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NUMERICAL INVESTIGATION S OF FRP PRESTRESSED BRIDGE GIRDERS By RAYMON WILLIAM NICKLE B.S., Colorado State University, 2009 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Civil Engineering 2015
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ii This thesis for the Master of Science degree by Raymon William Nickle has been approved for the Civil Engin eering Program by Nien Yin Chang Chair Yail Jimmy Kim, Advisor Kevin Rens April 23, 2015
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iii N ickle, Raymon William (M.S., Civil Engineering) Numerical Calibration of FRP Prestressed Girders for Structural Reliability Thesis directed by Associate Professor Yail Jimmy Kim ABSTRACT This t hesis addresses the calibration methodology for Load Factor and Resistance Design (LRFD) flexural resistance factors for the use of fiber reinforced polymers (FRP) as a concrete prestress ing material and estimated long term deflection multipliers for FRP prestressed girders Existing bridges, utilizing precast prestressed construction with a range of girder types, spans, and spacing were redesigned using Aramid (AFRP) and Carbon (CFRP) pre stressing tendons. Monte Carlo Simulations were utilized based on the trail designs, to provide reliability based statistical distributions for structure capacity using the recommended design methodology from the American Concrete Institute (ACI) Report 4 40.4R 04. Capacity distributions were used in two independent calibration procedures for resistance factors. R esistance factors were calibrated for target reliability ind ices of 2.5, 3.0 and 3.5; for compression controlled and tension controlled failure ty pes. Recommended resistance factors were found to be dependent on failure type considered for the beam and not FRP material Recommended long term deflection multipliers of FRP prestressed girders were provided for AFRP and CFRP prestressed girders for com posite and non composite construction. The Form and content of this abstract are approved. I recommend its publication. Approved: Y. Jimmy Kim
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iv TABLE OF CONTENTS List of Tables ................................ ................................ ................................ ................................ ................... viii List of Figures ................................ ................................ ................................ ................................ ..................... x List of Abbreviations ................................ ................................ ................................ ................................ ..... xi Chapter 1 Introduction ................................ ................................ ................................ ................................ ................. 1 1.1 Overview ................................ ................................ ................................ ................................ ............. 1 1.2 Research Significance ................................ ................................ ................................ .................... 2 1.3 Objective ................................ ................................ ................................ ................................ .............. 3 1.4 Scope ................................ ................................ ................................ ................................ ..................... 4 1.5 Thesis Outline ................................ ................................ ................................ ................................ ... 5 2 Literary Review ................................ ................................ ................................ ................................ .......... 7 2.1 Overvi ew ................................ ................................ ................................ ................................ ............. 7 2.2 FRP Prestressed Concrete ................................ ................................ ................................ ............ 7 2.2.1 FRP Prestressing Materials ................................ ................................ ................................ 8 2.2.2 FRP Material Strength ................................ ................................ ................................ .......... 9 2. 2.3 Advantages FRP Reinforcement ................................ ................................ .................... 10 2.2.4 Disadvantages FRP Reinforcement ................................ ................................ .............. 11 2.2.5 Existing Structures and Trial Applications ................................ ................................ 13 2.3 Advancement of Structural Design Philosophy ................................ ................................ 15 2.4 Probability in Structural Engineering ................................ ................................ ................... 17 2.4.1 Uncertainty in Design ................................ ................................ ................................ ......... 17
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v 2.4.2 Basic Reliability ................................ ................................ ................................ .................... 19 2.4.3 Probability of Failure ................................ ................................ ................................ .......... 21 2.5 Reliability Approach for this Report ................................ ................................ ...................... 24 2.6 Existing FRP Prestressing Resistance Factors ................................ ................................ ... 25 3 Capacity Modeling ................................ ................................ ................................ ................................ ... 29 3.1 Overview ................................ ................................ ................................ ................................ ........... 2 9 3.2 Simulation P rocedure ................................ ................................ ................................ .................. 29 3.3 Trial Spans ................................ ................................ ................................ ................................ ........ 31 3.3.1 Span Selection ................................ ................................ ................................ ....................... 31 3.3.2 Trial Design ................................ ................................ ................................ ............................ 33 3.4 Resist ance Model ................................ ................................ ................................ ........................... 38 3.5 Capacity Prediction ................................ ................................ ................................ ....................... 39 3.5.1 Balanced Ratio ................................ ................................ ................................ ....................... 41 3.5.2 Tension controlled Section ................................ ................................ .............................. 41 3.5.3 Compression controlled Section ................................ ................................ ................... 42 3.6 Random Variable Parameters ................................ ................................ ................................ .. 44 3.6.1 Material and Geometry Variables ................................ ................................ .................. 44 3.6.2 Calculation of AFRP and CFRP Parameters ................................ ............................... 46 3.6.3 Prestress Losses ................................ ................................ ................................ ................... 50 3.7 Distribution of Resistance ................................ ................................ ................................ .......... 52 3.8 Analysis Parameters ................................ ................................ ................................ ..................... 53
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vi 3.8.1 Existing Analysis Parameter ................................ ................................ ............................ 54 3.9 Calibration Parameters ................................ ................................ ................................ ............... 56 4 Calibration ................................ ................................ ................................ ................................ .................. 58 4.1 Overview ................................ ................................ ................................ ................................ ........... 58 4.2 Load Model ................................ ................................ ................................ ................................ ....... 58 4.2.1 Dead Load ................................ ................................ ................................ ................................ 59 4.2.2 Live Load ................................ ................................ ................................ ................................ 60 4.3 Target Reliability Calibration ................................ ................................ ................................ ... 61 4.1.1 Reliability Index ................................ ................................ ................................ .................... 62 4.1. 2 Verification of the Reliability Index Calculations ................................ ................... 64 4.1.3 Target Reliability Index ................................ ................................ ................................ ..... 66 4.1.4 Target Reliability Results ................................ ................................ ................................ .. 67 4.4 Direct Calculation Method ................................ ................................ ................................ ......... 78 4.5 Recommended Resistance Factors ................................ ................................ ......................... 80 5 Long Term Deflection Factors ................................ ................................ ................................ ............ 83 5.1 Overview ................................ ................................ ................................ ................................ ........... 83 5.2 Design Applicability ................................ ................................ ................................ ...................... 83 5.3 Simulation Methodology ................................ ................................ ................................ ............. 84 5.4 Determination of Multipliers ................................ ................................ ................................ .... 85 5.5 Random Variables ................................ ................................ ................................ ......................... 89 5.6 Recommended Factors ................................ ................................ ................................ ................ 90
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vii 6 Conclusions, recommendations and future research ................................ ............................... 92 6.1 Conclusions ................................ ................................ ................................ ................................ ...... 92 6.2 Future Research ................................ ................................ ................................ ............................. 92 6.2.1 Material Properties and Reporting ................................ ................................ ............... 93 6.2.2 Analysis Factors ................................ ................................ ................................ .................... 93 6.2.3 Serviceability ................................ ................................ ................................ ......................... 93 6.2.4 FRP Relaxation Losses ................................ ................................ ................................ ....... 94 6.2.5 FRP Harping and Debonding ................................ ................................ ........................... 94 6.2.6 Selecting Target Reliability Indices ................................ ................................ .............. 94 6.2.7 Prestressed Slabs ................................ ................................ ................................ ................. 95 6.3 Conclusion ................................ ................................ ................................ ................................ ........ 95 Notati ons ................................ ................................ ................................ ................................ ........................... 96 Bibliography ................................ ................................ ................................ ................................ ..................... 98 Appendix A Calculated Reliability Indices for Varying Resistance Factors ................................ ......... 101 B Sample VBA code for reliability index calculations ................................ ............................... 113 C Sample Span General Layouts and Typical Sections ................................ ............................. 123
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viii LIST OF TABLES Table 2 1: Material properties for Common CFRP Materials ................................ ................................ ..... 8 2 2: Material properties for Common AFRP Mater ials ................................ ................................ .... 9 2 3: Harping stresses in FRP tendons at varying Saddle Radius ................................ ................ 12 2 4: Probability of failure based on Reliability Index ................................ ................................ ..... 23 2 5: Existing Strength Reduction Factors (Recreated from ACI 440.4) ................................ .. 26 2 6: Comparison of All beams, recreated from Burke and Dolan, 2001 ................................ 27 2 7: Comparison of A FRP beams, recreated from Burke and Dolan, 2001 ............................ 27 2 8: Comparison of CFRP beams, recreated from Burke and Dolan, 2001 ............................ 27 3 1: Sample Span Layout Properties ................................ ................................ ................................ ...... 32 3 2: Tendon Properties for Trial Spans ................................ ................................ ................................ 33 3 3: Allowable tendon Jacking Stresses (ACI 440.4) ................................ ................................ ....... 34 3 4: Allowable Concrete Stresses (ACI 440.4) ................................ ................................ ................... 35 3 5: AFRP Trial Design Values for Capacity Simulation ................................ ................................ 36 3 6: CFRP Trial Design Values for Capacity Simulation ................................ ................................ 37 3 7: Flexural Capacity Equations for FRP Prestressed Girders ................................ .................. 40 3 8: Geometry Random Variable Parameters ................................ ................................ .................... 45 3 9: Concrete Random Variable Parameters ................................ ................................ ...................... 46 3 10: AFRP Sample sets for COV Calculation ................................ ................................ ...................... 47 3 11: CFRP Sample sets for COV Calculation ................................ ................................ ...................... 47 3 12: FRP Random Variable Parameter ................................ ................................ ............................... 48 3 13: Comparison of Reliability Index with varying AFRP COV ................................ ................. 49 3 14: Random Variable Loss Parameters ................................ ................................ ............................ 51 3 15: Best Fit Comparison for Resistance Distribution ................................ ................................ 53
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ix 3 16: Analysis Factors Considered for Resistance Model ................................ ............................. 54 3 17: FRP Random Variable by Girder Type ................................ ................................ ...................... 56 3 18: Resistance Random Variable Including Analysis Factors ................................ .............. 57 4 1: Load Factors for Calibration ................................ ................................ ................................ ............ 59 4 2: Dead Load and Live Load Random Variable Parameters ................................ ..................... 60 4 3: Live Load bias factors by Span Length for 75 year live loads ................................ ............ 61 4 4: Statistical Parameters for Steel Reinforced Sections ................................ ............................. 64 4 5: Reliability Indices based on Resistance Factors from the LRFD Code. ........................... 65 4 6: Calibrated Resistan T = 3.5 ................................ ................................ ................ 75 4 T = 3.0 ................................ ................................ ................ 76 4 T = 2.5 ................................ ................................ ................ 77 4 9: Direct calculated Resistance factors using k = 2.0 ................................ ................................ 79 4 10: Recommended Strength Reduction Factors ................................ ................................ ........... 82 5 1: Steel Prestressed Beam Recommended Deflection Multipliers ................................ ........ 84 5 2: Deflection Multipliers Variable Summary ................................ ................................ .................. 88 5 3: Random Variable Factors for Deflection Simulation ................................ ............................. 89 5 4: Recommended Long Term Multipliers for AFRP prestres sed Girders ........................... 90 5 5: Recommended Long Term Multipliers for CFRP prestressed Girders ........................... 91
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x LIST OF FIGURES Figure 2 1: Stress Strain relationship between various prestressing materials ............................ 11 2 = 0.1, = 0.2, and = 0.3 .............................. 20 2 log = 1.0, and = 0.1, = 0.2, and = 0.3 ............................. 20 2 4: Normally distributed load and resistance functions ................................ .............................. 22 2 5: Failure limit state distribution function ................................ ................................ ...................... 23 2 6: Calculated reliability index vs. Sample size ................................ ................................ ............... 25 3 1: Probability Distribution of Resistance (Sample span 8, compression controlled) ... 52 4 1: Calculated Reliability Index vs. Span length, Steel Reinforced Concrete ....................... 66 4 T = 4.0 ................................ ............. 67 4 T = 3.5 .............................. 68 4 T = 3.0 .............................. 68 4 T = 2.5 .............................. 69 4 T = 3.5 ........................... 69 4 T = 3.0 ........................... 70 4 T = 2.5 ........................... 70 4 9: S T = 3.5 ........................... 71 4 10: Span 25 Calibration Curves for Resistance Factor Sel T = 3.0 ......................... 71 4 T = 2.5 ......................... 72 4 T = 3.5 ........................... 73 4 T = 3.0 ........................... 73 4 14: T = 2.5 ........................... 74 4 15: Calculated Resistance factors vs. Span length ................................ ................................ ........ 80 4 16: Calculated Resistance factor vs Span/Depth ratio ................................ ............................... 81
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xi LIST OF ABBREVIATIONS AASHTO American Association of State Highway Transportation Officials AASHTO LRFD AASHTO LRFD Bridge Design Specifications AFRP Aramid Fiber Reinforcing Polymers AREMA American Railway, Engineer, and Maintenance Association ASD Allowable Stress Design CFRP Carbon Fiber Reinforcing Polymers FOSM First Order Second Moment FRP Fiber Reinforced Polymers GFRP Glass Fiber Reinforcing Polymers JCSS Probabilistic Model Code LFD Load Factor Design LRFD Load and Resistance Factor Design MCS Monte Carlo Simulation PCI Precast/Prestressed Concrete Institute RV Random Vari able FRP Guaranteed Tensile Strength Strength of FRP bars or tendons, described as the mean less 3 standard deviations. FRP Design Strength See Guaranteed Tensile Strength ksi Kips per square inch ft feet in inch FS Factor of Safety FVR Fiber Volume Ratio
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1 1 I ntroduction 1.1 Overview In recent years the desire to use high performance materials in structural applications has increased. Fiber reinforced polymer ( FRP ) materials are manmade composites, cons isting of aramid, glass, or carbon fibers within a binding resin, typically epoxy or polyester based. The use of FRP is desirable due to its corrosive, non (ACI 440.4R, 2004) FRP materials can be used for a variety of structural application, but have shown significant promise as structural reinforcement for concrete elements, as a direct replacement for traditional metallic m aterials FRP materials as concr ete reinforcement can be used fo r a variety of applications. Most commonly, FRP composites are used as an external wrap to strengthen or repair existing columns or beams. Less often, FRP is employed as a mild reinforcement alternative in new construction, taking advantage of non conductive and non corrosive properties. Recent developments have seen FRP materials applied as a prestressing alternative, fully utilizing the high tensile strengths. Currently usage of FRP materi als is growing and are considered commonplace for strengthening applications. However, for new construction, the lack of consistent material properties, the lack of support in common design codes, and a limited understanding of the performance, both long term and short term, has resulted in some reluctance within the industry to utilize FRP material s except in extreme cases.
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2 1.2 Research Significance The concept of using fiberous materials as reinforcement in concrete was introduced in the 1930s. However, it was not until 1978 when material technology caught up to theory, and German contractor Strabag Bau and chemical producer Bayer introduced the first glass fiber reinforced polymer (GFRP) tendon and anchor system for use in prestressed concrete Over the next 12 year s the partnership between Stabag Bau and Bayer installed the product in numerous bridges across Germany and Austria, by 1990, installation of the material ceased. In the mid 1980s, AKZO and HBG a Dutch chemical producer and contrac tor respectively developed the first aramid fiber reinforcing polymer (AFRP) prestressing tendon. At the same time the Japanese were researching FRP production methods and applications in concrete structures. It was not until the late 1980 s that extensiv e research, funded by the Florida Department of Transportation, developed new prestressing anchorage systems for the use in bridge and marine environments, which resulted in the construction of the first FRP prestressed bridge in Rapid City, South Dakota. The first major breakthrough in carbon fiber reinforced polymers (CFRP) for bridges came with the construction of the Beddington Trail Bridge in Calgary Alberta in 1992 and the subsequent 1996 release by ACI Committee of the Art Report on Fibe r Reinforced Plastic Reinforcement for Concrete the Bridge Street Bridge in Sou thfield, Michigan was completed. T his structure represents the current benchmark for CFRP application and design With over 80 years of history, it is only in the last 15 years that the use of FRP materials has become truly feasible for bridge applications. In part due to the continuously increasing requirement to make structures last longer with current American Association of State Highway and Transportati on Officials (AASHTO) LRFD Bridge Design Specifications which
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3 requires that structures be designed for a 75 year design life; but also in the development of cost effective production techniques and the introduction of CFRP materials, which bring the cost and strength of FRP materials closer to traditional steel reinforcement. As the need for FRP materials increases, it is critical that designers and owner s have a realistic understanding of the performance and a level of comfort with the design methodolo gy. In order to increase the understanding and use of FRP materials, it is critical that FRP design methods are comparable and compatible, with those for steel reinforced structures. (ACI Committee 440, 2004) The recommendations in the present study suggest calibrated flexural resistance factors for FRP prestressed bridge girders designed in accordance with the AASHTO LRFD Bridge Design Specification (AASHTO LRFD) for commonly used bridge girder sections Addit ionally, traditional Precast Prestressed Concrete Institute (PCI) deflection multipliers were investigated for FRP prestressed applications and recommended values are provided. 1.3 Objective The main objective of this research is to provide revised LRFD resis tance factors and updated PCI deflection multipliers for FRP prestressed girders. Detailed objectives include: 1) Identify, model and design twenty five (25) spans based on existing precast girder designs. a. Design prestressing reinforcing to for three materi als: i. Steel Prestressing Strand ( Per the design drawings) ii. AFRP P restressing Strand iii. CFRP P restressing Strand
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4 2) Develop FRP flexural strength designs based on ACI 440.4 recommendations, and identifying tension controlled design vs. compression controlled 3) Calib rate tension controlled and compression controlled resistance factors, using Monte Carlo numerical modeling for Steel, AFRP and CFRP designs. 4) Calibrate PCI deflection multipliers based on Monte Carlo simulation for concrete and reinforcing m aterials. 5) Provide recommended tension and com pression resistance factors to update Table 3.1 of ACI 440.4 R 04 and for use in AASHTO LRFD. 6) Provide recommended deflection multipliers to update Table 4.1 of ACI 440.4 R 04 1.4 Scope The scope of this study consists of the development of a probabilistic based numerical model to determine the flexural reliability of FRP prestressed bridge girders, designed to meet AASHTO LRFD and ACI 440.4 requirements. For FRP prestressed girders there are four main pieces to consider: the L oad and Resistance Factor Design (LRFD) methodology, the loading model, the FRP prestressing tendon, and structure geometry. For the purpose of this study, the numerical model was developed to predict the reliability in nominal loading and material propert ies based on the existing AASHTO LRFD Design S pecifications The FRP material models are intended to account for the variability between different manufacturers for products are classification as AFRP or CFRP, using two specific products as the baseline ma terial. Structure size and geometry is based on twenty five (25) sample spans based on existing designs utilizing traditional materials Modeled structures were selected to encompass varying span ranges, girder depths, and girder types.
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5 The flexural modeling and simulation method closely follows the approach used for the initial calibration of the AASHTO LRFD code, as outlined by Nowak (1999) in the National Cooperative Highway Research Program Report 368 (NCHRP 368 ) The simulation utilizes updated resistance and nominal loading models as developed by Kwon (2011) Atadero (2006) Okeil (2002) Gohsn (1986) and Riberio (2013) to account for material and loading variations in concrete, steel, FRP, live load, and dead load. Resistance is modeled using the recommendations of ACI 440.4 and prestress losses and deflections per AASHTO LRFD. The long term deflection simulation presented in Appendix A, f ollows the methodology of Martin (1977) for simplified multipliers. Material properties and associated inst antaneous prestressing losses are simulated via Monte Carlo methodology for reliability similar to flexural analysis. The goal is to simulate sufficient data to provide confidence in the proposed resistance factors and multipliers applicable to most FRP pr estressed girders designed to meet the reliability requirements of AASHTO LRFD and ACI 440.4. 1.5 Thesis Outline The contents of this thesis are briefly outlined below: Chapter 2: presents a literary review of existing research on the calibration of LRFD design codes, FRP prestressing, design of FRP elements, live loads and dead load models, an d the behavior of FRP elements: Chapter 3: provides a detailed description of the modelin g program, including sample structure summaries, capacity evaluation, component variability, simulation process, program setup, and procedure. This chapter includes detailed processes for both flexural reliability and deflection evaluations.
