.831.5010.137 BSOX070 4.006.000.58-0.292.210.651.5010.137 BSOX071 4.006.000.58-0.632.541.333.5030.317 BSOX072 4.006.000.58-0.632.540.723.5030.317 BSOX073 4.006.000.58-0.632.540.923.0025.577 BSOX074 4.006.000.58-0.632.540.623.0025.577 BSOX075 4.006.000.58-0.632.540.962.5020.737 BSOX076 4.006.000.58-0.632.540.632.5020.737 BSOX077 4.006.000.58-0.632.541.152.0015.747 BSOX078 4.006.000.58-0.632.540.782.0015.747 BSOX079 4.006.000.58-0.632.541.131.5010.487 BSOX080 4.006.000.58-0.632.540.621.5010.487 BSOX081 4.006.000.58-0.172.080.603.5027.657 BSOX082 4.006.000.58-0.172.080.383.5027.657 BSOX083 4.006.000.58-0.172.080.733.0023.407 BSOX084 4.006.000.58-0.172.080.483.0023.407 BSOX085 4.006.000.58-0.172.080.802.5019.087

PAGE 344

344 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders BSOX086 4.006.000.58-0.172.080.622.5019.087 BSOX087 4.006.000.58-0.172.081.002.0014.637 BSOX088 4.006.000.58-0.172.080.652.0014.637 BSOX089 4.006.000.58-0.172.080.861.509.977 BSOX090 4.006.000.58-0.172.080.591.509.977 BSOX091 4.006.000.580.002.250.913.5028.667 BSOX092 4.006.000.580.002.250.573.5028.667 BSOX093 4.006.000.580.002.250.683.0024.237 BSOX094 4.006.000.580.002.250.593.0024.237 BSOX095 4.006.000.580.002.250.992.5019.717 BSOX096 4.006.000.580.002.250.732.5019.717 BSOX097 4.006.000.580.002.250.942.0015.067 BSOX098 4.006.000.580.002.250.762.0015.067 BSOX099 4.006.000.580.002.250.841.5010.187 BSOX100 4.006.000.580.002.250.721.5010.187 BSOX101 4.006.000.58-0.292.540.983.5030.317 BSOX102 4.006.000.58-0.292.540.613.5030.317 BSOX103 4.006.000.58-0.292.540.883.0025.577 BSOX104 4.006.000.58-0.292.540.603.0025.577 BSOX105 4.006.000.58-0.292.540.922.5020.737 BSOX106 4.006.000.58-0.292.540.632.5020.737 BSOX107 4.006.000.58-0.292.541.412.0015.747 BSOX108 4.006.000.58-0.292.541.102.0015.747 BSOX109 4.006.000.58-0.292.541.081.5010.487 BSOX110 4.006.000.58-0.292.540.951.5010.487 BSOX111 4.006.000.58-0.632.880.893.5032.057 BSOX112 4.006.000.58-0.632.880.523.5032.057 BSOX113 4.006.000.58-0.632.880.743.0026.977 BSOX114 4.006.000.58-0.632.880.613.0026.977 BSOX115 4.006.000.58-0.632.880.752.5021.787 BSOX116 4.006.000.58-0.632.880.572.5021.787 BSOX117 4.006.000.58-0.632.881.322.0016.417 BSOX118 4.006.000.58-0.632.881.212.0016.417 BSOX119 4.006.000.58-0.632.881.011.5010.767 BSOX120 4.006.000.58-0.632.880.971.5010.767

PAGE 345

345 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders BSOX121 4.006.000.58-0.172.420.973.5029.627 BSOX122 4.006.000.58-0.172.420.583.5029.627 BSOX123 4.006.000.58-0.172.420.853.0025.017 BSOX124 4.006.000.58-0.172.420.603.0025.017 BSOX125 4.006.000.58-0.172.420.962.5020.307 BSOX126 4.006.000.58-0.172.420.762.5020.307 BSOX127 4.006.000.58-0.172.421.142.0015.467 BSOX128 4.006.000.58-0.172.420.802.0015.467 BSOX129 4.006.000.58-0.172.421.021.5010.367 BSOX130 4.006.000.58-0.172.420.781.5010.367 BSOX131 4.006.000.580.002.580.973.5030.547 BSOX132 4.006.000.580.002.580.553.5030.547 BSOX133 4.006.000.580.002.580.793.0025.757 BSOX134 4.006.000.580.002.580.523.0025.757 BSOX135 4.006.000.580.002.580.832.5020.877 BSOX136 4.006.000.580.002.580.742.5020.877 BSOX137 4.006.000.580.002.581.152.0015.837 BSOX138 4.006.000.580.002.581.262.0015.837 BSOX139 4.006.000.580.002.580.991.5010.527 BSOX140 4.006.000.580.002.580.851.5010.527 BSOX141 4.006.000.58-0.292.881.003.5032.057 BSOX142 4.006.000.58-0.292.880.723.5032.057 BSOX143 4.006.000.58-0.292.881.043.0026.977 BSOX144 4.006.000.58-0.292.880.773.0026.977 BSOX145 4.006.000.58-0.292.880.922.5021.787 BSOX146 4.006.000.58-0.292.880.702.5021.787 BSOX147 4.006.000.58-0.292.881.452.0016.417 BSOX148 4.006.000.58-0.292.881.322.0016.417 BSOX149 4.006.000.58-0.292.881.141.5010.767 BSOX150 4.006.000.58-0.292.881.031.5010.767 BSOR0014.006.000.580.171.420.773.5022.937 BSOR0024.006.000.580.171.420.453.5022.937 BSOR0034.006.000.580.171.420.753.0019.597 BSOR0044.006.000.580.171.420.483.0019.597 BSOR0054.006.000.580.171.420.862.5016.107

PAGE 346

346 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders BSOR0064.006.000.580.171.420.642.5016.107 BSOR0074.006.000.580.171.420.712.0012.537 BSOR0084.006.000.580.171.420.592.0012.537 BSOR0094.006.000.580.171.420.761.508.827 BSOR0104.006.000.580.171.420.791.508.827 BSOR0114.006.000.580.001.581.013.5024.247 BSOR0124.006.000.580.001.580.643.5024.247 BSOR0134.006.000.580.001.580.753.0020.647 BSOR0144.006.000.580.001.580.553.0020.647 BSOR0154.006.000.580.001.581.042.5016.927 BSOR0164.006.000.580.001.580.712.5016.927 BSOR0174.006.000.580.001.580.662.0013.127 BSOR0184.006.000.580.001.580.552.0013.127 BSOR0194.006.000.580.001.580.761.509.177 BSOR0204.006.000.580.001.580.841.509.177 BSOR0214.006.000.58-0.291.880.793.5026.317 BSOR0224.006.000.58-0.291.880.643.5026.317 BSOR0234.006.000.58-0.291.880.543.0022.317 BSOR0244.006.000.58-0.291.880.423.0022.317 BSOR0254.006.000.58-0.291.880.922.5018.237 BSOR0264.006.000.58-0.291.880.872.5018.237 BSOR0274.006.000.58-0.291.880.822.0014.057 BSOR0284.006.000.58-0.291.880.562.0014.057 BSOR0294.006.000.58-0.291.880.811.509.677 BSOR0304.006.000.58-0.291.880.771.509.677 BSOR0314.006.000.58-0.632.211.143.5028.417 BSOR0324.006.000.58-0.632.210.653.5028.417 BSOR0334.006.000.58-0.632.210.723.0024.027 BSOR0344.006.000.58-0.632.210.533.0024.027 BSOR0354.006.000.58-0.632.210.852.5019.557 BSOR0364.006.000.58-0.632.210.692.5019.557 BSOR0374.006.000.58-0.632.210.772.0014.967 BSOR0384.006.000.58-0.632.210.772.0014.967 BSOR0394.006.000.58-0.632.210.941.5010.137 BSOR0404.006.000.58-0.632.210.901.5010.137

PAGE 347

347 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders BSOR0414.006.000.580.171.750.653.5025.457 BSOR0424.006.000.580.171.750.723.5025.457 BSOR0434.006.000.580.171.750.613.0021.627 BSOR0444.006.000.580.171.750.583.0021.627 BSOR0454.006.000.580.171.750.852.5017.697 BSOR0464.006.000.580.171.750.682.5017.697 BSOR0474.006.000.580.171.750.782.0013.677 BSOR0484.006.000.580.171.750.812.0013.677 BSOR0494.006.000.580.171.750.791.509.477 BSOR0504.006.000.580.171.750.811.509.477 BSOR0514.006.000.580.001.920.863.5026.597 BSOR0524.006.000.580.001.920.583.5026.597 BSOR0534.006.000.580.001.920.713.0022.547 BSOR0544.006.000.580.001.920.533.0022.547 BSOR0554.006.000.580.001.920.832.5018.407 BSOR0564.006.000.580.001.920.712.5018.407 BSOR0574.006.000.580.001.920.842.0014.177 BSOR0584.006.000.580.001.920.912.0014.177 BSOR0594.006.000.580.001.920.801.509.747 BSOR0604.006.000.580.001.920.691.509.747 BSOR0614.006.000.58-0.292.210.803.5028.417 BSOR0624.006.000.58-0.292.210.623.5028.417 BSOR0634.006.000.58-0.292.210.643.0024.027 BSOR0644.006.000.58-0.292.210.483.0024.027 BSOR0654.006.000.58-0.292.210.882.5019.557 BSOR0664.006.000.58-0.292.210.822.5019.557 BSOR0674.006.000.58-0.292.211.122.0014.967 BSOR0684.006.000.58-0.292.210.992.0014.967 BSOR0694.006.000.58-0.292.210.911.5010.137 BSOR0704.006.000.58-0.292.210.851.5010.137 BSOR0714.006.000.58-0.632.541.063.5030.317 BSOR0724.006.000.58-0.632.540.793.5030.317 BSOR0734.006.000.58-0.632.540.943.0025.577 BSOR0744.006.000.58-0.632.540.623.0025.577 BSOR0754.006.000.58-0.632.540.802.5020.737

PAGE 348

348 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders BSOR0764.006.000.58-0.632.540.652.5020.737 BSOR0774.006.000.58-0.632.541.422.0015.747 BSOR0784.006.000.58-0.632.540.962.0015.747 BSOR0794.006.000.58-0.632.541.031.5010.487 BSOR0804.006.000.58-0.632.541.021.5010.487 BSOR0814.006.000.58-0.172.080.833.5027.657 BSOR0824.006.000.58-0.172.080.533.5027.657 BSOR0834.006.000.58-0.172.080.923.0023.407 BSOR0844.006.000.58-0.172.080.523.0023.407 BSOR0854.006.000.58-0.172.080.862.5019.087 BSOR0864.006.000.58-0.172.080.642.5019.087 BSOR0874.006.000.58-0.172.080.892.0014.637 BSOR0884.006.000.58-0.172.080.752.0014.637 BSOR0894.006.000.58-0.172.080.911.509.977 BSOR0904.006.000.58-0.172.080.811.509.977 BSOR0914.006.000.580.002.251.073.5028.667 BSOR0924.006.000.580.002.250.713.5028.667 BSOR0934.006.000.580.002.250.873.0024.237 BSOR0944.006.000.580.002.250.603.0024.237 BSOR0954.006.000.580.002.251.012.5019.717 BSOR0964.006.000.580.002.250.862.5019.717 BSOR0974.006.000.580.002.251.042.0015.067 BSOR0984.006.000.580.002.250.902.0015.067 BSOR0994.006.000.580.002.251.011.5010.187 BSOR1004.006.000.580.002.250.961.5010.187 BSOR1014.006.000.58-0.292.541.143.5030.317 BSOR1024.006.000.58-0.292.540.693.5030.317 BSOR1034.006.000.58-0.292.540.923.0025.577 BSOR1044.006.000.58-0.292.540.603.0025.577 BSOR1054.006.000.58-0.292.541.032.5020.737 BSOR1064.006.000.58-0.292.540.722.5020.737 BSOR1074.006.000.58-0.292.541.282.0015.747 BSOR1084.006.000.58-0.292.541.022.0015.747 BSOR1094.006.000.58-0.292.541.081.5010.487 BSOR1104.006.000.58-0.292.541.071.5010.487

