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Structure analysis and optimal design of curved bridge bearing considering temperature variations

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Title:
Structure analysis and optimal design of curved bridge bearing considering temperature variations
Creator:
Wang, Wanting ( author )
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
Publication Date:
Language:
English
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1 electronic file (124 pages). : ;

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Subjects / Keywords:
Bridges -- Bearings ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Review:
This thesis presents structure analysis and optimal design of bearings on the 8th Avenue Viaduct. Finite element two-dimensional structural analysis and one-dimensional transient heat conduction analysis were discussed. After a general description of Finite Element Method (FEM), three-dimensional models of the entire viaduct as well as a typical pot bearing structure in accordance with the shop drawings and as-built plans were developed using ANSYS software. A simplified beam line analysis was also conducted using SAP2000 software to obtain the bearings' vertical pressure, rotation, and transverse and longitudinal movements of the curved structure condition due to temperature variation effects. These initial values were input into the more complex three dimensional model in ANSYS for structural analysis of the pot bearing. Finally, some optimal design methods to relieve bearing grind and fracture issues were proposed.
Thesis:
Thesis (M.S.)--University of Colorado Denver. Civil engineering
Bibliography:
Includes bibliographic references.
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System requirements: Adobe Reader.
General Note:
Department of Civil Engineering
Statement of Responsibility:
by Wanting Wang.

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|University of Colorado Denver
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|Auraria Library
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All applicable rights reserved by the source institution and holding location.
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913746162 ( OCLC )
ocn913746162

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S TRUCTURE A NALYSIS AND O PTIMAL D ESIGN OF C URVED BRIDGE BEARING S CONSIDERING T EMPERATURE V ARIATION EFFECT S B y WANTING WANG B.S., Jilin University, 2012 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in p artia l fulfillment O f the requirements for the degree of Master of Science Civil Engineering Program 2015

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ii 2015 WANTING WANG ALL RIGHTS RESERVED

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iii This thesis for the Master of Science degree by Wanting Wang H as been approved for the Civil Engineering Program B y Chengyu Li Kevin L. Rens, Chair Frederick Rutz 2 3 April 2015

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iv Wanting, Wang (M.S., Civil Engineering) Structure Analysis and Optim al Design of Curved Bridge Bearing s Considering Temperature Variation Effect s Thesis directed by Professor Chengyu Li A BSTRACT This thesis presents structure analysis and optimal design of bearing s on the 8 th Avenue Viaduct. Fi nite element two dimension al structural analysis and one dimension al transient hea t conduction analysis were discussed After a general description of Finite Element Method (FEM) three dimension al models o f the entire viaduct as well as a typical pot bear ing structure in accordance with the shop drawings and as built plans were developed using ANSYS software A simplified beam line analysis was also conducted using SAP 2000 software to obtain the bearing s vertical pressure rotation and transverse and lon gitu d inal movements of the curved structure condition due to temperature variation effect s. These initial values were input into the more complex three dimensional model in ANSYS for structural analysis of the pot bearing Finally, some optimal design meth ods to relieve bearing grind and fracture issues were proposed. The form and content of this abstract are approved. I recommend its publication. Approved: Chengyu Li

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v ACKNOWLEDGEMENTS I really express my sincere gratitude to my advisor, Dr. Chengyu Li w hose knowledge and expertise guided me to finish my resea r ch work step by step. Besides professional skills of complex structural modeling, bridge analysis and design, bearing s and expansion joints accumulation through this project, it is Dr. Chengyu Li th at let me walk into the bridge engineering field, indeed, and he is a real bridge specialist in my mind. I express my thanks to Dr. Kevin Rens. With his support and recommendation I have participate d in the Denver Bridge Group Internship Program (UCD CCD Internship ) since my first semester M y practical experience has been accumulated and my oral and written com m unication skills have been improved greatly, which are very important and invaluable as an international student. Thank you Dr. Frederick Rutz for reviewing my research and giving me many valuable suggestions. Mr. Dick Miles from Bowman Construction Supply not only provided me much significant information, but also invited me to visit his warehouse. All the experience discussing with him toget her can deepen my recognition of bridge structure and spark my inspiration of my research work. Thanks to my best friend, greatly, Samir Mizyed. This academic project is an area await ing exploration and unlocking, and many difficulties and problems rem ain to be

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vi overc o me and resolve d, which calls for teamwork and enlightenment among professors and classmates in the daily work. It is fortunate to work with a comparable friend and we can learn from each other. From my educational experience in University of Colorado Denver till now, I learned a lot about the predomin a nt American teaching philosophy in detail through lectures and seminars, which is different from graduate education in China. All the invaluable help coming from my professors, classmates and fri ends give s me confidence and encouragement to continue and strive for my educational career in the future.

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vii TABLE OF CONTENTS CHAPTER I INTRODUCTION 1 Introduction of existing structure ................................ ................................ .......... 1 Problem s tatement ................................ ................................ ............................... 10 Objective and approach of r esearch ................................ ................................ .... 13 II. LITERATURE REVIEW 15 General ................................ ................................ ................................ ................ 15 Thermal movements ................................ ................................ ............................ 15 Bearing structural analysis ................................ ................................ .................. 22 Bearing optimal design ................................ ................................ ....................... 23 III. BRIDGE MODELING USING ANSYS 25 Model details ................................ ................................ ................................ ....... 25 Finite e lement m odels ................................ ................................ ......................... 26 Center line ................................ ................................ ................................ .... 30 Substructure ................................ ................................ ................................ 30 Piers and abutments ................................ ................................ .............. 30 Superstructure ................................ ................................ .............................. 34 Girders with diaphragms, diagonal braces, and stiffeners .................... 34 Deck ................................ ................................ ................................ ...... 40 Bearing ................................ ................................ ................................ ......... 41

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viii Expansion joint ................................ ................................ ............................ 47 Load combination ................................ ................................ ........................ 47 Boundary c ondition ................................ ................................ ..................... 47 Connections ................................ ................................ ................................ .. 47 IV. FINITE ELEMENT MODELING OF POT BEARINGS 50 AASHTO provisions concerning temperature range ................................ .......... 50 single gird er method of curved bridge ....... 51 F inite element analysis of pot bearing ................................ ................................ 61 V. OPTIMAL DESIGN OF POT TYPE BEARINGS 73 Bearing orientation improvement ................................ ................................ ....... 73 Bearing material improvement ................................ ................................ ........... 73 Bearing structure improvement ................................ ................................ ........... 75 Edge guided pot bearing with slider plate ................................ ................... 75 Center guided pot bearing ................................ ................................ ............ 77 VI. CONCLUSIONS 80 Recommendations ................................ ................................ ............................... 81 REFERENCES 83 APPENDIX A 86 ANSYS program ................................ ................................ ................................ 86 APPENDIX B 97 T heory and background ................................ ................................ ...................... 97

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ix LIST OF TABLES TABLE 1. Element types and material property of bridge structure ................................ ... 26 2. .................. 29 3. station (from #pier2 to #pier6) ................................ .......... 31 4. ................................ ........................... 31 5. ................................ ................................ .............. 45 6. ................................ .......................... 46 7. Material properties of the structure in SAP2000 ................................ ............... 52 8. Section property of FSEC2 through section designer ................................ ........ 53 9. Vertical reaction and max torsion in envelop ................................ ..................... 56 10. Horizontal and longitudinal m ovement under load combination ...................... 57 11. Comparison between lanes located on center line ................................ ............. 60 12. Element types, material property and dimension of fine bearings ..................... 61 13. ................................ ............ 65 14. Coordinates transfer from SAP2000 to ANSYS ................................ ................ 66 15. Maximum contact stress of PTFE ................................ ................................ ...... 75

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x LIST OF FIGURES FIGURE 1. Plan view of West 8 th Avenue Viaduct through satellite image ........................... 2 2. Bridge horizontal curve I ................................ ................................ ..................... 2 3. Bridge horizontal curve II ................................ ................................ .................... 3 4. Girders with diaphragms from Abutment #1 to Pier #6 ................................ ....... 4 5. Girders with diaphragms from Pier #6 to Abutment #20 ................................ ..... 4 6. Expansion bearings on one side of p iers ................................ .............................. 5 7. Guided bearings ................................ ................................ ................................ ... 6 8. Unguided bearings ................................ ................................ ............................... 6 9. Hammerhead piers ................................ ................................ ............................... 7 10. Strut type abutments ................................ ................................ ............................ 7 11. Common bearing types ................................ ................................ ........................ 8 12. Actual pot type bearing ................................ ................................ ...................... 10 13. Sectional view of the pot type bearing ................................ .............................. 10 14. Plastic material present at sliding interface ................................ ........................ 11 15. Grinding between guide bar and base plate ................................ ....................... 12 16. Guide bar fracture ................................ ................................ .............................. 13 17. Temperature distribution along depth ( F=1.8 C+ 32)(From[15].) ..................... 17 18. Displaced shape and radial and tangential displacements of deck .................... 18 19. Displaced shape and radial displacements of dec k ................................ ............ 19

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xi 20. Maximum normalized radial displacements ................................ ...................... 19 21. Profile line for the bridge plan(From[6].) ................................ .......................... 26 22. Profile line drawn through ANSYS ................................ ................................ ... 27 23. Spiral curve calculation ................................ ................................ ...................... 28 24. Piers elevation in the bride plan ( P ier # 2 P ier # 6) (From[6].) ............................ 32 25. Elevation of A butment # 20 in the bridge plan (From[6].) ................................ 32 26. West abutment ................................ ................................ ................................ .... 33 27. East abutment ................................ ................................ ................................ ..... 33 28. Pier#7 ................................ ................................ ................................ ................. 33 29. Pier#11(fixed at the bearing) ................................ ................................ .............. 34 30. Total graph of substructures through ANSYS ................................ ................... 34 31. Stiffener gap distribution, flange thickness variation (From[6].) ...................... 35 32. D 3 in the bridge plan (From[6].) ................................ ................................ ...... 36 33. D 5 in the bridge plan (From[6].) ................................ ................................ ...... 36 34. stribution (From [6].) ................................ ................................ ................................ .............................. 36 35. Girder dimension and deck layout (From[6].) ................................ ................... 37 36. D 4 through ANSYS ................................ ................................ .......................... 38 37. D 2 through ANSYS ................................ ................................ .......................... 38 38. D 1 through ANSYS ................................ ................................ .......................... 38 39. D 3 through ANSYS ................................ ................................ .......................... 39