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6 Chapter 4: out lines the results of FRP calibration p roviding summary t ables for capacity reliability and calibrated resistance factors. Chapter 5: presents Monte Carlo Simulations applied to PCI long term deflections factors for FRP prestressed girders. Chapter 6: pres ents study conclusions, recommendations, and opportunities for further research. References Appendix A : Calculated Reliability indices for Vary Resistance Factors Appendix B: Sample Visual Basic Code for Reliability Index Calculations Appendix C: Sample Span General Layouts and Typical Sections
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7 2 Literary Review 2.1 Overview In an effort to increase the durability of concrete structures, it has been proposed to use composite materials instead of standard steel reinforcement. Prominent forms of composite m aterials investigated for civil structural applications include : carbon fiber reinforcing polymers (CFRP), aramid fiber reinforced polymers (AFRP), and glass fiber reinforcing ir high strength, lightweight, noncorrosive, non conducting (ACI 440.4R, 2004) CFRP and AFRP currently show the most promise as prestressing materials H owever, due to the superior material pro perties of CFRP, significantly more research has been conducted on CFRP materials CFRP and AFRP has been successfully demonstrated in numerous aspects of bridge engineering including, barrier reinforcement, slab reinforcement, beam flexural reinforcement, and beam prestressing and post tensioning applications as well as ground anchoring systems. 2.2 FRP Prestressed Concrete FRP materials are composites consisting of te nsion carrying synthetic fibers and a binding resin to confine and protect the fiberous materials. Determining the strength of the FRP material depends on the direct properties of the unidirectional fibers and those of the resins. Fibers can consist of Carbon, Aramid, or Glass and bindi ng resins are typically either Epoxy or Vinyl ester based. For the purposes of prestressing concrete, only AFRP and CFRP materials are recommended for construction (ACI 440.4R, 2004)
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8 2.2.1 FRP Prestressing M aterials CFRP Prestres sing: Mitsubishi Leadline TM tendons are the most commonly used product in CFRP prestressing applications. These tendons consist of a ridged rod made of high tensile strength carbon fibers woven together and coated in epoxy. Leadline Tendons can come in a v ariety of finishes including smooth, indented or ribbed. Carbon Fiber Composite Cables (CFCC) made by Tokyo Rope are typically used for post tensioning applications but can also be used in prestressing operations. These two products represent the current standard for Carbon fiber reinforcement. Table 2 1 provides a summary of two common commercially available CFRP materials. Table 2 1 : Material properties for Common CFRP Materials Material Characteristics Mitsubishi. Leadline Tendons Tokyo Rope CFCC (1 x 7) Fiber Volume Ratio 0.65 0.65 Guaranteed Tensile Strength (ksi) 3 26 271 In plane S hear Strength (ksi) 10.3 Long. Compressive Strength (ksi) 210 Elastic Modulus (ksi) 20,600 1 9,900 Maximum Elongation (%) 1.9 1.5 Resin Epoxy Epoxy Prestressing with carbon fiber offers some significant advantages, due significantly to its high tensile strength, light weight and corrosion resistant properties, but carbon fiber does have drawbacks; notably its high cost, lack of duc tility and minimal shear strength. Design of a concrete section prestressed with CFRP is similar to steel strand prestressing at the ultimate limit states, the differences in material properties lead to some major differences at service limit states.
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9 AFRP Prestressing: There are three common manufactures of AFRP tendons. ARAPREE is the most commonly tested AFRP materials and is a product of AKZO Chemicals in the Netherlands. ARAPREE tendons are available in either flat or round bars and can be used for post tensioning and prestressing applications. FiBRA tendons, developed by Mitsui Construction of Japan, are unique in their ability to be fabricated into flexible road, which can be coils and easily transported. Sumitomo Con struction of Japan has developed a spirally bound AFRP tendon called Technora Technora tendons have been developed with an advanced anchorage which allows up to 19 m ulti bar tendons to be stressed together. These t hree products represent the current stan dard for Aramid fiber reinforcement. Table 2 2 provides a summary of the three materials Table 2 2 : Material properties for Common AFRP Materials Material Characteristics AKZO ARAPREE Mitsui FiBRA Sumitomo Technora Fiber Volume Ratio 0.45 0.65 0.65 Guaranteed Tensile Strength (ksi) 1 74 1 81 2 46 In plane S hear Strength (ksi) 0.7 Long. Compressive Strength (ksi) 48.6 Elastic Modulus (ksi) 9,000 9,400 7,800 Maximum Elongation (%) 2.4 2.85 3.75 Resin Epoxy Epoxy Vinyl ester 2.2.2 FRP Material Strength In the past, FRP tendon strength has been reported in multiple fashions based on the preference of the manufacturer, with most Japanese manufactures reporting guaranteed tendon strength, or design strength, and European and North Americans reporting mean ultimate strength, or the 95 percent inclusion strength. Due to the tendency for FRP tendons to exhibit l inear stress strain performance and not provide a strain hardening zone similar
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10 to steel, it can be a challenge for designers to select a design strength that will reliably provide capacity that will not be exceeded over the life of the structure. Th e latest release of ACI 440.6 (2008) Section 8.11, requires that FRP bar ultimate str ength be reported based on the guaranteed tensile strength of the bar, or the mean ultimate strength minus th re e standard deviations. The designer should be aware of the reported FRP strength and confirm that tendon strength used for design is, in fact, th e guaranteed tensile strength. The simulations and calibrations in this report utilized the guaranteed tensile strength for design 2.2.3 Advantages FRP R einforcement FRP materials offer one of the biggest advancements in reinforced and prestressed concrete perf ormance. The material properties of FRP provide numerous advantages in both structure repair and new construction. FRP materials are not susceptible to the same kind of corrosion that is common in steel reinforcing; this allows for extended service lives o f structures that may be capable of those specified in the AASHTO LRFD code. Additionally, FRPs a re non conductive making them non susceptible to damage due to stray currents, which is of particular use in the expanding electric rail industry. As a prestr essing material, the lower modulus of elasticity of FRP results in reduced losses due to concrete deformation, creep and shrinkage ; allowing for lower initial stressing which is of particular use when addressing FRP disadvantages. FRP is a linear elastic material, displaying a linear relationship between the induced strain and the internal stress of the bar or tendon. This relationship, compared to steel pre stressing strand, is indicated in F igure 2 1 Th e linear nature of the material leads to predic t able performance when loads are induced, providing a near linear response in concrete beam s until cracking, and then continuing to p erform linearly after cracking (ACI
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11 440.4R, 2004) While this makes design calculations mor e straight forward it leads to some confusion when designers attempt to provide ductility within a system. When steel will deform based on the load without failure, the internal st r ess of an FRP tendon will continue to rise until the tendon ruptures or th e concrete crushes, resulting in no ductile failures. Figure 2 1 : Stress Strain relationship between various prestressing materials 2.2.4 Disadvantages FRP R einforcement Shear Rupture: Due to the unidirectional nature of FRP materials, the tendons do not perform well when transverse stresses are applied. This property makes the tendons susceptible to failure from shear and flexural cracking which create openings in the concrete that need to be bridged by the tendons The o pen crack causes differential deflections applying shear stresses to the tendon F ailure caused by dowel action is called shear tendon rupture. A steel tendon can handle this shear without effecting performance of the beam in flexure because the steel has simila r capacity in all directions. FRP tendon s exhibit shear strength that is typically less than 5% of tensile strength; it is unable to handle significant shear in conjunction to tension. Research
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12 resisting capacity of beams prestressed with FRP tendons is about 15 percent less than that of beams with steel tendons, regardless of the shear fail ure mode (Park & Naaman, 1999) Harping Stress Increased: FRP tendons are sensitive to direction a l changes under stress. The linear elastic nature of FRP causes significant increases in internal stress at the location of bends or curves, which detrimentally affects the performance when using harped tendons in prestressed girders. For solid or stranded tendons, the stress increase at har ped points can be described by E quation 2 1 Eq. 2 1 Where E f is the elasticity of the fibers, R t is the radius of the tendon, and R is the radius of curvature of the harping saddle Table 2 3 provides stress increases in ksi for commercially available carbon Leadline and Aramid Fibra tendons for possible harping radius. Because the stress increase from the bend radius is considered in the tendon stress limits, it is cr itical to account for these stresses to ensure long term performance and limit the chances of ruptured strands during jacking operations. Table 2 3 : Harping stresses in FRP tendons at varying Saddle Radius CFRP Leadline AFRP Fibra R (in) f h R t = .39 in E = 20,600 ksi % Allowable (f u = 326 ksi) .65f u = 212 ksi f h R t = .39 in E = 9,400 ksi % Allowable (f u = 181 ksi) .5f u = 91 ksi 1 4017 1896% 1833 2025% 4 1004 474% 458 506% 6 670 316% 306 338% 12 335 158% 153 169% 24 167 79% 76 84% 60 67 32% 31 34% 120 33 16% 15 17%
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13 Creep Rupture: FRP tendons are susceptible to a phenomenon known as creep rupture. This is described as the failure of the tendon when subject to high levels of sustained stress. Both CFRP and AFRP tendons are susceptible to creep rupture; the failure of the tendon is different between the materials. CFRP tendons tend to fail with littl e or no warning once the creep threshold has been reach. AFRP tendons follow a typical creep failure pattern in which deflections or internal strains will increase slowly with time, providing a visible in dictor of failure. Based on 100 year design lives, C FRP sustained load should be limited to 70% of that ultimate capacity and AFRP should be limited to 55%. These stress limits for sustained load form the basis of the limiting jacking stresses allowed for FRP tendons (Table 3 3 ). 2.2.5 Existing Structures and Trial Applications The first major breakthroughs in FRP bridges within North America came with the construction of the Beddington Trail Bridge in Calgary Alberta in 1992 and the subsequent of the Art Report on Fiber Reinforced Plastic Reinfo rcement for Concrete Structures (2004) In 2001 the Bridge Street Bridge in Southfield, Michigan was completed T his bridge represents the current benchmark for carbon fiber reinforceme nt applications and design. Bridge Street Bridge: The Bridge Street Bridge project was completed in 2001 and consisted of the design and construction of 2 adjacent parallel bridges. Structure A was built first using traditional AASHTO I girders and standa rd reinforcing materials. Structure B, the first multi span CFRP concrete structure in the United States, was constructed using an existing substructure and a revolutionary Double Tee beam design that utilized CFRP materia ls for all reinforcement. The 3 sp an bridge, utilized four double tees in a side by side configuration with each span designed as simple spans. The deck was cast composite
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14 with beams. The purpose of this project was service life of highway br idges, thereby reducing construction related safety concer ns and annual maintenance costs (Grace, Roller, Navaree, Nacey, & Bonus, 2004) The beam design is a culmination of research on CFRP prestressing to the time of construc tion. It uses internally bonded prestressed Leadline tendons and external post tensioned CFCC ropes to increase ductility. Additionally, all shear reinforcement is formed out of Leadline rods and the four adjacent beams were transversely post tensioned wit h CFCC to increase their shear performance. As part of the project the bridge was wired for constant monitoring by the design team to ensure the structure is performing as designed and to confirm design assumptions. All 12 beams were monitored during fab rication and stressing operations. Only six beams, four beams in the first span and a single interior beam in the middle and end span, were wired for the full 5 year monitoring program. The beams were wired to monitor the concrete strains throughout the be ams, strains in the post tensioned longitudinal and transverse strands and deflections. Three years after the bridge opened to the public, the original design team performed a series of load test to confirm load distribution assumptions on the Bridge Stree t B ridge and validate the use of the AASHTO design code provisions. Using the existing monitoring equipment they were able to determine based on controlled loading that a single beam would see no more than 58.9% of the lane load applied. This is in alignment with the AASHTO calculated value of 60% for this structure and beam spacing, showing that existing provi sion apply to CFRP structures as well as traditional bridge methodology (Grace, Roller, Navaree, Nacey, & Bonus, 2004)
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15 2.3 Advancement of Structural Design Philosophy The use of factors to increase loads and decrease calculated capacity has been the basis Allowable Stress Design (ASD), used factors of safety to reduce the allowable stress within a system to pr ovide safe designs. Factors of S afet y were often selected based on the amount of experience and comfort levels that engineers had with materials structural systems and applications The use of a single safety factor resulted in significant variations in the reliability of structures design ed using allowable stress as it simplified the design to a point where variations in loading were not considered. Working stress design is being phased out of most design methodology and is generally used to supplement modern design methodology Equation 2 2 describes the g eneral approach to ASD, where F A llow is the allowable stress, f is the calculated stress, and FS is the factor of safety for the system. Eq. 2 2 Experience with ASD, and recognition of its disadvantages lead to the development of Load Factor Design (LFD). LFD is still largely in use today and is the preferred design method in the American Railway Engineering and Maintenance Association (AREMA) design manual. LFD started the implementation of multiple load factors to begin accommodating the native variability in different types of loads. In addition to load factors, strength reduction factors were also provided. Stren gth reduction factors in LFD w ere selected to provide a level of safety sim ilar to, and in many cases, identical to those used in allowable stress design. The biggest differenc e between and LFD design and LRFD design is in the determination of the load and resistance factors. LFD factors were selected based on
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16 past experience ; LR FD factors are calibrated based on the statistical variability of each load, and each resisting element. Load and Resistance Factor Design has been in use in the structural engineering community since the American Institute of Steel Constructors (AISC) rel eased the first LFRD edition of the AISC Manual of Steel Construction in 1986. The new LRFD code was a major shift in structural design as it resulted in two major changes; first, the implementation of probabilistic analysis of uncertainty and second the use of limit states to describe failure types. In LRFD approaches, different load factors are assigned to each type of load considered in the design, including dead, live, wind, and earthquake; additionally, calibrated resistance factors are assigned to t he type of force being resisted. This results in different factors for shear and moment capacity, as well as separate factors for steel, reinforced concrete, or prestressed concrete construction. Equation 2 3 describes the general equation for LFRD design. Eq. 2 3 Where R n and described the nominal system resistance and associated resistance factor, and and Q describe the load and the load factor (Atadero & Karbhari, 2006) The inherent flexibility of the LRFD approach allows for an LRFD code to be simply adapted for new and un expected situations. This includes new load and demand models, such as scour or physical attack, and new materials for resistance, such as FRP as desc ribed in this report. Research resulting in changes to either side of Equation 2 3 can be performed independently from the rest of the code, and reliability levels of the system can be maintained assuming consistent target reliabilit ies avoiding wholesale changes to the documents. The advantages of the LRFD approach make LRFD Codes practical for future use, and consequently LRFD is the target philosophy for FRP reinforcing and prestressing.
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17 2.4 Probability in Structural Engineering Modern Load and Resist ance Factor Design (LRFD) based codes use the theory of probability, founded in statistical analysis, to provide levels of structural reliability to mitigate the risk of structural failure. Probabilistic design, while not directly used when design ing to LRFD c odes, form the basis for load factors and resistance factors. These factors can be determined using a variety of probabilistic levels. A fully probabilistic design, level III, is extremely complex, requiring detailed knowledge of every available random var iable in the design, the distribution and the correlation between each random variable. Due to the complexity, level III probabilistic design is impractical to use in design (Atadero & Karbhari, 2006) The First Order Second M oment (FOSM) method, a level II probabilistic approach, is far simpler. FOSM methodology only requires basic knowledge of the statistical distribution of the loading and resistance, and assumes the two are not statistically related. FOSM was used in the de velopment of the AASHTO LRFD Bridge Specifications, and consequently is the approach of this report (Barker & Puckett, 2007) 2.4.1 Uncertainty in Design Predicting the capacity of any structural system is subject to a significant amount of uncertainty. Uncertainty in structural systems comes from two main sources: loading and resistance. Random v ariation in loading and resistance means that any struct ure i s subject for a certain un known risk of failure. In order to minimize the risk of fa ilure, while maintaining a cost effective approach to design and constr uction, modern design codes use a combination of load and resistance factors to predict demand and ca pacity. Load and
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18 resistance factors are a simplification that allows designers to capture the natural unpredictability of each contributing element to any load or resistance. Each element is subje ct to some level of uncertainty; the four main categories w hich are commonly included in reliability calculations, as described by (Atadero & Karbhari, 2006) are: Modeling : u ncertainty is associated with the need to use simplified relationships or equation s to represent a natural sit uation. Modeling u ncertainty can often be mitigated through additional research or increased available data. Prediction : uncertainty is a function of the need to predict future conditions. This can be related to probable earthquake events, or the actual strength of the concrete placed in the field. This level of uncertainty is not always included in a reliability model; how ever, it is indirectly included, such as the use of the HL 93 loading model in AASHTO LRFD. Physical : uncertainty accounts for the variations in material properties, size, and in the loads applied to a structure. This form of uncertainty is largely accounted for in reliability analysis. Statistical : uncertainty is controlled by the need to approximate m ean, standard deviations, or distribution functions from limited data. The level of uncertainty can be included in research by performing multiple analyses through varied parameters. Risk can be described as the chances that the structure will fail, as inf luenced by the uncertainty described above As structural engineers, we have a responsibility to reduce risk to acceptable levels, while still maintaining constructible and economical engineer ed designs Though it is impossible for an engineer to see into the future and predict the unexpected the implementation of reliability based design, allow s for an engineer to reasonably rationalize and model uncertainty and manage risk.
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19 2.4.2 Basic Reliability Statistical distribution functions are used to describe the ran dom variable that make up the probability system used to describe the loading and resistance of the structural system. Normal and lognormal distribution functions are commonly used to describe load and resistance respectively. Normal distributions can be described by the ir mean and standard deviation, as given in E quations 2 4 and 2 5 respectively Eq. 2 4 Eq. 2 5 Additionally, it is convenient to calculate the Coefficient of Variation ( COV ) for the distribution, E quation 2 6 Eq. 2 6 Figure 2 2 shows the effects of varying standard deviations on the distribution of a normal set of data N ote that normal distributions ar e symmetrical about the mean. It is common to calculate the mean and standard deviation for all sets of data regardless of the distribution. This is done because most commonly used distributions can be described in terms of the mean, standard deviation, or COV of a normal distribution.
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20 Figure 2 2 = 0.1, = 0.2, and = 0.3 Lognormal distributions are skewed distribution s when a portion of the data lies on the positive side of the distribution curve Figure 2 3 Lognormal distributions commonly describe the resistance of a sample, as it captures the tendency for designers to be conservative in the selection of input da ta. Figure 2 3 log = 1.0, and = 0.1, = 0.2, and = 0.3 Lognormal distributions are described by the l ognormal mean, Equation 2 7 and the lognormal standard deviation, Equation 2 8
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21 Eq. 2 7 Eq. 2 8 In addition to the mean, standard deviation, and Coefficient of Variation it is also necessary to quantify the bias of a system. Bias is used to explain differences between what is specified, and what is actually provided With concrete, this is the tendency for the compressive strength to exceed what is specified, or bias can also be used to capture the long term growth of a certain load system such as the tendency for live load s to increase over time. Bias is defined by E quation 2 9 where is the experimental or measured, and is the specified or predicted variable Eq. 2 9 2.4.3 Probability of Failure For the purpose of structural desig n, where various service and strength (ultimate) limit states control the design of structures, the probability of failure can be defined using the strength limit state in which is the failure of the element occurring when the resistance distribution ( R ) is less than the load effect distribution ( Q ): Eq. 2 10 If both R and Q are described by independent normal distributions and plotted then the visual representation of E quation 2 10 is shown in F igure 2 4 Wh ere the probability of failure is defined by the failure region which is the area of overlap between the distribution function for resistance ( R ) and the distribution function for load effect ( Q )
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22 Figure 2 4 : Normally distributed load and resistance functions The probability of failure in this function is described using the relationship between R and Q described in E quation 2 11 : Eq. 2 11 Plotting the limit state function directly provides another way to look at the probability of failure. In the case of ( Figure 2 5 ), the failure point can be clearly identified as the region less than 0.
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23 Figure 2 5 : Failure limit state distribution function Measuring th e distance to the failure plane in terms of the number of standard deviations from the mean, provides a simple way to determine the reliability of the system. This measurement is called the reliability index, and is defined as Eq. 2 12 Using the reliability index, the pro bability of failure for the system can easily be determined for normal R and Q systems, or for lognormal systems. Table 2 4 tabulates common failu re probabilities for normal distributions, additional failure probabilities can be found in most statistical text s and are often called z tables Table 2 4 : Probability of failure based on Reliability Index Reliability Index, Probability of Failure, P F Reliability (1 P F ) Reliability Index, Probability of Failure, P F Reliability (1 P F ) 0 0.500 0.500 2.5 0.00621 0.99379 0.5 0.309 0.691 3.0 0.00135 0.99865 1.0 0.159 0.841 3.5 0.000233 0.999767 1.5 0.0668 0.9332 4.0 0.0000317 0.9999683 2.0 0.0228 0.9772 4.5 0.00000340 0.99999660
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24 2.5 Reliability Approach for this Report This report is based on the use of First Order Second Moment reliability method, which is dependent on knowledge of the distribution of each variable in the system. Due to the scal e of the study and the relatively recent introduction of FRP based prestressing, sufficient data to develop distribution curves is not available. In order to determine sufficient information for calibration, special sample techni ques are required. Modeling component random variables for this level of r eliability analysis is predominantly based on the use of Monte Carlo Simulation(s) (MCS). MCS is used predominately for its ability to generate a large number of data points when th ere is limited information on the system performance. The MCS procedure allows for the independent modeling of materials, geometry, and loading where there is significant research to support the individual distribution functions of each contributing proper ty. The simulation is capable of providing a mean, standard deviation and bias for material and geometr ical component s of the entire resisting system. This distribution for capacity can be used in tandem with a Monte Carlo sampling of demand on the system to determine ranges of failure based on the limit state function (Equation 2 10 ) The accuracy of a Monte Carlo S imu lation is heavily dependent on the number of samples selected, with increased sample sizes providing more accurate data. Figure 2 6 shows the variation of the reliability index relative to the sample size, indicating a stable statistical average but a conv erging standard deviation ( ) as sample size increases. At approximately 50,000 samples the simulation converges and the use of additional samples provides minimal benefit to the model.