PAGE 349

349 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders BSOR1114.006.000.58-0.632.881.113.5032.057 BSOR1124.006.000.58-0.632.880.753.5032.057 BSOR1134.006.000.58-0.632.881.053.0026.977 BSOR1144.006.000.58-0.632.880.783.0026.977 BSOR1154.006.000.58-0.632.880.942.5021.787 BSOR1164.006.000.58-0.632.880.612.5021.787 BSOR1174.006.000.58-0.632.881.562.0016.417 BSOR1184.006.000.58-0.632.881.352.0016.417 BSOR1194.006.000.58-0.632.881.281.5010.767 BSOR1204.006.000.58-0.632.881.091.5010.767 BSOR1214.006.000.58-0.172.420.853.5029.627 BSOR1224.006.000.58-0.172.420.573.5029.627 BSOR1234.006.000.58-0.172.420.823.0025.017 BSOR1244.006.000.58-0.172.420.633.0025.017 BSOR1254.006.000.58-0.172.421.052.5020.307 BSOR1264.006.000.58-0.172.420.722.5020.307 BSOR1274.006.000.58-0.172.421.192.0015.467 BSOR1284.006.000.58-0.172.420.982.0015.467 BSOR1294.006.000.58-0.172.421.101.5010.367 BSOR1304.006.000.58-0.172.421.051.5010.367 BSOR1314.006.000.580.002.580.823.5030.547 BSOR1324.006.000.580.002.580.633.5030.547 BSOR1334.006.000.580.002.580.863.0025.757 BSOR1344.006.000.580.002.580.553.0025.757 BSOR1354.006.000.580.002.581.042.5020.877 BSOR1364.006.000.580.002.580.782.5020.877 BSOR1374.006.000.580.002.581.232.0015.837 BSOR1384.006.000.580.002.581.142.0015.837 BSOR1394.006.000.580.002.581.031.5010.527 BSOR1404.006.000.580.002.581.011.5010.527 BSOR1414.006.000.58-0.292.881.203.5032.057 BSOR1424.006.000.58-0.292.880.783.5032.057 BSOR1434.006.000.58-0.292.881.043.0026.977 BSOR1444.006.000.58-0.292.880.813.0026.977 BSOR1454.006.000.58-0.292.880.972.5021.787

PAGE 350

350 Table B 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length Girders BSOR1464.006.000.58-0.292.880.632.5021.787 BSOR1474.006.000.58-0.292.881.642.0016.417 BSOR1484.006.000.58-0.292.881.392.0016.417 BSOR1494.006.000.58-0.292.881.201.5010.767 BSOR1504.006.000.58-0.292.881.111.5010.767

PAGE 351

351 T able B -2. Significant force values for all physical model tests. Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. SLAB0016.36-9.231.89-4.950.005.360.29-0.25-0.01 SLAB00289.03-71.4338.79-38.443.0151.894.19-2.5337.48 SLAB0030.90-0.370.67-0.15-0.040.440.12-0.200.16 SLAB00489.93-56.7735.29-20.591.1171.592.86-2.28-1.15 SLAB005117.43-89.3150.76-38.732.2094.435.63-4.1110.15 SLAB006120.32-84.0945.89-38.497.1280.207.26-3.9439.52 SLAB00749.58-60.2116.68-18.76-0.3255.151.59-1.11-2.43 SLAB00875.19-65.7725.58-22.200.9862.312.77-1.98-8.18 SLAB00930.59-43.575.56-13.89-0.3234.711.95-1.40-6.76 SLAB01062.27-74.6721.07-27.372.4865.123.27-2.1020.50 SLAB011112.02-13.1391.23-12.122.2723.812.65-1.0390.64 SLAB012278.93-30.52156.82-25.0714.60123.2215.40-5.79155.07 SLAB01377.52-14.1252.64-10.510.6432.101.38-0.7742.63 SLAB014268.89-92.24118.86-25.493.66161.275.61-2.5397.13 SLAB015174.19-16.8992.00-14.271.4293.332.86-0.9987.98 SLAB016247.44-43.01136.88-20.982.01120.6311.93-2.74127.88 SLAB017187.22-68.0382.92-18.781.83111.875.05-2.4767.59 SLAB018264.27-69.93127.51-33.029.38143.5610.58-3.40102.01 SLAB019163.26-109.7258.51-42.433.38138.325.19-1.9446.94 SLAB020201.03-93.9078.13-47.362.64158.769.84-2.8263.43 SLAB02197.34-61.4086.99-62.873.5914.394.46-1.2677.41 SLAB022207.12-109.54180.16-105.508.3435.298.64-2.19166.29 SLAB02377.68-74.9370.44-76.842.8719.463.45-1.3258.59 SLAB024196.41-115.00169.17-115.117.9673.3711.61-2.22114.16 SLAB025112.87-72.2794.37-74.944.9531.927.02-1.2975.37 SLAB026199.45-113.92165.85-112.2811.5261.9412.42-2.79156.05 SLAB027126.02-84.3491.23-87.054.9151.758.01-1.2967.66 SLAB028228.09-113.37166.81-113.886.2478.2212.87-1.98139.64 SLAB029254.94-91.77118.01-108.1711.04156.5414.48-7.3993.87 SLAB030323.63-103.00144.67-121.739.06190.1716.49-4.24122.88 SLAB03197.95-123.03102.11-121.494.2313.814.31-2.98101.98 SLAB032157.97-243.14169.26-224.087.7328.837.87-4.83161.16 SLAB033100.39-131.45101.29-137.314.2443.336.64-2.7579.26 SLAB034157.64-214.51168.21-218.977.1666.2710.04-4.09109.58 SLAB03587.08-168.1292.19-174.724.8149.506.30-3.7737.40

PAGE 352

352 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. SLAB036159.32-256.25163.96-253.538.2864.129.60-3.8290.48 SLAB037112.06-188.33110.86-198.526.7574.369.08-3.4266.60 SLAB038126.75-275.23122.79-273.096.2291.2614.49-5.1258.46 SLAB039162.23-146.6582.74-156.048.80149.2413.00-5.8314.97 SLAB040154.59-282.7896.29-289.8110.53117.6911.43-4.6618.90 SLAB04134.95-47.7014.03-18.610.1228.251.67-1.077.74 SLAB042141.33-78.4797.03-33.634.0275.679.52-3.6790.95 SLAB0437.77-14.131.58-5.78-0.2711.320.59-0.47-5.63 SLAB044124.97-77.1167.17-36.525.3570.4910.96-8.5665.86 SLAB04553.35-52.0724.47-20.020.7048.071.94-1.4215.76 SLAB046123.60-85.4661.31-35.381.1072.379.03-7.9315.23 SLAB04764.33-63.5813.58-19.430.1360.062.23-1.47-9.73 SLAB048138.08-87.6149.04-39.084.62105.3312.04-9.27-2.68 SLAB04940.34-54.7411.82-17.11-0.3845.813.11-2.543.81 SLAB05087.03-94.4925.12-32.540.6275.074.08-3.97-11.74 SLAB05193.01-15.0493.40-13.161.0127.111.48-0.5574.91 SLAB052212.76-24.14177.10-20.866.7561.0010.92-1.67167.17 SLAB053150.11-34.9093.27-31.391.9173.227.44-3.9581.91 SLAB054356.06-44.58176.92-35.9616.68209.1917.19-6.41147.05 SLAB055155.15-15.9694.17-13.884.3761.655.24-1.5691.53 SLAB056347.52-32.50175.07-29.1214.77181.6118.06-7.62171.18 SLAB057238.48-48.39104.01-31.345.14135.8910.35-6.9954.63 SLAB058306.92-42.63170.49-37.909.83160.6917.85-14.34126.43 SLAB059177.94-85.6076.10-48.093.27104.6612.70-2.2170.11 SLAB060274.20-127.3896.86-57.857.30178.6920.14-9.7576.84 SLAB061111.37-90.62103.33-93.113.8714.015.12-1.4789.18 SLAB062196.61-111.42194.63-108.555.2918.516.58-2.6292.79 SLAB063129.52-80.45106.54-82.816.1737.868.66-1.6271.32 SLAB064239.33-128.49196.36-129.057.8582.6815.12-2.74133.89 SLAB065132.14-104.06112.34-104.504.8438.548.14-1.5357.90 SLAB066195.61-109.96158.61-106.806.4857.0310.06-1.94141.62 SLAB067205.02-102.38133.83-99.067.5487.4511.70-1.72110.71 SLAB068268.48-119.65179.22-119.724.10142.7613.87-1.86121.94 SLAB069272.77-95.58136.66-117.9411.82157.0917.19-6.82110.27 SLAB070280.39-118.84158.04-135.918.59159.0017.41-6.05126.42

PAGE 353

353 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. SLAB071110.19-151.76114.86-159.724.3916.304.53-3.19114.64 SLAB072169.47-271.91178.14-282.597.7337.787.84-5.43177.80 SLAB073101.01-126.14101.48-126.934.7520.796.64-2.45100.44 SLAB074176.34-217.57183.54-221.468.7357.3011.36-4.39133.08 SLAB075103.88-137.23110.08-141.934.7040.007.21-3.2183.58 SLAB076166.32-256.25167.62-268.749.04122.949.64-4.33100.91 SLAB07791.96-180.9295.08-183.325.2161.328.48-3.8279.43 SLAB078142.48-292.15130.98-285.318.0391.8414.07-5.0897.93 SLAB079193.86-198.8987.74-208.5811.53148.7711.79-4.5049.77 SLAB080203.54-263.36122.74-284.028.68113.2012.58-4.7936.14 SLAB08183.75-27.0143.18-14.480.1343.092.32-1.8038.44 SLAB082175.44-21.52116.59-17.059.3460.429.34-2.98115.15 SLAB08368.12-36.7133.79-22.951.6347.672.57-1.3333.37 SLAB084236.43-62.90152.91-48.8615.52105.6320.55-15.7176.87 SLAB08584.79-60.3832.29-18.611.8560.994.37-4.03-4.71 SLAB086160.00-52.93104.03-36.394.8773.2813.11-7.3753.00 SLAB087112.59-103.3266.57-39.011.5168.317.30-5.6442.87 SLAB088169.92-101.1193.68-50.274.64109.6717.60-13.4830.44 SLAB089126.55-133.6332.38-36.722.79122.947.27-4.053.08 SLAB090141.69-131.8451.30-48.185.92125.899.81-5.5018.32 SLAB09191.50-11.8689.69-10.601.4022.131.76-0.9188.99 SLAB092244.19-20.20180.20-17.248.7872.5412.21-2.47169.36 SLAB093188.36-28.6696.14-19.173.3192.2510.86-4.0086.34 SLAB094362.14-37.15196.15-29.8515.12178.9816.14-3.53185.58 SLAB095233.24-43.38108.21-18.867.71127.168.99-3.63104.91 SLAB096305.83-27.26160.75-27.1313.06146.6515.97-7.93159.31 SLAB097239.02-49.16113.20-33.225.99132.3314.84-9.1789.21 SLAB098351.81-64.17185.71-40.4812.00175.1022.78-15.75138.82 SLAB099193.79-107.9573.12-44.995.66163.0415.52-6.8368.90 SLAB100278.17-129.91109.11-57.3411.21184.5618.63-9.6276.15 SLAB101108.02-104.97107.09-104.552.559.133.97-1.5676.70 SLAB102268.11-105.47243.13-98.905.0543.777.12-2.33194.47 SLAB103103.83-75.3493.28-81.163.5537.955.92-1.5264.51 SLAB104238.22-119.67211.77-117.7011.3063.9213.27-2.27139.12 SLAB105138.54-89.42114.56-92.935.5842.3210.01-1.6778.11