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xii 40. ................................ ............ 39 41. The first ten section of the whole superstructure starting from #pier6 .............. 41 42. Bearing orientation (From [6].) ................................ ................................ .......... 42 43. Guided bearing ................................ ................................ ................................ ... 43 44. Non guided bearing ................................ ................................ ........................... 43 45. Fixed bearing ................................ ................................ ................................ ..... 44 46. C onnection between pier and bearings ................................ .............................. 48 47. Connection among girders, bearings, and pier ................................ ................... 49 48. Contour Maps for T MaxDesign for Steel Girder Bridges ................................ ........ 51 49. Contour Maps for T MinDesign for Steel Girder Bridges ................................ ........ 51 50. Cross section of superstructure in SAP2000 ................................ ..................... 52 51. Total graph of structure through SAP2000 ................................ ........................ 53 52. Live load case through SAP2000 ................................ ................................ ....... 55 53. Deformed shape due to the defined load combination ................................ ....... 56 54. Guided bearing(full view) ................................ ................................ .................. 62 55. Top plate removal ................................ ................................ .............................. 62 56. PTFE removal ................................ ................................ ................................ .... 63 57. Piston removal ................................ ................................ ................................ ... 63 58. Neoprene removal ................................ ................................ .............................. 64 59. Von mises stress ( s teel) ................................ ................................ ...................... 67 60. Von mises stress ( s teel II) ................................ ................................ .................. 68

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xiii 61. Von mises stress (PTFE) ................................ ................................ .................... 68 62. Von mises stress ( p iston) ................................ ................................ .................... 69 63. Von mises stress ( n eoprene) ................................ ................................ ............... 69 64. Von mises stress ( p ot) ................................ ................................ ........................ 70 65. Deformation in y direction ( s teel) ................................ ................................ ...... 70 66. Deformation in y dir ection (PTFE) ................................ ................................ .... 71 67. Deformation in y direction ( n eoprene) ................................ .............................. 71 68. Lateral view of edge guided pot bearing with slider plate ................................ 76 69. PTFE bonded with slider plate ................................ ................................ ........... 77 70. Pot base together with neoprene and piston ................................ ....................... 77 71. Full view of center guided pot bearing ................................ .............................. 78 72. Guide bar on the center of pot base ................................ ................................ ... 79 73. Change in energy ................................ ................................ ............................... 97 74. Strain in two dimensions ................................ ................................ ................. 100 75. Stress definition ................................ ................................ ............................... 101 76. Discretization with triangles and quadrilaterals ................................ ............... 103 77. Mopping between physical element and parent element ................................ 105 78. Corresponding coordinates of physical element and parent element ............... 106 79. Discretization into elements and nodes ................................ ............................ 108

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1 C HAPTER I INTRODUCTION Introduction of existing structure Bridge s are subjected to thermal stress es that are induced by solar r adiation and ambient air temperature fluctuation T hermal effect s can cause excessive movement and stress, resulting in damage to bridge structure s Colorado is a state where temp erature varies greatly, with variations of 40 F to even 60 F between day and night. As a result, bridges in Colorado experience much more damage than those in a mild climate Damages due to thermal movement have been observed i n the West 8 th Avenue Viaduct including concrete pier cracking, steel guide system fracture and expansion joint delamination This research is to study the thermal effects on this large structure. The West 8 th Ave nue Viaduct i s located in Downtown De n ver betwe en Mariposa St reet on the east and Vallejo St reet on the wes t as shown in F igure 1. The viaduct was constructed in 1985 The tota l length of th e viaduct is 2 372 f ee t with four horizontal curve s and three vertical curve s as shown in F igure 2 and F igure 3 It overpass es light rail tracks railroa d tracks and three local street s. The superstructure cons ists of 19 continuous span s and the substructure consists of 18 piers and 2 abutments.

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2 Figure 1 Plan view of West 8 th Ave nue Viaduct through satellite image Figure 2 B ridge horizontal curve I

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3 Figure 3 B ridge horizontal curve II T he superstructure consists of two steel open box g irders running parall el along the longitudinal direction of the brid ge from Span 1 through Span 19 There is an additional steel plate girder between the two steel box girder s from S pan 1 through Span 5 F or S pan s 1 through 5, both transverse diaph rag ms an d diagonal braces are located inside the steel box girder s and between the box girders as shown in F igure 4. For S pan s 6 through 19, transverse diaph rag ms are located both inside the box girders and be t ween the two steel box girders as shown in F igure 5 but diagonal b races are located inside the box girder s. T he spacing between stiff e ners varies with the smalles t spacing of 1.75 f ee t near the piers and the l ar gest spacing of 6 f ee t towards the middle part of the span s Deck width v aries along the length of the bridge from 40 f ee t to 58 f ee t D eck slab thickness is 8.5 inches with a waterproof membrane and asphalt wearing surface A t raffic barrier is

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4 placed on each side of the traffic lanes and a sidewalk is located on the south side only Figure 4 G irders with diaph rag ms from A butment # 1 to P ier # 6 Figure 5 G irders with diaph rag ms from P ier # 6 to A butment # 20 A ll bearings are pot type bearing s, as shown in F igure 6. All bearings are expansion sliding b earings except those on P ier # 11 which are fixed. The expansion sliding bearings on the interior side are uni directional guided bearings as shown in

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5 F igure 7 while unguided bearings are used on the exterior side as shown in F igure 8 For S pan s 6 throu gh 19, there are four bearings on each pier two below each steel box girder For S pan s 1 through 5, an additional bearing is used to support the steel plate girder between the two steel box girders A t ransflex expansion joint is installed at each a butmen t. Figure 6 E xpansion bearings on one side of piers

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6 Figure 7 G uided bearings Figure 8 U nguided bearings All piers are h ammerhead piers as shown i n F igure 9 A butments are wall type abutments as shown in F igure 10 Piers and abutments are support ed on piles and drilled caissons

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7 Figure 9 H ammerhead piers Figure 10 Wall type abutments A b earing is an important structural connection component between the s uperstructure and the substructure Bearings t rans m it load, deformation and rotation

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8 from the superstructure to the substructure Hence, bridge bearing s are required to have adequate capacity to resis t the loads and accommodate the deformations T he restraint of the bearing to deformation and rotation should be small so that only a small force is tran s ferred from the superstructure to the substruct u re B earing s should b e designed so that their installa tion maintenance repair and replacement are easy and convenient C ommon bridge bearings include rocker bearing s el astomeric bearing s pot bearing s and spherical bearing s as shown in F igure 11 Since all bearings in the West 8 th A ve nue V iaduct are pot t ype bearings as shown in F igure s 12 and 13 this study focus es on pot type bearings Figure 11 C ommon bearing types A n expansion pot bearing typically includes the following components:

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9 Top plate: Top plate consists of an upper steel plate and a thin stainless steel plate welded to the upper plate with a mirror finished sur face as a sliding top surface. The upper plate is either welded or bolted to the girder flange PTFE: TEFLON sheets are used as a sliding top surface of the sliding bearing in combination with stainless steel as the sliding top. The low friction coeffici ent between PTFE and stainless steel proves a sliding surface that can satisfy the movement of the superstructure through relative sliding. Elastomeric Dis k : The e lastomeric dis k is locate d inside the pot below the piston It is subjected to a three dimensional stress state which increases the bearing capacity greatly I t also allows for rotation of the bea ring. Piston: The p iston i s locate d between the elastomeric dis k and the PTFE slider. Its raised bottom inlays inside the pot allows for flexible rotation of the bearing. Sealing Rings: The s eal inlay s at the top of the elastomer ic dis k It s external diameter is close to the inner diameter of the pot so escape and failure of the elastomer ic dis k can be prevented largely by the sealing ring hooping ability Pot: It is made from steel to restrain the defor mation of the elastomeric dis k Guide systems: G uide bar s are used to restrain movement in one or more directions. The center guided system consists of a single guide bar along the center line of the bearing assembly and the edge gui d ed system utilizes t wo guide bars on both side s of the pot plate.

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10 Figure 12 A ctual pot type bearing Figure 13 S ectional view of the pot type bearing Problem s tatement Since the West 8 th Ave nue Viaduct has a complicated geomet ry with four horizontal

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11 curve s and three vertical curve s the superstructure moves not only in the longitudinal direction but also in the transverse direction resulting in horizontal force s to the bearing in both directions A b ridge inspection was carrie d out by Stantec Co r poration in early November 2014 and the f ollowing problem s of the bearing s w ere re ported : 1. Plastic material was found on the expansion pot bearings from the peeling of T eflon as shown in F igure 14 which increase s the friction coeffi cient between the stainless steel and the PTFE. Due to the increase of friction coefficient, the horizontal force s transferred through bearings to the substructures increase. The peeling of Teflon may increase the possibility of direct contact between the stainless steel and the piston T he cause of the peeling and means to eliminate it are the major focus es of this study. Figure 14 P lastic mater ial present at sliding interface 2. Excessive horizontal movement of the sole plate with the guide bars exceed ed the gap between the guid e bars and base plate, result ing in steel to steel grinding between

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12 the guide bar and base plate as shown in F igure 15 Figure 15 G rinding between guide bar and base plate 3. Some of the guide bars were remo ved in order to r elease th e contact However, due to the removal of the guide bars, the gui de system is n o longer effective to guide the bridge movement 4. G uide bar s on the bearings on P ier s # 2, # 14 and # 19 are fractured as shown in F igure 16

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13 Figure 16 G uide bar fracture Ob jective and a pproach of r esearch 1. Construct a three dimens ion structural model of the bridge based on the as built plans using finite element software A NSYS to perform a structural analysis to study the thermal effect s. L oads specified in AASHTO LRFD Bridge Design Specifications includ ing dead load, live load and temperature gradient are applied. B earing movement and stress due to the complicated geometry of the bridge and the temperature variation are obtained 2. Establish a simplified model for the bridge superstructure with a single girder line method usi ng the SAP 2000 program to acquire initial value s including longitudinal and transverse movement, vertical pressure and rotation that are transfer red to the bearing s

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14 3. Set up a finite element model for the pot bearing based on the shop drawing to study h ow much reaction, movement and rotation the current structure is subject ed to through structural analysis using A NSYS software The reason for the guide bar fracture and Teflon peeling are investigated. 4. Propose possible optimal design methods for rehabili tating the exi s ting pot bearings i n the West 8 th Avenue Viaduct

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15 CHAPTER II L ITERATURE REVIEW General In 1960, Engineering News Record ( Troy T. Tindal Chai H. Yoo 1998 ) proposed that the thermal effect on bridge structures needs to be giv en attention by researchers because of the following reasons: 1. A n increasing number of bridges were subjected to structural failure or partial structural damage due to the influence of thermal effect. 2. Compared to movement and stress caused by dead and li ve load s thermal movement can actually occupy a higher percentage of the total movement and therefore cannot be overlooked. 3. Usually, it is more reasonable and eco nomical for engineers to design structure s to accommodate thermal effect s instead of ne glecting such effects and then having to constantly rehabilitate the structures Since 1960, the thermal effect on bridge structure s has gradually c o me into Following is a brief summary of the research Thermal movements In 197 7, Will, Johnson, and Matlock at t he Center for Highway Research developed two finite element codes to obtain structural response under thermal loading. The first code was a two dimension al heat equation to obtain temperature distribution along the cross s ection of the bridge, while surface temperature measured from field

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16 investigation was applied for boundary conditions instead of heat analysis through solar radiation, convection and wind speed. T he second code was a two dimension al elastic stress strain c ode to obtain structural response through a structural model. One of the conclusion s drawn from this research is that the t hermal response of a continuous bridge lies between a simple span bridge and a two s pan bridge and the prestress effect on thermal re sponse is inconsequential ( Will, Johnson, and Matlock, 1977 ) In 1991, Pentas et al. conducted research to characterize and quantify bridge temperature distributions of the east approach of US 190 over the Atchafalaya River Four thermocouple arrays were i nstrumented and each array included six thermocouples along the depth of the superstructure s cross section. M easurement s w ere taken once a month for two years. Pentas summarized the equation s of vertical temperature distribution at selected location s alon g the superstructure depth according to the field measurement data as follows : ( Troy T. Tindal Chai H. Yoo 1998 ) EQUATION 1 EQUATION 2 EQUATION 3 Shashi Moorty and Charles W. Roeder reported the vertical temperature gradient of a three span curved bridge in the north s outh direction as shown in F ig ure 17 The structure consisted of four steel girders and a concrete deck A fixed bearing was instrumented G uide bearings were installed on the interior piers and north abutment.