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25 Figure 2 6 : Calculated reliability inde x vs. Sample size Due to limitations in processing power and file size within Microsoft E xcel, samples of 10,000 were used for capacity simulations. To expand the sample size, multiple runs of 10,000 were completed to expan d the sample to 100,000 or more. T his was completed using the Data Table function within Microsoft Excel. MCS has been utilized to simulate the material and fabrication bias, mean and COV for AFRP and CFRP bridge girders, which is t hen fit to a distribution curve. T his model is referred to as the capacity, o r resistance model, later in th is report. A second model simulates the load side of the equation, the load model, and compares it to the simulated values of resistance based on the results of the capacity model. Using these two models calculations can be completed for reliability indices for each sample span, and resistance factors recommended. 2.6 Existing FRP Prestressing Resistance Factors ACI 440.4 recommends strength reduction factors for AFRP and CFRP prestressed beam s of 0.70 for Aramid and 0.85 for Carbon in tension controlled behavior, when tendon
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26 strain is above 0.005, and 0.65 for both materials for compression controlled behaviors when the net tensile strain is below 0.002 ( Table 2 5 ). The resistance factor is ass umed to vary linearly between st r ains of 0.002 and 0.005, between the specified tension and compression controlled strains. These factors are based on the reported relationships between sample beams and the recommend design flexure equations presented in A CI 440.4 based on the recommendations of Burke and D olan, (2001) Table 2 5 : Existing Strength Reduction Factors (Recreated from ACI 440.4) Tendon Type Strength Reduction Factor, Condition Aramid 0.70 Tension controlled behavior Carbon 0.85 Aramid or Carbon 0.65 Compression controlled behavior The current strength reduction factors are based on an empirical comparison of the calculated design capacities, using the equations presente d in section 3.5 of this report and physical tests of the ultimate capacity of the beams. 30 sample beams 8 AFRP samples, and 22 CFRP samples were investigated. Table 2 6 through Table 2 8 provides a summary of the combined tension and compression results for the initial selection of the strength reductions factors (Burke & Dolan, 2001)
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27 Table 2 6 : Comparison of All beams, recreate d from Burke and Dolan, 2001 M exp / M n = 1.00 = 0.90 = 0.85 = 0.80 = 0.75 = 0.70 = 0.65 = 0.60 = 0.55 1.097 1.219 1.290 1.371 1.462 1.567 1.687 1.828 1.994 0.196 0.218 0.231 0.245 0.261 0.280 0.301 0.327 0.356 COV = 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.494 1.004 1.259 1.514 1.770 2.025 2.280 2.535 2.790 It should be noted that Table s 2 6, 2 7 and 2 8 have been modified from the original presented tables to provide calculation of the Coefficient of Variation and have been expanded to include resistance factors of 0.65, 0.60 and 0.55, to provide a comparison with the existing factors u sed for FRP reinforced concrete members Table 2 7 : Comparison of AFRP beam s, recreated from Burke and Dolan, 2001 M exp / Mn = 1.00 = 0.90 = 0.85 = 0.80 = 0.75 = 0.70 = 0.65 = 0.60 = 0.55 1.022 1.135 1.202 1.277 1.362 1.459 1.572 1.703 1.857 0.237 0.263 0.278 0.296 0.316 0.338 0.364 0.394 0.430 COV = 0.232 0.232 0.232 0.232 0.232 0.232 0.232 0.232 0.232 = 0.091 0.513 0.725 0.936 1.147 1.358 1.570 1.781 1.992 Table 2 8 : Comparison of CFRP beams, recreated from Burke and Dolan, 2001 M exp / Mn = 1.00 = 0.90 = 0.85 = 0.80 = 0.75 = 0.70 = 0.65 = 0.60 = 0.55 1.124 1.249 1.322 1.405 1.499 1.606 1.729 1.874 2.044 0.177 0.197 0.208 0.222 0.236 0.253 0.273 0.295 0.322 COV = 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.700 1.265 1.547 1.829 2.111 2.393 2.675 2.958 3.240
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28 The final row s in Tables 2 6, 2 7 and 2 8 provides a base calculation for determining the statistical reliability of the system. Using the calculation with basic reliability methods, the beta can be interpreted as k, from equation 4 14 If reliability methods are applied, assuming an ideal value of beta is approximately 2.0, then phi factors fo r the selected samples would range between 0.80 and 0.70 for Carbon prestressed girders, would be approximately 0.55 for Aramid prestressed girders, or would average around 0.70 if both materials are considered together. These values indicate that the curr ent factors may not provide a level of reliability that is consistent with current LRFD design codes Additionally, it is important to note that aramid strength and carbon strength are reported by different methods in the current calibration. Aramid stren gth is based on mean tensile strength of the tendon, while the carbon girders are based on the design tensile strength. This variation explains the large discrepancy between the performances of the two materials. If the material strength is reported in the same manner, it is expected that the resistance factor for CFRP and AFRP prestressed concrete will converge.
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29 3 Capacity Modeling 3.1 Overview When determining the performance reliability of any system using the FOSM process it is imperative to have representative tests that provide a directly comparable result and statistical distributions for the system capacity. In the case of this study, there is very limited information on the performance of FRP prestressed bridge girders, consequently not provid ing a sufficient sample size to determine a statistical distribution for capacity. Past studies have developed a wealth of material knowledge which has allowed researchers to develop representative distributions for each material, or parameter, contributin g to the capacity of the element This chapter presents the simulation approach for the capacity model, the representative equations used to predict capacity, and the material and geometry distributions used in those equations. Monte Carlo Simulation meth ods of the capacity allow for the development of thousands of data points, which can be used to determine the statistical distribution of the capacity for first order determination of the reliability of the system. 3.2 Simul ation Procedure The Monte Carlo Si mulation for capacity was a multi stage process. The simulation was completed for all 25 trial span s for both AFRP and CFRP materials. The process for simulating the fabrication and materials distribution for FRP prestressed girders is described below.
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30 1. Se lection of Trial S pans: Select 20 30 representative structure for re design to FRP prestressing. 2. Trial Design: Full superstructure designs were completed for each structure to determine applied loads, based on AASHTO LRFD load requirements. Additionally, preliminary designs for each FRP type, were completed to determine the area of prestressing required meeting strength and service limits s tates based on AASHTO LRFD and ACI 440.4 requirements Design verifications were also completed for the original design s utilizing 270 ksi prestressing steel strands. Preliminary designs were performed using Bentley C ONSPAN 3. System Properties: Characterist ic geometry and material properties were collected from the preliminary design to provide a realistic basis for the calibration procedure. These properties were taken as the mean values within the simulation. 4. Statistical D istributions: Each characteristic property was assigned a statistical distribution based on previous research or performed calculations 5. Simulated Properties: Random variables were used to provide random samples for each property, distributed based on the individual probability. 6. Balanced Ratio: The balanced reinforcement ratio was calculated for each span based on the simulated properties. This was compared with the simulated span reinforcement ratio to determined tension controlled or compression controlled behavior. 7. Capacity: Equations w ere used to calculate the span capacity based on strain compatibility. 8. Bias: Parallel flexural resistance calculations completed for capacities with bias in property and without bias. T he bias for fabrication and materials is then taken as FM n /M n based on the simulated values.
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31 9. Monte Carlo Simulation : Repeat s teps 5 through 8 for 10,000 iterations to create a representative capacity sample control type ( t ension or c ompression), Coefficient of Variation and bias 10. Increase Sample Size: In order to ensure adequate sample size, step 9 is repeated 10 times for an overall sample size of 100,000. Final Fabrication and Materials average control type ( t ension or c ompression), Coefficient of Variation and bias are calculated. 11. Analysis Factors: The Fabrication and Material bias and Coefficient of Variation are combined with selected analysis factors for using the calibration model. 3.3 Trial Span s Trial spans were selected from a pool of existing structures designed to the AASHTO LRFD code. The goal of the trial span selection is to provide between 20 and 30 sample spans that represent a w ide range of prestressed girder applications, with variable spans, girder type s spacing, bridge widths, and skews. Selected bridges were limited to those with fact ory prestressing using bonded tendo ns and girder shapes that could be considered industry standard. See Appendix C for additional information on the selected trial spans. 3.3.1 Span Selection The basis of the trial designs is 25 spans, from 1 6 different structures that represent a broad range of precast prestressed structures today. Structures were selected from recently designed or constructed bridges from Colorado Florida, and Texas th at were designed to meet AASHTO LRFD. The selected struct ures are all girder slab bridge type construction with box girders, tub girder or U girders, and Bulb Tee or I Girders represented in the set. Spans range from 56 feet to 164 feet, with girder spacing from 4 f ee t
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32 when adjacent box girders are used, to as much as 18.5 feet for U girders and girder depths vary from 20 inches to 84 inches Additionall y, both simple span and simple span made continuous construction are represented The trial spans are shown in Table 3 1 the spans are ordered by span length Table 3 1 : Sample Span Layout Properties Span Span (ft) Structure Depth (in) Span/ Depth (in/in) Girder Type No. Girder Girder Spacing (ft) Bridge Width (ft) Skew (deg.) 1 56 25 26.9 B 20 4.1 82.0 90.0 2 56 25 26.9 B 20 4.1 82.0 90.0 3 87 80 13.1 U 4 13.7 54.6 90.0 4 107 44 29.2 BX 19 4.6 90.0 50.5 5 108 44 29.5 BX 19 4.6 90.0 50.5 6 109 68.5 19.1 U 4 18.0 71.0 85.5 7 109 68.5 19.1 U 4 18.0 71.0 85.5 8 111 50 26.6 BT 7 6.0 43.0 63.2 9 118.9 62 23.0 BT 8 6.5 51.5 63.9 10 120 80 18.0 U 4 13.7 54.6 90.0 11 121 62.5 23.2 U 3 9.0 28.8 82.3 12 124 71.5 20.8 I 16 9.8 157.2 90.0 13 124 71.5 20.8 I 16 9.8 157.2 90.0 14 125 46.75 [52.75] 32.1 [28.4] BX 12 5.4 67.0 90.0 15 125 80 18.8 U 7 18.5 126.5 64.7 16 126 74 [80] 20.4 [18.9] U 8 16.0 120.0 90.0 17 127 55.5 27.5 BX 12 5.4 67.0 90.0 18 127 80 19.1 U 7 18.5 126.5 64.7 19 128.5 74 20.8 U 8 16.0 120.0 90.0 20 132 54 29.3 BX 10 4.1 43.0 34.0 21 135 49 [53] 33.1 [30.6] BX 24 6.2 152.0 69.8 22 135 49 [53] 33.1 [30.6] BX 24 6.2 152.0 69.8 23 143 80 21.5 U 4 13.7 54.6 90.0 24 144 74 [80] 23.4 [21.6] U 8 16.0 120.0 90.0 25 163.6 92 21.3 BT 4 6.7 28.0 60.0
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33 3.3.2 Trial Design Trial designs were performed for each span utilizing the Bentley prestressed girder design software CONSPAN. C ONSPAN was selected because it is commonly used for bridge girder design in the state of Colorado where most of the sample structures are located The software package allows for the rapid design of prestressed concrete bridges to meet the AASHTO LRFD code, which also enables the user to define geometry and material properties. All bridges were modeled using their actual design geometry and girder shapes structures that were intended as precast made continuous were assumed fixed over the pier The exception to this were spans 14, 21, 22, and 24; which required deeper girder sections to meet stress requirements for the AFRP prestressing material, indicated in brackets in T able 3 1 Interior girders were selected for design and live load was distributed based on the AASHTO LRFD distribution factors. Te n dons for trial design are based on commercially available products. With Leadline tendons forming the basis for CFRP design, and Fibra that basis for AFRP design. Table 3 2 summarizes the nominal tendon properties used for the trail design. Table 3 2 : Tendon Properties for Trial Spans CFRP AFRP Steel PS Material 0.5 326 CF 0.75 181 AF 0.5 270 LL 0.6 270 LL Dia (in) 0.62 0.93 0.5 0.6 F.V.R 0.65 0.65 N/A N/A Aps (in2) 0.196 0.442 0.153 0.217 f u (ksi) 326 181 270 270 E (ksi) 20600 9400 28500 28500 Jacking limit 0.65 f u 0.50 f u 0.75 f u 0.75 f u Stress (ksi) 211.9 90.5 202.5 202.5 F jack (kips) 41.6 40.0 31.0 43.9
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34 In order to provide a degree of compatibility with the existing steel reinforced designs, tendon diameters were scaled to provide jacking forces similar to those provided by a 0.6 inch diameter 270 ksi low relaxation strand. This resulted in FRP tendon dia meters larger than those that are commercially available. For the purposes of simulation, it was assumed that the change in diameter ha d no effect on the nominal tensile properties of the bar. All tendons were prestressed, f jack to the maximum allowable a s defined by ACI 440.4 (2004) as tabulated in T able 3 3 In addition to strength checks, the software was used to verify the service limit state allowable stresses and release stresses, as limited per T able 3 4 The ass umption was made that all general strand layout would be similar to the referenced base design, using straight strands, debonded strands and harped strands to help control stresses. Note that stress increase d at the harp points were ignored, and the assum ption was made that the benefits of harping could be achieved by debonding alone at the final design stage. Table 3 3 : Allowable tendon Jacking Stresses (ACI 440.4) Allowable Jacking Stress CFRP 0.65 f pu AFRP 0.50 f pu Allowable Stress immediately after Transfer CFRP 0.60 f pu AFRP 0.40 f pu
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35 Table 3 4 : Allowable Concrete Stresses (ACI 440.4) Allowable Stress at Transfer (Before Losses) psi (a) Extreme fiber stress (Compression) 0.6f'ci (b) Extreme fiber stress (Tension) (c) Extreme fiber stress at tendon end (Tension) Allowable Stress at Service (After Losses) psi (a) Extreme fiber stress, P/S + DL (Compression) 0.45f'ci (b) Extreme fiber stress, P/S + Total Load (Compression) 0.6f'ci (c) Extreme fiber stress Precompressed Zone (Tension) Representative FRP properties were input into CONSPAN which then scaled the AASHTO tension equations based on the FRP properties. CFRP tendons were modeled as low relaxation strands and AFRP tendons were modeled as stress relieved strands. This resulted in conservative stresses and losses for CFRP strands, and un conservative stresses and losses for AFRP tendon s compared to the ACI 440.4 model. Losses were modeled using the AASHTO simplified method, with relaxation losses calculated based on the FRP representative strand type indicated above. From the trial design, k ey information was extracted for AFRP Spans (Table 3 5 ) and CFRP Spans (Table 3 6 ) a s presented below; these values are taken to be equal to the mean values for the simulated capacity model. Note that Area of prestressing ( A ps ) is reported based on the adjusted area, multiplied by the Fiber Volume Ratio ( FVR ), and not t he nominal area of the bar.
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36 Table 3 5 : AFRP Trial Design Values for Capacity Simulation
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37 Table 3 6 : CFRP Trial Design Values for Capacity Simulation Table 3 6 : C FRP Trial Design Values for Capacity Simulation
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38 3.4 Resistance Model The ability of the structure to carry the applied loads depends on the resistance of each load resisting element. In the case of this study, the positive moment resistance is the pertinent resistance, with the load carrying capacity being a function of the materials used and the geometry of the girder and composite slab. Nowak (1999) describes the resistance of the system using E quation 3 1 Eq. 3 1 Where R is the random resistance of the sys tem, R n is the nominal capacity. P the analysis factor, is the factor describing approximations due to the analysis method and the idealized stress or strain models, such as use of the equivalent stress block. F is the fabrication factor which includes the element geometry, nominal dimension and section properties, and M the material factor, described the materials of the system, such as ultimate strength and modulus of elasticity. Bias factors and C oefficients of V ariations ( COV ) can be used to describe each of the analysis fabrication, and material factors. The combination of these factors describes the system as a whole. Fabrication and material factors are often combined and can be calculated using Monte Ca rlo Simulations, however the analysis factor requires extensive verification of real world performance extensive elastic or finite element analysis to determine the performance of the simplified calculations used to describe the system. When the fabricat ion and materials factors are combined the bias of the resisting system can be described by E quation 3 2 Eq. 3 2
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39 Where R is the bias factor for the entire system, P is the bias factor for analysis variation, and FM describes the bias of the combined fabrication and materials. Similarly, the COV of the resistance system is described by E quation 3 3 Eq. 3 3 Where V R i s the COV for the entire system, V P is the COV describing analysis variation, and FM describes the COV of the combined fabrication and materials. The following sections in this chapter describe the process and individual properties which contribute to th e P F and M factors: Section 3.5 describe s the equations used to analyze and predict the strength of the section; variations in these equations compared to a full analysis contribute to the P factors. Section 3.6 addresses variation in the material and geometry of the section, addressing the calculation of the F and M factors. Section 3.7 describes the final selection of the P factors, due to the un availability of detailed analysis 3.5 Capacity Prediction Calibration is dependent on the COV and bias factor for the selected resisting system. Due to the lack of overall test information on a sufficient number of tr ia l bridge girders, it was determined that a Monte Carlo S imulation would be used to determine the fabrication and material bias and variation, V FM and FM for the FRP trial designs. The Monte Carlo S imulation allows for each material or property that contributes to the flexure capacity of
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40 the element to be individually simulated, and then combined using the recommended capacity equat ions. In order to provide reasonable calibrations for multiple structure types and failure modes, multiple simulations are required. Table 3 7 below summarizes the capacity equat ions utilized to model each tria l span Aggregate COV and bias factors will be provided for each case, and final flexural calibrations will be performed to provide resistance factors, based on the situational V FM FM Additionally, all c ompression or tension structures will be calibrated for each mat erial based on aggregate calibration factors (averaged across beam types) for tension and compression. Table 3 7 : Flexural Capacity Equations for FRP Prestressed Girders Material Girder Type Control Nominal Moment AFRP Box (BX or B) Tension ACI 440.4 (3 21) Compression ACI 440.4 (3 17) I (I or BT) Tension ACI 440.4 (3 21) Compression ACI 440.4 (3 17) Open Box (U) Tension ACI 440.4 (3 21) Compression ACI 440.4 (3 17) CFRP Box (BX or B) Tension ACI 440.4 (3 21) Compression ACI 440.4 (3 17) I (I or BT) Tension ACI 440.4 (3 21) Compression ACI 440.4 (3 17) Open Box (U) Tension ACI 440.4 (3 21) Compression ACI 440.4 (3 17) Tension controlled and compression performance is determined based on the trial area of FRP prestressing reinforcement in relation to the balanced reinforcement ratio ( r ) as defined by E quation 3 4 This property is dependent on the simulated concrete strength, girder dept h and area of FRP, so it will vary with each individual sample within the greater simulation.
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41 3.5.1 Balanced Ratio The balanced reinforcement ratio, b is critical in the prediction of beam failure type. Reinforcement in excess of the balanced ratio indicate s a comp ression failure will take place and reinforcement below the balanced ratio indicates a tension failure. The balanced ratio is calculated based on simultaneous failure of the section by concrete crushing and tendon ruptur ing This relationship is cal culated based on the assumption that concrete failure will occur at the ultimate concrete compression strain, cu = 0.003. Strain compatibility can be used to describe the failure point. Note that this calculation assumes that failure of the bottom tendon by rupture will result in the subsequent failure of vertically distributed tendons so the balanced condition is assumed to represent multiple rows of reinforcement as well as a single row. The balanced ratio is described by ACI 440.4 as : Eq. 3 4 Where 0.85 represents the equivalent stress block of the concrete in compression and pu pe represents the available strain the prestressing steel after all losses. 3.5.2 Tension C ontrolle d Section Tension controlled sections are described as sections where the prestressing tendon fails before the concrete is able to reach its crushing strength. The performance of the beam can be calculated with the use of strain compatibility and the use o f an equivalent stress block to simplify concrete compression, which has been shown by Dolan and Burke (1996) to result in less than 3 percent error compared to a full elastic analysis.
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42 Since multiple layers of reinforcement are provided within each beam, and the Monte Carlo Simulations are computer resource intensive, it was determ ined that a simplification of ACI 440.4 Equation ( 3 21 ) was necessary to allow for numerous iterations. To simplify the calculation, the assumption was made that the relationship between the individual layers could be described by the total area of prestressing material and the center of gravity of prestressing material, such that : Eq. 3 5 Where f i and A i represents the stress and area of reinforcing in each individual layer, and f cg and A ps represent the stress at the center of gravity of the reinforcing tendons and the total area of reinforcing respectively. Bas ed on this assumption, AC I 440.4 equation (3 21) becomes: Eq. 3 6 Where f ecg is the effective prestress at the center of gravity of the reinforcing and f mcg is the available capacity of the center of gravity. 3.5.3 Compression controlled Section Compression controlled sect ions are described as section s where the concrete fails in compression before rupture of the tendon. Compression controlled sections can be solve d by strain compatibility similar to tension controlled sections, w ith one significant difference, the stress in the tendon is unknown. In order to solve for capacity of the section, an iterative process is required, in which the compression block depth is assumed and then
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43 the tendon stress is solved. The tendon force is then compared to the compression force for force equilibrium in the section as described by ACI 440.4 as : Eq. 3 7 Because the compression block will usually extend beyond the dec k and into the girder flange, a weighted compression block width, b ave was used in the iterative process. b ave is calculated with each iteration modifying the reinforcement ratio for compression control led behavior. Manual calculations showed that E quation 3 7 would converge in three iterations, however to ensure convergence four iterations were used in the simulation Note that it is assumed the ultimate compressive strain is cu = 0.003. Upon converge nce of the equilibrium equation, capacity could be calculated using ACI 440.4 methodology : Eq. 3 8 Where : Eq. 3 9 And : Eq. 3 10
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44 3.6 Random Variable Parameters The capacity prediction equations provided in Section 3.5 are based on a number of variable s that can be simulated based on a statistical distribution. Each variable, from the strength of concrete to the width of the slab can be assigned a distribution function and Coefficient of Variation This allows the Monte Carlo S imulation to predict the variability of the entire system, resulting in a probability distribution function for the capacity of the girder in flexure in lieu of a reliability function of the individual property of the system. 3.6.1 Material and G eometry Variables Geometry : Geometrical variability within the system can be handled in a number of ways. Past research has defined set tolerances for items such as reinforcement placement, beam depths and widths, or slab thickness. For the purpose of this calibration the methods utilized by Okeil (2002) to define key geometry have been utilized T his assumes a constant COV for each property of 0.03 and a bias of 1.0 with the mean being the designed width, depth to reinforcing or height of the beam element. Area of FRP prestressing reinforcing has been var ied based on the recommendations from Atadero and Karbhari (2006) for fabricated FRP sheet s. While this recommendation is intended for sheet elements, its use for bar area is reasonable due to similarities in manufacturi ng processes and base materials of consist of a fiber resin matrix. Table 3 8 summarizes the geometr ic statistical parameter s for the simulation. It should be noted that geometric properties of the prestressed girders were not simulated, and that designed section properties are assumed.