PAGE 354

354 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. SLAB106245.02-122.84216.61-114.466.1563.9211.99-2.58124.65 SLAB107153.91-99.94123.43-100.754.2453.389.56-1.3792.49 SLAB108267.70-113.79200.78-122.514.9899.7014.74-1.99121.37 SLAB109272.36-95.57124.85-110.668.46160.1213.35-3.23104.81 SLAB110404.65-120.63186.17-136.8916.89237.3322.46-8.30158.34 SLAB111112.71-148.42116.28-153.334.0343.594.31-2.96111.67 SLAB112185.50-278.17194.73-285.527.4734.857.63-5.79178.81 SLAB11393.27-124.2297.64-129.083.6135.025.66-3.0982.78 SLAB114164.89-233.22166.84-237.308.0655.7811.33-4.94121.18 SLAB115107.53-150.62102.57-157.545.0339.106.46-3.2385.45 SLAB116157.85-260.05165.10-261.037.4261.9411.26-5.89139.86 SLAB117126.74-242.63112.51-245.627.12100.0314.24-4.5954.11 SLAB118190.47-319.45138.15-318.2910.1189.2714.59-5.0890.29 SLAB119155.15-168.0075.88-178.1310.35121.6211.59-4.7256.32 SLAB120217.87-303.42113.30-317.889.47143.7911.39-6.4255.43 SLAB121123.68-20.7261.92-16.752.2761.955.39-6.1234.99 SLAB122156.44-21.97113.92-21.354.8442.859.55-6.3686.05 SLAB12378.68-54.7746.16-26.54-2.5159.329.21-7.8511.23 SLAB124214.62-55.21142.33-47.2410.56101.4722.99-22.2289.18 SLAB125108.46-75.3765.22-25.103.7454.7414.99-14.5432.41 SLAB126174.06-84.8197.90-32.874.5592.6515.30-13.8286.18 SLAB127146.35-110.4189.54-49.029.4675.4718.99-17.4525.44 SLAB128226.61-112.81115.44-53.436.75111.6721.22-17.4319.43 SLAB129146.58-123.9945.15-49.562.46120.0411.20-9.09-14.58 SLAB130166.82-121.6558.14-50.232.93133.9324.99-16.9439.60 SLAB131128.73-36.48105.17-14.972.8273.774.32-2.5196.48 SLAB132232.05-27.27188.15-22.867.0076.3116.60-7.98176.14 SLAB133335.92-27.14144.24-23.794.86191.6917.49-11.8292.52 SLAB134354.55-43.20211.76-36.6714.79176.5625.32-15.08198.53 SLAB135298.82-45.00123.29-19.123.96191.9411.66-7.7291.69 SLAB136347.50-29.54189.11-27.7013.81167.6120.38-14.85155.88 SLAB137348.52-54.24165.27-32.5213.94214.9136.44-29.15116.59 SLAB138380.72-80.33202.13-42.0616.37187.6537.17-29.40124.80 SLAB139330.23-142.30112.77-75.6826.82232.1929.41-17.7095.15 SLAB140293.63-143.87113.61-67.287.02196.0424.80-21.8762.67

PAGE 355

355 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. SLAB141124.44-108.81120.89-107.243.2612.603.84-1.8979.57 SLAB142209.24-122.56213.09-117.045.7456.867.37-3.03174.24 SLAB143173.38-101.26144.93-105.865.6646.669.03-2.6093.28 SLAB144267.10-128.38236.32-126.9810.2564.7911.21-3.83224.53 SLAB145216.40-120.77175.76-119.606.9464.4610.00-2.3991.96 SLAB146330.97-123.43307.08-112.324.0288.2815.53-3.69210.08 SLAB147231.90-108.27169.05-101.9910.6897.3411.28-2.72111.33 SLAB148266.94-121.73227.23-120.708.9181.3816.27-2.96227.00 SLAB149368.38-116.13164.04-132.617.99213.8523.39-5.14139.51 SLAB150394.33-127.40179.81-147.1524.18240.9324.31-7.12153.11 BSXX001 134.92-41.4870.82-28.696.2768.437.81-5.1652.94 BSXX002 61.48-48.4626.44-18.960.8541.842.50-1.86-1.08 BSXX003 89.51-41.1057.94-21.992.3956.174.51-2.8852.30 BSXX004 86.36-69.8725.92-17.430.0569.742.59-1.67-5.54 BSXX005 138.20-67.0766.81-28.742.4773.206.86-3.8544.72 BSXX006 71.77-66.9631.09-21.070.9758.521.75-1.1626.92 BSXX007 119.09-72.8044.83-30.121.3794.544.23-2.359.31 BSXX008 48.98-68.9718.73-20.93-0.1762.431.51-0.7012.17 BSXX009 99.50-103.9929.43-33.371.54103.363.24-0.9915.96 BSXX010 86.25-110.9833.00-34.212.8799.804.32-0.7030.22 BSXX011 159.69-27.48154.96-22.244.4344.787.31-0.59145.85 BSXX012 109.95-23.35108.34-22.692.3322.264.11-0.28104.14 BSXX013 237.74-47.08131.25-27.992.88117.736.48-0.77129.17 BSXX014 194.99-35.16112.81-26.280.5582.717.96-2.35110.68 BSXX015 199.44-29.03126.95-25.5710.9973.1712.56-3.91109.39 BSXX016 158.43-16.14106.70-14.573.3261.045.69-0.46106.69 BSXX017 245.97-38.14140.49-27.346.08105.799.49-3.41116.55 BSXX018 153.99-33.1195.21-31.102.7777.507.50-3.1077.12 BSXX019 214.95-109.5892.24-52.194.78156.779.55-7.1470.86 BSXX020 176.06-104.0370.77-57.195.73145.777.40-6.5834.73 BSXX021 129.01-102.64133.93-100.616.0019.716.01-2.10133.81 BSXX022 100.05-88.19105.14-91.074.329.204.44-1.59103.30 BSXX023 126.55-100.65133.75-105.636.769.586.92-1.92121.55 BSXX024 93.46-82.6597.92-87.223.9631.364.23-1.6595.88 BSXX025 125.82-109.68133.58-109.656.0818.586.36-2.20132.50

PAGE 356

356 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. BSXX026 125.27-98.56131.38-96.235.7413.946.03-1.71126.12 BSXX027 123.60-108.21128.38-114.525.9869.106.52-2.19122.20 BSXX028 101.28-96.50104.32-95.134.5643.014.60-1.82104.17 BSXX029 113.19-100.42109.85-114.345.6340.757.00-1.42100.54 BSXX030 131.02-105.68102.51-110.363.8056.386.75-1.7378.77 BSXX031 113.66-195.82122.64-196.295.9226.396.64-7.6392.17 BSXX032 57.77-121.5360.98-125.393.177.483.37-4.8660.71 BSXX033 101.49-155.47109.13-159.525.3026.898.89-5.7777.80 BSXX034 62.26-115.8266.00-119.314.586.965.72-4.1062.52 BSXX035 91.19-174.4197.21-174.716.2424.446.79-6.5395.61 BSXX036 64.48-118.1068.84-120.944.649.575.57-5.0756.17 BSXX037 75.88-140.3683.83-145.217.0111.967.82-5.5466.14 BSXX038 71.89-137.0873.34-142.756.7715.247.39-5.6073.30 BSXX039 84.16-155.8173.67-161.327.3341.949.73-6.6470.31 BSXX040 71.81-159.4663.39-165.526.1042.6910.01-5.4610.94 BSXX041 94.83-37.0657.18-28.741.9171.818.08-5.9122.80 BSXX042 132.89-66.2276.84-31.650.6061.999.22-4.8066.87 BSXX043 121.57-64.3976.13-24.19-0.4266.459.17-7.7545.35 BSXX044 100.00-40.2949.90-30.012.1253.255.29-3.2649.78 BSXX045 140.28-30.1583.29-28.282.7670.4312.38-8.0845.99 BSXX046 123.77-36.7256.67-25.005.1280.587.58-6.1520.86 BSXX047 100.10-60.3964.75-38.63-2.1447.967.86-7.0662.92 BSXX048 166.78-79.0981.67-39.192.61111.3310.10-7.7325.03 BSXX049 129.69-113.4246.31-45.238.08103.878.94-3.5434.59 BSXX050 122.79-106.5340.89-42.834.24114.609.85-6.5839.83 BSXX051 167.44-22.00140.93-20.804.8346.556.19-1.92136.77 BSXX052 82.52-17.2084.16-17.171.8220.502.27-0.5666.90 BSXX053 253.15-23.15152.96-20.271.85108.319.72-1.99130.34 BSXX054 193.28-33.66102.66-28.605.5993.607.38-1.4980.39 BSXX055 244.98-34.12144.51-27.8310.67102.4913.06-7.3389.13 BSXX056 182.19-21.42121.45-19.010.6393.677.17-2.6178.57 BSXX057 271.43-46.55151.15-33.4811.03134.0814.54-6.75118.56 BSXX058 259.62-38.70158.63-36.6111.17115.5017.38-7.44123.38 BSXX059 213.21-103.58100.32-61.7711.42116.1214.32-13.7826.08 BSXX060 182.97-117.0784.13-58.146.09126.0110.57-11.07-23.72

PAGE 357

357 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. BSXX061 141.62-127.70149.92-131.745.6015.935.85-2.90134.34 BSXX062 100.51-109.92105.30-113.584.057.674.24-2.05104.91 BSXX063 148.43-112.38150.42-115.098.6514.969.76-2.90150.34 BSXX064 115.34-93.75119.46-98.725.166.545.86-1.8995.15 BSXX065 129.38-110.99137.32-111.585.6420.557.32-2.16137.21 BSXX066 113.64-123.65119.59-119.055.5217.155.88-2.08110.67 BSXX067 127.71-108.81133.25-116.805.6455.016.88-2.12131.49 BSXX068 123.85-112.88123.83-117.355.6731.286.96-2.29118.42 BSXX069 167.65-116.24125.06-115.971.77101.6610.77-2.61123.05 BSXX070 132.61-116.62108.04-117.801.76138.299.37-2.1096.32 BSXX071 96.26-192.71102.54-197.094.6218.345.01-7.5584.25 BSXX072 77.57-164.1282.16-168.233.6322.654.15-5.8081.37 BSXX073 109.11-150.47118.66-156.487.5615.359.27-7.26113.03 BSXX074 84.41-119.5791.50-123.395.8310.056.18-6.4784.95 BSXX075 104.20-165.16107.65-170.577.1629.218.32-7.0971.08 BSXX076 60.22-143.9563.50-146.505.129.905.28-5.6054.33 BSXX077 100.67-160.93108.48-169.937.5719.159.62-6.9992.83 BSXX078 74.13-143.6874.01-148.576.0717.066.98-5.7446.40 BSXX079 109.77-169.5079.41-168.619.8348.2810.99-7.2311.72 BSXX080 106.09-183.5681.72-182.5311.6248.5912.78-7.0081.60 BSXX081 195.75-34.74133.72-32.7111.0578.5412.73-4.68116.79 BSXX082 70.43-53.3245.40-28.09-1.6847.106.15-4.8110.68 BSXX083 176.96-48.56126.99-45.89-1.9765.4718.31-7.91116.02 BSXX084 72.41-55.9136.51-26.54-2.6560.085.17-3.290.16 BSXX085 158.70-42.6487.27-31.214.1584.5615.73-9.1148.44 BSXX086 81.39-49.1652.22-23.560.9933.497.23-6.2021.36 BSXX087 173.22-94.3793.37-47.496.7982.7418.03-12.1331.66 BSXX088 142.16-101.2761.76-40.174.4590.448.31-6.9444.27 BSXX089 133.49-111.9354.96-48.888.30102.5614.86-10.846.41 BSXX090 126.70-105.4344.00-44.595.05109.1513.30-7.8427.40 BSXX091 141.61-35.30141.31-29.824.7943.285.66-0.93141.28 BSXX092 131.95-23.21108.42-18.223.5055.694.36-1.9596.70 BSXX093 307.08-43.09190.27-37.5413.08118.9028.54-12.51145.77 BSXX094 222.80-30.35116.05-29.0011.23106.7516.27-6.50104.00 BSXX095 264.39-35.33185.87-30.944.27114.1720.34-7.82169.27