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17 Two expansion joints were located on both si des of the girder ends. ( Sh ashi Moorty, Charles W. Roeder 1991) Figure 17 T emperature distribution along depth ( F=1.8 C+32) ( f rom[15] ) A couple of conclusions were reached based on this research. For thermal movement of the c urved bridge, the displaced shape, radial displacement and tangential displacem ent of the deck w ith g uide bearing s oriented along the tangent i s shown in F igure 18. F ig ure 19 displays the same variables but with the bearings oriented along the chord Bearing s are usually orient ed along the chord, especially when the fixed bearing s ar e on a stiff support such as an abutment. It was also found that the radial displa cements increase with the increase of the angl e of curvature of the bridge as shown in F ig ure 2 0. ( Sh ashi Moorty, Charles W. Roeder 1991)

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18 Figure 18 Displaced s hape and r adial and t angential d isplacements of d eck when g uided b earing along t angent ( from[15] )

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19 Figure 19 Displaced s hape and r adial d isplacements of d eck when b earings a re o riented along c hord ( from[15] ) Figure 20 Maximum n ormalized r adial d isplacements v ersus a ngle of c urvature ( from[15] ) In 2002, Troy T. Tindal and Chai H. Yoo conducted research to explore t he effect of thermal load on co mposite steel highway br idge s Three kinds of bearing orientation s were examined sep a rately. B ridge models with varying width, depth, skew were

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20 established in di vidually T he influence of each parameter under different seasonal thermal loads within each bearing orientation through the finite element structural an alysis approach was plotted (this will be discussed later ) Finally, the equation to predict displacement and restrained force of the bearing was summarized through the regression analysis method (Troy T. Tindal,Chai H Y oo ,2002) Equations to calculate maximum bearing dis placements were formed for the three cases of bearing orientation as follows : (Troy T. Tindal, Chai H Yoo 2002) Traditional: EQUATION 4 Radia l from Corner: EQUATION 5 Radial from Center: EQUATION 6 W here : L is the span length W is the bridge width is the skew Equations to calculate maximum bearing forces were also developed for the same three bearing orientation cases as follows : (Troy T. Tindal, Chai H Yoo 2002) Traditional: EQUATION 7

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21 Where Radial from Corner: EQUATION 8 Where Radial from Center: EQUATION 9 Where To predict the temperature behavior of large scale bridges, Yong Xia and Bo Chen proposed using theoretical transient heat transfer analysis and field monitoring measurement s to determine the t emperature distribution and associated responses of Tsing Ma Bridge, which is a large span suspen s ion bridge with a complicated configuration. The a cquired temperature distribution of each component was applied to a global finite element model of the bridg e to calculate temperature induced displacement and strain. A finite element model of the deck, cross frame and tower segment was established A go od agreement was shown between temperature induced displacement at selected locations obtained through the mo del and the field measurement data, but the calcula ted temperature induced stress wa s negligible ( Yong X ia Bo Chen, Xiao qing Zhou, You lin Xu 2013) With the assumption of temperature distribution along th e longitudinal direction of bridge s at a fixed elevation the relationship between the temperature field of the

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22 cross section and time function can be expressed by a two dimensional heat flow equation as follow s : ( Yong X ia Bo Chen, Xiao qing Zhou, You lin Xu 2013) EQUATION 10 Bearing structural analysis According to M. Imbimbo and A. De Luca, the effect of the shape factor on the stress distributions and stress concentrations of a laminated elast omeric bearing under ax ial vertical loads was determined through the finite element approach. The analyses show that the edge effects decr ease with the increase of the shape factor which for a rubber confined pad is the ratio between the rface area and the lateral area ( M. Imbimbo, A. De Luca 1998) Junji Yoshida and Masato Abe proposed a co nstitutive model of a high damping rubber ( HDR ) bearing and construct ed a three dimensional finite element model of the HDR bearing throug h the constitutive model. T his model can be used to simulate complicated deformation s such as torsional and rotational deformation of the bearing. ( Junji Yoshida, Masato Abe, Yozo Fujino, Hiroshi Watanabe 2004) In Ioannis V. Kalpakidis and Michael C. Co study, the effects of cumulative travel on the mechanical properties of lead rubber bearings w ere presente d The post elastic stiffness which depends on the rubb c haracteristic strength which depends upon t he mechanical behavior of lead are two important properties that are affected by temperature variation, aging and loading history.

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23 T he result s of their study indicated that the strength of lead rubber bearings at thermal loading conditions is influenced g reatly by the previous cumulative travel if the amplitude of cyclic motion during travel is large because of strain hardening of lead ( Ioannis V. Kalpakidis, Michael C. Con stantinou, Andrew S. Whittaker 2010) Bearing optimal design In the research by Che nxi Xing and Hao Wang, a new multi functional bridge bearing was designed and numerical calculation and experimental analysis w ere conducted to verify the y and stable performance. It is well know n that a sliding isolation bearing can keep movable characteristics with the advantage of a low coefficient of friction existing between the stainless steel and Teflon, while a lead core rubber bearing is widely used to provide vertical pressure and horizontal shear. Xing and Wang combine d these two devices together to adapt the deformation by temperature variant, live load, and concrete creep, etc. Also, fi nite element analysis was performed by developing a solid linear elastic model of the bearing and the FE results indicate d relatively n o big difference occurred when changing the loading frequency and the vertical pressure ( Chen xi X ing Hao W ang Ai qun L i Ji rong W u 2012) A modified design process of a high damp ing rubber bridge bearing for long term performance was developed by Y os hito Itoh and Hausheng Gu in 200 6 According to this process an HDR bridge bearing can be designed following the current design m anual considering bridge loads and thermal load. E quivalent horizontal stiffness and equivalent

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24 damping ratio can be obtained through the finite element method F inally seismic analysis on the bridge pier with the aged HDR bearing was performed using the predicted results.

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25 CHAPTER III BRIDGE MODELING USING ANSYS A three dimensional structural model is developed to study the movements and stresses in the superstructure, bearings, substructure, and guide elements due to temperature fluctuation and gradient, dead load, and live load. The complex shape of the West 8 th Avenue Viaduct requires a comp rehensive three dim ensi onal FEM that properly simulates the behavior of the structure. P revious studies only modeled the structure as a single girder line. A sectional FEM is also not valid because the bridge is not symmetrical about Pier #11. Therefore, to properly model th e effect of temperature loads, the entire viaduct must be included in the finite element model. Model details Several diff erent types of elements are used in the FEM to simulate the behavior of the different structural elements in the v iaduct. The 8 noded elastic solid element s are used to model piers and abutments The 4 noded elastic shell elements are used to model the steel box girders, steel plate girders and concrete deck. The stiffeners located in the girders are also modeled using 4 noded elastic sh ell elements. The 2 noded elastic beam elements are used to model diaphragms and diagonal braces. T he expansion joints located at the abutments are modeled using 2 noded nonlinear spring elements. Finally, the expansion pot bearings are modeled using 8 nod ed elastic solid element s to simulate top plates and base plates with nonlinear contact elements, which can simulate the sliding and friction. Th e elements and material pro pert ies used in the model are listed in T able 1

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26 Table 1 : E lement types and material propert ies of bridge structure Element type Young's modulus (kips/ft 2 ) Poisson's ratio Density (lb/ft 3 ) Deck shell elements (4 node 181) 5.90E+0 5 0.2 150 Box /Plate girders shell elements (4 node 181) 4.176E+0 6 0.3 500 Stiff e ners shell elements (4 node 181) 4.176E+0 6 0.3 500 Diaphragms beam elements (2 node 188) 4.176E+0 6 0.3 500 Diagonal braces beam elements (2 node 188) 4.176E+0 6 0.3 500 Bearings solid elements (8 node 185) 4.176E+0 6 0.3 500 nonlinea r contact elements (contact 174) Expansion joints nonlinear spring elements ( spring damping14 ) S tiffness: Abutment #1: 15.99 kip/in Abutment #20: 12.68 kip/in Piers solid elements (8 node 185) 5.90E+0 5 0.2 150 Finite e lement m odels The Finite Element Model of the West 8 th Avenue Viaduct was developed based on the as built plans. Figure 21 P rofile line for the bridge plan ( from[6] ) Profile line The West 8 th Ave nue Vi aduct has four horizontal curves and three vertical curves as shown in F igure 2 1 Each horizontal curve consists of two spiral curve s and one circular curve in the middle. Horizontal curves are used to define the location of piers an d

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27 abutments. Each pier and abutment was given a key point, whose special coordinates were calculated in reference to a point of origin In the model, each horizontal curve was determined by placing key points along the profile line every 10 feet and the n connecting these key points with second order curved lines. For the straight line portions, only two key points were needed for each line (one at the beginning and one at the end). The resulting profile is shown in F igure 2 2 The elevation of each point was also calculated with a reference elevation in the model. Figure 22 P rofile lin e drawn through ANSYS Figure 23 shows t he equations to calculate the spiral curves are :

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28 Where the total arc length of spiral curve the t otal centering angle of spiral curve the corresponding arc length from initial point in the spiral curve the corresponding centering angle from initial point in the spiral curve Figure 23 S piral curve c a l culation K ey point coordinates of the profile line are l isted in T able 2 The horizontal curve e lement was calculated as follows : , , ,