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45 Table 3 8 : Geometry Random Variable Parameter s Variable Distribution Mean Bias COV Reference d, d cg Normal Nominal 1 0.03 Okeil, (2002) h, t, b Normal Nominal 1 0.03 Okeil, (2002) A ps frp Normal Nominal 1 0.05 Atadero, (2007) Concrete : Concrete i s the main compression resisting material in any concrete beam system, thus, selection of the concrete distribution is key to understanding the performance of the concrete and predicting the system reliability. A report by Szerszen and Nowak (2003 ) provided updated concrete parameters for building structures that signific antly increased the bias for in place concrete and reduced the COV While the report was based on building structures it is reasonable to assume that bridge concrete has prog ressed in parallel fashion. Szerszen and Nowak (2003) set the revised COV of concrete to 0.1, a reduction from 0.15 used in the initial calibration s of AASHTO LRFD and provide d an equation to describe the bias of the co ncrete: Eq. 3 11 and the minimum bias is limited to 1.14. This change results in a significant modification to the concrete bias factors, which in turn, affect the compression controlled calibrations. For example, 5000 psi concrete was previously assigned a bias of 0.805 and COV of 0.15; the updated model the COV and Bias for 5000 psi concrete to 0.1 and 1.15 respectively. Table 3 9 summarizes the bias and COV for common strengths used in the trial designs. Note that Bias for concrete is calculated using Equation 3 11.
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46 Ta ble 3 9 : Concrete Random Variable Parameters Variable Distribution Mean Bias COV Reference Concrete Strength Normal 3 1.40 0 .1 0 Kwon et al. 2010 4 1.23 5 1.16 7 1.14 8+ 1.14 Concrete Modulus Normal 1 .00 0.1 0 Atadero 2007 3.6.2 Calculation of AFRP and CFRP Parameter s AFRP and CFRP materials are used for the calibration of the trial spans as the main tension resisting system (reinforcement) For the simulation, i t was assumed the COV for ultimate tensile stress was the same as the COV for material modulus of elasticity. T he bias factor for the FRP modulus of elasticity wa s taken as 1.0 4 per Behnam and Eamon (2013) while the ultimate strength of the mat erial varied based on the nominal value selected for design Bias for CFRP and AFRP strength are extracted from a multiple studies on the tension performance of FRP tendons. Zhang, Benmokrane, Chennouf, Mukhopadhyaya and El Safty (2001) reported guaranteed ultimate tensile strength and tension test results for ARAPREE tendons indicating an AFRP bias of 1.04. Research performed by Elrefai, West and Soudki (2006) on anchor fatigue loading, provided bias information on CFRP tendons, and indicated a marginally higher bias in the CFRP tendon of 1. 05 AFRP sample sets for COV calculation are summarize d in T able 3 1 0 The aggregate factors are based on the average tests from two independent studies. In 1995 (Arockiasamy & Sandepudi) determined the modulus of e lasticity for nine AFRP ARAPEE tendons (Group 1). Additional tests by Piraye h Gar et al. (2014) investigated the direct tension ch aracteristics of six ARAPEE AFRP bars (Group 2).
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47 Table 3 10 : AFRP Sample sets for COV Calculation Sample 1 Sample 2 Aggregate n 9 6 15 Mean 6484.0 10088 8285.750 Stdev 541.0 451.0 496.005 COV 0.083 0.045 0.060 e (5%) 0.055 0.036 0.030 n min 65 35 47 CFRP sample sets for COV calculation are summarized in Table 3 1 1 Similar to the AFRP set, the aggregate factors are based on the average tests from two independent studies. CFRP tests were performed by Yan, Miller and Nanni (1999) on 24 CFRP samples from various manufactures in the United States (Group 1), additional testing on C FRP bars were performed by Mice l l i and Nanni (2001) which provided test s on an additional 2 1 CFRP bars (Group 2). Table 3 11 : CFRP Sample sets for COV Calculation Group 1 Group 2 Aggregate n 24 21 45 Mean 19375 17270 18323 Stdev 983.2 593.6 788.4 COV 0.051 0.034 0.043 e (5%) 0.020 0.015 0.013 n min 40 27 34 FRP tensile design strength is modeled in tension based on a Weibull distribution: Eq. 3 12
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48 Parameters u are then estimated from the sample mean, taken as the bias modified nominal material strength and COV for the material indicated (Zureik, Bennett, & Ellingwood, 2006) Eq. 3 13 Eq. 3 14 The modulus of elasticity FRP is distributed by a lognormal distribution as described by Atadero and Karbhari (2007) A summary of the FRP properties used for the Monte Carlo Simulation are provided in Table 3 1 2 Table 3 12 : FRP Random Variable Parameter Variable Distribution Mean Bias COV AFRP Modulus Lognormal 9400 ksi 1 .04 0.06 Design Strength AFRP Weibull 181 ksi 1.04 0.06 CFRP Modulus Lognormal 20600 ksi 1 .04 0.043 Design Strength CFRP Weibull 326 ksi 1.05 0.043 Sampling Error: Due to the small sample sizes used for calculation of the COV of the material, there wa s some concern of the error within the simulation. In order to understand this error, 5 percent error calculations were completed to determine the error range based on the sample size available. Error was calculated based on AS TM E122 (ASTM International, 2000) methodology, in which the sampling error is defined by E quation 3 15 Eq. 3 15
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49 Where e is the sampling error in relation to the sample mean, E/ ; V is the Coefficient of Variation and n is the sample size. The 1.96 constant is used to describe the 5 percent error range, this value wou ld be 3.0 for 0.3 percent error or near certainty. Error for the AFRP sample set is approximately 3% of the sample me an, or half of the calculated COV ; with the sample size less than one third of the required minimum for 5% error. Error for CFRP was significantly less, approximately 1.3% of the sample mean, or about one third of the sample COV ; and a sample size approxim ately 30% higher than that required to maintain a 5% error margin. The CFRP sample set is considered acceptable due to the number of samples colle cted exceeding th e required to limit the error to within 5% of that calculated. However the AFRP sample set was not large enough, this combined with only the ARAPREE fiber bars being represented in the sample lead to further investigation into the effects of a variable COV value. Table 3 1 3 summarizes the COV used and how the variation affe cts the overall reliability index of the system. Table 3 13 : Compar ison of Reliability Index with V arying AFRP COV COV AFRP Span 9 Span 20 COV Compression COV R = 0. 7 COV Tension COV R = 0. 7 0.015 0.0790 0.197 4.29 0.084 0.199 4.16 0.03 0.0800 0.197 4.29 0.087 0.200 4.14 0.045 0.0800 0.197 4.29 0.094 0.203 4.09 0.06 0.0800 0.197 4.29 0.101 0.206 4.03 0.075 0.0800 0.197 4.29 0.111 0.211 3.94 0.09 0.0790 0.197 4.29 0.121 0.217 3.84 0.105 0.0800 0.197 4.29 0.131 0.223 3.74 The variation test was completed for two of the sample spans, one that exhibited compression controlled performance and the other that pr edicted tension controlled behavior. As expected, the variation in the tension strength COV has negligibl e effect to the
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50 compression controlled section. However, the tension controlled section did see substantial variation in the range of the r esistance COV as the AFRP COV changes. As the COV R changed, the variation in the calculated reliability index also changed, resulting in a reliability index varying from 4.16 to 3.74, a range of 0.42. This possible presence of error should be considered in the final selection of resistance factors that de scribe the tension performance of AFRP tendons in flexure. 3.6.3 Prestress Losses Prestressing loss calculation s were provided within the simulation to provide modeling of the effective and transfer prestressing forces. Short term and long term concrete losses associated with elastic shortening, creep and shrinkage of the concrete elements are calculated using the AASHTO simplified method This approach is consistent with the recommendations of ACI 440.4 Elastic shortening losses are calculated with E quation 3 16 and long term losses are calculated using E quations 3 17 and E quation 3 18 Note that all parameter s in the system for geometry and material properties are simulated as described in the previous sections. Relative humidity H is taken as 60 percent a verage, with a COV of 5% (Vu & Stewart, 2000) Elastic Shortening Losses: Eq. 3 16
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51 Long Term Losses: Eq. 3 17 Eq. 3 18 Eq. 3 19 Relaxation losses ( a re material dependent based on either AFRP or CFRP tendon materials. Since empirical representations of the relaxation are not yet available, an average value was selected for the long term relaxation of the tendons based on the observed stress loss under sustained load. Average relaxation losses were selected from Table 2.2 in ACI 440.4, with 12% used for AFRP tendons and 2% used for CFRP tendons. COV of the relaxation losses was taken as 0.30 and was based on general observation of loss variation in prest ressed elements by the JCSS Probabilistic Model Code (Technical University of Denmark, 2005) Table 3 15 summarizes the variable parameters specific to the calculation of prestress losses. Table 3 14 : Random Variable Loss Parameters Variable Distribution Mean Bias COV Reference Relative Humidity (H) Normal 60 1 0.75 Vu & Stewart (2000) Relaxation AFRP Normal 12% f jack 1 0.3 ACI 440.4 / JC S S Relaxation CFRP Normal 2% f jack 1 0.3 ACI 440.4 / JC S S
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52 3.7 Distribution of Resistance The formulation of the Coefficient of Variation and bias factors for the resistance model are key to determining the reliability of the structures. However, because the COV and bias are developed from a variety of different distribution types including normal, lognormal and Weibull; it is not clear which distribution function will best fit the simulat ed capacity. In order to select a representative distribution function, the cumulative distribution of 10,000 MCS data points were plotted based on 112 bins, and the average and standard deviation were calculated. The resistance MCS mean and standard devi ation were transformed for use in normal, lognormal, and Weibull distributions and random simulation was used to create sample sets based on each distribution. The cumulative distribution for each representative distribution was created based on the same b in size used above (Figure 3 1). Figure 3 1 : Probability Distribution of Resistance (Sample span 8, compression controlled) Using the Chi Squared method to determine fit the ideal resistance distribution was selected The C hi Squared test can generally be described by E quation 3 20
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53 Eq. 3 20 Where S imulated refers to the MCS data set, and Distributed refers to the data created by the trial distribution. The test is performed at each of the 112 bins, resulting in a relative best fit value categorized as the percent difference between the bin quantity Table 3 1 5 shows the calculated difference, with t he lowest calculated 2 value being the best fit distribution. In all cases, lognormal appears to provide the best prediction of the distribution for flexural capacity. Table 3 15 : Best Fit Comparison for Resistance Distribution Span 8 Compression Controlled Span 20 Tension C ontrolled Distribution AFRP CFRP AFRP CFRP Normal 3.92% 4.11% 3.95% 4.13% Lognormal 0.36% 0.70% 3.58% 2.52% Weibull 43.35% 46.65% 36.67% 38.90% 3.8 Analysis Parameters Analysis parameters are used to capture differences in the simplified design model and the actual performance of the element. Traditionally, different factors have been utilized for different structure types, such as reinforced concrete, composite steel, and prest ressed concrete. For FRP prestressed concrete, there is a limited number of research bridges available to determine a realistic analysis factor, and there are few full elastic or finite models available to explain the behavior of the system in greater deta il. For this reason, multiple options were considered for use as the analysis parameters for FRP prestressed bridges.
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54 3.8.1 Existing Analysis Parameter Three sets of analysis factors were considered for use with prestressed FRP concrete calibrations. The base factors considered were based on those for steel prestressed concrete, also considered were factors used for the calibrations of FRP reinforced concrete, and finally, analysis factors were considered based on the initial prestressed concrete strength facto rs selections. Table 3 1 6 summarizes each set of factors considered. For final calibrations, the analysis factors based on existing FRP prestressed concrete girder tests were utilized. Table 3 16 : Analysis Factors Considered for Resistance Model Reference Type Analysis Bias P Analysis COV COV P Nowak Reinforced Steel 1.00 0 .06 Nowak Prestressed Steel 1.01 0 .06 Behnam and Eamon Reinforced FRP 0.89 0.16 Burke and Dolan Prestressed FRP 1.10 0.18 3.8.1.1 Nowak Prestressed and Reinforced Concrete In the initial calibration of the AASHTO LRFD code, Nowak (1999) identified analysis factor for reinforced concrete and prestressed concrete girders. Reinforced concrete beams in flexure were assi gned a bias as 1.00 and COV of 6 percent, similarly, prestressed girders in flexure has a bias of 1.01 and a COV of 6 percent. Comparison of reinforced and prestressed analysis factors indicate very little difference. This is reasonable as both flexure cal culations are dependent upon strain compatibility within the beam, and load balancing between the concrete compression and the steel tension element.
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55 3.8.1.2 FRP Reinforced Concrete Recent studies performed by Behnam and Eamon (2013) on the reliability performance of Ductile Hybrid FRP (DHFRP) reinforced concrete identified analysis factors as part of the calibration effort reporting that a bias of 0.89 and a COV of 0.16 would adequately predict the performance of FRP reinf orced concrete designs. Based on the similarity that was identified by Nowak between steel mildly reinforced and prestressed concrete, it is reasonable to assume that utilizing the same bias and COV for FRP reinforced and FRP prestressed concrete is repres en tative of the actual condition 3.8.1.3 Existing Strength Factor Selection Study The existing strength reduction factor study presented by Burke and Dolan (2001) provides a comparison of 30 sample beams and how they relate to the nominal capacity equations used for flexural design as presented in Section 3.5. A ssuming actual concrete and FRP tendon strengths were used in the check calculations, it is reasonable to use the relationship between the teste d FRP prestressed beam capacity and the nominal beam capacity as a measure for the analysis factors. Referencing the results from Table 2 6, for all tested beams at = 1.0 the analysis bias can be taken as the mean, P = 1.097 and COV P = 18 percent. The COV slightly exceeds t hose observed for FRP reinforced sections, but the bias is significantly larger than that observed for FRP reinforced sections, matching the trend found that prestressed sections perform better than their reinforced counter parts. Because these values were obse rved on FRP prestressed girders and show trends consistent with the existing FRP reinforced analysis factors and the relationship between mildly reinforced
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56 and prestressed concrete girders, P = 1.097 and COV P = 18 percent was used for final calibrati ons. 3.9 Calibration Parameters Final resistance model bias and COV were selected based on the calculated Fabrication and Material factors and the A nalysis factors for FRP reinforced concrete Final resistance random variables are provided based on failure, materials, and girder types in Table 3 18 Note that these values do not include modification for analysis factors. Table 3 17 : FRP Random Variable by Girder Type AFRP CFRP Girder Failure Type COV FM COV FM Box Compression 0.083 1.101 0.07 8 1.10 0 Tension 0.083 1.087 Combined 0.083 1.101 0.0 81 1. 093 Tub Compression 0.079 1.129 Tension 0.101 1.085 0.086 1. 092 Combined 0.099 1.089 0.086 1. 092 I/BT Compression 0.080 1.108 Tension 0.10 1 1.100 0.086 1. 100 Combined 0.087 1.105 0.086 1. 100 Upon review of the girder results, it was determined that there was insufficient information to proceed with analysis by girder type as the simulation run s did not provide factors for both tension controlled and compression controlled failures for each material and girder type. A cursory review of the MCS output by girder type indicates a stable COV for compression controlled sections, regardless of prestres sing material of approximately 0.08. Bias for compression controlled section s had slight variation, indicating a range between 1.10 and 1.13. Tension control led behavior varied by material type. AFRP prestressed girder simula ted a COV of 0.101 for Tub or I type girders, and CFRP runs
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57 indicated slightly reduced COV of 0.86 for Tub and I girders. Tension controlled bias was approximately 1.09 regardless of girder type or material type for tension controlled behavior. Due to the lack of tension and compressi on controll ed information for each girder type, factors for calibrations were progressed based on the indicating failure type and the prestressing material. Table 3 1 9 provide s the base FM variable s from the MCS and the final resistance factors, modified with the analysis factors using E quation 3 2 and 3 3 Table 3 18 : Resistance Random Variable Including Analysis Factors Fabrication/Materials Analysis Resistance Failure Type C OV FM FM COV P P COV R R AFRP Compression 0.082 1.10 5 0.1 8 0 1.097 0.198 1.212 Tension 0.101 1.087 0.18 0 1.097 0.207 1.193 Combined 0.090 1. 097 0.18 0 1.097 0.201 1.204 CFRP Compression 0.07 8 1.10 0 0.18 0 1.097 0.196 1.207 Tension 0.086 1. 092 0.18 0 1.097 0.199 1.198 Combined 0.08 4 1. 094 0.18 0 1.097 0.199 1.200
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58 4 Calibration 4.1 Overview Chapter 2 of this document provided a n introduction to general reliability theory, outlining the groundwork for resistance factor calibration. Chapter 3 developed statistical distributions used to describe the resistance of AFRP or CFRP prestressed bridge girders. This chapter describes the load model used for calibration and the calibra tion procedures utilized. Calibrations are completed using a target beta method in which the calculated reliability index is compared to the desired reliability index, and varied for convergence. This process is then verified using a second direct calculat ion method. Calibration results are then provided based on long term construction and temporary works for both AFRP and CFRP prestressed girders, defined by different target reliability indices. A complete analysis of the results, including relationships b etween girder types, span ranges and span to depth ratios are provided. 4.2 Load Model Barker and Pucket (2007) defines loading on a bridge structure as either permanent or transient. Permanent loading, consist s of the gra vity induced loading that is unlikely to change over the service life of the bridge, specifically girders, deck, wearing surface, and parapets. Transient loading which is also based on gravity accelerations consists of those forces that are variable in nat ure. Transient loads can include pedestrians, vehicles, railways, winds and water. For the purposes of this calibration, gravity loads that induce flexure into girder elements are critical For the calibration loading, the AASHTO LRFD (2012) design code b asic load
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59 I load case, the main source of loadin g is permanent dead load and transient live load. The Strength I limit state is an ultimate condition, so load factors are used to account for th e variability within the permanent and transient loads Strength I load factors were selected for the calibration as they represent the original method used to model the representative structures. Additionally, AASHTO LRFD is the current baseline for all bridge designs within the United States, so the use of fa miliar load factors helps ensure the compatibility of the calibration with existing design codes. Load factors used within the calibration are provided in Table 4 1 Table 4 1 : Load Factors for Calibration Load Factor, i Cast in Place Dead Load (DC) 1.25 Wearing Surface (DW) 1.50 Live Load + Impact (LL+IM) 1.75 4.2.1 Dead Load Gravity induced dead loads within the Strength I limit state can be sep arated into two main categories; the component load (DC) and the w earing c ourse load (DW). AASHTO LRFD defines a maximum load factor for DC as 1.25 and for DW as 1.5. Both DC and DW loading categories were calculated based on the nominal dimensions provided in the base design drawings from the sample spans, using commonly accep ted unit densities for the materials (i.e. 150 pounds per cubic foot for concrete). Random variable simulation is based on normal distributions for all dead loads. For elements that are detail ed for precast made continuous construction self weight calcula tions of the girders
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60 and deck system are calculated as a simple span, additional loading from barriers and the wearing surface are calculated as continuous. The selection of bias and COV for dead load is based on the recommendations by Nowak (1999) however, loading from factory constructed elements and cast in place elements have been consolidated, using an average COV and bias for random variable generation. Table 4 2 summarizes the dead load random variable parameters for MCS simulation. 4.2.2 Live Load Live loads within the Strength I limit state are typically considered for truck and lane live load (LL), as well as truck impact load (IM). For both LL and IM, a 1.75 load factor is specified for design. Both LL and IM load ings were calculated based on the recommendations of AASHTO LRFD. Nominal loading was based on the HL 93 load combination, which consist s of a 0.64 kips per linear foot lane load, and either a single, or double, 72 kips design truck. Impact is taken as a c onstant 33% increase in the truck portion of the live load. Note that the use of 33% impact force is conservative; calculations performed in NCHRP 386 indicate a mean impact force equal to only 10% 15% of the applied live load Live load and impact is dist ributed to the design girders based on the girder distribution factors recommended in AASHTO LRFD. Table 4 2 : Dead Load and Live Load Random Variable Parameters Variable Distribution Mean Bias COV Reference Cast in Place Dead Load (DC) Normal As Calculated 1 0.10 Nowak (1999) Wearing Surface (DW) Normal As Calculated 1 0.25 Nowak (1999) Live Load + Impact (LL+IM) Normal As Calculated Varies 0.18 Nowak (1999)
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61 The selection of bias and COV for live load is based on the recommendations by Nowak (1999) The live load bias factors are allowed to vary by span length, however the COV is held at a constant 18% (Table 4 2 ). Table 4 3 summarizes the live load bias parameters for MCS calculations. The reader should note that the live load bias factors provided in Table 4 3 are based on a 75 year design life; reduced factors are av ailable for shorter design lives. Table 4 3 : Live Load B ia s Factors by Span Length for 75 year L ive L oads Span (ft) Bias Span (ft) Bias 50 1.321 120 1.291 60 1.321 130 1.269 70 1.311 140 1.237 80 1.315 150 1.229 90 1.311 160 1.239 100 1.312 170 1.242 4.3 Target Reliability Calibration The procedure described below is utilized for the calibration of the girders prestressed with both AFRP and CFRP tendons. Additionally, verification calibrations are performed on the steel prestressed designs and reinforced design to ensure calculated reliability factors are approximately 4.0, and 3.5 respectively the calculated result from the initial calibration of the AASHOT LRFD 1. Statistical D istributions: Each characteristic load was assigned a statistical distribution based on previous research. Load factors are based on AASHTO LRFD. 2. Select a T rial : Select ed a trial resistance factor for use. For this study, the resistance factor was varied between 1.0 and 0.4, at intervals of 0.05.