PAGE 358

358 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. BSXX096 201.26-23.84140.56-20.45-4.4681.5920.63-11.61118.94 BSXX097 276.31-86.00174.55-41.157.70188.6625.15-14.04138.28 BSXX098 266.12-54.69161.78-42.356.60121.6419.97-12.39112.49 BSXX099 285.11-125.14123.57-91.4115.80169.9927.29-22.488.37 BSXX100 242.84-115.6895.58-68.515.30154.3817.77-16.135.28 BSXX101 175.29-120.69183.47-120.756.6628.427.02-3.18171.81 BSXX102 103.20-103.74107.79-102.903.2313.423.56-1.83101.83 BSXX103 150.80-123.21161.51-120.546.4623.459.19-2.70155.43 BSXX104 106.55-93.13111.16-97.334.316.015.76-1.9893.60 BSXX105 177.69-115.82169.13-115.556.1638.558.43-2.47169.13 BSXX106 108.15-118.62113.44-115.254.9919.385.74-1.82111.63 BSXX107 199.86-124.50166.11-122.188.9952.2810.32-2.35157.83 BSXX108 115.77-106.01117.19-110.055.6325.227.21-1.82112.97 BSXX109 143.13-125.98135.11-121.305.0050.0810.10-2.1194.09 BSXX110 145.11-117.75130.88-115.144.4267.8211.09-2.55105.40 BSXX111 93.45-201.4298.49-208.104.6512.975.13-6.4185.64 BSXX112 69.03-153.1572.67-158.233.799.193.86-5.2767.26 BSXX113 118.44-160.08128.92-165.417.1515.658.51-8.54102.07 BSXX114 82.36-135.1089.61-139.915.4012.116.01-8.0180.20 BSXX115 108.51-184.00118.89-188.607.1516.918.76-7.17117.30 BSXX116 67.11-148.3871.46-151.444.8836.075.65-6.2556.65 BSXX117 97.91-184.60107.68-192.457.6799.339.38-7.7292.77 BSXX118 93.13-155.0899.62-161.447.6757.178.86-6.1384.56 BSXX119 124.76-200.3782.62-208.3011.0953.8211.85-7.5382.45 BSXX120 69.96-189.0458.07-192.608.2133.6910.96-6.317.38 BSXX121 123.77-21.1193.86-16.073.1036.955.41-1.8778.98 BSXX122 73.78-30.3344.60-18.381.6243.081.62-0.6430.95 BSXX123 202.88-93.87144.04-67.2112.9365.6624.15-12.4685.19 BSXX124 145.12-43.5276.18-34.166.6281.616.70-3.8637.05 BSXX125 236.50-47.20110.34-40.960.10129.0013.16-6.3798.45 BSXX126 111.44-56.8463.02-30.362.8154.298.81-4.8836.82 BSXX127 189.80-82.06153.22-58.553.2964.5325.18-14.9468.02 BSXX128 179.99-98.1487.21-44.56-1.33124.889.34-5.8658.55 BSXX129 170.61-122.0083.63-75.507.78107.0816.04-5.9865.82 BSXX130 123.36-116.4440.07-49.271.53103.057.56-2.7335.57

PAGE 359

359 Table B -2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. BSXX131 156.94-27.48113.97-24.191.4963.286.82-2.35103.36 BSXX132 98.09-23.4487.02-16.921.5232.692.11-1.1551.97 BSXX133 252.65-35.51180.76-30.543.87118.9320.90-4.79147.35 BSXX134 185.75-27.90111.05-19.402.9291.337.96-1.14107.32 BSXX135 323.82-40.20197.65-33.638.38162.0418.21-2.72181.10 BSXX136 204.78-30.05144.32-26.160.0179.6611.44-2.33124.15 BSXX137 263.28-64.06195.07-43.738.21133.1424.92-11.76132.76 BSXX138 227.56-41.23155.15-32.8410.0891.0817.98-5.52116.58 BSXX139 232.95-131.78101.00-98.387.32156.4016.89-11.9951.67 BSXX140 239.81-113.45101.66-80.289.77162.4916.11-9.2749.62 BSXX141 157.31-139.16163.57-128.474.9330.325.73-3.17151.02 BSXX142 96.76-108.32101.26-111.853.1517.593.47-2.0697.37 BSXX143 151.40-132.04161.59-129.047.5261.088.27-3.73148.36 BSXX144 118.61-103.97125.71-106.885.6710.896.37-2.02110.58 BSXX145 193.91-123.18185.62-122.818.5937.368.66-2.98171.80 BSXX146 129.62-114.88137.14-114.285.4322.966.50-2.26135.92 BSXX147 241.72-120.54193.05-116.779.05141.2512.81-2.37188.63 BSXX148 137.61-114.09127.32-119.675.7826.597.83-2.41127.28 BSXX149 133.33-140.69135.16-135.912.9246.7012.29-2.3679.80 BSXX150 139.93-131.57115.09-116.398.2786.699.16-2.4594.81 BSOX001 35.53-46.6811.19-16.90-0.1041.960.98-0.6110.90 BSOX002 87.13-68.3843.98-40.465.9545.626.18-3.5740.93 BSOX003 4.08-0.744.47-0.540.840.970.84-0.134.47 BSOX004 52.09-61.6029.34-22.32-0.3742.312.61-1.82-6.89 BSOX005 43.59-38.9810.82-19.950.8233.562.02-1.278.39 BSOX006 94.08-87.9336.08-34.691.5272.794.69-2.72-2.82 BSOX007 7.06-9.952.57-5.860.225.420.33-0.09-4.54 BSOX008 53.97-52.3111.83-16.000.9845.691.86-1.56-5.31 BSOX009 35.51-52.636.31-13.040.8739.281.52-0.833.05 BSOX010 69.66-77.9215.66-26.361.6469.724.06-1.6210.03 BSOX011 107.40-7.6082.45-6.061.6329.161.64-0.5280.53 BSOX012 245.83-30.70163.33-18.7214.30114.3015.21-3.56130.64 BSOX013 81.52-10.9168.17-7.690.7237.331.38-0.7038.15 BSOX014 307.99-52.60139.58-20.854.10224.0310.66-3.04115.45 BSOX015 177.28-13.1185.81-8.021.6091.542.47-1.4873.79

PAGE 360

360 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. BSOX016 255.79-22.69144.89-18.279.50118.3913.12-4.95144.29 BSOX017 166.11-60.5971.96-21.651.10104.753.06-1.2052.52 BSOX018 237.50-92.43125.30-48.938.87126.879.35-2.47112.28 BSOX019 165.28-76.6958.10-45.135.01112.615.01-1.4652.67 BSOX020 241.31-108.4089.48-55.1313.74174.0915.51-3.9574.32 BSOX021 103.13-53.7092.68-54.824.5013.155.52-1.4676.73 BSOX022 205.82-112.45170.90-111.161.1454.959.41-1.74136.18 BSOX023 94.48-68.6982.11-70.974.4815.424.89-1.3581.33 BSOX024 205.47-128.16153.74-134.286.84188.3214.25-2.04113.34 BSOX025 105.72-97.5494.10-100.935.0921.006.45-1.3766.61 BSOX026 176.52-124.23147.68-120.968.0755.9111.39-1.7980.88 BSOX027 106.92-80.0393.53-82.645.6240.437.91-1.2562.75 BSOX028 197.51-119.26152.66-114.204.0972.1411.20-1.53114.89 BSOX029 161.74-95.3887.38-105.5011.5088.0512.36-2.7071.06 BSOX030 253.49-106.63143.52-118.589.03140.2016.61-5.58101.33 BSOX031 68.44-127.7772.87-130.122.5611.622.93-3.1867.01 BSOX032 112.38-210.91121.09-216.833.9125.074.31-5.29115.78 BSOX033 69.06-120.9873.67-126.272.4073.734.58-2.6661.51 BSOX034 144.90-163.50150.49-168.134.0445.257.57-5.33150.43 BSOX035 83.51-148.9886.57-145.393.7125.274.05-3.8282.46 BSOX036 127.52-202.09130.01-209.575.3141.976.33-5.14121.93 BSOX037 79.89-151.9082.20-149.384.2135.985.36-3.6674.36 BSOX038 145.38-165.69120.47-176.154.5048.797.50-4.6593.31 BSOX039 155.11-130.0770.60-142.9910.88123.1410.88-4.6031.42 BSOX040 221.10-185.93101.62-205.1614.08166.0915.79-5.4562.62 BSOX041 27.18-32.3014.21-16.650.0924.341.32-1.09-2.44 BSOX042 164.90-52.4185.45-34.0910.6281.6112.42-5.6381.67 BSOX043 23.76-23.274.09-10.43-0.2923.690.76-0.61-10.14 BSOX044 181.01-85.8378.19-42.624.86103.1713.46-8.7945.88 BSOX045 48.19-36.6019.28-19.580.3535.301.78-0.7717.52 BSOX046 141.37-81.5273.78-40.714.1892.7410.29-8.1413.47 BSOX047 53.84-56.2515.71-18.650.4355.561.21-1.162.76 BSOX048 121.06-97.4142.23-37.981.28103.977.56-6.39-2.47 BSOX049 72.01-81.6612.73-25.57-1.1982.294.76-2.44-7.47 BSOX050 122.90-107.8529.57-37.041.69103.314.62-2.02-21.32

PAGE 361

361 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. BSOX051 122.97-10.0587.89-9.801.0839.814.41-1.2584.60 BSOX052 329.96-35.65187.71-30.9615.56144.1519.32-3.04183.27 BSOX053 126.61-37.8868.54-16.900.0559.013.36-1.1965.16 BSOX054 242.46-25.80137.23-20.557.04116.288.68-2.63119.34 BSOX055 153.61-52.7595.46-21.910.6571.696.58-4.3165.73 BSOX056 238.45-32.27154.44-26.687.1184.9416.76-11.38128.03 BSOX057 177.84-66.9696.12-26.693.26111.2611.36-6.8879.02 BSOX058 323.21-60.36160.29-40.798.34169.9715.25-8.97146.20 BSOX059 224.70-129.1188.49-42.386.44159.0412.53-7.1077.57 BSOX060 259.98-119.12102.14-57.5710.33170.4517.21-8.7482.03 BSOX061 102.68-82.5798.16-85.563.358.654.34-1.2485.87 BSOX062 192.49-114.18198.10-109.904.3426.237.30-2.23161.08 BSOX063 88.77-88.5982.09-92.682.5117.475.30-1.6056.79 BSOX064 205.98-121.29188.74-120.246.0464.9513.52-2.74142.30 BSOX065 119.12-103.84104.62-112.823.5390.948.66-1.28-11.88 BSOX066 209.04-113.12176.50-110.405.32168.3410.25-1.9529.87 BSOX067 159.81-105.05131.98-107.494.9757.809.37-1.9582.05 BSOX068 253.50-112.28182.10-108.186.34101.4411.69-2.02143.45 BSOX069 293.58-110.26160.42-118.127.59155.7518.27-5.85128.37 BSOX070 286.83-106.16154.46-119.5212.36160.8618.71-5.55115.99 BSOX071 79.21-151.1684.07-157.243.1632.303.56-3.0884.07 BSOX072 131.99-260.43139.66-244.674.5926.225.29-6.66125.54 BSOX073 91.20-108.6594.03-112.905.0739.117.27-2.4458.58 BSOX074 130.73-185.82142.15-190.077.3849.909.18-5.63134.92 BSOX075 96.85-132.9899.34-136.194.1629.595.94-3.5889.97 BSOX076 137.17-195.38134.27-198.945.8678.729.25-5.46111.91 BSOX077 88.12-166.9184.49-168.925.5257.487.21-3.8374.03 BSOX078 133.87-191.76113.53-198.265.3365.708.35-4.9824.66 BSOX079 139.97-150.0960.40-160.699.44120.7510.07-4.398.89 BSOX080 229.86-189.42118.20-206.2810.64250.5112.25-5.86-28.56 BSOX081 39.69-65.7323.13-19.71-0.3740.912.14-1.892.10 BSOX082 202.87-45.06116.74-37.6914.0788.1215.76-6.7772.29 BSOX083 62.29-39.4119.65-16.28-0.0654.051.16-0.9019.10 BSOX084 111.11-66.8968.67-28.15-0.2060.946.05-4.6640.85 BSOX085 90.07-68.0644.04-25.39-0.7946.285.83-4.94-7.68