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29 Ta ble 2 : K ey point coordinates of a complete curve following profile line Station Ls/ft KP x/ft y/ft z/ft D elta (circular curve) TS.Sta.5+37.50 537.5 90.4 61 90.4 0 0 Sta.5+40.00 540 2.5 62 2.5 0 0.001091 Sta.5+50.00 550 12.5 63 12.49995 0 0.027271 Sta.5+60.00 560 22.5 64 22.49969 0 0.088356 Sta.5+70.00 570 32.5 65 32.49906 0 0.184347 Sta.5+80.00 580 42.5 66 42.4979 0 0.315239 Pier # 3 Sta.5+83.00 583 45.5 67 45.49742 0 0.361312 Sta.5+90.00 590 52.5 68 52.49603 0 0.48103 Sta.6+00.00 600 62.5 69 62.49331 0 0.681717 Sta.6+10.00 610 72.5 70 72.48955 0 0.917294 Sta.6+20.00 620 82.5 71 82.48461 0 1.187756 Sta.6+30.00 630 92.5 72 92.4783 0 1.493097 Sta.6+40.00 640 102.5 73 102.4705 0 1.833309 Sta.6+50.00 650 112.5 74 112.461 0 2.208384 Sta.6+60.00 660 122.5 75 122.4496 0 2.618314 SC.Sta.6+62.50 662.5 125 76 124.9465 0 2.726242 Sta.6+70.00 670 7.5 77 7.499923 0 0.029452 0.007853979 Sta.6+80.00 680 17.5 78 17.49902 0 0.160348 0.018325951 Sta.6+90.00 690 27.5 79 27.4962 0 0.395944 0.028797922 Sta.7+00.00 700 37.5 80 37.49036 0 0.736216 0.039269894 Sta.7+10.00 710 47.5 81 47.48041 0 1.181126 0.049741866 Pier # 4 Sta.7+17.38 717.38 54.88 82 54.8498 0 1.576548 0.057470181 Sta.7+20.00 720 57.5 83 57.46526 0 1.73062 5 0.060213838 Sta.7+30.00 730 67.5 84 67.4438 0 2.384653 0.070685809 Sta.7+40.00 740 77.5 85 77.41495 0 3.143138 0.081157781 Sta.7+50.00 750 87.5 86 87.37761 0 4.005998 0.091629753 Sta.7+60.00 760 97.5 87 97.33069 0 4.973137 0.102101725 Sta.7+70.00 77 0 107.5 88 107.2731 0 6.044449 0.112573697 Sta.7+80.00 780 117.5 89 117.2037 0 7.219817 0.123045668 Sta.7+90.00 790 127.5 90 127.1215 0 8.499112 0.13351764 Sta.8+00.00 800 137.5 91 137.0254 0 9.882194 0.143989612 Sta.8+10.00 810 147.5 92 146.9142 0 11. 36891 0.154461584 Sta.8+20.00 820 157.5 93 156.7869 0 12.9591 0.164933555 Sta.8+30.00 830 167.5 94 166.6424 0 14.65259 0.175405527 Sta.8+40.00 840 177.5 95 176.4796 0 16.44919 0.185877499 Sta.8+50.00 850 187.5 96 186.2975 0 18.3487 0.196349471 Pier # 5 Sta.8+51.75 851.75 189.25 97 188.0136 0 18.69168 0.198182066 Sta.8+60.00 860 197.5 98 196.095 0 20.35092 0.206821442 Sta.8+70.00 870 207.5 99 205.8709 0 22.45563 0.217293414 Sta.8+80.00 880 217.5 100 215.6243 0 24.66259 0.227765386 CS.Sta.8+86.11 886. 11 223.61 101 221.5721 0 26.06127 0.234163761 Sta.8+90.00 890 3.89 102 3.889998 0 0.002641 Sta.9+00.00 900 13.89 103 13.88993 0 0.033673 Sta.9+10.00 910 23.89 104 23.88963 0 0.09961 Sta.9+20.00 920 33.89 105 33.88893 0 0.200452 Sta.9+30.00 930 43. 89 106 43.88768 0 0.336196 Sta.9+40.00 940 53.89 107 53.88571 0 0.506838 Sta.9+50.00 950 63.89 108 63.88285 0 0.712374 Sta.9+60.00 960 73.89 109 73.87894 0 0.952801 Sta.9+70.00 970 83.89 110 83.87381 0 1.228111

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30 Center line The profile line and ce nter line are the two major lines needed to define each structure element location. While the bridge structure s orientation can be determine d through the profile line this line does not run through the center of either the superstructure or the substruct ure. Therefore, it is much more convenient to establish a center line and use it as a reference when creating the superstructure eleme nts. According to the bridge plan s the profile line is 7 inches to the nort h of the center line before Pier #4 and the n g radually change s to 3 .9 f ee t to the south of the center line at Pier #6 and then remain s at that distance throughout the rest of the bridge The change in distance between the profi le line and center line between Pier #4 to Pier #6 was assumed to be linear Substructure Piers and abutments A center point at the bottom of each pier and abutment was determined in reference to the profile line A local coordinate system was created at that point. A reference area based on the d i mension s given in the design pla n s was draw n This area was then extruded to a distance equal to the thickness of the pier or abutment in order to create volum e T he top surface of the pier was approximated as flat The dimension s of the piers and abutments are shown in T able s 3 and 4, a s well as shown in F igure s 24 and 25 Figure s 26 27 28 29 and 3 0 show some of the models that were created through ANSYS.

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31 Ta bl e 3 : P s and station s (from P ier # 2 to P ier # 6) PIER TABLE Pier Sta. Elevation A B C D E F 2 4+83.000 5225.0 46.68 46.82 46.96 46.77 46.71 3 5+83.000 5221.0 50.63 50.62 50.57 50.34 50.34 4 7+17.375 5221.0 53.84 53.63 53 .28 52.96 52.78 5 8+51.750 5221.0 54.03 53.82 53.48 53.26 53.03 6 9+86.125 5221.0 52.18 52.30 52.30 52.32 52.21 Table 4 : P s (from P ier # 2 to P ier # 6) Pier Dimension G H J K L M N P 2 3 4 5 5 1/2 6

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32 Figure 24 P iers elevation s in the brid g e plan ( P ier # 2 P ier # 6) ( f rom[6] ) Figure 25 E levation of A butment # 20 in the bridge plan ( f rom[6] )

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33 Figure 26 W est abutment Figure 27 E ast abutment Figure 28 Pier # 7

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34 Figure 29 P ier # 11 (fixed at the bearing) Figure 30 T otal graph of substructures through ANSYS Superstructure Girder s with diaph rag m s diagonal brace s and stiff e ner s There are seven different thicknesses of the girder top flanges and three different thickness es of the bottom plate, as shown in F igure 3 1 There are also three different widths of the top flanges in each span 10 inches n ear the center, 16 inches near the piers

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35 and 12 inches i n between Moreover, there are three different kinds of stiff e ners weld ed to the girder including top only bottom only and top and bottom. A top stiff e ner means there is a gap between the top plat e and the top edge of the stiff e ner A b ottom stiff e ner means there is a gap between the base plate and the stiff e ner bottom edge. A t op and bottom s t iff e ner means a gap exists between both the top plate and top edge and bottom plate and bottom edge of the stiff e ner. Figure 31 Stiff e ner gap distribution, flange thickness variation ( from[6] ) Furthermore, five kinds of transverse diaph rag m s exist in the drawing, two of which are shown in F igure s 3 2 and 33 A t the bottom of the open box girder, small WT se ction girders were welded to the bottom flange varying from one WT section to three WT section s running parallel to each other, depending on the location, as shown in F igure 34

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36 Figure 32 D 3 in the bridge plan ( f rom[6] ) Figure 33 D 5 in the bridge plan ( f rom[6] ) Figure 34 St iff e s and WT girder distribution ( f rom [6] ) Consequently, it is not appropriate to simply create one girder section and then

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37 sweep ing it along the profile line due to the fact that in every few feet the section change s Instead, each different section was draw n sep a rately and a linear connection is made between each two adjacent sections following the center line. T here are mo re than 30 kinds of different section s of the girder and almost 800 sections in total that have been connected linearly. Due to the difficulty in making the girders curved, straight lines are used to connect each section though the location of each sectio n still follows the horizontal and vertical curves. S ince the length of each of these straight lines is small the error is very minor Figure 35 shows girder dimen sion and deck layout, while some modeling examples in ANSYS a re shown in F igure 36 37 38 39 and 4 0 Figure 35 G irder dimension and deck layout ( f rom[6])

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38 Figure 36 D 4 through ANSYS Figure 37 D 2 through ANSYS F i gure 38 D 1 through ANSYS

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39 Figure 39 D 3 through ANSYS Figure 40 G s from P ier # 6 to P ier # 7 After the girder sections have been created they need to be connected in the model. To do so a profile line and a center line were made specific ally for girders in order to connect each adjacent section and key point s following these two lines. W hen ever a stiff e ner appear s or any change occurs, there is a corresponding key point in the center

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40 line that is used as a reference point. Since the secti ons are single piece s, they are assigned with the correct orientation along the center line and all these single pieces are merged together. The front end of the section with respect to the reference point on the center line was drawn and a local coordinat e system was placed. T hen the adjacent coordinate system was move d and activate d. T he section s o ther end was drawn by regarding the following key point on the center line as the new reference po int. The other sections were created similarly and merged to create the entire bridge model. D iaph rag m and diagonal braces between corresponding stiff e ners were added according to bridge plan s. Both the diagrams and diagonal braces are modeled as 2 node beam element s Deck The concrete deck slab is 8.5 inch es thick on the top of the girder s with a waterproofing membrane and a 2 inch asp h alt wearing surface. A sidewalk is located on the south side of the deck and tw o barriers are located on the outsides of the traffic lane s T he d eck is considered as a structural ele ment but the waterproofing membrane and asp h alt wearing surface are considered as applied loads The d eck is modeled using shell elements. E ach deck section is attached to the girder section as shown in F igure 4 1

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41 Figure 41 T he first ten section s of the whole superstructure starting from P ier # 6 Bearing s Bearing s are used to transfer load and movement from supers tructure to substructure. There are two types o f pot bearings expansion bearings ( both gui ded and non guided ) and fixed bearings In general, fixed bearing s should be located on the middle pier to reduce longitudinal movement at both ends A guided bearing should be located in the interior position and a non guided bearing should be located in the exterior position so that transverse movement of the superstructure can be control l ed to some extent. Bearing orientation is another important issue. In th e West 8 th Avenue Viaduct, fixed bearings on Pier #11 are placed p erpendicular to the pier cap an d all other guided and non guided sliding bearings are oriented along an individual chord between the sliding bearing and the fixed bearing on Pier #11 with an assumption that the superstructure only move s in the longitudinal direction towards the fixed be aring as

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42 shown in F igure 4 2 Figure 42 B earing orientation ( f rom [6] ) The b earing is modeled with a top plate attached to the bottom of the girder and a base plate attached to the top of the pier, as well as two guide bars attached to the bottom of the top plate for the guided bearing as shown in F igure 43 A fin ite element model includes a top plate, Teflon, piston, neop r ene and pot. A nonlinear contact element is used to simulate the sliding and friction between stainless steel and Teflon A contact element is a conne ctio n only subjected to compression but not tension in ANSYS so is the most suitable element to simulate bearing under vertical loads, transverse and longitudi n al movements, and rotation In this model, contact was defined with a coeffic ient of friction of 0.08 between the bottom face of the top plate and the top face of the base plate, as well as between the side s of the guide bar and the side s of the base p late. A 0.125 in ch contact surface offset is specified between the guide bar and base plate so that the guide bar cannot go through the base p late when the top plate is subjected to transverse displacement.