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62 3. Simulated Properties: Random variables were used to provi de random samples for each load and distributed based on the individual probability. Mean loading is calcu lated from the code analysis of the trial span. 4. Resistance Calculation: Use d the resistance COV and bias factors calculated in C hapter 3 to calculat e the overall resistance. 5. Reliability Index: Calculate d the sample reliability index using numerical calcul ations and simulated loading 6. Monte Carlo Simulation: Repeat ed steps 3 through 5 for 10,000 iterations t o create a representative average reliability index for the trail span 7. Increase Sample Size: In order to ensure adequate sample size, step 6 wa s repeated 1 0 times for an overall sample size of 1 00,000. 8. Calibrate Resistance Factor: Perform ed comparative analysis of the calculated reliability indices calculated to the target reliability index Select ed based on 100% inclusion of sample mod els, s elected value is rounded down to 0.05 for simplicity in design 9. Final Selection of Resistance Factors: Using the output from the target reliability index approach, final resistance factors were calculated based on methods consistent with the LRFD design code. 4.1.1 Reliability Index Calculation of the reliability index for calibration was completed using a n expanded version of equation 2 12, presented in Chapter 2 For FRP prestressed concrete the lo ad was normally distributed and the resistance is described using a lognormal distribution. For normally distributed load and resistance variable s the expanded reliability index wa s given in E quation 4 1
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63 Eq. 4 1 For a normal load and a lognormal resistance distribution function pattern, equation 4 1 must be adjusted to account for statistical means and coefficients of variations of the lognormal function and adjust ed for a lognormal loading distribution; equation 4 4 is given as ( Nowak 1999 ). Eq. 4 2 Defining the mean resistance, provided via MCS simulation, as the minimum load effect required, can be expressed as: Eq. 4 3 Eq. 4 4 Where is the simulated load effect, given as the un factored total load : Eq. 4 5
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64 F actors A and k were used to transform the mean resistance and resistance standard deviation to a lognormal distribution. Eq. 4 6 Eq. 4 7 4.1.2 Verification of the Reliability Index Calculations In order to confirm that the reliability index model is properly calculating the system reliability, trial runs were completed for the sample spans for standard steel prestressed concrete and reinforced co ncrete. The verification runs were completed using accepted bias and COV factors (Nowak, 1999) and the current specified resistance factors (AASHTO LRFD), for verification, all runs were considered tension controlled Table 4 4 summarizes the parameters used for verification. Table 4 4 : Statistical Parameters for Steel Reinforced Sections Steel Prestressed Girder Steel Reinforced Girder R 1.05 1.12 COV R 0.075 0.135 1.00 0.90 The calculated reliability indices for steel prestressed and reinforced concrete are presented in Table 4 5 and graphically by span length in Figure 4 1 Note that the data from Table 4 5 has been normalized across similar span lengths to clarify Figure 4 1. The avera ge reliability index of prestressed concrete is 4.2, and the minimum value is 3.7, exceeding 4.0 in most cases, consistent with calibrations performed by Nowak (1999) for prestressed concrete girders in simple spans. Ca librations for steel reinforced girders
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65 show similar trends, with the average being approximately 3.7, in excess of 3.5, consistent with initial AASHTO LRFD calibrations. Table 4 5 : Reliability Indices B ased on Resistance Factors from the LRFD Code Bridge Span (ft) Type Structure Depth (in) Prestressed = 1.0 Reinforced = 0.90 1 56 B 25 4.43 3.82 2 56 B 25 4.44 3.82 3 87 U 80 4.43 3.83 4 107 BX 44 3.68 3.41 5 108 BX 44 3.68 3.40 6 109 U 68.5 4.19 3.67 7 109 U 68.5 4.37 3.77 8 111 BT 50 4.38 3.78 9 118.9 BT 62 4.38 3.76 10 120 U 80 4.32 3.74 11 121 U 62.5 4.12 3.62 12 124 I 71.5 4.36 3.75 13 124 I 71.5 4.36 3.75 14 125 BX 46.75 4.43 3.79 15 125 U 80 4.28 3.70 16 126 U 74 4.07 3.59 17 127 BX 55.5 4.28 3.70 18 127 U 80 4.26 3.69 19 128.5 BT 71 4.34 3.73 20 132 BX 54 4.21 3.66 21 135 BX 49 4.19 3.64 22 135 BX 49 4.18 3.63 23 143 U 80 4.10 3.60 24 144 U 74 4.25 3.67 25 163.6 BT 92 4.17 3.63
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66 Figure 4 1 : Calculated Reliability Index vs. Span length, Steel Reinforced Concrete 4.1.3 Target Reliability Index Based on current calibrations for the AASHTO LRFD code, a baseline target reliability index of 3.5 wa s selected for calibrations. In addition to calibrating to 3.5, cali brations were also performed to 3.0 and 2.5. With T = 2.5 being applicable to temporary construction and existing structures, and T = 3.5 used for new construction. Target reliability indices of 2.5, 3.0, and 3.5 correspond to chances of failure of approximately 0.6%, 0.1%, and 0.02% respectively. For the implementation of the target reliability method, the calculated reliability index, based on the trial resistance factor, *, is compared to the target index using Equation 4 8 This approach is a basic modification of the approach used by Atadero and Karbhari (2006) for the selection of resistance factors for FRP strengthening. Eq. 4 8 By minimizing the results of Equation 4 8 the user can quickly compare resistance factors for various reliability indices and levels of safety. Figure 4 2 provides a sample of the
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67 graphs used for calibration of the individual spans t he low point of the curve represents the ideal resistant factor This particular set of curves represents trial span one f or a tension controlled (a) and compression controlled section s (b) Figure 4 2 : Calibration Curves for Resistance Factor Selection T = 4.0 Review of the curves indicates reveals that compression controlled reliability is similar for both AFRP and CFRP prestressed sections, however there is a small difference in tension controlled behavior. 4.1.4 Target Reli ability Results Full calibration curves based on the target reliability method are provided for each type of girder, span 1 is presented for box girders, span 10 for U girders, and span 2 5 for I girders Additionally curves are provided for span 4 a box girder structure which represents the controlling structure for the calibration Note that for all figures, figure (a) represents tension controlled behavior and (b) compression controlled behavior. It should be noted that calibrations all use load factors as specified in Table 4 1, and do not consider increased or reduced load factors for alternate design lives or loading conditions.
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68 Span 1 is an adjacent box girder structure that represents the shortest span in the trail. The span to depth ratio is relati vely high at 26.9, which is typical for adjacent box girder construction. Curves are presented for target reliability indices of 3.5, 3.0 and 2.5 Figures 4 3 through 4 5 indicate CFRP tension factors of 0.80, 0.90, 1.00 and AFRP tension factors of 0.81, 0 0.83, 0.92, and 1.00 for both materials. Figure 4 3 : T = 3.5 Figure 4 4 : T = 3.0
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69 Figure 4 5 : T = 2.5 Span 10 represents an intermediate span at 120 fee t utilizing U girders. The span to depth ratio is relatively is 18.0. Curves are presented for target reliability indices of 3.5, 3.0 and 2.5. Figures 4 6 through 4 8 indicate CFRP tension factors of 0.8 0 0. 89 1.00 and AFRP tension factors of 0.7 7 0. 87, 0.9 6 are approximately 0.82, 0.91, and 1.00 for both materials. Figure 4 6 : T = 3.5
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70 Figure 4 7 : T = 3.0 Figure 4 8 : T = 2.5 Span 25 is a Colorado Department of Transport ation Bulb Tee girder bridge that represents the longest span in the trail, at 164 feet. The span to depth ratio is 21.3. Curves are presented for target reliability indices of 3.5, 3.0 and 2.5. Figures 4 9 through 4 11 indicate CFRP tension factors of 0. 79 0.88 0.97 and AFRP tension factors of 0. 76 0.85 0.95 Compression factors are approximately 0.80, 0.89, and 0.98 for both materials.
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71 Figure 4 9 : Span 25 Calibration Curves for Res T = 3.5 Figure 4 10 : Span 25 T = 3.0
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72 Figure 4 11 : Span 25 T = 2.5 Span 4 is an adjacent box girder structure with an intermediate span of 107 feet The span to depth ratio is 29.2 and a high skew over 50 degrees is also considered Span 4 represented the controlling structure in the study, providing a calculated reliability index that was consistently 5% below the average of the set Curves are presented for target reliability indices of 3.5, 3.0 and 2.5. Figures 4 12 through 4 14 in dicate CFRP tension factors of 0. 79 0.8 6 0.9 5 and AFRP tension factors of 0.7 5 0.8 4 0.9 4 2.5 respectively. Compression factors are approximately 0. 79 0.8 7 and 0.95 for both materials.
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73 Figure 4 12 : Span 4 T = 3.5 Figure 4 13 : Span 4 T = 3.0
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74 Figure 4 14 : Span 4 Calibration Curves for T = 2.5 Full results of calibrated resistance factors using the target reliability method are provided for T = 3.5, 3.0, and 2.5 in Table 4 6 Table 4 7 and Table 4 8 respectively. The values provided are based on interpola ted curves and their intercept at or near zero A cursory review indicates some trends in the information; as the reliability target increases the resistance factors decrease, as the span increases the resistance factors generally reduce, CFRP provides sim ilar performance levels in both compression and tension controlled failure modes, and CFRP slightly out performs AFRP in tension. Additional analysis of the results can be found in S ection 4.5. Full reliability index calculation at trial resistance factors can be found in Appendix A.
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75 Table 4 6 : T = 3.5 CFRP AFRP Span Span (ft) Span/Depth (in/in) Girder Type T C T C 1 56 26.9 B 0.81 0.83 0.79 0.83 2 56 26.9 B 0.82 0.83 0.80 0.83 3 87 13.1 U 0.82 0.83 0.80 0.83 4 107 29.2 BX 0.79 0.79 0.75 0.79 5 108 29.5 BX 0.79 0.79 0.76 0.79 6 109 19.1 U 0.80 0.80 0.77 0.80 7 109 19.1 U 0.81 0.82 0.78 0.82 8 111 26.6 BT 0.81 0.82 0.78 0.82 9 118.9 23.0 BT 0.81 0.82 0.77 0.82 10 120 18.0 U 0.80 0.82 0.77 0.82 11 121 23.2 U 0.79 0.81 0.77 0.81 12 124 20.8 I 0.81 0.82 0.77 0.82 13 124 20.8 I 0.81 0.82 0.78 0.82 14 125 32.1 BX 0.81 0.83 0.78 0.83 15 125 18.8 U 0.80 0.81 0.77 0.81 16 126 20.4 U 0.80 0.81 0.78 0.81 17 127 27.5 BX 0.80 0.82 0.77 0.82 18 127 19.1 U 0.80 0.81 0.78 0.81 19 128.5 20.8 U 0.81 0.82 0.78 0.82 20 132 29.3 BX 0.80 0.81 0.77 0.81 21 135 33.1 BX 0.80 0.81 0.76 0.81 22 135 33.1 BX 0.79 0.80 0.76 0.80 23 143 21.5 U 0.79 0.80 0.76 0.80 24 144 23.4 U 0.79 0.80 0.77 0.80 25 163.6 21.3 BT 0.79 0.80 0.76 0.80
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76 Table 4 7 : T = 3.0 CFRP AFRP Span Span (ft) Span/Depth (in/in) Girder Type T C T C 1 56 26.9 B 0.90 0.92 0.88 0.92 2 56 26.9 B 0.90 0.92 0.88 0.92 3 87 13.1 U 0.90 0.92 0.88 0.92 4 107 29.2 BX 0.86 0.87 0.84 0.87 5 108 29.5 BX 0.86 0.87 0.84 0.87 6 109 19.1 U 0.88 0.90 0.86 0.90 7 109 19.1 U 0.90 0.91 0.87 0.91 8 111 26.6 BT 0.90 0.91 0.87 0.91 9 118.9 23.0 BT 0.90 0.91 0.87 0.91 10 120 18.0 U 0.89 0.91 0.87 0.91 11 121 23.2 U 0.88 0.89 0.85 0.89 12 124 20.8 I 0.89 0.91 0.87 0.91 13 124 20.8 I 0.89 0.91 0.87 0.91 14 125 32.1 BX 0.90 0.91 0.87 0.91 15 125 18.8 U 0.89 0.90 0.86 0.90 16 126 20.4 U 0.88 0.90 0.86 0.90 17 127 27.5 BX 0.89 0.90 0.86 0.90 18 127 19.1 U 0.88 0.90 0.86 0.90 19 128.5 20.8 U 0.89 0.90 0.87 0.90 20 132 29.3 BX 0.88 0.90 0.86 0.90 21 135 33.1 BX 0.88 0.89 0.85 0.89 22 135 33.1 BX 0.87 0.89 0.85 0.89 23 143 21.5 U 0.87 0.89 0.85 0.89 24 144 23.4 U 0.87 0.88 0.86 0.89 25 163.6 21.3 BT 0.88 0.89 0.85 0.89
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77 Table 4 8 : T = 2.5 CFRP AFRP Span Span (ft) Span/Depth (in/in) Girder Type T C T C 1 56 26.9 B 1.00 1.00 0.97 1.00 2 56 26.9 B 1.00 1.00 0.97 1.00 3 87 13.1 U 1.00 1.00 0.97 1.00 4 107 29.2 BX 0.95 0.95 0.94 0.95 5 108 29.5 BX 0.95 0.95 0.94 0.95 6 109 19.1 U 0.98 1.00 0.95 1.00 7 109 19.1 U 1.00 1.00 0.96 1.00 8 111 26.6 BT 1.00 1.00 0.96 1.00 9 118.9 23.0 BT 1.00 1.00 0.96 1.00 10 120 18.0 U 1.00 1.00 0.96 1.00 11 121 23.2 U 0.97 1.00 0.95 1.00 12 124 20.8 I 1.00 1.00 0.96 1.00 13 124 20.8 I 1.00 1.00 0.96 1.00 14 125 32.1 BX 1.00 1.00 0.96 1.00 15 125 18.8 U 0.98 1.00 0.95 1.00 16 126 20.4 U 0.98 1.00 0.96 1.00 17 127 27.5 BX 0.98 1.00 0.95 1.00 18 127 19.1 U 0.98 1.00 0.95 1.00 19 128.5 20.8 U 0.99 1.00 0.96 1.00 20 132 29.3 BX 0.97 1.00 0.95 1.00 21 135 33.1 BX 0.97 0.99 0.95 0.99 22 135 33.1 BX 0.97 0.99 0.95 0.99 23 143 21.5 U 0.96 0.98 0.95 0.98 24 144 23.4 U 0.96 0.97 0.95 0.98 25 163.6 21.3 BT 0.97 0.98 0.95 0.98
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78 4.4 Direct Calculation Method In order to verify the results determined by the target reliability calculation T = 3.5 traditional calculation of the resistance factor will be performed. The direct calculation is completed assuming both th e load and resistance follow an approximately normal distribution, so the reliability index is described by Equation 4 1. Using the following approximation for the square root term (Barker & Puckett, 2007) : Eq. 4 9 Substituting Equation 4 9 into Equation 4 1, and setting the reliability index equa tion to the target index, Equation 4 1 becomes: Eq. 4 10 Because it is know that R and Q are function of the nominal load or capacity and the bias (Equation 4 3), and that standard deviation is a function o f the COV and the mean (Equation 2 6). Equation 4 10 can be expanded to: Eq. 4 11 Considering the relationship between Equation 4 1 1 and Equation 2 3, the load factors and the res istance factors can be taken as: Eq. 4 12 Eq. 4 13
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79 Equations 4 9 through 4 1 3 were developed assuming that both the resistance and the load described by a normal distribution. Since our system is described by a normal loading and a lognormal resistance, k is substituted for resulting in the modification to equation 4 12 (Nowak 1999): Eq. 4 14 Knowing that the original AASHTO LRFD code was calibrated using k = 2.0 for the load factors, Equation 4 1 4 can be further simplified to provide direct comparison with the existing code, Eq. 4 15 Table 4 9 : Direct calculated Resistance factors using k = 2.0 Failure Type COV R R AFRP Compression 0.198 1.212 0.73 Tension 0.207 1.193 0.70 Combined 0.201 1.204 0.72 CFRP Compression 0.196 1.207 0.73 Tension 0.199 1.198 0.71 Combined 0.199 1.200 0.72
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80 4.5 Recommended Resistance Factors A review of the results from sections 4.3 and 4.4 indicate that compression performance of the girders is very similar regardless of the material used for reinforcement. This is consistent with observations of FRP reinforced concrete from ACI 440.1 (2006) Table 4 9 calculated factors appear to be conservative when compared to the calibrated factors but are within approximately 0.05 of the actual factors. For this reason it was determined that factors would be based on t he calibrated factors, selected for 100% inclusion of the trial spans, and rounded to 0.05 for consistency with current design codes. The calculated resistance factors showed general trends to decrease as the span increased (Figure 4 15), and the decreases as the span to depth ratios increased (Figure 4 16). This relationship is also applicable to the reliability index, and is consistent with similar trends were observed in steel reinforced sections presented in Figure 4 1. Figure 4 15 : Calculated Resistance factors vs. Span length
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81 In addition the decreasing trends between span length and span to depth ratio, the similarity between the CFRP and AFRP tension controlled resistance factor was limited to a rang e of less than 0.05 in all cases. This minor difference between the two materials, while consistent, can be ignored and an all inclusive tension controlled based factor can be selected for use. This is consistent with the current factors for steel reinforc ed sections, as similar to the idea that the resistant factors would not change when using different grades of steel, or when using prestressing bar, in lieu of wire strands, for steel prestressed concrete. Figure 4 16 : Calcul ated Resistance factor vs Span/Depth ratio The observation of decreasing reliability of the structure combined with the variation in the simulated values and reliability results lead to a conservative selection of recommended resistance factors, with the author targeting 100% inclusion of the data. Final recommended reduction factors are presented in Table 4 10 It is recommended that tension controlled behavior and compression controlled behavior be separated as currently recommended in ACI 440.4. Tension controlled sections with a net tensile strain of 0.005 or
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82 more should use th e tension controlled factor and sections with a tensile strain of 0.002 or lower should utilize the compression controlled recommendations. The strength reduction factor should be varied linearly when tensile strains are between 0.002 and 0.005. Table 4 10 : Recommended Strength Reduction Factors Material Strain limit Flexure Condition FRP* 0.75 Tension Controlled 0.80 Compression Controlled *Only Aramid and Carbon fiber tendons are recommended for prestressed applications
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83 5 Long Term Deflection Factors 5.1 Overview While Chapters 2 through 4 of this report investigated the effects of reliability on the flexural resistance factors for AFRP and CFRP prestressed girder, Chapter 5 is dedicated to the service level deflections of FRP prestressed girders. The Precast Concr ete Institute (PCI) Design Handbook for Precast and Prestressed Concrete recommends the use of multipliers for the estimation of long term deflections for use with prestressed concrete beams or girders The factors provide a convenient way to quickly estim ate the long term deflections caused by the effects of prestressing as well as load effects. This chapter presents the Monte Carlo S imulation approach used to update the deflection multipliers as provided by Martin (1977) for AFRP and CFRP prestressed concrete. The use of MCS for the determination of deflection factors aligns the statistical based methods of LRFD calibrations with service level factors based on the simplifications of material properties. MCS based multip liers are recommended to account for the material and geometrical distributions of the beam and slab system properties. 5.2 Design Applicability Current m ultipliers utilized for the deflections calculations for bridge girders and precast beams are based on a practical analysis of typical material performance as observed through the experience of the originator. Martin (1977) recommends two sets of factors to account for prestressed beams without composite topping and with c omposite topping, presented in Table 5 1 these same multipliers can be found in the 7 th Edition of the PCI Design Handbook (2014) These values are further broken down into long term and
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84 erection level factors to a llow the user a simple way to predict beh avior during key stages of the structure life. Table 5 1 : Steel Prestressed Beam Recommended Deflection Multipliers Without Composite Topping With Composite Topping Erection (1) Deflection G irder S elf W eight 1 .85 1. 85 (2) Camber Due to P restress 1.8 0 1.8 0 Final (3) Deflection Girder S elf W eight 2. 7 0 2.40 (4) Camber Due to P restress 2. 45 2. 2 0 (5) Deflection Superimposed Dead load 3.00 3.00 (6) Deflection Composite T opping 2. 3 0 The continued use of the above factors is a testament in itself to the simplicity and effectiveness of the approach. However, Martin is careful to address the inherent unpredictability of long term deflections and cambers, specifically noting that ti me Regardless, d esigners continue to use these factors for basic applications for simple span prestressed, despite the availability of more accurate methods such as the ACI 209 model and the CEB FIP model Due to the simplicity of the approach and long standing use, it is important to ensure the values are reasonable for FRP prestressed concrete as well as steel reinforced concrete. 5.3 Simulation Methodology Similar to the applications used in Cha pter 3 and Chapter 4, MCS forms the basis of the deflection multiplier simulations. The process for simulating the individual factors is briefly described by the following steps.