PAGE 362

362 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. BSOX086 159.24-80.0293.53-38.453.3790.1712.77-10.0910.02 BSOX087 101.87-81.5539.68-33.48-0.1465.203.91-3.66-0.27 BSOX088 165.76-109.5080.33-49.99-1.2987.1013.77-12.1820.28 BSOX089 127.20-127.7638.61-43.273.96114.8210.16-5.1832.72 BSOX090 162.04-130.9751.37-44.4615.15122.5918.41-4.9747.23 BSOX091 93.30-13.4992.99-11.82-0.6521.042.28-0.7974.54 BSOX092 225.25-32.02183.24-24.175.3073.6914.14-1.86176.48 BSOX093 248.82-36.99110.01-25.734.46139.0510.59-4.5794.17 BSOX094 359.37-45.46183.71-34.259.70180.2516.14-9.43152.41 BSOX095 185.23-28.1696.99-15.241.60107.468.67-5.0077.08 BSOX096 277.13-36.37165.48-31.9711.78134.1717.23-9.95139.20 BSOX097 237.08-63.86126.10-30.416.32142.9811.24-7.3972.34 BSOX098 343.61-69.63186.05-49.6112.16170.4124.91-18.67132.58 BSOX099 200.77-97.2480.85-48.204.44156.3713.85-7.8472.91 BSOX100 268.86-167.31113.32-68.955.24203.3017.90-11.3690.44 BSOX101 111.58-114.26106.87-113.913.0023.624.05-1.6881.29 BSOX102 216.26-121.51220.46-114.875.7432.237.55-2.5197.24 BSOX103 133.32-118.82107.79-123.905.0943.5310.55-1.8769.94 BSOX104 228.12-133.81189.30-129.687.5178.9013.92-3.31133.77 BSOX105 120.20-90.88106.16-93.654.6730.067.91-1.3769.70 BSOX106 271.00-121.92216.83-118.4811.8989.8113.77-2.33176.14 BSOX107 150.45-111.07126.19-110.444.5260.269.97-1.8478.04 BSOX108 280.22-123.42200.32-122.318.89100.4413.90-2.01138.78 BSOX109 220.79-95.36112.11-109.6012.44123.3915.45-4.4292.33 BSOX110 286.69-131.59156.28-133.1914.97153.3620.78-8.13118.30 BSOX111 70.02-166.9873.93-171.453.0421.683.17-3.6568.33 BSOX112 128.78-267.57135.65-269.834.7137.045.00-6.19104.12 BSOX113 77.87-137.2884.05-137.414.0321.714.52-3.3383.81 BSOX114 151.22-193.19150.64-197.257.4239.838.64-6.44131.63 BSOX115 91.27-148.6496.40-150.904.0715.285.19-3.7079.18 BSOX116 138.39-219.40142.25-228.077.2939.318.13-6.02113.82 BSOX117 91.83-157.6685.89-163.395.0454.036.61-4.2980.47 BSOX118 185.32-200.80139.38-210.456.2577.1810.71-5.4474.71 BSOX119 143.10-132.1279.57-148.738.70104.819.31-4.1633.35 BSOX120 196.64-217.0399.44-203.8810.53144.5311.55-5.8960.17

PAGE 363

363 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. BSOX121 60.28-18.4636.40-12.920.0836.732.17-2.0415.11 BSOX122 144.51-20.48114.91-19.861.9977.858.74-6.3370.44 BSOX123 104.74-44.2062.36-25.625.8848.2812.14-11.2822.35 BSOX124 219.76-62.57165.94-44.916.9991.4526.46-21.64104.92 BSOX125 151.60-84.9269.22-36.357.3082.5918.47-16.32-21.46 BSOX126 249.92-110.27126.28-35.265.79129.2919.27-12.3566.37 BSOX127 114.69-75.2862.51-30.750.4456.6814.99-12.7356.19 BSOX128 249.27-123.02140.41-59.835.56142.0825.43-23.2269.59 BSOX129 156.02-140.1338.74-43.444.40143.6612.48-6.4231.77 BSOX130 190.69-138.1470.27-64.609.39148.0221.42-13.6459.74 BSOX131 96.52-13.5298.29-10.800.9422.072.52-1.5973.29 BSOX132 217.10-30.97187.12-25.64-3.3680.0511.75-4.34165.98 BSOX133 224.29-30.99105.60-21.896.87123.2810.74-6.6291.95 BSOX134 382.22-56.18225.60-46.4518.01202.4431.56-16.55215.83 BSOX135 255.87-55.32122.96-21.233.08136.727.69-9.40102.33 BSOX136 370.92-32.90206.91-34.8411.47164.5114.14-11.03190.74 BSOX137 247.25-47.01151.80-34.040.44124.8012.43-17.50146.71 BSOX138 308.24-106.59199.37-69.5812.20147.0836.91-28.92125.24 BSOX139 248.67-113.6092.52-50.324.73184.6119.18-19.0426.72 BSOX140 320.25-185.97101.96-101.187.79228.5319.90-15.3756.90 BSOX141 121.05-103.38121.37-107.842.6610.713.61-2.11105.76 BSOX142 254.24-131.32264.63-123.864.2583.988.74-3.69211.65 BSOX143 156.00-98.45132.86-100.655.0046.4911.00-2.0483.85 BSOX144 328.29-121.01240.89-119.3411.24146.7214.36-3.01171.70 BSOX145 187.29-111.08160.85-108.334.5845.389.57-2.20121.31 BSOX146 367.99-125.59273.19-117.019.6097.8414.46-3.56187.94 BSOX147 202.82-109.19163.02-104.505.3860.819.92-1.83107.71 BSOX148 275.27-115.39219.69-111.618.01108.0814.20-3.50134.30 BSOX149 299.36-118.47140.55-127.3617.76175.8925.11-8.08116.43 BSOX150 296.18-114.75189.44-114.6615.84167.7821.04-4.59131.15 BSOR0016.36-9.231.89-4.950.005.360.29-0.25-0.01 BSOR00289.03-71.4338.79-38.443.0151.894.19-2.5337.48 BSOR0030.90-0.370.67-0.15-0.040.440.12-0.200.16 BSOR00489.93-56.7735.29-20.591.1171.592.86-2.28-1.15 BSOR005117.43-89.3150.76-38.732.2094.435.63-4.1110.15

PAGE 364

364 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. BSOR006120.32-84.0945.89-38.497.1280.207.26-3.9439.52 BSOR00749.58-60.2116.68-18.76-0.3255.151.59-1.11-2.43 BSOR00875.19-65.7725.58-22.200.9862.312.77-1.98-8.18 BSOR00930.59-43.575.56-13.89-0.3234.711.95-1.40-6.76 BSOR01062.27-74.6721.07-27.372.4865.123.27-2.1020.50 BSOR011112.02-13.1391.23-12.122.2723.812.65-1.0390.64 BSOR012278.93-30.52156.82-25.0714.60123.2215.40-5.79155.07 BSOR01377.52-14.1252.64-10.510.6432.101.38-0.7742.63 BSOR014268.89-92.24118.86-25.493.66161.275.61-2.5397.13 BSOR015174.19-16.8992.00-14.271.4293.332.86-0.9987.98 BSOR016247.44-43.01136.88-20.982.01120.6311.93-2.74127.88 BSOR017187.22-68.0382.92-18.781.83111.875.05-2.4767.59 BSOR018264.27-69.93127.51-33.029.38143.5610.58-3.40102.01 BSOR019163.26-109.7258.51-42.433.38138.325.19-1.9446.94 BSOR020201.03-93.9078.13-47.362.64158.769.84-2.8263.43 BSOR02197.34-61.4086.99-62.873.5914.394.46-1.2677.41 BSOR022207.12-109.54180.16-105.508.3435.298.64-2.19166.29 BSOR02377.68-74.9370.44-76.842.8719.463.45-1.3258.59 BSOR024196.41-115.00169.17-115.117.9673.3711.61-2.22114.16 BSOR025112.87-72.2794.37-74.944.9531.927.02-1.2975.37 BSOR026199.45-113.92165.85-112.2811.5261.9412.42-2.79156.05 BSOR027126.02-84.3491.23-87.054.9151.758.01-1.2967.66 BSOR028228.09-113.37166.81-113.886.2478.2212.87-1.98139.64 BSOR029254.94-91.77118.01-108.1711.04156.5414.48-7.3993.87 BSOR030323.63-103.00144.67-121.739.06190.1716.49-4.24122.88 BSOR03197.95-123.03102.11-121.494.2313.814.31-2.98101.98 BSOR032157.97-243.14169.26-224.087.7328.837.87-4.83161.16 BSOR033100.39-131.45101.29-137.314.2443.336.64-2.7579.26 BSOR034157.64-214.51168.21-218.977.1666.2710.04-4.09109.58 BSOR03587.08-168.1292.19-174.724.8149.506.30-3.7737.40 BSOR036159.32-256.25163.96-253.538.2864.129.60-3.8290.48 BSOR037112.06-188.33110.86-198.526.7574.369.08-3.4266.60 BSOR038126.75-275.23122.79-273.096.2291.2614.49-5.1258.46 BSOR039162.23-146.6582.74-156.048.80149.2413.00-5.8314.97 BSOR040154.59-282.7896.29-289.8110.53117.6911.43-4.6618.90

PAGE 365

365 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. BSOR04134.95-47.7014.03-18.610.1228.251.67-1.077.74 BSOR042141.33-78.4797.03-33.634.0275.679.52-3.6790.95 BSOR0437.77-14.131.58-5.78-0.2711.320.59-0.47-5.63 BSOR044124.97-77.1167.17-36.525.3570.4910.96-8.5665.86 BSOR04553.35-52.0724.47-20.020.7048.071.94-1.4215.76 BSOR046123.60-85.4661.31-35.381.1072.379.03-7.9315.23 BSOR04764.33-63.5813.58-19.430.1360.062.23-1.47-9.73 BSOR048138.08-87.6149.04-39.084.62105.3312.04-9.27-2.68 BSOR04940.34-54.7411.82-17.11-0.3845.813.11-2.543.81 BSOR05087.03-94.4925.12-32.540.6275.074.08-3.97-11.74 BSOR05193.01-15.0493.40-13.161.0127.111.48-0.5574.91 BSOR052212.76-24.14177.10-20.866.7561.0010.92-1.67167.17 BSOR053150.11-34.9093.27-31.391.9173.227.44-3.9581.91 BSOR054356.06-44.58176.92-35.9616.68209.1917.19-6.41147.05 BSOR055155.15-15.9694.17-13.884.3761.655.24-1.5691.53 BSOR056347.52-32.50175.07-29.1214.77181.6118.06-7.62171.18 BSOR057238.48-48.39104.01-31.345.14135.8910.35-6.9954.63 BSOR058306.92-42.63170.49-37.909.83160.6917.85-14.34126.43 BSOR059177.94-85.6076.10-48.093.27104.6612.70-2.2170.11 BSOR060274.20-127.3896.86-57.857.30178.6920.14-9.7576.84 BSOR061111.37-90.62103.33-93.113.8714.015.12-1.4789.18 BSOR062196.61-111.42194.63-108.555.2918.516.58-2.6292.79 BSOR063129.52-80.45106.54-82.816.1737.868.66-1.6271.32 BSOR064239.33-128.49196.36-129.057.8582.6815.12-2.74133.89 BSOR065132.14-104.06112.34-104.504.8438.548.14-1.5357.90 BSOR066195.61-109.96158.61-106.806.4857.0310.06-1.94141.62 BSOR067205.02-102.38133.83-99.067.5487.4511.70-1.72110.71 BSOR068268.48-119.65179.22-119.724.10142.7613.87-1.86121.94 BSOR069272.77-95.58136.66-117.9411.82157.0917.19-6.82110.27 BSOR070280.39-118.84158.04-135.918.59159.0017.41-6.05126.42 BSOR071110.19-151.76114.86-159.724.3916.304.53-3.19114.64 BSOR072169.47-271.91178.14-282.597.7337.787.84-5.43177.80 BSOR073101.01-126.14101.48-126.934.7520.796.64-2.45100.44 BSOR074176.34-217.57183.54-221.468.7357.3011.36-4.39133.08 BSOR075103.88-137.23110.08-141.934.7040.007.21-3.2183.58