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43 Figure 43 G uided bearing Figure 44 Non guided bearing

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44 Figure 45 F ixed bearing Even though the modeling of a non gu i ded bearing and a fixed bearing look very similar, as shown in F igure s 44 an d 45 the difference is significant. Contact relation with a coefficient of fri c tion of 0.08 between the top plate and base plate is used to simulate sliding for a non guided bearing, while a fixed attachment between the t op plate and base plate is made fo r the fixed bearing. The bearing dimension s used in the modeling are listed in T ables 5 and 6

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45 Table 5 : Guided bearings dimension s

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46 Table 6 : s

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47 Expansion joint E xpansion joints are modeled using spring elements to simulate the expansion joint stiffness. The spring stiffness was provided by the expansion joint supplier, Bowman Construction Supply, Inc. The spring stiffness is a s follows: Abutment #1: 15.99kip/in Abutment #20: 12.68kip/in Load combination According to AASHTO LRFD Bridge Desi gn Specifications, th e following S trength I load combination is used: Q=1.25DC+1.5DW+1.75LL+1.2TU Boundary c ondition The piers are assumed to be fixed at the foundation level This assumption is valid due to the piers being supported on concrete cap s, piles and caissons. Co nnection s The s ubstructure and superstructure were modeled separately and connected with bearings T he expansion bearing on Pier #12 as shown in F igure 46 is used as an example First a new local coordinate was created using three key points: bearing cent er point of Pier #12 with the same elevation of Pier #11 (point of origin) bearing center point of Pier #11 (x axis) and bearing center point of Pier #12 (xy plane) After that a new key point was created with the same elevation of the bearing center po int o f Pier #12 but towards

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48 Pier #11 a long the x axis of this new local coordinate system and the expansion bearing orientation can be determined through the line connection between the bearing center point of Pier #12 and the new key point. After that, t he bottom area of the base plate was drawn for the bearing and the top area of the pier was divided by that area. By t his way the bottom of the base plat was attached to the top of the pier. Figure 46 C onn ection between pier and bearings For the connection between the bearing and girder since the superstructure and subs tructure were created before the bearing, the thickness of the bearing needed to be exactly the same as the gap between the bottom of the girder and the top of the pier. This is complicated due to the fact that the bottom surface of the girders is tilted gradually instead of being flat which means all fo u r corners of the top area of the top plate may have differen t elevations However, this difficulty was overcome by using

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49 variable thickness sole plates. Figure 47 shows the connection among girders, bearings, and pier Figure 47 Connection among girders, bearings, and pier

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50 C HAPTER I V FINITE ELEMENT MODELING OF POT BEARING S The focus of this study is on the bearing behaviors under thermal movements. A refined finite element model was developed for th e pot bearing. A simplified girder line anal ysis was performed with the SAP 2000 program to obt ain the reactions and movements which were used as the applied loads in the bearing model. AASHTO p rovisions concerning temperature range Since the West 8 th A venue Viaduct is a composite system with a concrete deck and steel girders the temperature ra n ge for the s tate of Colorado is T maxdesign = 110 F and T mindesign = 20 F based on F igure s 48 and 49 from AASHTO LRFD Bridge Design Specifications 3.12.2 T herma l movement can be de termined using the following equation. Where L = expansion length (in.) = coefficient of thermal expansion (in./in./F)

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5 1 Figure 48 Contour Maps for T MaxDesign f or Steel Girder Bridges with Concrete Decks ( f rom[1]) Figure 49 Contour Maps for T MinDesign for Steel Girder Bridges with Concrete Decks ( from[1] ) Initial values acquirement through single girder method of curved bridge A simpli fied single girder line model of the superstructure was developed using the SAP 2000 program to obtain the initial value s such as the vertical p ressure and rotation transferr ed to the bearing, as well as the transverse and longitudinal m ovement of the bear ings. T hese initial values were appl ied to the refined modeling of the pot bearing to

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52 investigate guide bar fracture. T he details of the material propert ies are shown in T able 7 For cross sections, a utility called section designer in SAP2000 was used to calculate section propert ies The results are shown in F igure 5 0 and T able 8 Since transverse moment of inertia which causes horizontal movement has no relationship with stiff e ners and diaph rag ms, all of the stiff e ners and diaphragms were neglected for th e simplified model. A total of five sections were created in SAP 2000. Figure 57 and T able 8 show the details of a section in the sixth span as an example. Table 7 : Material propert ies of the structure in SAP 2000 Young's modulus (kips/ft 2 ) Poisson's ratio Coefficient of Thermal Expansion Yield Strength (ksi) Compressive Strength (psi) Steel(A992) 4.176E+0 6 0. 3 6.5E 06 50 Concrete 5.191 E+0 5 0. 2 5.5E 06 4000 Figure 50 Cross section of superstructure in SAP 2000

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53 Table 8 : Section propert ies of FSEC2 through section designer Property Data(FSEC2) Cross section (axial) area 4.7219 Shear modulus about 3 axis 3.1187 Torsional constant 21.8995 Shear modulus about 2 axis 30.6222 Moment of i nertia about 3 axis 12.1764 Plastic modulus about 3 axis 140.369 Moment of i nertia about 2 axis 614.9954 Plastic modulus about 2 axis 297.5499 Shear area in 2 direction 0.5461 Radius of g yration about 3 axis 1.6058 Shear area in 3 direction 3.3917 Radius of g yration about 2 axis 11.4125 In the model, Pier #11 was defined as fix ed while all of the other bearings were defined as rollers Figure 5 1 shows th e total graph of the structure in SAP 2000. Figure 51 T otal graph of structure through SAP 2000 For load case s and load combination s dead load, secondary dead load, wearing surface dead load live load and temperature variation were included. The f ollowing

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54 S trength I limit state specified in AASHTO LRFD Bridge Design Specifications is used. Q=1.25DC+1.5DW+1.75LL+1.2TU Dead load ( g ravity): z direction of gravity multiplier need s to be defined as 1 S econdary dead load (barriers an d sidewalk): For this simplified model, the barriers and sidewalk w ere appl ied along the entire length of deck as a uniform distributed load as follows: Wearing surface: Wearing surface was al so applied a s a distributed load as follows: Live load: T wo traffic lanes were defined based on the offset of each lane center line from the frame center line A n HL93 live load was selected as the vehicle load whose axle loads considering impact coefficient 1.33 are shown in F igure 5 2 A simplified hand calculation for a joint reaction was made for calibration as follows:

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55 Figure 52 L ive load case through SAP 2000 Temperature variation: According to AASHTO specification, T maxdesign e quals 110 F and T mindesign equals 20 F in the s tate of Colorado A simplified hand calculation for longitudinal movement of the be aring was made for calibration as follows: All the vertical reaction s and movements transferr ed to the bearing s are su mmarized in T ables 9 and 1 0 Figure 53 shows a deformed shape due to the S trength I load combination.

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56 T able 9 : Vertical reaction and max torsion in envelop e Figure 53 D eformed shape due to the defined load combinat ion R eaction (kip) max torsion in envelope (kip ft) Dead L oad ( g ravity) Secondary Dead L oad Wearing Surface Live Load Temp Variation Load Comb ination Load Combination B earing # 1 417.386 68.49 67.1 220.28 0.02 1093.498 0 B earing # 2 1186.731 194.71 1 90.74 338.43 0.02 2605.142 134.1475 B earing # 3 1412.047 231.93 277.19 369.54 3042.449 354.8231 B earing # 4 1651.986 270.31 264.79 386.23 3475.957 601.1969 B earing # 5 1540.006 261.2 255.87 391.14 3319.804 630.3857 B earing # 6 1384.078 264.32 25 8.92 394.04 0.01 3138.457 932.5016 B earing # 7 1297.206 263.86 258.48 391.85 3024.78 961.0997 B earing # 8 1313.909 265.19 259.78 390.76 3047.385 1473.1471 B earing # 9 1275.696 257.96 252.7 383.41 2967.109 1467.551 B earing # 10 1230.554 248.71 243 .63 369.63 0.04 2861.671 1603.3421 B earing # 11 1242.237 251.68 245.98 382.48 2904.778 1360.2675 B earing # 12 1236.605 249.94 244.84 368.21 0.05 2869.887 571.3407 B earing # 13 1236.055 249.85 244.75 376.75 0.01 2883.799 571.185 B earing # 14 1238.17 7 250.27 245.16 379.97 2893.257 571.2149 B earing # 15 1231.463 248.91 243.83 380.31 0.03 2881.786 734.491 B earing # 16 1225.884 247.79 242.73 380.11 0.02 2871.362 984.96 B earing # 17 1231.411 248.91 243.83 379.26 0.02 2879.813 454.6243 B earing # 18 1251.658 253 247.83 376.95 2912.235 3.6057 B earing # 19 1209.291 244.43 239.44 363.4 0.02 2812.293 3.9141 B earing # 20 346.221 69.98 68.55 227.47 1021.16 0

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57 Table 10 : Horizontal and longitu din al movement under load combination Unit/in D ead L oad ( g ravity) Secondary Dead L oad Wearing Surface Live Load Temperature Variation Load Combination B earing # 1 U1 5.3791 0.1168 0.1144 0.2912 4.7272 13.2238 U2 1.3445 0.2833 0.2775 0.7287 0.2161 3.4671 B earing # 2 U1 4.847 0.1871 4.4083 11. 8891 U2 0.9451 0.4693 0.0873 2.4091 B earing # 3 U1 4.2889 0.09 4.0464 10.4907 U2 0.6069 0.235 0.0592 1.5134 B earing # 4 U1 3.6122 0.0354 3.5504 8.7716 U2 0.3713 0.0463 0.2276 0.8902 B earing # 5 U1 3.0357 0.0008 3.3 0376 7.4394 U2 0.3291 0.0084 0.3347 0.8236 B earing # 6 U1 2.5127 0.0007 2.5144 6.158 U2 0.3693 0.0057 0.376 0.9183 B earing # 7 U1 1.9887 0.0007 1.9903 4.8742 U2 0.386 0.003 0.386 0.9512 B earing # 8 U1 1.4641 0.0006 1.4652 3.5881 U2 0.3861 0.0024 0.3832 0.9482 B earing # 9 U1 0.9431 0.0006 0.9435 2.3115 U2 0.3186 0.0022 0.3177 0.7838 B earing # 10 U1 0.4612 0.0003 0.4616 1.1305 U2 0.1893 0.0008 0.1884 0.4646 B earing # 11 U1 0 0 0 0 U2 0 0 0 0 B earing # 12 U1 0.4514 0.0004 0.4515 1.106 U2 0.2088 0.0008 0.2087 0.5117 B earing # 13 U1 0.9022 0.0008 0.9028 2.2117 U2 0.4172 0.0005 0.4175 1.0229 B earing # 14 U1 1.3526 0.001 1.3541 3.316 U2 0.6 28 0.0008 0.6264 1.5388 B earing # 15 U1 1.8046 0.0012 1.8069 4.4242 U2 0.8338 0.0017 0.8301 2.0432 B earing # 16 U1 2.2832 0.0113 2.2793 5.6111 U2 0.9928 0.0104 0.9733 2.4379 B earing # 17 U1 2.7873 0.0235 2.7714 6.8597 U2 1.1689 0.081 1.0221 2.91 B earing # 18 U1 3.3075 0.0179 3.2699 8.1103 U2 1.4442 0.227 1.0226 3.6614 B earing # 19 U1 3.8075 0.0179 3.7693 9.3348 U2 1.7734 0.4242 1.0211 4.5628 B earing # 20 U1 4.1837 0.0068 0.0066 0.0 176 4.1502 10.259