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8 5 1. Sample Span Properties : Isolate key information from 25 trail spans and simul ations performed in Chapter 2. Collection of geometry and material properties will for m the basis for the statistical means for simulation 2. Statistical D istributions: Each characteristic property was assigned a statistical distribution based on previous re search or performed calculations. 3. Simulated Properties: Random variables were used to provide random samples for each property, distributed based on the individual probability. 4. Calculate Sample Multipliers: Using the methodology and equations presented by Martin (1977), calculate d the deflection multipliers based on the simulated system properties. 5. Monte Carlo Simulation : Repeat ed steps 5 through 8 for 10 0 ,000 iterations to create a representative deflection multiplier based on the average of the simulated multipliers. 6. Deflection Multipliers : Provide d recommended deflection multiplier for CFRP and A FRP Prestressed girder. Identified factors for non composite girders, thin composite topping construction, and reinforced composite slab construction. 5.4 Determination of Multipliers Martin (1977) proposed a simple method for determining multipliers to predicting long term deflections of prestressed structures. That method is outlined in this section with minor modificati ons to adjust the original method with updated equations. I nitial calculations for the prestressed concrete deflection multipliers were based on an equation from ACI 318 71, describing the long term deflections of mildly reinforced concrete beams. I n revie w ing the current release of ACI 318 11, it was determined that the previous equation was no longer in use and that an updated version was used to describe
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86 the relationship between long term deflections and immediate deflections. Equation 5 1 provides the basis for the multiplier deflections. Eq. 5 1 Where is a constant that describes the effect compared to time, take n as 2.0 for long term effects (5+ years) and 1.0 for short term effects (Les s than 3 months), and the compression reinforcement ratio at mid span. For long term deflections from dead load, the base factor is modified by the strength gain in the concrete, based on the release m odulus of elasticity and the 28 day elasti city. Eq. 5 2 Upward displacements for the final camber or deflection are defined as a function of the prestressing force. Initial displacements are calculated at release then modified for the final effective prestressing force ; the long term prestr essing factor can be defined using Equation 5 3. Where P 0 represents the prestressing force immediately following release, and P represents the final effective prestressing force. Eq. 5 3 Erection cambers and deflections can be defined as a portion of the final factor multipliers that are described above. Martin utilized a constant 50% to define the amount of deflection at erection; however for this calculation a variable is used. The deflection from dead load at the time of erection can be defined as: Eq. 5 4 The camber component during construction is a multiplier using the erection dead load fa ctor as a basis, combined with the average prestressing force assuming a linear
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87 relationship between pretensioning release and the final effective force. The upward component from prestressing at erection is defined as: Eq. 5 5 For the application of long term super imposed dead loads, the base factor can be used to describe the long term effects. Eq. 5 6 The use of composite construction is common for precast beams and girders. Because composite construction modifies the stiffness of the system the factors provided in E quations 5 2 through 5 6 must be modified. Equations 5 7 and 5 8 provide composite mult ipliers for the load and prestressing effects, respectively. Eq. 5 7 Eq. 5 8 For long term deflection of the topping, Eq. 5 9
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88 It is important to note, that these factor s are intended to be additive to the initial calculated deflections, so to use them as direct multipliers, they must be increased by 1.0 in most cases. Due to the numbe r of factors, it can be difficult to keep track of how the factors relate to the recommended values. Table 5 2 is provided to reflect the format of Table 5 1 but has been updated to reflect the factor variables instead of the actual factors. Table 5 2 : Deflection Multipliers Variable Summary Without Composite Topping With Composite Topping Erection (1) Deflection G irder S elf W eight 1+ de 1+ de (2) Camber Due to P restress 1+ p e 1+ p e Final (3) Deflection Girder S elf W eight 1+ df 1+ df (4) Camber Due to P restress 1+ pf 1+ pf (5) Deflection Superimposed Dead load 1+ sd 1+ sd (6) Deflection Composite T opping 1+ t The factors provided above assume that the composite topping is constructed as a thin, 2 to 3 inch, composite slab that does not change the reinforcement ratio of the system. However, for most bridge girder s with composite slabs the slab system contain s a substantial amount of reinforcement, resu lting in composite compression reinforcement ra tios between .003 and .0006 For this reason, an additional set of multiplier ; was calculated in which the increased reinforcement ratio is accounted for ; this new set of data
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89 5.5 Random Variables The deflection prediction equations provided in Section 5.4 are based on a number of variables t hat can be simulated based on statistical distribution s Similar to the methods used for capacity prediction in Chapter 3, each individual variable from the compre ssive strength of concrete to the effective prestres s force is assigned a mean, COV, and distribution function based on existing literature This allows the Monte Carlo S imulation to predict the variability of the entire system, resulting in simulated mult ipliers that capture the natural variability of the girders. Table 5 3 : Random Variable Factors for Deflection Simulation Variable Distribution Mean Bias COV Reference ci Normal Based on Material Varies 0.1 Kwon et al, 2010 c Normal Based on Material Varies 0.1 Kwon, 2010 E ci Normal 1 0.1 Kwon, 2010 E c Normal 1 0.1 Kwon, 2010 E FRP Log n ormal Based on Material 0.2 Atadero, 2008 E s Logn ormal 28500 ksi 1.04 0.024 Kwon, 2010 A ps,FRP Normal Based on Material 0.05 Atadero, 2008 A ps Normal 28500 ksi 0.0125 Kwon, 2010 d, d e b Normal As Designed 0.03 Okeil, 2002 For the application of the erection factors the simulation allowed the deflection adjustment to randomly vary from 40% of the long term deflection to 60% of the long term deflections. This corresponds to construction times between about 30 from precasting to as long as 6 months prior to construction.
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90 5.6 Recommended Factors Use of Monte Carlo simulation methods have allowed for the modeling of numerous situations that are possible in the performance of individual girders, accounting for variations in reinforcement placement, material strengths, and even time from precasting t o erection. The MCS models predicted factors for all 25 trial spans, for both CFRP and AFRP, the results were then averages across all spans to create representative factors for general use. In addition to the non composite factors and the factors for gird ers with composite topping, additional factors were provided for composite reinforced slabs that are common in bridge applications. The results of the simulation run are provided in Table 5 3 for AFRP prestressed girders, and Table 5 4 for CFRP girders. T able 5 4 : Recommended Long Term Multipliers for AFRP prestressed Girders Without Composite Topping With Un Reinforced Composite Topping With Reinforced Structural Slab Erection (1) Deflection (downward) Girder Self weight 1. 85 1. 85 1. 85 (2) Camber (upward) Prestress 1.7 0 1 .70 1. 70 Final (3) Deflection (downward) Girder Self weight 2.70 2.35 2. 35 (4) Camber (upward) Prestress 2. 00 1.90 1.85 (5) Deflection (downward) Superimposed Dead load 2.90 2. 90 2. 85 (6) Deflection (downward) Composite topping 2.05 2.0 5
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91 Table 5 5 : Recommended Long Term Multipliers for CFRP prestressed Girders Without Composite Topping With Un Reinforced Composite Topping With Reinforced Structural Slab Erection (1) Deflection (downward) Girder Self weight 1. 85 1. 85 1. 85 (2) Camber (upward) Prestress 1. 80 1. 80 1.80 Final (3) Deflection (downward) Girder Self weight 2.70 2.35 2. 30 (4) Camber (upward) Prestress 2.55 2.25 2. 20 (5) Deflection (downward) Superimposed Dead load 2.90 2.90 2. 85 (6) Deflection (downward) Composite topping 2.05 2.0 5 Review of the simulation results indicate a significant difference in the long term prestress camber results, investigating this variation, it was determined that the reduction in prestressed camber when using AFRP beams was a direct result of the additional losses from relaxation. When the effective prestressing force was input into Equations 5 3 or 5 5, the factors were proportionally decrea sed.
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92 6 Conclusions, recommendations and future research 6.1 Conclusions With over 80 years of history, it is only in the last 15 years that the use of FRP materials has become truly feasible for bridge applications. In part due to the ever increasing requirement to make structures last longer, with current American Association of State Highway and Transportation Officials (AASHTO) LRFD Bridge Design Specifications, requiring that structures be designed for a 75 year design life; but also in the development of cost effective production techniques, and the introduction of CFRP ma terials, which bring the cost and strength of FRP materials closer to traditional steel reinforcement. The recommendations of ACI 440.4 provide comprehensive recommendations on design methodology, predictive equations, and recommendations for strength an d service limits states. The recommendations presented in the present study suggest calibrated flexural resistance factors for FRP prestressed bridge girders for use in accordance with the AASHTO LRFD Bridge Design Specification (AASHTO LRFD) for commonly used bridge girder sections. Additionally, traditional Precast Prestressed Concrete Institute (PCI) deflection multipliers were investigated for FRP prestressed applications and recommended values are provided. 6.2 Future Research Reliability based design is h eavily dependent on knowledge of the performance of the materials used to resist specified design loading. And thus this report is only as accurate as the Coefficient of Variation s, Bias Factors, and strength models that are available. Further research sho uld be considered based on the following recommendations
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93 6.2.1 M aterial Properties and Reporting The most limiting portion of the provided simulations and recommend strength reduction factors is the availability of existing research on the materials and their pe rformance. Further research should be considered to develop a baseline or database of FRP material properties and tests. Additionally, standards should be developed for testing procedures and reporting of c ommercially available materials, requiring that gu aranteed tensile strength and standard deviation are reported for all composite materials. Unlike steel bars where the strength of the material is normalized across manufacturers, FRP composites can vary widely from one manufacturer to another. While this study suggest that the material type does not directly affect the reliability of the structure, normalization of the material would allow designers to more confidently specify FRP reinforcement without requiring extensive research prior to construction. 6.2.2 A nalysis Factors Limited data is currently available for the analytical performance of the predictive equations used for flexural design of FRP prestressed girders. This report uses existing journal articles to extrapolate analysis factors, but a detailed finite analysis of multiple structures, or physical testing intended to verify the nominal equations These factors have a significant impact on the reliability analysis, and should be updated in future calibrations. 6.2.3 Serviceability This study assumes there is no relationship between ultimate flexural strength and the effects of crack width on the flexural performance of the element. Prior research has shown that FRP reinforcement is susceptible to down failure when cracking occurs i n flexural
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94 regions; however, this is not considered in this calibration. The effects of serviceability failures on the ultimate performance FRP prestressed concrete should be investigated to verify the recommended design equations and confirm the analysis factors indicated in section 5.2.2. 6.2.4 FRP Relaxation Losses Loss parameters in this report are based on observed losses as presented in ACI 440.4, and assumed distributions based on general models describing prestress losses. Additional research into the statistical distribution of relation losses, as well as the development of empirical equations to describe the relaxation of FRP tendons would help to simplify the design procedure and more consistently predict the overall long term losses. 6.2.5 FRP Harping and Debonding Harping and debonding of prestressing tendons is important to designing and fabricating cost effective designs with many steel reinforced prestressed girder utilizing both methods to help minimize overall structure depth There is very little i nformation on the performance of FRP tendons using debonding and limited information harping performance. Future research is required to determine the performance of debonded tendons, and determine effective harping methods that minimize the stress gains a t the harp locations. 6.2.6 Selecting Target Reliability Indices This study assumes that the reliability indices used in the original calibration of the AASHTO LRFD code are applicable to FRP prestressed girders. While the use of 3.5 and 2.5
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95 are generally accept ed for new construction and temporary construction respectively, it may be necessary to consider different reliability indices to ensure ductility or design life. 6.2.7 Prestressed Slabs This study investigates the reliability of prestressed bridge girders; howe ver, trends in the reliability results indicated a reduction in reliability as the span to depth ratio increased. Prestressed thin slabs should be considered independently and the current recommended factors should be considered with care in relatively thi n elements. 6.3 Conclusion This work has provided updated LRFD resistance factors for the flexural design of FRP prestressed bridge girder. Additionally, simulations have been done to update the current PCI long term deflection multipliers for FRP materials. T he simulations and calibrations are based on the latest available material; however, there is significant room for additional research to refine the current approach. The methods utilized in this report align the methodology for flexural design of FRP pres tressed concrete with the current LRFD codes, founding the recommendation in reliability analysis in lieu of limited experience.
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96 NOTATIONS A s = Area of tensile reinforcement (in 2 ) A s = Area of compression reinforcement (in 2 ) A ps = Area of prestressing ( in 2 ) b = Width of compression Face (in) c = Depth to neutral axis (in) COV = Coefficient of Variation COV FM = Coefficient of Variation Fabrication and Materials COV P = Coefficient of Variation Analysis COV R = Coefficient of Variation Resistance DC = Component Dead Load DW = Wearing Surface and Utility Dead Load E f = Modulus of elasticity of fibers (ksi) E p = Modulus of elasticity of the tendon (ksi) E c = Modulus of elasticity of concrete (ksi) f = Stress (ksi) f h = Induced stress from tendon harping hold downs f mcg = Available Stress at the CG of the prestressing reinforcement (ksi) f ecg = Effective pres tress at the CG of the prestressing reinforcement (ksi) = Concrete Compressive Strength (ksi) = Concrete Compressive Strength at Rele ase (ksi) IM = Live Load Impact FS = Factor of Safety LL = Live Load M n = Nominal Moment N = Number of Samples = Mean Resistance R = Resistance R h = Radius of Curvature for the harping saddle R n = Nominal Resistance R t = Radius of the tendon = M ean Load Q = Load Q n = Nominal Load = Statistical Average x n = Nominal Value (Compared to the Statistical Average) = Statistical Average LOG = Lognormal Average = Lognormal Standard Deviation = Bias = Reliability Index (Safety Index) 1 = Concrete Compression block factor
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97 cu = Concrete Ultimate Compressive Strain (0.003) pu = Available ultimate strain in prestressing material pe = Effective strain in prestressing after losses FM = Fabrication and Material Bias P = Analysis Bias R = Re sistance Bias b = Balanced Reinforcement Ratio cg = Reinforcement Ratio based on depth to the CG of prestressing reinforcement = Standard Deviation
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98 BIBLIOGRAPHY AASHTO LRFD Bridge Design Specifications. (2012). Washington DC: American Association of State Highway and Transportation Officials. ACI Committee 318. (2011). Building Code Requirements for Structural Concrete. Farmington Hills: American Concrete Institute. ACI Committee 440. (2004). Prestressin g Concrete Structure with FRP Tendons. American Concrete Institute. ACI Committee 440. (2006). Guide for the Design and Construction of Structural Concrete Reinforced with FRP Bars. American Concrete Institute. ACI Committee 440. (2008). Specification for Carbon and Glass Fiber Reinforced Polymer Bar Materials for Concrete Reinforcement. Farmington Hills: American Concrete Institute. Arockiasamy, M., & Sandepudi, K. (1995). Active Deformation Control of Bridges prestressed iwht Aramid Fiber Reinforced Plast ic Cables. Boca Raton: Center for Infrastructure and Constructed Facilities. ASTM International. (2000). Standard Practice for Calculating Sample Size to Estimate With a Specified Tolerable Error. West Conshohocken: ASTM International. Atadero, R. A., & Ka rbhari, V. M. (2007). Calibration of resistance factors for reliability based design of externally bonded FRP composites. Composites 665 679. Atadero, R., & Karbhari, V. M. (2006). Development of Resistance Factors for LRFD Design for RFRP Strengthening o f Reinforced Concrete Bridges. La Jolla: University of California, Snadiego. Barker, R. M., & Puckett, J. A. (2007). Design of Highway Bridges An LRFD Approach. New Jersey: John Wiley & Sons, Inc.
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99 Behnam, B., & Eamon, C. (2013). Resistance Factors for Duct ile FRP Reinforced Concrete Flexural Members. Journal of Composites for Construction 566 573. Burke, C. R., & Dolan, C. W. (2001). Flexural Design of Prestressed Concrete Beams using FRP Tendons. PCI Journal 76 87. Dolan, C., & Burke, C. (1996). Flexural Strength and Design of FRP Prestressed Beams. Advanced Composite Materials in Bridges 383 390. Elrefai, A., West, J., & Soudki, K. (2006). Performance of CFRP tendon anchor assembly under fatigue loading. Composite Structures 352 360. Ghosn, M., & Moses F. (1986). Reliability Calibration of Brdige Design Code. Journal of Structural Engineering 745 763. Grace, N. F., Roller, J. J., Navaree, F. C., Nacey, R. B., & Bonus, W. (2004). Load Testing a CFRP Reinforced Bridge. Concrete International 1 7. Kwon, O. S., Kim, E., Orton, S., Salim, H., & Hazlett, T. (2011). Calibration of the Live Load Factor in LRFD Design Guidelines. Jefferson City: Missouri Department of Transportation. Martin, L. D. (1977). A Rational Method foro Estimating Camber and Defle ction of Precast Prestressed Members. PCI Journal 100 108. Micelli, F., & Nanni, A. (2001). Mechanical Properties and Durability of FRP Rods. Center for Infrastructure Engineering Studies. University of Missouri Rolla. Nowak, A. S. (1999). Calibration of LRFD Bridge Design Code. Washington, D.C.: Transportation Research Board. Okeil, A., El Tawil, S., & Shahawy, M. (2002). Flexural Reliability of Reinfroced Concrete Bridgeg GIrders Strengthened with CFRP Laminates. Journal of Bridge Engineering 290 299.
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100 P ark, S. Y., & Naaman, A. E. (1999). Shear Behavior of Concrete Beams Prestressed with FRP Tendons. PCI Journal 74 85. Pirayeh Gar, S., Hurlebaus, S., Mander, J., Cummings, W., Prouty, M., & Head, M. (2014). Sustainability of Transportation Structures usin g Composite Materials to Support Trand and Growth. College Station: Texas A&M Transportation Institute. Ribeiro, S., & Diniz, S. (2013). Reliability based design recommendations for FRP reinforced concrete. Engineering Structures 273 283. Szerszwn, M., & Nowak, A. (2003). Calibration of design code for buildings (ACI 318): Part2 Reliability analysis and resistance factors. ACI Structural Journal. Technical University of Denmark. (2005, April). Probabilistic Model Code. Retrieved from http://www.jcss.byg.dt u.dk/Publications/Probabilistic_Model_Code.aspx Vu, K., & Stewart, M. (2000). Structural reliability of concrete bridges including improved chloride induced corrosion models. Structural Safety. Yan, X., Miller, B., & Nanni, A. (1999). Characterization of C FRP Rods used as Near Surface Mounted Reinforcement. Structural Faults and Repair Conf. 1 12. Zhang, B., Benmokrane, B., Chennouf, A., Mukhopadhyaya, P., & El Eafty, A. (2001). Tensile Behavior of FRP Tendons for Prestressed Ground Anchors. Journal of Composites for Construction 85 93. Zureik, A., Bennett, R., & Ellingwood, B. (2006). Statistical Characterization of Fiber Reinforced Polymer Composite Material Properties for Structural Design. ASCE Journal of Structural Engineering.