PAGE 366

366 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. BSOR076166.32-256.25167.62-268.749.04122.949.64-4.33100.91 BSOR07791.96-180.9295.08-183.325.2161.328.48-3.8279.43 BSOR078142.48-292.15130.98-285.318.0391.8414.07-5.0897.93 BSOR079193.86-198.8987.74-208.5811.53148.7711.79-4.5049.77 BSOR080203.54-263.36122.74-284.028.68113.2012.58-4.7936.14 BSOR08183.75-27.0143.18-14.480.1343.092.32-1.8038.44 BSOR082175.44-21.52116.59-17.059.3460.429.34-2.98115.15 BSOR08368.12-36.7133.79-22.951.6347.672.57-1.3333.37 BSOR084236.43-62.90152.91-48.8615.52105.6320.55-15.7176.87 BSOR08584.79-60.3832.29-18.611.8560.994.37-4.03-4.71 BSOR086160.00-52.93104.03-36.394.8773.2813.11-7.3753.00 BSOR087112.59-103.3266.57-39.011.5168.317.30-5.6442.87 BSOR088169.92-101.1193.68-50.274.64109.6717.60-13.4830.44 BSOR089126.55-133.6332.38-36.722.79122.947.27-4.053.08 BSOR090141.69-131.8451.30-48.185.92125.899.81-5.5018.32 BSOR09191.50-11.8689.69-10.601.4022.131.76-0.9188.99 BSOR092244.19-20.20180.20-17.248.7872.5412.21-2.47169.36 BSOR093188.36-28.6696.14-19.173.3192.2510.86-4.0086.34 BSOR094362.14-37.15196.15-29.8515.12178.9816.14-3.53185.58 BSOR095233.24-43.38108.21-18.867.71127.168.99-3.63104.91 BSOR096305.83-27.26160.75-27.1313.06146.6515.97-7.93159.31 BSOR097239.02-49.16113.20-33.225.99132.3314.84-9.1789.21 BSOR098351.81-64.17185.71-40.4812.00175.1022.78-15.75138.82 BSOR099193.79-107.9573.12-44.995.66163.0415.52-6.8368.90 BSOR100278.17-129.91109.11-57.3411.21184.5618.63-9.6276.15 BSOR101108.02-104.97107.09-104.552.559.133.97-1.5676.70 BSOR102268.11-105.47243.13-98.905.0543.777.12-2.33194.47 BSOR103103.83-75.3493.28-81.163.5537.955.92-1.5264.51 BSOR104238.22-119.67211.77-117.7011.3063.9213.27-2.27139.12 BSOR105138.54-89.42114.56-92.935.5842.3210.01-1.6778.11 BSOR106245.02-122.84216.61-114.466.1563.9211.99-2.58124.65 BSOR107153.91-99.94123.43-100.754.2453.389.56-1.3792.49 BSOR108267.70-113.79200.78-122.514.9899.7014.74-1.99121.37 BSOR109272.36-95.57124.85-110.668.46160.1213.35-3.23104.81 BSOR110404.65-120.63186.17-136.8916.89237.3322.46-8.30158.34

PAGE 367

367 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. BSOR111112.71-148.42116.28-153.334.0343.594.31-2.96111.67 BSOR112185.50-278.17194.73-285.527.4734.857.63-5.79178.81 BSOR11393.27-124.2297.64-129.083.6135.025.66-3.0982.78 BSOR114164.89-233.22166.84-237.308.0655.7811.33-4.94121.18 BSOR115107.53-150.62102.57-157.545.0339.106.46-3.2385.45 BSOR116157.85-260.05165.10-261.037.4261.9411.26-5.89139.86 BSOR117126.74-242.63112.51-245.627.12100.0314.24-4.5954.11 BSOR118190.47-319.45138.15-318.2910.1189.2714.59-5.0890.29 BSOR119155.15-168.0075.88-178.1310.35121.6211.59-4.7256.32 BSOR120217.87-303.42113.30-317.889.47143.7911.39-6.4255.43 BSOR121123.68-20.7261.92-16.752.2761.955.39-6.1234.99 BSOR122156.44-21.97113.92-21.354.8442.859.55-6.3686.05 BSOR12378.68-54.7746.16-26.54-2.5159.329.21-7.8511.23 BSOR124214.62-55.21142.33-47.2410.56101.4722.99-22.2289.18 BSOR125108.46-75.3765.22-25.103.7454.7414.99-14.5432.41 BSOR126174.06-84.8197.90-32.874.5592.6515.30-13.8286.18 BSOR127146.35-110.4189.54-49.029.4675.4718.99-17.4525.44 BSOR128226.61-112.81115.44-53.436.75111.6721.22-17.4319.43 BSOR129146.58-123.9945.15-49.562.46120.0411.20-9.09-14.58 BSOR130166.82-121.6558.14-50.232.93133.9324.99-16.9439.60 BSOR131128.73-36.48105.17-14.972.8273.774.32-2.5196.48 BSOR132232.05-27.27188.15-22.867.0076.3116.60-7.98176.14 BSOR133335.92-27.14144.24-23.794.86191.6917.49-11.8292.52 BSOR134354.55-43.20211.76-36.6714.79176.5625.32-15.08198.53 BSOR135298.82-45.00123.29-19.123.96191.9411.66-7.7291.69 BSOR136347.50-29.54189.11-27.7013.81167.6120.38-14.85155.88 BSOR137348.52-54.24165.27-32.5213.94214.9136.44-29.15116.59 BSOR138380.72-80.33202.13-42.0616.37187.6537.17-29.40124.80 BSOR139330.23-142.30112.77-75.6826.82232.1929.41-17.7095.15 BSOR140293.63-143.87113.61-67.287.02196.0424.80-21.8762.67 BSOR141124.44-108.81120.89-107.243.2612.603.84-1.8979.57 BSOR142209.24-122.56213.09-117.045.7456.867.37-3.03174.24 BSOR143173.38-101.26144.93-105.865.6646.669.03-2.6093.28 BSOR144267.10-128.38236.32-126.9810.2564.7911.21-3.83224.53 BSOR145216.40-120.77175.76-119.606.9464.4610.00-2.3991.96

PAGE 368

368 Table B 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Horiz. Assoc. Slam Max Horiz. Max Horiz. Min Vert. Assoc. BSOR146330.97-123.43307.08-112.324.0288.2815.53-3.69210.08 BSOR147231.90-108.27169.05-101.9910.6897.3411.28-2.72111.33 BSOR148266.94-121.73227.23-120.708.9181.3816.27-2.96227.00 BSOR149368.38-116.13164.04-132.617.99213.8523.39-5.14139.51 BSOR150394.33-127.40179.81-147.1524.18240.9324.31-7.12153.11

PAGE 369

369 APPENDIX C PHYSICAL MODEL DATA SLAMMING FORCES The following tables are a list of all physical model tests performed and the significant variables and values associated with each test. The tables are divided into variables and forces. The tests can be differentiated by the individual case prefix and reference number All flat plate slamming cases contain the prefix SLAM. Table C -1 contains the relevant fluid and structure parameters for all tests. Table C -2 contains the measured significant forces and moments for all tests. All dimensions are in feet, all forces are in pounds, and all times are in seconds.

PAGE 370

370 Table C 1. Structure and fluid parameters for all physical model tests. Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length SLAM0014.006.000.080.252.420.503.5029.63 SLAM0024.006.000.080.252.420.953.5029.63 SLAM0034.006.000.080.252.420.503.0025.01 SLAM0044.006.000.080.252.420.793.0025.01 SLAM0054.006.000.080.252.420.692.5020.30 SLAM0064.006.000.080.252.420.922.5020.30 SLAM0074.006.000.080.252.420.762.0015.46 SLAM0084.006.000.080.252.421.072.0015.46 SLAM0094.006.000.080.252.420.781.5010.36 SLAM0104.006.000.080.252.420.971.5010.36 SLAM0114.006.000.080.172.500.483.5030.09 SLAM0124.006.000.080.172.501.003.5030.09 SLAM0134.006.000.080.172.500.523.0025.38 SLAM0144.006.000.080.172.500.913.0025.38 SLAM0154.006.000.080.172.500.672.5020.59 SLAM0164.006.000.080.172.500.952.5020.59 SLAM0174.006.000.080.172.500.762.0015.65 SLAM0184.006.000.080.172.501.092.0015.65 SLAM0194.006.000.080.172.500.701.5010.45 SLAM0204.006.000.080.172.501.031.5010.45 SLAM0214.006.000.080.082.580.443.5030.54 SLAM0224.006.000.080.082.581.003.5030.54 SLAM0234.006.000.080.082.580.373.0025.75 SLAM0244.006.000.080.082.580.913.0025.75 SLAM0254.006.000.080.082.580.492.5020.87 SLAM0264.006.000.080.082.580.992.5020.87 SLAM0274.006.000.080.082.580.702.0015.83 SLAM0284.006.000.080.082.581.202.0015.83 SLAM0294.006.000.080.082.580.641.5010.52 SLAM0304.006.000.080.082.581.041.5010.52 SLAM0314.006.000.080.002.670.473.5030.99 SLAM0324.006.000.080.002.671.913.5030.99 SLAM0334.006.000.080.002.670.453.0026.11 SLAM0344.006.000.080.002.670.953.0026.11 SLAM0354.006.000.080.002.670.482.5021.14

PAGE 371

371 Table C 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length SLAM0364.006.000.080.002.670.902.5021.14 SLAM0374.006.000.080.002.670.772.0016.00 SLAM0384.006.000.080.002.671.362.0016.00 SLAM0394.006.000.080.002.670.661.5010.59 SLAM0404.006.000.080.002.671.011.5010.59 SLAM0414.006.000.08-0.082.750.433.5031.42 SLAM0424.006.000.08-0.082.751.063.5031.42 SLAM0434.006.000.08-0.082.750.473.0026.46 SLAM0444.006.000.08-0.082.751.003.0026.46 SLAM0454.006.000.08-0.082.750.522.5021.40 SLAM0464.006.000.08-0.082.750.902.5021.40 SLAM0474.006.000.08-0.082.750.922.0016.17 SLAM0484.006.000.08-0.082.751.402.0016.17 SLAM0494.006.000.08-0.082.750.751.5010.66 SLAM0504.006.000.08-0.082.751.091.5010.66 SLAM0514.006.000.08-0.172.830.503.5031.85 SLAM0524.006.000.08-0.172.831.103.5031.85 SLAM0534.006.000.08-0.172.830.563.0026.80 SLAM0544.006.000.08-0.172.830.993.0026.80 SLAM0554.006.000.08-0.172.830.552.5021.66 SLAM0564.006.000.08-0.172.830.902.5021.66 SLAM0574.006.000.08-0.172.830.782.0016.33 SLAM0584.006.000.08-0.172.831.442.0016.33 SLAM0594.006.000.08-0.172.830.841.5010.73 SLAM0604.006.000.08-0.172.831.151.5010.73 SLAM0614.006.000.080.252.080.433.5027.69 SLAM0624.006.000.080.252.080.853.5027.69 SLAM0634.006.000.080.252.080.413.0023.41 SLAM0644.006.000.080.252.080.753.0023.41 SLAM0654.006.000.080.252.080.562.5019.08 SLAM0664.006.000.080.252.081.622.5019.08 SLAM0674.006.000.080.252.080.632.0014.63 SLAM0684.006.000.080.252.080.882.0014.63 SLAM0694.006.000.080.252.080.661.509.97 SLAM0704.006.000.080.252.080.801.509.97

PAGE 372

372 Table C 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length SLAM0714.006.000.080.172.170.383.5028.19 SLAM0724.006.000.080.172.170.743.5028.19 SLAM0734.006.000.080.172.170.483.0023.82 SLAM0744.006.000.080.172.170.833.0023.82 SLAM0754.006.000.080.172.170.472.5019.40 SLAM0764.006.000.080.172.170.752.5019.40 SLAM0774.006.000.080.172.170.542.0014.85 SLAM0784.006.000.080.172.170.822.0014.85 SLAM0794.006.000.080.172.170.641.5010.08 SLAM0804.006.000.080.172.170.811.5010.08 SLAM0814.006.000.080.082.250.353.5028.68 SLAM0824.006.000.080.082.250.743.5028.68 SLAM0834.006.000.080.082.250.353.0024.23 SLAM0844.006.000.080.082.250.773.0024.23 SLAM0854.006.000.080.082.250.532.5019.71 SLAM0864.006.000.080.082.250.822.5019.71 SLAM0874.006.000.080.082.250.562.0015.06 SLAM0884.006.000.080.082.251.012.0015.06 SLAM0894.006.000.080.082.250.571.5010.18 SLAM0904.006.000.080.082.250.861.5010.18 SLAM0914.006.000.080.002.330.413.5029.16 SLAM0924.006.000.080.002.330.863.5029.16 SLAM0934.006.000.080.002.330.403.0024.62 SLAM0944.006.000.080.002.330.743.0024.62 SLAM0954.006.000.080.002.330.512.5020.01 SLAM0964.006.000.080.002.330.892.5020.01 SLAM0974.006.000.080.002.330.642.0015.26 SLAM0984.006.000.080.002.331.092.0015.26 SLAM0994.006.000.080.002.330.661.5010.27 SLAM1004.006.000.080.002.330.881.5010.27 SLAM1014.006.000.08-0.082.420.423.5029.63 SLAM1024.006.000.08-0.082.420.863.5029.63 SLAM1034.006.000.08-0.082.420.363.0025.01 SLAM1044.006.000.08-0.082.420.753.0025.01 SLAM1054.006.000.08-0.082.420.522.5020.30