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58 From the structura l analysis performed through SAP 2000 we can draw the following conclusions : 1. I f the live load effect was not considered transverse and longitudinal movement of the bearings was mostly caused by dead load (gravit y) and temperature variation condition due to the curvature effect. A o ne dimension equation from AASHTO specification s was used to calculate the thermal movement A ll the movement due to temperature variation was along the axial direction of the frames du e to thermal expansion and contraction. Consequently, the simplified thermal movement calculated i s not very accurate and is not a reflection of the actual ther mal movement 2. Transverse and longitudinal movement of the bearings caused by the wearing surfa ce and secondary dead load was minor and can be neglected. 3. Generally speaking, if the bridge structure was modeled through the s ingle line g irder method, the lanes are actually located on the center line, but their loads are multipl ied by the correct of fset load coefficient. If the bridge structure was simulated through the grillage method, then the lanes are located on the actual eccentric situation defined, giving more accuracy. 4. The most interesting and curious part of this analysis was the live loa d effect. T he live loads were applied on the bridge superstructure in two different ways The first was that two lanes were located on the center line without offset loads coefficient through SAP 2000 and the second way was that two lanes were defined on t he a ctual eccentric situation of 11.42 f ee t for the first lane and 3.58 f ee t for the second lane These two ways

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59 gave much different results for the transverse and longitudinal movement of the bearings, especially for the exterior ones S ubsequently max i mum torsion in the envelope changed greatly due to the lane offset even though all the other loads a ffecting the displacement and reaction values such as dead load, wearing surface and temperature variation, were not changed The movement reaction and max imum torsion values from the un factored live loads for both ways are compared in T able 1 1 T here are two possible reasons for this huge variation in results The first is that SAP 20 0 0 does not allow applying the offset load s coefficient when we plac e the live loads at the center A modified single girder m odel might solve this problem. T he second is that the live load location indeed has a great effect on the horizontal and longitudinal movement of the bearings and causes a large amount of torsion transferr ed to them as well

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60 Table 11 : Comparison between lanes located on center line and lanes located on actu al position

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61 F in ite element analysis of pot bearing A refined fin ite element model was developed using ANSYS based on the shop drawing s to investigate h ow much reaction, movement and rotation the current bearings are subjected to as well as determining how these problems were caused and how to make an optimal design for rehabili t ation to change this situation. The actual structure of the pot bearing from top to bo ttom usually is comprised of a top p late attach ed to the guide bar s a PTFE a piston, a neoprene, and a pot. Solid elements (8 node 185) were simulated for all the assemblies. T he dimensions and material propert ies of each layer are shown in T able 1 2 while F igure s 54 55 56 57 and 58 show how the pot bea ring is constructed, layer by layer. Table 12 : E lement types, material propert ies and dimension of fine bearings Young's modulus (kips/ft 2 ) Poisson's ratio Density (lb/ft 3 ) H eight (in) D iameter (in) Top plate 4.176E+0 6 0. 3 50 0 Guide bar 4.176E+0 6 0.3 500 1.5 TFPE 1.08 E+0 4 0. 46 138.24 0.125 9.91 Piston 4.176E+0 6 0.3 500 9.91 Neoprene 1.25E+04 0. 48 75 0.5 9.91 Pot 4.176E+0 6 0.3 500 1.5/1

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62 Figure 54 Guided bearing (full view) Figure 55 T op plate removal

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63 Figure 56 PTFE removal Figure 57 P iston removal

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64 Figure 58 N eoprene removal In this refin ed model, a standard contact relation was set up between the stainless steel and PTFE, and a coupling (z direction) was set up between the top plate and center point of PTFE. A bonded condition was defined between the bottom face of the PTFE and the top fa ce of the piston A coupling (z direction) was defined between the bottom face of the piston and the top face of the neoprene, and between the bottom face of the neoprene and the top face of the pot. A coupling (x direction and y direction) was set up betw een both the side of the piston and the surrounding pot, and the side of the neoprene and the surrounding pot Finally a standard contact condition was defined between the side of the gu ide bar and the side of the pot with the same coefficient of friction a s before T he difference here is that the value of 0.125 in ches for the initial contact surface offset should be provided between the guide bar and the pot so that the guide bar cannot penetrat e into the base plate when the top plate is subjected to hori zontal displacement.

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65 Typically we regard the concave, hard, large, and rough low order sur face as the target surface, while the conve x soft, small, smooth high order surface is regarded as the contact surface. A g uided bearing on Pier # 1 4 was used as an e xample to make a re fine structural analysis. The bearing s orientation and girder corresponding section s orientation can be calculated through the global coordinates of two points along the bearing s edge line or the girder center line as shown in T abl e 13 Table 13 : C oordinates for determining the bearing s orientation Node Numbers X (ft) Y(ft) Z(ft) Bearing 1546 1950.6 43.753 33.630 1549 1952.2 43.753 34.319 Girder 1540 1950 43.753 35.081 1541 1950.7 43.753 35.425 1542 1950.4 43.753 35.217 Another issue is how to transfer the correct horizontal and longitudinal movement, vertical pressure and torsion from SAP 2000 to ANSYS. Since the solid el ement in ANSYS software doesn t have rotational freedom, torsion is simulated by two equal and opposite reactions at the edge line s of the bearing with a force arm. Table 1 4 shows how

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66 the coordinates are transfer r ed from SAP 2000 to ANSYS. The a ctual force arm in ANSYS is : The longitudinal distance of two equal and opposite reactions is : And actual reaction is : Table 14 : C oordinates transfer from SAP 2000 to ANS YS H orizontal movement (in) Longitudinal movement (in) Vertical restraint (kip) Torsion (kip ft) Reaction (kip) SAP 2000 U 2 =1.5388 U 1 = 3.316 379.970 571.2149 358.5 A NSYS Z = 1.5388 X= 3.316 379.970 571.2149 355.53 Since Pier #14 is close to Pier #11 and they are al ong a straight line the bearing s orientation is almost the same as the girder s orientation. Thus, the reaction s variation is negligible While the bearings on Pier #14 are used as an example to show how to calculate a simu lated reaction for each bearing all the other bearings structural analysis will be taken in the same way. Contact is a complicated non linear problem and was considered in the modeling ; h owever, m aterial nonlinearity was not considered. F our key points of the top plate were defined with spring elements to avoid excess rigid displacement and stress concentration

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67 Coupling relationship w as used to simulate the co ntact pair to simp lif y the complicated contact problem. But the contact between the top plate and PTFE cannot be substituted since the outside top plate didn t have forces when it was subjected to slide. T he contact between the guide bar and the pot base cannot be substituted as well since the initial offset exi sts. V on mises stress es can be plotted after running the model for analytical results These values are shown in F igure s 59 60 6 1 6 2 63 and 64 The deformatio ns are plotted in F igure s 65 66 and 67 Figure 59 V on M ises stress ( s teel)

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68 Figure 60 V on M ises stress ( s teel II ) Figure 61 V on M ises stress ( PTFE )

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69 Figure 62 V on M ises stress ( p iston ) Figure 63 V on M ises stress ( n eoprene )

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70 Figure 64 V on M ises stress ( p ot ) Figure 65 Deformation in y direction ( s teel)

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71 Figure 66 Deformation in y direction ( PTFE ) Figure 67 Deformation in y direction ( n eoprene ) The following conclusion s from this analysis can be drawn: 1. Even though the simulated torque caused relative stress concentration, it ca n be seen that the mid part of the guide bar was subjected to a ve ry high value of stress

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72 T his is a n indication of why the guide bar fracture d on Pier #14 2. Since the couple between the top plate and the center point of PTFE was made for restraint to control rigid displacement, the stress concentration was generated on that point. 3. The linear spring element need s to be defined at the four key points of the top plate instead of vertical restraint to release the stress concentration, or most of the stress will be transferred to the support directly. 4. The piston and the pot deformed due to the horizontal movement of the top plate of the bearing. The top plate deformation in the y dir ection was mostly caused by the torque.

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73 C HAPTER V OPTIMAL DESIGN OF POT TYPE BEARING S Based on the results from this study, some pre liminary solutions to solve the bearing issues are proposed here along with possible studies that can be done for future research Bearing orientation improvement The bearing orientation is an important issue of the whole structure. For a curve d bridge, there are two common methods for bearing orientation. T he first is that bearings are oriented parallel to the tangent line of the curved line, which means that the bearing is orien t ed in the same direction as the girder direction. The second method is tha t the fixed bearings are placed perpendicular to the pier cap and all the other guided bearings an d non guided bearings are oriented along individual chord s towards the fixed bearing s The bearing orientation in the West 8 th Avenue Viaduct is in the chord d irection. For future studies we can assign different orientations for the bearing and perform structural analysis for each case separately. A relationship can be summarized between the curvature of the bridge and the bearing orientation and equations can be developed to determine the optimal bearing orientation that will result in minimum stress. Bearing material improvement The problem of PTFE peeling was observed The pe e ling not only increase s the coefficient of f r iction between the stainless steel and PTFE but also increase s the

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74 possibility of direct contact between the stainless steel and the piston The analysis results indicated that e xcessive edge loading may be the major rea son for the pe e ling W hen loads are applied along a continuous girder, th e support s of the girder will bear a large shear force that is transferred from the bearing to the substructure. Because the top plate of the bearing together with guide bars have longitudinal movement due to the curvature of the bridge, the top plate of t he bearing cannot transfer the load to the PTFE directly since they are not concentric compone nts. Thus, the load transferred to the PTFE edge become s much larger and exceed s the edge contact stress, which causes peeling or grind ing Worst of all if direc t contact occurs between the stainless steel and the piston it would be very likely to cause steel corrosion due to moist ure penetration, result ing in sli ding expansion bearings behaving as fixed bearing s and causing much more forces to the substructures For rehabilitation consideration we can extend the PTFE s shape to a larger ellipse over the base pot of the bearing, which can relieve the PTFE s heavy grind, or we can substitute it with another kind of polymeric material that has better a brasion r esis tance tha n the current PTFE. In the future research, different contact pair s friction can be defined to simulate different polymeric material s in the bearing model to find the most suitable one