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101 A Calculate d Reliability Indices for Varying Resistance Factors Table A 1 = 1.0 Steel CFRP AFRP 1 Control 1.00 1.00 1.00 Control 1.00 1.00 1.00 Span 1 4.48 Tension 2.50 2.57 2.51 Compression 2.39 2.57 2.50 2 4.48 Tension 2.50 2.57 2.51 Compression 2.39 2.57 2.50 3 4.46 Tension 2.52 2.59 2.52 Tension 2.41 2.58 2.51 4 3.80 Tension 2.26 2.33 2.27 Compression 2.16 2.33 2.26 5 3.79 Tension 2.26 2.33 2.27 Compression 2.16 2.33 2.26 6 4.25 Tension 2.38 2.46 2.39 Tension 2.28 2.45 2.38 7 4.42 Tension 2.47 2.54 2.47 Tension 2.36 2.53 2.46 8 4.43 Tension 2.47 2.54 2.48 Compression 2.36 2.54 2.46 9 4.40 Tension 2.46 2.53 2.47 Compression 2.35 2.53 2.45 10 4.35 Tension 2.44 2.52 2.45 Tension 2.34 2.51 2.44 11 4.18 Tension 2.35 2.42 2.36 Compression 2.25 2.42 2.35 12 4.38 Tension 2.45 2.52 2.46 Tension 2.34 2.52 2.45 13 4.38 Tension 2.45 2.52 2.46 Tension 2.34 2.52 2.45 14 4.42 Compression 2.48 2.55 2.49 Compression 2.35 2.53 2.46 15 4.29 Tension 2.41 2.49 2.42 Tension 2.31 2.48 2.41 16 4.12 Tension 2.40 2.47 2.41 Tension 2.32 2.50 2.43 17 4.30 Tension 2.41 2.49 2.42 Compression 2.31 2.48 2.41 18 4.28 Tension 2.41 2.48 2.41 Tension 2.30 2.47 2.40 19 4.34 Tension 2.43 2.50 2.44 Transition 2.33 2.50 2.43 20 4.23 Compression 2.38 2.45 2.39 Compression 2.26 2.44 2.37 21 4.19 Compression 2.37 2.44 2.38 Compression 2.25 2.42 2.35 22 4.17 Compression 2.37 2.44 2.38 Compression 2.24 2.41 2.34 23 4.11 Tension 2.34 2.41 2.35 Tension 2.24 2.41 2.34 24 4.24 Tension 2.32 2.40 2.33 Tension 2.29 2.46 2.39 25 4.18 Tension 2.36 2.43 2.37 Tension 2.25 2.43 2.36
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102 Table A 2 = 0.95 Steel CFRP AFRP 0.95 Control 0.95 0.95 0.95 Control 0.95 0.95 0.95 Span 1 5.08 Tension 2.75 2.82 2.76 Compression 2.63 2.82 2.74 2 5.08 Tension 2.75 2.83 2.76 Compression 2.63 2.82 2.74 3 5.05 Tension 2.76 2.84 2.77 Tension 2.64 2.83 2.76 4 4.39 Tension 2.51 2.58 2.52 Compression 2.40 2.58 2.51 5 4.38 Tension 2.51 2.58 2.52 Compression 2.40 2.58 2.50 6 4.86 Tension 2.63 2.71 2.64 Tension 2.52 2.70 2.63 7 5.01 Tension 2.71 2.79 2.72 Tension 2.60 2.78 2.71 8 5.02 Tension 2.72 2.79 2.73 Compression 2.60 2.79 2.71 9 5.00 Tension 2.71 2.78 2.72 Compression 2.59 2.78 2.70 10 4.95 Tension 2.69 2.77 2.70 Tension 2.58 2.76 2.69 11 4.78 Tension 2.60 2.68 2.61 Compression 2.49 2.67 2.60 12 4.98 Tension 2.70 2.78 2.71 Tension 2.58 2.77 2.69 13 4.98 Tension 2.70 2.78 2.71 Tension 2.58 2.77 2.69 14 5.01 Compression 2.73 2.81 2.74 Compression 2.59 2.78 2.70 15 4.89 Tension 2.66 2.74 2.67 Tension 2.55 2.73 2.66 16 4.71 Tension 2.65 2.72 2.66 Tension 2.56 2.75 2.67 17 4.89 Tension 2.66 2.74 2.67 Compression 2.55 2.73 2.66 18 4.87 Tension 2.66 2.73 2.66 Tension 2.54 2.72 2.65 19 4.94 Tension 2.68 2.76 2.69 Transition 2.57 2.75 2.68 20 4.83 Compression 2.63 2.71 2.64 Compression 2.50 2.69 2.62 21 4.78 Compression 2.62 2.70 2.63 Compression 2.49 2.67 2.60 22 4.76 Compression 2.62 2.69 2.63 Compression 2.48 2.66 2.59 23 4.71 Tension 2.59 2.66 2.60 Tension 2.48 2.66 2.59 24 4.83 Tension 2.57 2.65 2.58 Tension 2.53 2.71 2.64 25 4.77 Tension 2.61 2.68 2.62 Tension 2.49 2.68 2.60
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103 Table A 3 = 0.90 Steel CFRP AFRP 0.9 Control 0.90 0.90 0.90 Control 0.90 0.90 0.90 Span 1 5.70 Tension 3.01 3.09 3.02 Compression 2.88 3.08 3.00 2 5.70 Tension 3.01 3.09 3.02 Compression 2.88 3.08 3.00 3 5.66 Tension 3.03 3.11 3.03 Tension 2.90 3.10 3.02 4 5.01 Tension 2.77 2.85 2.78 Compression 2.65 2.84 2.77 5 5.01 Tension 2.77 2.85 2.78 Compression 2.65 2.84 2.76 6 5.49 Tension 2.90 2.98 2.91 Tension 2.77 2.97 2.89 7 5.64 Tension 2.98 3.06 2.99 Tension 2.85 3.05 2.97 8 5.65 Tension 2.98 3.06 2.99 Compression 2.85 3.05 2.97 9 5.62 Tension 2.97 3.05 2.98 Compression 2.84 3.04 2.96 10 5.57 Tension 2.96 3.04 2.96 Tension 2.83 3.03 2.95 11 5.41 Tension 2.87 2.94 2.87 Compression 2.74 2.94 2.86 12 5.61 Tension 2.96 3.04 2.97 Tension 2.84 3.03 2.96 13 5.61 Tension 2.96 3.04 2.97 Tension 2.84 3.03 2.96 14 5.64 Compression 2.99 3.07 3.00 Compression 2.84 3.04 2.96 15 5.52 Tension 2.93 3.01 2.93 Tension 2.80 3.00 2.92 16 5.34 Tension 2.91 2.99 2.92 Tension 2.82 3.01 2.93 17 5.52 Tension 2.93 3.00 2.93 Compression 2.80 3.00 2.92 18 5.50 Tension 2.92 3.00 2.93 Tension 2.79 2.99 2.91 19 5.57 Tension 2.95 3.03 2.95 Transition 2.82 3.02 2.94 20 5.46 Compression 2.89 2.97 2.90 Compression 2.76 2.95 2.88 21 5.41 Compression 2.88 2.96 2.89 Compression 2.74 2.94 2.86 22 5.39 Compression 2.88 2.96 2.89 Compression 2.73 2.93 2.85 23 5.33 Tension 2.85 2.93 2.86 Tension 2.73 2.92 2.85 24 5.46 Tension 2.84 2.91 2.85 Tension 2.78 2.97 2.90 25 5.40 Tension 2.87 2.95 2.88 Tension 2.75 2.94 2.86
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104 Table A 4 = 0.85 Steel CFRP AFRP 0.85 Control 0.85 0.85 0.85 Control 0.85 0.85 0.85 Span 1 6.36 Tension 3.29 3.37 3.30 Compression 3.15 3.36 3.28 2 6.36 Tension 3.29 3.37 3.30 Compression 3.15 3.36 3.28 3 6.31 Tension 3.30 3.39 3.31 Tension 3.16 3.37 3.29 4 5.68 Tension 3.05 3.13 3.06 Compression 2.92 3.12 3.04 5 5.67 Tension 3.05 3.13 3.06 Compression 2.92 3.12 3.04 6 6.16 Tension 3.18 3.26 3.18 Tension 3.04 3.25 3.17 7 6.30 Tension 3.26 3.34 3.26 Tension 3.12 3.33 3.24 8 6.32 Tension 3.26 3.34 3.27 Compression 3.12 3.33 3.25 9 6.29 Tension 3.25 3.33 3.26 Compression 3.11 3.32 3.24 10 6.23 Tension 3.23 3.32 3.24 Tension 3.10 3.31 3.22 11 6.08 Tension 3.14 3.23 3.15 Compression 3.01 3.22 3.13 12 6.27 Tension 3.24 3.32 3.25 Tension 3.10 3.31 3.23 13 6.27 Tension 3.24 3.32 3.25 Tension 3.10 3.31 3.23 14 6.29 Compression 3.27 3.35 3.28 Compression 3.11 3.32 3.24 15 6.18 Tension 3.20 3.29 3.21 Tension 3.07 3.28 3.19 16 6.01 Tension 3.19 3.27 3.20 Tension 3.08 3.29 3.21 17 6.18 Tension 3.20 3.29 3.21 Compression 3.07 3.28 3.19 18 6.16 Tension 3.20 3.28 3.20 Tension 3.06 3.27 3.19 19 6.23 Tension 3.22 3.31 3.23 Transition 3.09 3.30 3.21 20 6.13 Compression 3.17 3.26 3.18 Compression 3.03 3.23 3.15 21 6.07 Compression 3.16 3.24 3.17 Compression 3.01 3.22 3.14 22 6.05 Compression 3.16 3.24 3.17 Compression 3.00 3.21 3.13 23 5.99 Tension 3.13 3.21 3.14 Tension 3.00 3.20 3.12 24 6.12 Tension 3.12 3.20 3.12 Tension 3.05 3.25 3.17 25 6.07 Tension 3.15 3.23 3.16 Tension 3.02 3.22 3.14
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105 Table A 5 : = 0.80 Steel CFRP AFRP 0.8 Control 0.80 0.80 0.80 Control 0.80 0.80 0.80 Span 1 7.06 Tension 3.58 3.67 3.59 Compression 3.43 3.66 3.57 2 7.06 Tension 3.58 3.67 3.59 Compression 3.43 3.66 3.57 3 7.00 Tension 3.60 3.68 3.60 Tension 3.45 3.67 3.58 4 6.38 Tension 3.35 3.43 3.35 Compression 3.20 3.42 3.33 5 6.38 Tension 3.34 3.43 3.35 Compression 3.20 3.42 3.33 6 6.87 Tension 3.47 3.56 3.48 Tension 3.32 3.54 3.46 7 7.00 Tension 3.55 3.64 3.56 Tension 3.40 3.62 3.54 8 7.02 Tension 3.55 3.64 3.56 Compression 3.40 3.63 3.54 9 6.99 Tension 3.54 3.63 3.55 Compression 3.39 3.62 3.53 10 6.93 Tension 3.53 3.62 3.54 Tension 3.38 3.60 3.52 11 6.79 Tension 3.44 3.53 3.45 Compression 3.29 3.51 3.43 12 6.97 Tension 3.54 3.62 3.54 Tension 3.39 3.61 3.52 13 6.97 Tension 3.54 3.62 3.54 Tension 3.39 3.61 3.52 14 6.99 Compression 3.56 3.65 3.57 Compression 3.39 3.62 3.53 15 6.88 Tension 3.50 3.59 3.51 Tension 3.35 3.57 3.49 16 6.72 Tension 3.49 3.57 3.49 Tension 3.37 3.59 3.50 17 6.89 Tension 3.50 3.59 3.51 Compression 3.35 3.57 3.49 18 6.87 Tension 3.49 3.58 3.50 Tension 3.34 3.57 3.48 19 6.93 Tension 3.52 3.61 3.53 Transition 3.37 3.59 3.51 20 6.83 Compression 3.47 3.55 3.48 Compression 3.31 3.53 3.44 21 6.77 Compression 3.46 3.54 3.46 Compression 3.30 3.51 3.43 22 6.75 Compression 3.45 3.54 3.46 Compression 3.29 3.50 3.42 23 6.69 Tension 3.43 3.51 3.43 Tension 3.28 3.50 3.41 24 6.82 Tension 3.41 3.50 3.42 Tension 3.33 3.55 3.46 25 6.77 Tension 3.44 3.53 3.45 Tension 3.30 3.52 3.43
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106 Table A 6 = 0.75 Steel CFRP AFRP 0.75 Control 0.75 0.75 0.75 Control 0.75 0.75 0.75 Span 1 7.80 Tension 3.90 3.99 3.91 Compression 3.74 3.97 3.88 2 7.80 Tension 3.90 3.99 3.90 Compression 3.74 3.97 3.88 3 7.73 Tension 3.91 4.00 3.92 Tension 3.75 3.98 3.89 4 7.13 Tension 3.66 3.75 3.67 Compression 3.51 3.73 3.65 5 7.12 Tension 3.66 3.75 3.67 Compression 3.51 3.73 3.64 6 7.63 Tension 3.79 3.88 3.79 Tension 3.63 3.86 3.77 7 7.75 Tension 3.86 3.96 3.87 Tension 3.70 3.94 3.85 8 7.77 Tension 3.87 3.96 3.87 Compression 3.71 3.94 3.85 9 7.74 Tension 3.86 3.95 3.86 Compression 3.70 3.93 3.84 10 7.67 Tension 3.84 3.93 3.85 Tension 3.68 3.92 3.83 11 7.54 Tension 3.75 3.84 3.76 Compression 3.60 3.83 3.74 12 7.72 Tension 3.85 3.94 3.86 Tension 3.69 3.92 3.83 13 7.72 Tension 3.85 3.94 3.86 Tension 3.69 3.92 3.83 14 7.73 Compression 3.88 3.97 3.89 Compression 3.70 3.93 3.84 15 7.63 Tension 3.81 3.90 3.82 Tension 3.66 3.89 3.80 16 7.47 Tension 3.80 3.89 3.81 Tension 3.67 3.90 3.81 17 7.64 Tension 3.81 3.90 3.82 Compression 3.65 3.89 3.80 18 7.62 Tension 3.80 3.90 3.81 Tension 3.65 3.88 3.79 19 7.68 Tension 3.83 3.92 3.84 Transition 3.67 3.91 3.82 20 7.59 Compression 3.78 3.87 3.79 Compression 3.61 3.84 3.75 21 7.52 Compression 3.77 3.86 3.78 Compression 3.60 3.83 3.74 22 7.49 Compression 3.77 3.86 3.77 Compression 3.59 3.82 3.73 23 7.44 Tension 3.74 3.83 3.75 Tension 3.58 3.81 3.72 24 7.57 Tension 3.72 3.81 3.73 Tension 3.63 3.87 3.78 25 7.53 Tension 3.76 3.85 3.77 Tension 3.60 3.83 3.74
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107 Table A 7 = 0.70 Steel CFRP AFRP 0.7 Control 0.70 0.70 0.70 Control 0.70 0.70 0.70 Span 1 8.60 Tension 4.23 4.33 4.24 Compression 4.06 4.31 4.21 2 8.60 Tension 4.23 4.33 4.24 Compression 4.06 4.31 4.21 3 8.51 Tension 4.24 4.34 4.25 Tension 4.07 4.32 4.22 4 7.93 Tension 4.00 4.09 4.00 Compression 3.83 4.07 3.98 5 7.92 Tension 3.99 4.09 4.00 Compression 3.83 4.07 3.98 6 8.44 Tension 4.12 4.22 4.13 Tension 3.95 4.20 4.10 7 8.55 Tension 4.20 4.30 4.21 Tension 4.03 4.28 4.18 8 8.57 Tension 4.20 4.30 4.21 Compression 4.03 4.28 4.18 9 8.53 Tension 4.19 4.29 4.20 Compression 4.02 4.27 4.17 10 8.47 Tension 4.18 4.27 4.19 Tension 4.01 4.25 4.16 11 8.35 Tension 4.09 4.19 4.10 Compression 3.92 4.17 4.07 12 8.52 Tension 4.18 4.28 4.19 Tension 4.01 4.26 4.17 13 8.52 Tension 4.18 4.28 4.19 Tension 4.01 4.26 4.17 14 8.52 Compression 4.21 4.31 4.22 Compression 4.02 4.27 4.17 15 8.43 Tension 4.15 4.25 4.16 Tension 3.98 4.22 4.13 16 8.27 Tension 4.14 4.23 4.14 Tension 3.99 4.24 4.14 17 8.44 Tension 4.15 4.24 4.16 Compression 3.98 4.22 4.13 18 8.42 Tension 4.14 4.24 4.15 Tension 3.97 4.22 4.12 19 8.48 Tension 4.17 4.26 4.18 Transition 4.00 4.24 4.15 20 8.39 Compression 4.12 4.21 4.13 Compression 3.94 4.18 4.09 21 8.32 Compression 4.11 4.20 4.11 Compression 3.92 4.17 4.07 22 8.29 Compression 4.10 4.20 4.11 Compression 3.91 4.15 4.06 23 8.24 Tension 4.07 4.17 4.08 Tension 3.91 4.15 4.06 24 8.37 Tension 4.06 4.15 4.07 Tension 3.96 4.20 4.11 25 8.33 Tension 4.09 4.19 4.10 Tension 3.93 4.17 4.08
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108 Table A 8 = 0.65 Steel CFRP AFRP 0.65 Control 0.65 0.65 0.65 Control 0.65 0.65 0.65 Span 1 9.45 Tension 4.59 4.70 4.60 Compression 4.41 4.67 4.57 2 9.46 Tension 4.59 4.69 4.60 Compression 4.41 4.67 4.57 3 9.35 Tension 4.60 4.70 4.61 Tension 4.42 4.68 4.58 4 8.78 Tension 4.36 4.46 4.36 Compression 4.18 4.43 4.34 5 8.78 Tension 4.36 4.45 4.36 Compression 4.18 4.43 4.33 6 9.30 Tension 4.48 4.58 4.49 Tension 4.30 4.56 4.46 7 9.41 Tension 4.56 4.66 4.57 Tension 4.37 4.64 4.54 8 9.43 Tension 4.56 4.66 4.57 Compression 4.38 4.64 4.54 9 9.40 Tension 4.55 4.65 4.56 Compression 4.37 4.63 4.53 10 9.33 Tension 4.54 4.64 4.55 Tension 4.35 4.62 4.51 11 9.22 Tension 4.45 4.55 4.46 Compression 4.27 4.53 4.43 12 9.38 Tension 4.55 4.65 4.55 Tension 4.36 4.62 4.52 13 9.38 Tension 4.55 4.65 4.55 Tension 4.36 4.62 4.52 14 9.37 Compression 4.57 4.67 4.58 Compression 4.37 4.63 4.53 15 9.29 Tension 4.51 4.61 4.52 Tension 4.33 4.59 4.49 16 9.14 Tension 4.50 4.60 4.50 Tension 4.34 4.60 4.50 17 9.30 Tension 4.51 4.61 4.52 Compression 4.33 4.59 4.49 18 9.28 Tension 4.50 4.60 4.51 Tension 4.32 4.58 4.48 19 9.34 Tension 4.53 4.63 4.54 Transition 4.34 4.61 4.51 20 9.25 Compression 4.48 4.58 4.49 Compression 4.28 4.54 4.44 21 9.18 Compression 4.47 4.57 4.47 Compression 4.27 4.53 4.43 22 9.15 Compression 4.46 4.56 4.47 Compression 4.26 4.52 4.42 23 9.10 Tension 4.43 4.54 4.44 Tension 4.25 4.51 4.41 24 9.23 Tension 4.42 4.52 4.43 Tension 4.30 4.56 4.46 25 9.19 Tension 4.46 4.56 4.46 Tension 4.27 4.53 4.43
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109 Table A 9 = 0.60 Steel CFRP AFRP 0.6 Control 0.60 0.60 0.60 Control 0.60 0.60 0.60 Span 1 10.38 Tension 4.98 5.09 4.99 Compression 4.78 5.06 4.95 2 10.38 Tension 4.98 5.09 4.99 Compression 4.78 5.06 4.96 3 10.26 Tension 4.99 5.10 5.00 Tension 4.79 5.07 4.96 4 9.71 Tension 4.75 4.85 4.75 Compression 4.55 4.83 4.72 5 9.71 Tension 4.74 4.85 4.75 Compression 4.55 4.82 4.72 6 10.24 Tension 4.87 4.98 4.88 Tension 4.67 4.95 4.85 7 10.34 Tension 4.95 5.06 4.96 Tension 4.75 5.03 4.92 8 10.37 Tension 4.95 5.06 4.96 Compression 4.75 5.03 4.93 9 10.33 Tension 4.94 5.05 4.95 Compression 4.74 5.02 4.92 10 10.25 Tension 4.93 5.03 4.93 Tension 4.73 5.01 4.90 11 10.15 Tension 4.84 4.95 4.85 Compression 4.64 4.92 4.82 12 10.31 Tension 4.93 5.04 4.94 Tension 4.74 5.01 4.91 13 10.31 Tension 4.93 5.04 4.94 Tension 4.74 5.01 4.91 14 10.29 Compression 4.96 5.07 4.97 Compression 4.74 5.02 4.91 15 10.22 Tension 4.90 5.00 4.91 Tension 4.70 4.98 4.87 16 10.07 Tension 4.88 4.99 4.89 Tension 4.71 4.99 4.89 17 10.23 Tension 4.90 5.00 4.91 Compression 4.70 4.98 4.87 18 10.21 Tension 4.89 5.00 4.90 Tension 4.69 4.97 4.86 19 10.27 Tension 4.92 5.02 4.93 Transition 4.72 5.00 4.89 20 10.19 Compression 4.87 4.97 4.88 Compression 4.66 4.94 4.83 21 10.10 Compression 4.85 4.96 4.86 Compression 4.65 4.92 4.82 22 10.07 Compression 4.85 4.96 4.86 Compression 4.63 4.91 4.80 23 10.02 Tension 4.82 4.93 4.83 Tension 4.63 4.90 4.80 24 10.16 Tension 4.81 4.92 4.82 Tension 4.68 4.96 4.85 25 10.12 Tension 4.84 4.95 4.85 Tension 4.65 4.92 4.82
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110 Table A 10 = 0.55 Steel CFRP AFRP 0.55 Control 0.55 0.55 0.55 Control 0.55 0.55 0.55 Span 1 11.38 Tension 5.40 5.52 5.41 Compression 5.19 5.49 5.37 2 11.38 Tension 5.40 5.52 5.41 Compression 5.19 5.49 5.37 3 11.25 Tension 5.41 5.52 5.42 Tension 5.20 5.49 5.38 4 10.72 Tension 5.17 5.28 5.18 Compression 4.96 5.25 5.14 5 10.71 Tension 5.17 5.28 5.18 Compression 4.96 5.25 5.14 6 11.26 Tension 5.30 5.41 5.30 Tension 5.08 5.38 5.27 7 11.35 Tension 5.37 5.49 5.38 Tension 5.16 5.45 5.34 8 11.38 Tension 5.37 5.49 5.38 Compression 5.16 5.46 5.34 9 11.33 Tension 5.36 5.48 5.37 Compression 5.15 5.45 5.33 10 11.26 Tension 5.35 5.46 5.36 Tension 5.14 5.43 5.32 11 11.17 Tension 5.26 5.38 5.27 Compression 5.05 5.35 5.23 12 11.32 Tension 5.36 5.47 5.37 Tension 5.14 5.44 5.33 13 11.32 Tension 5.36 5.47 5.37 Tension 5.14 5.44 5.33 14 11.29 Compression 5.38 5.50 5.39 Compression 5.15 5.44 5.33 15 11.22 Tension 5.32 5.43 5.33 Tension 5.11 5.40 5.29 16 11.09 Tension 5.31 5.42 5.32 Tension 5.12 5.42 5.30 17 11.24 Tension 5.32 5.43 5.33 Compression 5.11 5.40 5.29 18 11.22 Tension 5.31 5.43 5.32 Tension 5.10 5.39 5.28 19 11.28 Tension 5.34 5.45 5.35 Transition 5.13 5.42 5.31 20 11.20 Compression 5.29 5.40 5.30 Compression 5.07 5.36 5.25 21 11.11 Compression 5.28 5.39 5.29 Compression 5.05 5.34 5.23 22 11.07 Compression 5.27 5.39 5.28 Compression 5.04 5.33 5.22 23 11.03 Tension 5.25 5.36 5.25 Tension 5.04 5.33 5.22 24 11.16 Tension 5.23 5.34 5.24 Tension 5.09 5.38 5.27 25 11.13 Tension 5.27 5.38 5.28 Tension 5.06 5.35 5.24
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111 Table A 11 = 0.50 Steel CFRP AFRP 0.5 Control 0.50 0.50 0.50 Control 0.50 0.50 0.50 Span 1 12.48 Tension 5.87 5.99 5.88 Compression 5.63 5.95 5.83 2 12.48 Tension 5.87 5.99 5.88 Compression 5.63 5.95 5.83 3 12.33 Tension 5.87 5.99 5.88 Tension 5.64 5.96 5.84 4 11.82 Tension 5.63 5.75 5.64 Compression 5.41 5.72 5.60 5 11.82 Tension 5.63 5.75 5.64 Compression 5.41 5.72 5.60 6 12.38 Tension 5.76 5.88 5.77 Tension 5.53 5.84 5.73 7 12.45 Tension 5.83 5.96 5.84 Tension 5.60 5.92 5.80 8 12.48 Tension 5.84 5.96 5.85 Compression 5.61 5.92 5.80 9 12.44 Tension 5.83 5.95 5.84 Compression 5.60 5.91 5.79 10 12.35 Tension 5.81 5.93 5.82 Tension 5.58 5.90 5.78 11 12.29 Tension 5.73 5.85 5.74 Compression 5.50 5.81 5.69 12 12.43 Tension 5.82 5.94 5.83 Tension 5.59 5.91 5.79 13 12.43 Tension 5.82 5.94 5.83 Tension 5.59 5.91 5.79 14 12.38 Compression 5.85 5.97 5.85 Compression 5.59 5.91 5.79 15 12.33 Tension 5.78 5.90 5.79 Tension 5.55 5.87 5.75 16 12.20 Tension 5.77 5.89 5.78 Tension 5.57 5.88 5.76 17 12.35 Tension 5.78 5.90 5.79 Compression 5.55 5.87 5.75 18 12.32 Tension 5.78 5.90 5.79 Tension 5.55 5.86 5.74 19 12.38 Tension 5.80 5.92 5.81 Transition 5.57 5.89 5.77 20 12.31 Compression 5.75 5.87 5.76 Compression 5.51 5.83 5.71 21 12.22 Compression 5.74 5.86 5.75 Compression 5.50 5.81 5.69 22 12.18 Compression 5.74 5.85 5.74 Compression 5.49 5.80 5.68 23 12.13 Tension 5.71 5.83 5.72 Tension 5.48 5.79 5.68 24 12.26 Tension 5.70 5.81 5.70 Tension 5.53 5.85 5.73 25 12.24 Tension 5.73 5.85 5.74 Tension 5.50 5.82 5.70
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112 Table A 12 = 0.40 Steel CFRP AFRP 0.4 Control 0.40 0.40 0.40 Control 0.40 0.40 0.40 Span 1 15.05 Tension 6.95 7.09 6.96 Compression 6.68 7.04 6.91 2 15.05 Tension 6.95 7.09 6.96 Compression 6.68 7.04 6.91 3 14.86 Tension 6.95 7.09 6.96 Tension 6.68 7.04 6.91 4 14.41 Tension 6.72 6.85 6.73 Compression 6.46 6.81 6.68 5 14.40 Tension 6.72 6.85 6.73 Compression 6.45 6.81 6.68 6 14.99 Tension 6.85 6.98 6.86 Tension 6.58 6.94 6.80 7 15.03 Tension 6.92 7.06 6.93 Tension 6.65 7.01 6.88 8 15.07 Tension 6.92 7.06 6.93 Compression 6.65 7.01 6.88 9 15.03 Tension 6.91 7.05 6.92 Compression 6.64 7.00 6.87 10 14.93 Tension 6.90 7.03 6.90 Tension 6.63 6.99 6.85 11 14.90 Tension 6.81 6.95 6.82 Compression 6.55 6.90 6.77 12 15.02 Tension 6.91 7.04 6.91 Tension 6.64 7.00 6.86 13 15.02 Tension 6.91 7.04 6.91 Tension 6.64 7.00 6.86 14 14.95 Compression 6.93 7.06 6.94 Compression 6.64 7.00 6.86 15 14.91 Tension 6.87 7.00 6.88 Tension 6.60 6.96 6.83 16 14.80 Tension 6.86 6.99 6.87 Tension 6.61 6.97 6.84 17 14.94 Tension 6.87 7.01 6.88 Compression 6.60 6.96 6.83 18 14.91 Tension 6.86 7.00 6.87 Tension 6.59 6.95 6.82 19 14.98 Tension 6.89 7.02 6.90 Transition 6.62 6.98 6.84 20 14.91 Compression 6.84 6.98 6.85 Compression 6.56 6.92 6.79 21 14.80 Compression 6.83 6.96 6.83 Compression 6.54 6.90 6.77 22 14.75 Compression 6.82 6.95 6.83 Compression 6.53 6.89 6.75 23 14.71 Tension 6.79 6.93 6.80 Tension 6.53 6.88 6.75 24 14.85 Tension 6.78 6.92 6.79 Tension 6.58 6.94 6.80 25 14.84 Tension 6.82 6.95 6.83 Tension 6.55 6.91 6.77
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113 B Sample VBA code for reliability index calculations Beta Iteration Runs Sub Phi_All() ' K_All Macro ' Phi = 1.0 Sheets("Bridge _Simulation").Select Range("AE8").Select ActiveCell.FormulaR1C1 = "1.0" 'Run Full Model Application.Run "R un_Beta" Sheets("Summary").Select Calculate Range("D3:S32").Select Selection.Copy Sheets("Summary_Combined").Select Range("D3").Select Selection.PasteSpecial Paste:=xlPasteValues, Operation:=xlNone, SkipBlanks :=False, Transpose:=False Application.CutCopyMode = False 'Phi = 0.95 Sheets("Bridge _Simulation").Select Application.CutCopyMode = False Range("AE8").Select ActiveCell.FormulaR1C1 = "0.95" 'Run Full Model Application.Run "Run_Beta" Sheets("Summary").Select Calculate Range("D3:S32").Select Selection.Copy Sheets("Summary_Combined").Select Range("D35").Select Selection.PasteSpecial Paste:=xlPasteValues, Operation:=xlNone, S kipBlanks :=False, Transpose:=False Application.CutCopyMode = False 'Phi = 0.90 Sheets("Bridge _Simulation").Select Application.CutCopyMode = False Range("AE8").Select ActiveCell.FormulaR1C1 = "0.90" 'Run Full Model Application.Run "Run_Beta" Sheets("Summary").Select Calculate Range("D3:S32").Select Selection.Copy Sheets("Summary_Combined").Select Range("D67").Select Selection.PasteSpecial Paste:=xlPasteValues, Operation:= xlNone, SkipBlanks :=False, Transpose:=False Application.CutCopyMode = False 'Phi = 0.85 Sheets("Bridge _Simulation").Select Application.CutCopyMode = False Range("AE8").Select ActiveCell.FormulaR1C1 = "0.85" 'Run Ful l Model