PAGE 373

373 Table C -1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length SLAM1064.006.000.08-0.082.420.822.5020.30 SLAM1074.006.000.08-0.082.420.692.0015.46 SLAM1084.006.000.08-0.082.421.172.0015.46 SLAM1094.006.000.08-0.082.420.641.5010.36 SLAM1104.006.000.08-0.082.420.911.5010.36 SLAM1114.006.000.08-0.172.500.383.5030.09 SLAM1124.006.000.08-0.172.500.783.5030.09 SLAM1134.006.000.08-0.172.500.433.0025.38 SLAM1144.006.000.08-0.172.500.903.0025.38 SLAM1154.006.000.08-0.172.500.472.5020.59 SLAM1164.006.000.08-0.172.500.882.5020.59 SLAM1174.006.000.08-0.172.500.692.0015.65 SLAM1184.006.000.08-0.172.501.212.0015.65 SLAM1194.006.000.08-0.172.500.851.5010.45 SLAM1204.006.000.08-0.172.501.041.5010.45 SLAM1214.006.000.080.251.750.393.5025.56 SLAM1224.006.000.080.251.750.663.5025.56 SLAM1234.006.000.080.251.750.513.0021.63 SLAM1244.006.000.080.251.750.753.0021.63 SLAM1254.006.000.080.251.750.532.5017.69 SLAM1264.006.000.080.251.750.792.5017.69 SLAM1274.006.000.080.251.750.502.0013.67 SLAM1284.006.000.080.251.750.732.0013.67 SLAM1294.006.000.080.251.750.611.509.47 SLAM1304.006.000.080.251.750.761.509.47 SLAM1314.006.000.080.171.830.293.5026.11 SLAM1324.006.000.080.171.830.693.5026.11 SLAM1334.006.000.080.171.830.423.0022.09 SLAM1344.006.000.080.171.830.693.0022.09 SLAM1354.006.000.080.171.830.442.5018.05 SLAM1364.006.000.080.171.830.812.5018.05 SLAM1374.006.000.080.171.830.532.0013.92 SLAM1384.006.000.080.171.830.822.0013.92 SLAM1394.006.000.080.171.830.751.509.61 SLAM1404.006.000.080.171.830.581.509.61

PAGE 374

374 Table C 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length SLAM1414.006.000.080.081.920.353.5026.65 SLAM1424.006.000.080.081.920.743.5026.65 SLAM1434.006.000.080.081.920.383.0022.54 SLAM1444.006.000.080.081.920.733.0022.54 SLAM1454.006.000.080.081.920.422.5018.40 SLAM1464.006.000.080.081.920.772.5018.40 SLAM1474.006.000.080.081.920.462.0014.17 SLAM1484.006.000.080.081.920.692.0014.17 SLAM1494.006.000.080.081.920.571.509.74 SLAM1504.006.000.080.081.920.741.509.74 SLAM1514.006.000.080.002.000.503.5027.17 SLAM1524.006.000.080.002.000.753.5027.17 SLAM1534.006.000.080.002.000.363.0022.98 SLAM1544.006.000.080.002.000.713.0022.98 SLAM1554.006.000.080.002.000.492.5018.75 SLAM1564.006.000.080.002.000.822.5018.75 SLAM1574.006.000.080.002.000.472.0014.40 SLAM1584.006.000.080.002.000.752.0014.40 SLAM1594.006.000.080.002.000.501.509.86 SLAM1604.006.000.080.002.000.811.509.86 SLAM1614.006.000.08-0.082.080.423.5027.69 SLAM1624.006.000.08-0.082.080.863.5027.69 SLAM1634.006.000.08-0.082.080.313.0023.41 SLAM1644.006.000.08-0.082.080.773.0023.41 SLAM1654.006.000.08-0.082.080.422.5019.08 SLAM1664.006.000.08-0.082.080.822.5019.08 SLAM1674.006.000.08-0.082.080.492.0014.63 SLAM1684.006.000.08-0.082.080.822.0014.63 SLAM1694.006.000.08-0.082.080.591.509.97 SLAM1704.006.000.08-0.082.080.781.509.97 SLAM1714.006.000.08-0.172.170.403.5028.19 SLAM1724.006.000.08-0.172.171.013.5028.19 SLAM1734.006.000.08-0.172.170.373.0023.82 SLAM1744.006.000.08-0.172.170.763.0023.82 SLAM1754.006.000.08-0.172.170.482.5019.40

PAGE 375

375 Table C 1. Continued Test case Struct. Width Struct. Length Struct. Thick. Struct. Clear. Water Depth Wave Height Wave Period Wave Length SLAM1764.006.000.08-0.172.170.912.5019.40 SLAM1774.006.000.08-0.172.170.592.0014.85 SLAM1784.006.000.08-0.172.171.102.0014.85 SLAM1794.006.000.08-0.172.170.661.5010.08 SLAM1804.006.000.08-0.172.170.841.5010.08

PAGE 376

376 Table C -2. Significant force values for all physical model tests. Test case Vert. MaxVert. Min Quasi Max Quasi Min Slam Max Pressure Max SLAM0014.41-13.523.21-9.712.40 0.03 SLAM002157.09-78.13107.43-46.2458.400.22 SLAM00322.75-29.7017.17-20.1613.210.13 SLAM004152.04-122.28120.41-89.8983.180.44 SLAM00554.75-37.7241.58-35.7229.380.16 SLAM006135.74-87.3687.38-55.5573.470.31 SLAM007100.99-103.6047.34-41.4374.650.16 SLAM008188.66-150.4993.81-80.07127.290.31 SLAM00939.59-50.5231.14-46.7119.580.20 SLAM01055.23-114.1043.41-73.5363.020.29 SLAM01156.30-22.8237.22-20.6822.140.10 SLAM012216.19-92.89142.94-64.2282.900.18 SLAM013104.44-65.2669.59-40.0350.080.19 SLAM014238.97-87.80152.64-73.7398.160.55 SLAM01588.94-39.4567.21-27.9837.680.20 SLAM016203.12-106.35138.35-59.12112.390.59 SLAM017115.25-114.0884.02-50.9279.370.17 SLAM018327.88-142.51164.99-97.14189.950.26 SLAM01965.86-67.7143.74-55.6227.850.19 SLAM02086.80-142.6264.91-91.6358.450.26 SLAM02192.46-25.7461.53-20.2633.670.11 SLAM022188.06-80.25125.22-67.8165.470.21 SLAM02389.98-33.3758.30-19.5338.340.13 SLAM024276.40-109.09181.73-56.3895.360.48 SLAM025106.11-36.4967.55-23.6245.140.18 SLAM026296.94-88.70155.98-60.78150.170.50 SLAM027139.49-54.12112.41-42.5343.840.20 SLAM028331.70-102.88182.84-86.08199.090.37 SLAM02978.00-77.4064.20-60.4447.080.17 SLAM030121.61-153.9394.57-107.2381.100.37 SLAM031102.65-35.7378.71-32.3325.530.04 SLAM032155.19-89.51129.92-76.1133.740.11 SLAM033124.56-44.93103.23-38.5238.920.15 SLAM034313.28-100.68212.86-67.58123.430.35 SLAM035167.06-45.31102.99-39.9271.690.10

PAGE 377

377 Table C 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Slam Max Pressure Max SLAM036247.96-113.38192.43-75.5177.130.25 SLAM037218.99-80.58120.02-49.55106.830.21 SLAM038270.23-109.63200.87-82.7983.940.30 SLAM039133.57-108.3884.25-56.8875.690.16 SLAM040153.39-135.67116.85-111.9793.830.29 SLAM04154.69-89.4048.49-91.7027.270.03 SLAM042114.36-142.42113.29-139.0445.300.15 SLAM04385.88-80.1960.26-81.9732.710.05 SLAM044285.02-125.00191.74-123.68101.060.30 SLAM045104.65-103.3667.30-102.0738.760.05 SLAM046230.17-132.57172.08-127.47109.420.25 SLAM04788.51-110.3272.82-107.6155.980.10 SLAM048400.02-148.88174.46-126.67226.170.28 SLAM04995.99-118.1970.50-97.8263.450.15 SLAM050125.40-150.33111.35-141.4989.690.19 SLAM05163.09-111.6466.30-112.2110.720.01 SLAM052113.46-153.55120.86-147.9342.140.06 SLAM05382.19-101.0789.60-103.4812.100.02 SLAM054218.70-128.88174.84-129.2859.870.14 SLAM05588.00-120.9189.47-122.2144.310.05 SLAM056206.27-142.25170.15-138.2776.290.10 SLAM05788.00-105.4865.74-104.6641.170.05 SLAM058241.47-141.01163.43-138.7180.320.12 SLAM05961.46-110.5759.40-105.8020.420.08 SLAM060151.62-171.43112.30-142.2983.320.17 SLAM0619.64-18.678.05-17.336.690.12 SLAM062149.89-79.5698.33-73.9864.490.17 SLAM0631.59-1.661.89-1.431.240.00 SLAM064100.68-76.7381.58-57.0056.450.27 SLAM06533.34-40.6917.69-20.1122.270.09 SLAM06696.46-78.9072.33-53.7558.980.22 SLAM06758.26-79.0228.51-31.2052.920.09 SLAM06899.21-123.4960.13-57.9679.480.19 SLAM06928.55-37.5618.60-28.8828.150.48 SLAM07053.19-87.7137.10-56.6444.290.18

PAGE 378

378 Table C 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Slam Max Pressure Max SLAM07139.66-20.7535.35-18.7119.060.49 SLAM072165.20-96.54106.28-54.4676.810.13 SLAM07369.03-45.5158.82-24.0031.290.19 SLAM074138.40-50.75105.32-36.9854.150.19 SLAM07560.25-31.3839.21-23.2427.620.14 SLAM076147.59-62.71109.10-49.7881.020.32 SLAM07780.51-67.0940.27-29.8054.880.51 SLAM078241.11-109.37115.32-76.13153.370.19 SLAM07955.20-74.5541.24-42.9734.200.23 SLAM080103.43-95.7168.82-74.5442.410.21 SLAM08170.94-23.0159.41-21.3121.300.16 SLAM082162.81-78.29120.84-62.1251.740.07 SLAM08375.48-18.9547.14-15.9730.580.11 SLAM084209.40-88.83159.31-47.4271.520.29 SLAM085135.59-45.4679.87-27.2862.860.19 SLAM086222.66-102.74140.21-56.2489.270.31 SLAM087127.29-51.1593.38-32.5749.360.21 SLAM088269.01-86.62166.79-75.79130.490.29 SLAM08983.49-67.1262.13-54.7858.180.17 SLAM090113.94-148.0794.06-95.5184.500.22 SLAM091100.35-28.1770.01-27.7334.520.07 SLAM092142.11-73.49121.40-64.1931.560.12 SLAM093135.99-27.5182.65-22.0953.440.11 SLAM094231.92-78.55157.28-48.4284.170.26 SLAM095125.76-33.9495.18-30.2241.970.10 SLAM096216.61-84.44156.49-65.03105.350.45 SLAM097154.28-60.96111.07-43.6157.650.17 SLAM098235.98-105.22172.99-81.1188.150.25 SLAM099128.84-95.4389.30-57.1163.450.23 SLAM100151.23-120.69105.83-87.3977.840.21 SLAM10169.22-93.6451.85-94.9537.270.04 SLAM102104.37-136.00103.14-132.4236.780.06 SLAM10392.00-83.1261.64-83.2640.030.06 SLAM104241.15-109.59151.01-111.0790.220.30 SLAM10586.41-96.8863.79-96.8834.330.05