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75 Table 15 : Maximum contact stress of PTFE at the Service Limit State (ksi) ( f rom[1]) M aterial Average Contact Stress (ksi) Edge Contact Stress (ksi) Perma nent Loads All L oads Perma nent Loads All L oads Unconfined PTFE: Unfilled sheets Filled sheets with Maximum Filler Content 1.5 2.0 2.5 4.5 2.0 3.5 3.0 5.5 Confined Sheet PTFE 3.0 4.5 3.5 5.5 Woven PTFE Fiber over a Metallic Substrate 3.0 4.5 3.5 5.5 Reinforced Woven PTFE over a Metallic Substrate 4.0 5.5 4.5 7.0 Bearing structure improve ment For the current bearing structure in the West 8 th Avenue Viaduct, steel grind and guide bar fracture problems were caused by an un suitable bearing structure used on the curved bridge. For curved bridge structures, the theoretical direction of m ovement is a chord projected from the midpoint of the deck at the expansion bearing line to the midpoint of the deck at a fixed bearing line. However, a nalysis results indicate d that the expansion bearings are also subjected to transverse movement together with longitudinal movement T he original guide bars cannot provide enough rotation and come in contact with the bas e plate due to the transverse movement. Since spherical bearing s are much more expensive than pot bearin gs and elastomeric bearings, two new kinds of pot bearings (an e dge guided pot bearing with a slider plate and a c enter guided pot bearing ) may solve this problem Edge guided pot bearing with slider plate Figure s 68 69 and 7 0 show a model for an edge guided pot bearing with a slider

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76 pl ate. The re are two major difference s between this new type of bearing and the original bearing i n the West 8 th Avenue Viaduct. The first is that the depth of the guide bars becomes shorter so that the guide bars cannot hit the base pot wh en it is subjected to transverse movement. The second difference is that there is a new slider plate between the PTFE and the piston, and the PTFE is bonded to this plate Since the slider plate is assigned on the top of the piston and the contact between them is only throu gh steel steel friction, the top plate together with the guide bars can bear a certain amount of transverse movement along the slider plate. Since the slider plate is a square or a rectangle and its side length is larger than the d iameter of the piston, th is new type of bearing can be subjected to more rotation than the original bearing. Figure 68 Lateral view of edge guided pot bearing with slider plate

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77 Figure 69 PTFE bonded with slider plate Figure 70 P ot base together with neoprene and piston Center guided pot bearing The center guided system consists of a single guide bar along the center line of the bearing assembly as shown in F igure s 7 1 and 7 2 T he top plate of the be aring has a

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78 corresponding gr oove for the guide bar. The biggest benefit of the center guided pot bearing is that the top plate of the bearing can bear a relatively large in plane rotation because the guide bar together with the piston can rotate in the bas e pot. However, the load is transferred from the superstructure to the substructure through the center guide b ar. This not only cause s heavy stress concentration, but also the vertical and horizontal load the bearing can be subjected to is limited. Fig ure 71 F ull view of center guided pot bearing

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79 Figure 72 Guide bar on the center of pot base

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80 C HAPTER VI CONCLUSIONS Following are the conclusions of this research. 1. The b earing finite element model indi cated that the mid part of the guid e bar was subjected to much higher str ess, which is the reason for the fracture of the guide bar on Pier #14 2. Since the couple between the top plate and the center point of the PTFE was made for restraint to control ri gid displacement, the stress concentration was generated on that point. 3. The piston and the pot were deformed due to the horizontal movement of the top plate of the bearing, while t he top plate deformation in the y direction was mostly caused through torqu e. 4. For the West 8 th Avenue Viaduct, since the theoretical direction of movement is a chord line from the midpoint of the deck at the expansion bearing line to the mid point of the deck at the fixed bearing line, the fixed bearings are assigned perpendicula r to the corresponding pier cap an d all the other guided and non guided bearings are oriented along the ir individual chord s towards the fixed bearing s 5. E xcessive edge loading is the major reason for PTFE peeling. The top plate of the bearings together wit h the guide bars have longitudinal movement due to t he curvature of the bridge, but the top plate cannot transfer its load to the PTFE directly since they are not concentric components. Thus, the load transferring to the PTFE edge

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81 become s much larger and e xceed s edge contact stress, which causes peeling or grind ing There is a possibility of direct contact between the stainless steel and the piston which would cause sli ding expansion bearings not being able to slide and generating much larger stress. 6. In be aring rehabilitation we can extend the PTFE s shape to a larger ellipse over the base pot of the bearing, which can relieve the PTFE s heavy grind, or we can substitute another kind of polymeric material with better a brasion r esistance than the curre nt P TFE. In future research, we can define different values for the contact pair s friction to simulate different polymeric material in the fine bearing model to find the most suitable one 7. For the current bearing structure in the West 8 th Avenue Viaduct, steel grind and even guide bar fracture problems are due to a n un suitable bearing stru cture being used for this curved bridge. The original guide bars cannot provide enough rotation and are too deep so the guide bar becomes in contact wi th the ba se pot when there is a large enough transverse movement. 8. Either an edge guided pot bearing or a center guide bearing can be used to replace the existing bearing s i n the West 8 th Avenue Viaduct Recommendations The following rec ommendations are proposed for future research. 1. The modified single girder method can be conducted for the simplified model with the actual location of two lanes. It can check whether the live load really does have a

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82 large effect on the transverse and longitudinal movemen t of bearings and results in large torsion being transferred to the bearings as well. 2. Since the ANSYS m odel movements on bearings w ere determined through the simplified model using SAP 2000 Once the thr ee dimensional ANSYS model is complete, the bridge can be analyzed with all the loads to get much more accurate results. 3. Only Pier #14 was analyzed in this study However, the structural analysis of all the other bearings will be performed in the same wa y as the guided bearing on Pier #14. Additional analytical results of other bearings can provide more information about how the curvature and thermal variation will affect the transverse and longitudinal movement s of the bearings. 4. Several solutions need to be p roposed for rehabilitation of the bearing with both qualitative analysis and q uantitative discussion All the se solutions, such as the orientation, materials and new structures, need to be structural ly analyzed and compared with scientific data to m a ke sure that they can fix the bearing problems.

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83 REFERENCES [1] AASHTO LRFD bridge design specif ication 6 th ed ition The American Association of State Highway and Transportation Officials Washington, D C .,2012 [2] Ahmet Yakut, Joseph A. Yura.(2002). Temperature Test Methods Journal of Bridge Engineering, 7(1). 50 56. [3] Ahmet Yakut, Joseph A. Yura. (2002). Journal of S tructural Engineering, 128(8).986 994. [4] Low Temperature Journal of Cold Regions Engineering, 4(3).113 132. [5] Chen xi XING, Hao WANG, Ai qun LI, Ji rong WU( 2012) verification of a new multi J ournal of Zhejiang Univ Sci A (Appl Phys & Eng) 13(12). 904 914 [6]City and County of Denver, Department of Public Works, Design Engineering Divis ion,(1993), Project No.B83 050 West 8 th Avenue Viaduct Between Mariposa and Vallejo Streets Denver, Colorado [7] Thermal Actions for Concrete Bridge J ournal of Structural Engineering,119(8).2313 2331. [8] Ioannis V. Kalpakidis, Michael C. Constantinou, Andrew S. Whittaker.(2010). of Large Cumulative Travel on the Behavior of Lead Journal of Structural Engineering,136(5).491 501. [9]Jacob Fish, Ted Belytschk o, A First Course in Finite Elements John Wiley&Sons Ltd, The Atrium, Southern Gate, Chichester, England,2007 [10] Jhon A. Henao Sepulveda, Manuel Toledo Quiones, and Yi Jia. (2005, May). Instrumentatio n and Measurement Technology Conference, Canada. [11] Junji Yoshida, Masato Abe, Yozo Fujino, Hiroshi Watanabe.(2004). Dimensional Finite Element Analysis of High Damping Rubber

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84 Journal of Engineering Mechanics,130(5).607 620. [12] M. Imbim bo, A. De Luca.(1998). Computers & Structures,68.31 39. [13] Pentas, Herodotos A, Avent, R. Richard, Gopu, Vijaya, K. A, and Rebello, Keith J., (1994) "Field Study of Bridge Temperatures in Compos ite Bridges," Transportation Research Record 1460. [14] Pentas, Herodotos A, Avent, R. Richard, Gopu, Vijaya, K. A, and Rebello, Keith J., (1995) "Field Study of Longitudinal Movements in Composite Bridges," Transportation Research Record 1476. [15] Shashi Moorty, Charles W. Roeder.(1991). Dependent Bridge Journal of Structural Engineer ing,118(4).1090 1105. [16] T.I.Campbell, M.J.Fatemi, D.G.Manning.(1993). Journal of Structural Engineering,119(11).3169 3177 [17] Troy T. Tindal and Chai H. Yoo.(1998). Bridges and Including Bearing Orientation, American Iron and Steel Institute [18] Troy T. Tindal and Chai H. Yoo.(2003 Thermal Effects on Skewed Steel Highway Journal of Bridge Engineering,8(1).57 65. [19] Will, K. M., Johnson, C. P., and Matlock, H. (1977) "Analytical and Experimental Investigation of the Thermal Response of Highway B ridges," Research Report 23 2, Center for Highway Research, University of Texas at Austin [20] Yongda Fu, John T. DeWolf.(2001). Journal of Bridge Engineering,6(1).23 29. [21] Yong X i a Bo Chen, Xiao qing Zhou, You Field monitoring and Structural Control And Health Monitoring, 20.560 575 [22] Yoshito ITOH, Haosheng GU, Kazuya SATOH, Yoshihisa YAMAMO TO,(2006). Term Deterioration of High Damping Rubber Bridge Structural Eng./Earthquake Eng, 23(2).215 227.

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85 [23] Yoshito ITOH, Haosheng GU, Kazuya SATOH, Yukihiro KUTSUNA,(2006). Structural Eng./Earthquake Eng, 23(1).17 31.