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114 Application.Run "Run_Beta" Sheets("Summary").Select Calculate Range("D3:S32").Select Selection.Copy Sheets("Summary_Combined").Select Range("D99").Select Selection.PasteSpecial Paste:=xlPasteValues, Opera tion:=xlNone, SkipBlanks :=False, Transpose:=False Application.CutCopyMode = False 'Phi = 0.80 Sheets("Bridge _Simulation").Select Application.CutCopyMode = False Range("AE8").Select ActiveCell.FormulaR1C1 = "0.80" 'Run Full Model Application.Run "Run_Beta" Sheets("Summary").Select Calculate Range("D3:S32").Select Selection.Copy Sheets("Summary_Combined").Select Range("D131").Select Selection.PasteSpecial Paste:=xlP asteValues, Operation:=xlNone, SkipBlanks :=False, Transpose:=False Application.CutCopyMode = False 'Phi = 0.75 Sheets("Bridge _Simulation").Select Application.CutCopyMode = False Range("AE8").Select ActiveCell.FormulaR1C1 = "0.75" 'Run Full Model Application.Run "Run_Beta" Sheets("Summary").Select Calculate Range("D3:S32").Select Selection.Copy Sheets("Summary_Combined").Select Range("D163").Select Selection.PasteSpecial Pas te:=xlPasteValues, Operation:=xlNone, SkipBlanks :=False, Transpose:=False Application.CutCopyMode = False 'Phi = 0.70 Sheets("Bridge _Simulation").Select Application.CutCopyMode = False Range("AE8").Select ActiveCell.FormulaR1C1 = "0.70" 'Run Full Model Application.Run "Run_Beta" Sheets("Summary").Select Calculate Range("D3:S32").Select Selection.Copy Sheets("Summary_Combined").Select Range("D195").Select Selection.PasteSpecial Paste:=xlPasteValues, Operation:=xlNone, SkipBlanks :=False, Transpose:=False Application.CutCopyMode = False 'Phi = 0.65 Sheets("Bridge _Simulation").Select Application.CutCopyMode = False Range("AE8"). Select ActiveCell.FormulaR1C1 = "0.65" 'Run Full Model Application.Run "Run_Beta"
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115 Sheets("Summary").Select Calculate Range("D3:S32").Select Selection.Copy Sheets("Summary_Combined").Select Range( "D227").Select Selection.PasteSpecial Paste:=xlPasteValues, Operation:=xlNone, SkipBlanks :=False, Transpose:=False Application.CutCopyMode = False 'Phi = 0.60 Sheets("Bridge _Simulation").Select Application.CutCopyMode = False Range("AE8").Select ActiveCell.FormulaR1C1 = "0.60" 'Run Full Model Application.Run "Run_Beta" Sheets("Summary").Select Calculate Range("D3:S32").Select Selection.Copy Sheets("Summary_Combined").Select Range("D259").Select Selection.PasteSpecial Paste:=xlPasteValues, Operation:=xlNone, SkipBlanks :=False, Transpose:=False Application.CutCopyMode = False 'Phi = 0.55 Sheets("Bridge _Simulation").Select Application.CutCopyMode = False Range("AE8").Select ActiveCell.FormulaR1C1 = "0.55" 'Run Full Model Application.Run "Run_Beta" Sheets("Summary").Select Calculate Range("D3:S32").Select Selection.Copy Sheets("Summary_Combined ").Select Range("D291").Select Selection.PasteSpecial Paste:=xlPasteValues, Operation:=xlNone, SkipBlanks :=False, Transpose:=False Application.CutCopyMode = False 'Phi = 0.50 Sheets("Bridge _Simulation").Select Application.CutCopyMode = False Range("AE8").Select ActiveCell.FormulaR1C1 = "0.50" 'Run Full Model Application.Run "Run_Beta" Sheets("Summary").Select Calculate Range("D3:S32").Select Selection.Copy Shee ts("Summary_Combined").Select Range("D323").Select Selection.PasteSpecial Paste:=xlPasteValues, Operation:=xlNone, SkipBlanks :=False, Transpose:=False Application.CutCopyMode = False 'Phi = 0.40 Sheets("Bridge _Simulation").Select Application.CutCopyMode = False Range("AE8").Select ActiveCell.FormulaR1C1 = "0.40" 'Run Full Model Application.Run "Run_Beta" Sheets("Summary").Select
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116 Calculate Range("D3:S32").Select Selection.Copy Sheets("Summary_Combined").Select Range("D355").Select Selection.PasteSpecial Paste:=xlPasteValues, Operation:=xlNone, SkipBlanks :=False, Transpose:=False Application.CutCopyMode = False 'Phi = 0.30 Sheets(" Bridge _Simulation").Select Application.CutCopyMode = False Range("AE8").Select ActiveCell.FormulaR1C1 = "0.30" 'Run Full Model Application.Run "Run_Beta" Sheets("Summary").Select Calculate Range( "D3:S32").Select Selection.Copy Sheets("Summary_Combined").Select Range("D387").Select Selection.PasteSpecial Paste:=xlPasteValues, Operation:=xlNone, SkipBlanks :=False, Transpose:=False Application.CutCopyMode = False End S ub Sample Material Run CFRP Sub Full_Run_CFRP() ' Full_Run_COV1 Macro Run Full Model ' Set CFRP Parameters 'CFRP Sheets("CFRP_Analysis").Select Range("F90:J114").Select Application.CutCopyMode = False Selection.Copy Range( "F4").Select Selection.PasteSpecial Paste:=xlPasteValues, Operation:=xlNone, SkipBlanks:=False, Transpose:=False Sheets("Bridge _Simulation").Select Range("E4").Value = "CFRP" ' Set to Compression Factors Sheets("Bridge _Simulation").Select Range("G4").Value = "Compression" 'Begin Compression Calibration Sheets("Bridge _Simulation").Select Range("C4").Value = "1" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("C7:J57").Value = Sheets( "Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "2" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("M7:T57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Va lue Sheets("Bridge _Simulation").Select Range("C4").Value = "3" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("W7:AD57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value
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117 Sheets("Bridge _Simulation"). Select Range("C4").Value = "4" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("AG7:AN57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "5" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("AQ7:AX57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "6" ActiveSheet.Calculate Sheets("CFRP_Br idge Results_C").Range("BA7:BH57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "7" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("BK7:BR57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "8" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("BU7:CB57").Value = Sheets("Bridge _Simulation").Range("AQ1 1:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "9" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("CE7:CL57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _S imulation").Select Range("C4").Value = "10" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("CO7:CV57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "11" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("CY7:DF57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "12" ActiveSheet.Calculate Sh eets("CFRP_Bridge Results_C").Range("DI7:DP57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "13" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("DS7 :DZ57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "14" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("EC7:EJ57").Value = Sheets("Bridge _Simulati on").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "15" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("EM7:ET57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "16" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("EW7:FD57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "17" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("FG7:FN57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value
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118 Sheets("Bridge _Simulation").Select Range("C4").Value = "18" ActiveSheet .Calculate Sheets("CFRP_Bridge Results_C").Range("FQ7:FX57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "19" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C ").Range("GA7:GH57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "20" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("GK7:GR57").Value = Sheets("Bri dge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "21" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("GU7:HB57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "22" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("HE7:HL57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "23" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("HO7:HV57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Sel ect Range("C4").Value = "24" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("HY7:IF57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "25" Act iveSheet.Calculate Sheets("CFRP_Bridge Results_C").Range("II7:IP57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value ' ' Set to Tension Factors Sheets("Bridge _Simulation").Select Range("G4").Value = "Tension" 'Begin Tension Calibration Sheets("Bridge _Simulation").Select Range("C4").Value = "1" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("C7:J57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Sim ulation").Select Range("C4").Value = "2" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("M7:T57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "3 ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("W7:AD57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "4" ActiveSheet.Calculate Sheets("CF RP_Bridge Results_T").Range("AG7:AN57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value
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119 Sheets("Bridge _Simulation").Select Range("C4").Value = "5" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("AQ7:AX57").V alue = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "6" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("BA7:BH57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "7" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("BK7:BR57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "8" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("BU7:CB57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Selec t Range("C4").Value = "9" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("CE7:CL57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "10" Active Sheet.Calculate Sheets("CFRP_Bridge Results_T").Range("CO7:CV57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "11" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("CY7:DF57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "12" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("DI7:DP57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "13" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("DS7:DZ57").Value = Sheets("Bridge _Simulation").Range("AQ11 :AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "14" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("EC7:EJ57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _S imulation").Select Range("C4").Value = "15" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("EM7:ET57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "16" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("EW7:FD57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "17" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("FG7:FN57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "18" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("F Q7:FX57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value
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120 Sheets("Bridge _Simulation").Select Range("C4").Value = "19" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("GA7:GH57").Value = Sheets("Bridge _Simulation ").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "20" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("GK7:GR57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value She ets("Bridge _Simulation").Select Range("C4").Value = "21" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("GU7:HB57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Ran ge("C4").Value = "22" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("HE7:HL57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "23" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("HO7:HV57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "24" ActiveSheet.Calculate Sheets("CFRP_B ridge Results_T").Range("HY7:IF57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "25" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_T").Range("II7:IP57").Valu e = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value ' ' Set to Average Factors Sheets("Bridge _Simulation").Select Range("G4").Value = "Combined" 'Begin Ave Calibration Sheets("Bridge _Simulation").Select Range("C4").Value = "1" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("C7:J57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "2" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("M7:T57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "3" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range(" W7:AD57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "4" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("AG7:AN57").Value = Sheets("Bridge _Simul ation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "5" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("AQ7:AX57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value
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121 Sheets("Bridge _Simulation").Select Range("C4").Value = "6" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("BA7:BH57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "7" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("BK7:BR57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "8" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("BU7:CB57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "9" ActiveSheet.Calculate Sheets("CFRP_ Bridge Results_Ave").Range("CE7:CL57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "10" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("CO7:CV57") .Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "11" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("CY7:DF57").Value = Sheets("Bridge _Simulation"). Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "12" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("DI7:DP57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Shee ts("Bridge _Simulation").Select Range("C4").Value = "13" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("DS7:DZ57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "14" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("EC7:EJ57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "15" ActiveShe et.Calculate Sheets("CFRP_Bridge Results_Ave").Range("EM7:ET57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "16" ActiveSheet.Calculate Sheets("CFRP_Bridge R esults_Ave").Range("EW7:FD57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "17" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("FG7:FN57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "18" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("FQ7:FX57").Value = Sheets("Bridge _Simulation").Range("A Q11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "19" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("GA7:GH57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value
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122 Sheets("Bridge _Simulation").Select Range("C4").Value = "20" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("GK7:GR57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").V alue = "21" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("GU7:HB57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "22" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("HE7:HL57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "23" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave"). Range("HO7:HV57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "24" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("HY7:IF57").Value = Sheets("Bridge _Simulation").Range("AQ11:AX61").Value Sheets("Bridge _Simulation").Select Range("C4").Value = "25" ActiveSheet.Calculate Sheets("CFRP_Bridge Results_Ave").Range("II7:IP57").Value = Sheets("Bridge _Simulation").Range("AQ 11:AX61").Value End Sub
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123 C Sample Span General Layouts and Typical Sections Figure B 1 : Span 1 and Span 2 1 Calibrated Structure General Layout Figure B 2 : Span 1 and Span 2 Calibrated Structure Elevation 1 For multi span structures, trail span numbers are indicated from left to right. Trial span numbers are based on ordered spans from shortest to longest.
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124 Figure B 3 : Span 1 and Span 2 Calibrated Structure Typical Section Figure B 4 : Span 3, Span 23 and Span 10 Calibrated Structure General Layout Figure B 5 : Span 3, Span 23 and Span 10 Calibrated Structure Elevation
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125 Figure B 6 : Span 3, Span 23 and Span 10 Calibrated Structure Typical Section Figure B 7 : Span 4 and Span 5 Calibrated Structure General Layout
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126 Figure B 8 : Span 4 and Span 5 Calibrated Structures Elevation Figure B 9 : Span 4 and Span 5 Calibrated Structure Typical Section Figure B 10 : Span 6 and Span 7 Calibrated Structure General Layout
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127 Figure B 11 : Span 6 and Span 7 Calibrated Structure Elevation Figure B 12 : Span 6 and Span 7 Calibrated Structure Typical Section
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128 Figure B 13 : Span 8 Calibrated Structure General Layout Figure B 14 : Span 8 Calibrated Structure Elevation
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129 Figure B 15 : Span 8 Calibrated Structure Typical Section Figure B 16 : Span 9 Calibrated Structure General Layout Figure B 17 : Span 9 Calibrated Structure Elevation
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130 Figu re B 18 : Span 9 Calibrated Structure Typical Section Figure B 19 : Span 11 Calibrated Structure General Layout
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131 Figure B 20 : Span 11 Calibrated Structure Elevation Figure B 21 : Span 11 Calibrated Structure Typical Section
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132 Figure B 22 : Span 12 and Span 13 Calibrated Structure General Layout Figure B 23 : Span 12 and Span 13 Calibrated Structure Elevation
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133 Figure B 24 : Span 12 and Span 13 Calibrated Structure Partial Typical Section Figure B 25 : Span 14 Calibrated Structure General Layout
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134 Figure B 26 : Span 14 Calibrated Structure Elevation Figure B 27 : Span 14 Calibrated Structure Typical Section
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135 Figure B 28 : Span 15 and Span 18 Calibrated Structu re General Layout Figure B 29 : Span 15 and Span 18 Calibrated Structure Elevation Figure B 30 : Span 15 and Span 18 Calibrated Structure Typical Section
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136 Figure B 31 : Span 16 and Span 24 Calibrated Structure General Layout Figure B 32 : Span 16 and Span 24 Calibrated Structure Elevation Figure B 33 : Span 16 and Span 24 Calibrated Structure Typical Section
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137 Figure B 34 : Span 17 Calibrated Structure General Layout Figure B 35 : Span 17 Calibrated Structure Elevation Figure B 36 : Span 17 Calibrated Structure Typical Section
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138 Figure B 37 : Span 19 Calibrated Structure General Layout Figure B 38 : Span 19 Calibrated Structure Elevation
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139 Figure B 39 : Span 19 Calibrated Structure Typical Section Figure B 40 : Span 20 Calibrated Structure General Layout
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140 Figure B 41 : Span 20 Calibrated Structure Elevation Figure B 42 : Span 20 Calibrated Structure Typical Section
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141 Figure B 43 : Span 2 1 and Span 22 Calibrated Structure General Layout Figure B 44 : Span 2 1 and Span 22 Ca librated Structure Elevation Figure B 45 : Span 2 1 and Span 22 Calibrated Structure Typical Section
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142 Figure B 46 : Span 25 Calibrated Structure General Layout Figure B 47 : Span 25 Calibrated Structure Elevation
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143 Figure B 48 : Span 25 Calibrated Structure Typical Section