PAGE 379

379 Table C 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Slam Max Pressure Max SLAM106230.42-119.27127.81-115.91110.480.19 SLAM10773.87-92.6961.58-93.9150.450.07 SLAM108320.68-133.60139.36-120.87192.620.29 SLAM10988.57-107.8568.54-90.2045.320.09 SLAM110139.17-168.97100.95-131.0167.570.15 SLAM11153.06-85.9356.49-88.976.730.01 SLAM112104.13-140.7296.97-139.1743.370.05 SLAM11372.65-84.2477.69-85.329.840.01 SLAM114217.32-119.06174.28-119.8869.410.13 SLAM11576.43-104.7281.68-104.3824.420.02 SLAM116208.55-126.45152.60-129.2169.980.09 SLAM11773.23-107.9567.33-106.3129.470.04 SLAM118157.08-129.99142.81-137.0872.490.20 SLAM11993.51-122.5771.35-108.5534.210.08 SLAM120162.17-155.49117.98-130.8968.800.12 SLAM1211.64-10.391.84-9.212.890.03 SLAM12243.69-39.8536.06-37.9313.360.19 SLAM1230.96-4.131.50-3.182.270.01 SLAM12462.60-51.2246.81-30.1732.330.25 SLAM12515.04-21.188.47-17.528.440.46 SLAM12692.43-78.4558.80-58.3749.280.24 SLAM1276.92-15.413.36-15.136.020.08 SLAM12855.85-74.2538.56-42.3837.280.17 SLAM12925.58-41.4014.25-24.3424.950.47 SLAM13033.76-44.6122.60-40.4927.080.18 SLAM13114.93-14.669.20-15.216.730.14 SLAM132121.45-85.1297.88-58.4357.370.23 SLAM13321.80-22.7717.82-15.8311.770.10 SLAM134103.27-45.0377.55-26.7840.940.13 SLAM13530.55-36.0024.05-20.4217.380.11 SLAM136101.14-83.8990.85-56.9251.450.24 SLAM13770.69-96.1447.73-35.9256.880.11 SLAM138134.22-77.4680.84-53.1774.920.18 SLAM13965.24-88.3447.49-65.0835.100.21 SLAM14043.33-39.8931.46-34.8619.080.16

PAGE 380

380 Table C 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Slam Max Pressure Max SLAM14163.12-22.5248.51-17.4322.310.12 SLAM142142.45-61.73103.77-51.5143.190.16 SLAM14378.49-21.5056.17-16.0527.840.10 SLAM144172.64-56.67143.14-39.8666.700.21 SLAM14599.36-31.2162.46-21.1437.460.09 SLAM146210.21-107.51137.86-54.1572.880.32 SLAM147132.88-52.9781.54-33.2768.490.20 SLAM148259.12-95.80138.26-70.40122.720.23 SLAM14979.36-66.8962.15-55.2558.040.13 SLAM15083.31-90.4578.39-75.7166.250.18 SLAM15163.09-111.6466.30-112.2110.720.01 SLAM152129.37-70.24127.22-60.2120.030.05 SLAM153117.89-29.1581.45-29.4950.270.18 SLAM154237.53-97.63172.39-52.6887.140.28 SLAM155135.02-43.2084.12-38.1357.290.13 SLAM156203.39-75.13162.91-64.0158.460.15 SLAM157143.78-58.8796.24-39.2856.620.15 SLAM158212.04-103.89149.45-90.3187.090.25 SLAM159100.98-78.4869.04-44.3551.860.14 SLAM160132.89-122.48102.97-79.4582.830.16 SLAM16162.45-79.6857.86-82.5614.170.02 SLAM162125.73-141.47121.69-124.8141.440.06 SLAM16380.25-73.8748.16-75.0036.850.05 SLAM164299.58-111.72172.81-113.35131.840.29 SLAM16560.26-85.0649.98-90.9835.110.05 SLAM166181.97-141.44118.09-122.5674.550.10 SLAM16782.32-85.0857.71-81.7032.830.06 SLAM168326.50-152.44129.63-110.08273.260.33 SLAM169102.40-111.2172.44-86.6358.240.16 SLAM170118.64-146.8983.19-110.5658.620.13 SLAM17160.57-81.8364.33-84.659.620.01 SLAM172142.21-158.89143.35-151.2440.690.06 SLAM17354.80-80.6658.14-84.8424.800.03 SLAM174176.48-112.49149.75-114.7337.880.07 SLAM17568.68-98.0872.80-100.8530.370.05

PAGE 381

381 Table C 2. Continued Test case Vert. MaxVert. Min Quasi Max Quasi Min Slam Max Pressure Max SLAM176151.87-123.93132.85-123.9440.520.05 SLAM17787.97-99.7476.02-96.2766.960.08 SLAM178146.12-117.93132.29-122.5934.240.11 SLAM179120.23-120.7678.56-110.3149.610.10 SLAM180144.76-162.36102.43-127.3875.480.15

PAGE 382

382 LIST OF REFERENCES American Association of State Highway and Transportation Officials (2008). Guide Specifications for Bridges Vulnerable to Coastal Storms. AASHTO, Washington, DC. Bea, R. G., Iversen, R., and Xu, T. (2001). Wavein -deck forces on offshore platforms. J. Offshore Mech Arctic Eng. 123, 10-21. Bea, R. G., Xu, T., Stear, J., and Ramos, R. (1999). Wave forces on decks of offshore platforms. J. Wate rw ay, Port, Coast al, Ocean Eng., 125(3), 136-144. Cardone, V. J., and Cox, A. T. (1992). Hindcast study of Hurricane Andrew, offshore Gulf of Mexico. J oint Industry Project Rep Ocean -weather Inc., Cos Cob, Conn. Cuomo, G., Allsop, W., and McConnell, K. (2003). Dynamic wave loads on coastal structures: analysis of impulsive and pulsating wave loads. Proc., Coastal Structures 2003, COPRI, Portland, Or. Cuomo, G., Tirindelli, M., and Allsop, W. (2007). Wavein -de ck loads on exposed jetties. Coastal Eng., 54, 657-679. Da Costa, S. L., and Scott, J. L. (1988). Wave impact force on the Jones Island east dock, Milwaukee, Wisconsin. Proc., Oceans 88, 31, 1231-1238. Dean, R. G., Torum, A., and Kjeldsen, S. P. (1985). Wave forces on a pile in the surface zone from the wave crest to wave trough. Proc., Int. Symp. on Separated Flow Around Marine Structures Norwegian Institute of Technology, Trondheim, Norway. Dean, R. G., and Dalrymple, R. A. (2002). Coastal Processes with Engineering Applications Cambridge University Press, Cambridge, U.K. Denson, K. H. (1978). Wave forces on causewaytype coastal bridges. Misc. Rep. Water Resources Research Institute, Mississippi State University, Starkville, Miss. Denson, K. H. (1980). Wave forces on causewaytype coastal bridges: Effects of angle of wave incidence and crosssection shape. Technical Rep. No. MSHDRD -80-070, Water Resources Research Institute, Mississippi State University, Starkville, Miss. Douglas, S. L., Chen, Q., and Olsen, J. M. (2006). Wave forces on bridge decks. Draft Report CTEREC, University of South Alabama, Mobile, Ala. El Ghamry, O. A. (1963). Wave forces on a dock. Research Technical Report HEL -9-1, Hydraulic Engineering Laboratory, Berkele y, Cal. Faltinsen, O., Kjarland, O., Nottveit, A., and Vinje, T. (1977). Water impact loads and dynamic response of horizontal circular cylinders in offshore structures. Proc., Offshore Technology Conf., SPE, Richardson, Tex.

PAGE 383

383 Finnigan, T. D., and Petrauskas, C. (1997). Wavein deck forces. Proc. 6th Int. Offshore and Polar Engineering Conf. ISOPE, Golden, Colo. French, J. A. (1970). Wave uplift pressures on horizontal platforms. Proc. Civil Eng. in the Oceans Conf., ASCE. Imm, G. R., OConnor, J. M., and Stahl, B. (1994). South Timbalier 161A: A successful application platform requalification technology. Proc., Offshore Technology Conf., SPE, Richardson, Tex. Isaacson, M., and Bhat, S. (1996). Wave forces on a horizontal plate. Int J. Offsho re Polar Eng ., 6(1), 19-26. Kaplan, P. (1992). Wave impact forces on offshore structures: Reexamination and new interpretations. Proc. Offshore Tech. Conf., SPE, Richardson, Tex. Kaplan, P., Murray, J. J., and Yu, W. C. (1995). Theoretical analysis of wave impact forces on platform deck structures. Proc. 14th Int. Conf. Offshore Mech. and Arctic Eng., ASME, New York. Kjeldsen, S. P., and Myrhaug, D. (1979). Breaking waves in deep water and resulting wave forces. Proc., Offshore Technology Conf., S PE, Richardson, Tex. Kjeldsen, S. P., and Hasle, E. K. (1985). Ekofisk jacket Model experiments. Rep. No. NHL 85-0295, Norwegian Hydrodynamics Laboratory, Trondheim, Norway. Kjeldsen, S. P., Torum, A., and Dean, R. G. (1986). Wave forces on vertical piles caused by 2 and 3dimensional breaking waves. Proc., Coastal Engineering Conf., ASCE, New York. Marin, J. M. (2009). Wave loading on a horizontal plate. Masters Thesis, University of Florida, Gainesville, Florida. Morison, J. R., OBrien, M. P., Johnson, J. W., and Schaaf, S. A. (1950). The force exerted by surface waves on piles. P etrol. Trans. 189, 149-154. McConnell, K. J., Allsop, N. W. H., Cuomo, G., and Cruickshank, I. C. (2003). New guidance for wave forces on jetties in exposed locations. Proc., 6th Int. Conf. on Coastal and Port Engineering in Developing Countries COPEDEC, Columbo, Sri Lanka. Murray, J. J., Kaplan, P., and Yu, W. C. (1995). Experimental and analytical studies of wave impact forces on Ekofisk platform structures. Proc., Offshore Tech. Conf., SPE, Richardson, Tex.. Overbeek, J., and Klabbers, I. M. (2001). Design of jetty deck for extreme vertical wave loads. Technical Report. Delta Marine Consultants, Netherlands. Payne, P. R. (1981). The virtual mass of a rectangular flat plate of finite aspect ratio. Ocean Eng., 8(5), 541-545.

PAGE 384

384 Schumacher, T., Higgins, C., Bradner, C., Cox, D., and Yim, S. (2008). Largescale wave flume experiments on highway bridge superstructures exposed to hurricane wave forces. Proc., 6th National Seismic Conf. on Bridges and Highways Charleston, SC. Stear, J., and Bea, R. G. (1997). Ultimate limit state capacity analysis of two Gulf of Mexico platforms. Proc., Offshore Technology Conf., SPE, Richardson, Tex. Suchithra, N., and Koola, P. M. (1995). A study of wave impact on horizontal slabs. Ocean Eng., 22(7), 687-697. Sulisz, W., Wilde, P., and Wisniewski, M. (2005). Wave impact on elastical ly supported horizontal deck. Jo. Fluids and Structures 21, 305-319. Tirindelli, M., Cuomo, G., Allsop, N. W. H., and McConnell, K. J. (2002). Exposed jetties: Inconsistencies and gaps in design methods for waveinduced forces. Proc., 28th Int. Conf. on Coastal Engineering, ASCE, Cardiff, UK. Tirindelli, M., Cuomo, G., Allsop, N. W. H., and Lamberti, A. (2003). Wavein -deck forces on jetties and related structures. Proc., 13th Int. Offshore and Polar Eng. Conf., ISOPE, Honolulu, Haw. Vannan, M. T., Thompson, H. M., Griffin, J. J., and Gelpi, S. L. (1994). An automated procedure for platform strength assessment. Proc., Offshore Technology Conf., SPE, Richardson, Tex. U.S. Army Corp of Engineers (2006). Coastal Engineering Manual. Manual No. EM 1110-21100, U.S. Army Corp of Engineers, Coastal and Hydraulics Laboratory, Vicksburg, Miss. U.S. Department of Transportation (2006). Louisiana receives $863 million for new I-10 bridge and repairs to hurricane-damaged roads. Memo DOT 19 -06, U.S. Department of Transportation. Wang, H. (1970). Water wave pressure on horizontal plate. J. of the Hydraulics Division, 96(HY10), 1997-2017. Weggel, J. R. (1997). Breakingwave loads on vertical walls suspended above mean sea level. J. Waterway, Port, Coastal, Ocean Eng., 123(3), 143-148. Yu, Y., (1945). Virtual masses of rectangular plates and parallelpipeds in water. J. Applied Physics, 16, 724-729.

PAGE 385

385 BIOGRAPHICAL SKETCH Justin Marin was born in 1981 in New Orleans, Louisiana, the son of two civil engi neers. He moved to south Florida in 1995 and attended high school in Ft. Lauderdale. He began attending the University of Florida in August 1999 and graduated with a B.S. in civil engineering in May 2003. Remaining at the University of Florida throughout graduate school, he received his M.S. in coastal and oceanographic engineering in May 2009 and his Ph.D. in coastal and oceanographic engineering in December 2010. His further graduate studies included expanding the work on wave loading to bridge decks and more complex shapes. While at the University of Florida, he was part of the Coastal Engineering Lab field dive team, installing and retrieving instruments from the field and inspecting bridge foundations for scour effects. His favorite color is grey.