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86 APPENDIX A A NSYS program !!!!!!!!!!!!!!!!!!West 8th Avenue Viaduct Stress and Movement Study !!!!!!!!!By Lisa Wang !!!!!!!!!Unit lb & ft FINISH /CLEAR /TITLE,STRUCTURAL ANALYSIS OF FINE BEARING MODEL *AFUN,DEG /NERR,1e6,1e6 /vup, 1,z !!!!!!! input up_steel vsel,none fini /aux15 ~satin,up_steel,sat,model,solids Fini /PREP7 /facet vptn,all cm,up_steel,volu !!!!!!! input pot vs el,none fini /aux15 ~satin,pot,sat,model,solids Fini /PREP7 /facet vptn,all cm,pot,volu !!!!!!! input tfpe

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87 vsel,none fini /aux15 ~satin,tfpe,sat,model,solids Fini /PREP7 /facet cm,tfpe,volu !!!!!!! input piston vsel,none fini /aux15 ~satin,piston,sat ,model,solids Fini /PREP7 /facet vptn,all cm,piston,volu !!!!!!! input neop vsel,none fini /aux15 ~satin,neop,sat,model,solids Fini /PREP7 /facet vptn,all cm,neop,volu MPTEMP,,,,,,,, MPTEMP,1,0 ET,3,SOLID185,,,,,,,, !Steel properties MPDATA,EX,1,,4. 176e9

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88 MPDATA,PRXY,1,,0.3 MPDATA,DENS,1,,500 !Neoprene properties MPDATA,EX,2,,1.25e7 MPDATA,PRXY,2,,0.48 MPDATA,DENS,2,,75 !TFPE properties MPDATA,EX,3,,1.08e7 MPDATA,PRXY,3,,0.46 MPDATA,DENS,3,,138.24 cmsel,s,up_steel cmsel,a,pot cmsel,a,piston vatt, 1,,3 esize,0.1 vsweep,all cmsel,s,tfpe vatt,3,,3 esize,0.1 vsweep,all cmsel,s,neop vatt,2,,3 esize,0.03 vsweep,all bbaseplate_dep=1.5/12 bbaseplate_wid=11.625/12 bbaseplate_leng=11.625/12 btopplate_leng=28.5/12 btopplate_wid=19.125/12 btopplate_dep=(3. 39 2.08)/12 btot_dep=3.39/12 guidebar_wid=1.5/12

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89 guidebar_dep=1.5/12 guidebar_leng=28.5/12 gap=0.125/12 pot_ID=9.91/12 bpot_dep=1/12 neopnene_dep=0.5/12 PTFE_dep=0.125/12 piston_dep= (btot_dep btopplate_dep) (bbaseplate_dep bpot_dep ) neopnene_dep PTF E_dep !!!!!!!!!!CONTACT PAIR local,18,0,,, 23.3 wpcsys, 1,0 !csys,4 csys,18 ET,4,CONTA173,,0 ET,5,TARGE170 MP,MU,10,0 FKN=0.1 R,10,,,FKN cmsel,s,up_steel eslv aslv asel,s,,,89 asel,a,,,196 nsla,s,1 TYPE,5 MAT,10 REAL,10 ESURF cmsel,s,tfpe eslv !aslv a sel,s,,,266 !asel,r,loc,z,btot_dep btopplate_dep nsla,s,1

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90 TYPE,4 MAT,10 REAL,10 ESURF ET,7,CONTA173,,0 !!!!!CONTACT OFFSET KEYOPT,7,5,1 R,20,,,FKN cmsel,s,pot eslv aslv asel,s,,,230 asel,a,,,18 nsla,s,1 TYPE,7 MAT,10 REAL,20 ESURF cmsel,s,up_steel eslv aslv asel,s,,,167 asel,a,,,7 nsla,s,1 TYPE,5 MAT,10 REAL,20 ESURF R,30,,,FKN cmsel,s,pot eslv aslv asel,s,,,233 asel,a,,,21 nsla,s,1 TYPE,7 MAT,10 REAL,30

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91 ESURF cmsel,s,up_steel eslv aslv asel,s,,,148 asel,a,,,4 nsla,s,1 TYPE,5 MAT,10 REAL,30 ESURF ET ,6,CONTA173,,0 !!!!!BONDED KEYOPT,6,12,3 R,40,,,FKN cmsel,s,piston eslv aslv asel,s,,,274 !asel,r,loc,z,btot_dep btopplate_dep ptfe_dep nsla,s,1 TYPE,5 MAT,10 REAL,40 ESURF cmsel,s,tfpe eslv aslv asel,s,,,265 !asel,r,loc,z,btot_dep btopplate_dep ptfe_d ep nsla,s,1 TYPE,6 MAT,10 REAL,40 ESURF R,50,,,0.1 cmsel,s,pot eslv

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92 aslv asel,s,,,227 asel,a,,,231 nsla,s,1 TYPE,5 MAT,10 REAL,50 ESURF cmsel,s,piston eslv aslv asel,s,,,267 asel,a,,,269 nsla,s,1 TYPE,4 MAT,10 REAL,50 ESURF wpcsys, 1,0 csys,4 cmsel,s,p ot eslv nsle,s,1 nsel,r,loc,z, 0.001,0.001 d,all,ux,,,,,uy,uz !/eof cmsel,s,neop aslv lsla nslv,s,1 nsel,r,loc,z,0.0833,0.0834 cmsel,s,piston eslv ceintf,,uz cmsel,s,neop aslv lsla

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93 nslv,s,1 nsel,r,loc,z,0.04167,0.04168 cmsel,s,pot eslv ceintf,,uz wpcsys 1,1 csys,1 cmsel,s,neop aslv lsla nslv,s,1 nsel,r,loc,x,pot_id/2 0.001,pot_id/2+0.001 cmsel,s,pot eslv ceintf,,ux,uy cmsel,s,tfpe aslv lsla nsel,s,,,2870 cmsel,s,up_steel eslv ceintf,,all et,10,combin39 r,100, 1e5, 1,0,0,0.1,1e4 a1=nx(1995) b1=ny(19 95) c1=nz(1995) *get,nnum,node,,num,maxd nsel,none n,nnum+1,a1,b1,c1 ntemp=nnum+1 nsel,a,node,,1995 type,10 real,100 e,1995,ntemp

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94 nsel,s,node,,ntemp d,all,uz allsel a1=nx(2049) b1=ny(2049) c1=nz(2049) *get,nnum,node,,num,maxd nsel,none n,nnum+1,a1,b1,c1 n temp=nnum+1 nsel,a,node,,2049 type,10 real,100 e,2049,ntemp nsel,s,node,,ntemp d,all,uz allsel a1=nx(1136) b1=ny(1136) c1=nz(1136) *get,nnum,node,,num,maxd nsel,none n,nnum+1,a1,b1,c1 ntemp=nnum+1 nsel,a,node,,1136 type,10 real,100 e,1136,ntemp nsel,s,nod e,,ntemp d,all,uz allsel a1=nx(2218) b1=ny(2218) c1=nz(2218) *get,nnum,node,,num,maxd nsel,none n,nnum+1,a1,b1,c1 ntemp=nnum+1

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95 nsel,a,node,,2218 type,10 real,100 e,2218,ntemp nsel,s,node,,ntemp d,all,uz cmsel,s,neop eslv nsle wpcsys, 1,0 csys,4 nsel,r,lo c,z,0,0.04168 cmsel,s,pot eslv ceintf,,uz !!!!!!LOAD HMOV= 1.5388/12 LMOV= 3.316/12 T_FORCE=358500 FORCE=2893257 lsel,s,,,156 lsel,a,,,158 lsel,a,,,371 lsel,a,,,372 lsel,a,,,374 lsel,a,,,376 DL,ALL,,Ux,HMOV lsel,s,,,373 lsel,a,,,56 lsel,a,,,299 lsel,a, ,,99 lsel,a,,,60 lsel,a,,,56 lsel,a,,,379 lsel,a,,,141 lsel,a,,,93

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96 DL,ALL,,Uy,LMOV cmsel,s,up_steel aslv asel,r,loc,z,0.283 !SFA,all,,PRES,1087690 SFA,all,,PRES,142850 nsel,all F,1898,Fz, T_FORCE F,1788,Fz, T_FORCE !/eof save,prosolve,db !!!!!!LOAD ST EP /SOLU TIME,1 ANTYPE,STATIC NSUBST,50 AUTOS,ON nropt,full,,on NEQIT,25 LNSRCH,ON PRED,ON,,ON OUTPR,ALL,ALL OUTRES,ALL,ALL !EQSLV,SPARSE,1E 5 ALLSEL SOLVE

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97 APPENDIX B T heory and background The finite element method (FEM) is a numerical approach to s olve partial differential equations approximately instead of classical analytical methods for arbitrary shapes It can solv e practical problems such as stress analysis, heat transfer, and electromagnetics through computer simulation. (Fish & Belytschko, 20 07) Finite element thermal analysis ( one dimension al transient hea t conduction) Figure 73 C hange in energy Change in Energy=Incoming Flux Outgoing Flux + source of heat in time as shown in F igure 73 EQUATION 11 Divide by dx

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98 Where dU is the change in energy in time Strong form EQUATIO N 12 B.C. I.C. Weak form: multiply with w and integrate from 0 to l, and apply integration by parts for B.C. The final weak form is as follows; EQUATION 13 Finite element discretization

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99 EQUATION 14 Where Finite element structure ana lysis (two dimensional elasticity) The basic assumptions are as follows: 1. A ll types of deformation are limited to the linear elastic range 2. E ach triangle or square element after mesh generation only has small displacement and small rotation Strain in two dimensions

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100 Figure 74 S train in two dimension s Shear strain Normal strain Normal Figure 74 shows that

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101 Str ess definition Figure 75 S tress definition Stress tension is as shown F igure 75 where: i -N ormal direction of the surface where force is applied j -D irection of force T o verify moment equation, we have T he stress can be represented by a vector For plane strain

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102 EQUATION 15 Where E If there is no relation bet w een the force in one direction and the displacement in the others, and A more concise way to write this Equilibrium principle of virtual work Apply a small displacement field and look at the change of internal ene r gy and weak form external force Internal work If we have External work

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103 Where body force; surface tractions Then weak form is as follows Discretizatio n and interpolation in two dimension Figure 76 D iscretiz a tion with triangles and quadrilaterals Figure 76 shows that e ach shape is charact er rized by : Number of elements Number of nodes that belong to element

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104 Connectivity (c ounter clockwise) Coordinate of nodes Element# Node1 Node2 Node3 8 1 3 2 ... ... ... ... We need to solve for the nodal displacements (2d DOF ) To discretize the weak form ---use interpolation This will be done with shape function s Node# x y 1 ... n

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105 Figure 77 M opping between physical e lement and parent element Shape function We need to derive 3 shape functions as shown in figure 77 , In the s ame way In element e

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106 EQUATION 16 Ma pping between physical and parent element Figure 78 C orresponding coordinates of physical element and parent element Map ping with shape fu n ction in figure 78 Computing derivatives of displacement Reminder

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107 we know that EQUATION 17 The Jacobi an of the ma pping will be EQUATION 18 Then we have

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108 Computing the [B] matrix Reminder of weak form Where Figure 79 D iscretization into elements and nodes The weak form following figure 79 is rewritten EQUATION 19

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109 Consider the integral In the parent element, we can write Where H ere j is the ratio between physical and parent element T he weak form becomes Interpolations Similarly T hen weak form becomes Interpolations

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110 EQUATION 20 Where Th en The final weak form is EQUATION 21 Proceed to assembly

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111 Where