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STRUCTURAL SYSTEMS RESEARCH PROJECT Report No. SSRP– 06/ 13 DEVELOPMENT OF RESISTANCE FACTORS FOR LRFD DESIGN FOR FRP STRENGTHENING OF REINFORCED CONCRETE BRIDGES by REBECCA ATADERO VISTASP M. KARBHARI Final Report Submitted to the California Department of Transportation Under Contract No. 59A0401. May 2006 Department of Structural Engineering University of California, San Diego La Jolla, California 92093 0085 University of California, San Diego Department of Structural Engineering Structural Systems Research Project Report No. SSRP– 06/ 15 DRAFT Development of Resistance Factors for LRFD Design for FRP Strengthening of Reinforced Concrete Bridges by Rebecca Atadero Graduate Student Researcher Vistasp M. Karbhari Professor of Structural Engineering Final Report Submitted to the California Department of Transportation Under Contract No. 59A0401. Department of Structural Engineering University of California, San Diego La Jolla, California 92093 0085 May 2006 Technical Report Documentation Page 1. Report No. FHWA/ CA/ ES 2006/ 11 2. Government Accession No. 3. Recipient’s Catalog No. 4. Title and Subtitle Development of Load and Resistance Factor Design for FRP Strengthening of Reinforced Concrete Structures 5. Report Date 5/ 26/ 2006 6. Performing Organization Code 7. Author( s) Rebecca A. Atadero and Vistasp M. Karbhari 8. Performing Organization Report No. UCSD / SSRP 06/ 13 9. Performing Organization Name and Address Department of Structural Engineering School of Engineering 10. Work Unit No. ( TRAIS) University of California, San Diego La Jolla, California 92093 0085 11. Contract or Grant No. 59A0401 12. Sponsoring Agency Name and Address California Department of Transportation 13. Type of Report and Period Covered Final Report – 5/ 26/ 2006 Engineering Service Center 1801 30th St., West Building MS 9 Sacramento, California 95807 14. Sponsoring Agency Code 15. Supplementary Notes Prepared in cooperation with the State of California Department of Transportation. This report is the first of multiple reports. 16. Abstract Externally bonded fiber reinforced polymer ( FRP) composites are an increasingly adopted technology for the renewal of existing concrete structures. In order to encourage the further use of these materials, a design code is needed that considers the inherent material variability of the composite, as well as the variations introduced during field manufacture and environmental exposure while in service. Load and Resistance Factor Design ( LRFD) is a reliability based design methodology that provides an ideal framework for these considerations and is compatible with existing trends in civil engineering design codes. This investigation studies the application of LRFD to FRP strengthening schemes, with an emphasis on wet layup, carbon fiber composites applied to reinforced concrete T beam bridge girders. Models to describe variation in the existing structural materials and the structural loading are drawn from the literature. Techniques for reliability analysis are discussed and existing work on externally bonded FRP reliability is surveyed. Stochastic variation in the FRP is characterized based on tensile testing of several sets of field manufactured wet layup composites. A general design procedure applicable to many different situations is proposed using a composite specific resistance factor to consider material variability, a set of Application Factors to account for deviations introduced through field manufacture, and an environment and service life specific factor for FRP degradation. Preliminary resistance factors for design of FRP strengthening are calibrated over a range of design scenarios. FRP degradation is considered based on existing durability models, and continued degradation of the structure due to general corrosion of the reinforcing steel is included. The girders used for calibration are selected as representative examples from a sample of California bridge plans. The reliability has been evaluated using simulation and first order reliability methods. An example of the proposed design procedure, using the calibrated resistance factors, is provided. The results of this work bring to light the many variables affecting the reliability of strengthened members, and the need for continuing research to better describe these variables. Two variables of particular significance, requiring extensive further study, are the state of the existing structure when strengthening is applied and the loads acting on the structure. 17. Key Words FRP, LRFD, Design of Strengthening 18. Distribution Statement No restrictions. This document is available to the public through the National Technical Information Service, Springfield, Virginia 22161. 19. Security Classification ( of this report) Unclassified 20. Security Classification ( of this page) Unclassified 21. No. of Pages ~ 420 22. Price Form DOT F 1700.7 ( 8 72) Reproduction of completed page authorized DISCLAIMER The opinions expressed in this report are those of the authors and do not represent positions of the California Department of Transportation. iii TABLE OF CONTENTS Disclaimer ............................................................................................................................... . iii Table of Contents ...................................................................................................................... iv List of Figures ........................................................................................................................ xvii List of Tables......................................................................................................................... .. xx Abstract ............................................................................................................................... .. xxv Chapter 1. Introduction .............................................................................................................. 1 1.1 Overview ................................................................................................................ 1 1.2 FRPs for Strengthening of Civil Structures ............................................................ 1 1.2.1 Fiber Reinforced Polymer Composites ............................................................ 1 1.2.2 Strengthening and Repair of Civil Structures................................................... 2 1.2.3 Advantages of FRPs for Strengthening............................................................ 4 1.2.4 Disadvantages of FRPs for Strengthening ....................................................... 5 1.3 Design Code for FRP Strengthening ...................................................................... 5 1.3.1 Need for a Design Code ................................................................................... 5 1.3.2 Uncertainty in Structural Design...................................................................... 7 1.3.3 Design Philosophies as the Basis for Design Codes ........................................ 7 1.3.3.1 Working Stress Design............................................................................ 8 1.3.3.2 Load and Resistance Factor Design ........................................................ 8 1.3.3.3 Advantages of LRFD............................................................................. 10 1.3.4 Current Design Guidelines for FRP Strengthening........................................ 11 1.4 Problem Statement and Research Objectives ....................................................... 13 1.4.1 Problem Description....................................................................................... 13 1.4.2 Research Objectives ....................................................................................... 15 iv 1.4.3 Research Approach ........................................................................................ 16 1.4.4 Outline of the Report...................................................................................... 19 Chapter 2. Background for Structural Reliability, LRFD and Design Uncertainty.................. 22 2.1 Introduction........................................................................................................... 22 2.2 Structural Reliability Methods.............................................................................. 22 2.2.1 Uncertainty and Risk...................................................................................... 22 2.2.2 Evaluation of Structural Reliability................................................................ 24 2.2.2.1 Effect of Uncertainty ............................................................................. 24 2.2.2.2 Deterministic Safety Factors ................................................................. 25 2.2.2.3 Basic Reliability Problem...................................................................... 26 2.2.2.4 The Reliability Index............................................................................. 29 2.2.2.5 Methods of Computing the Reliability Index........................................ 32 2.2.2.5.1 First Order, Second Moment Reliability Index ........................... 32 2.2.2.5.2 First and Second Order Reliability Methods ( FORM and SORM) …………….................................................................................................... 33 2.2.2.5.3 Monte Carlo Simulation ( MCS)................................................... 34 2.2.2.5.4 Other Techniques......................................................................... 35 2.2.2.6 Levels of Reliability Methods ............................................................... 35 2.2.2.7 Component vs. System Reliability ........................................................ 36 2.2.2.8 Time dependent Reliability ................................................................... 37 2.2.2.9 Limitations of Reliability Methods ....................................................... 38 2.2.2.10 Reliability Methods Used for This Report .......................................... 39 2.3 Previous Development of LRFD .......................................................................... 41 2.3.1 Steel................................................................................................................ 41 v 2.3.2 Loads.............................................................................................................. 42 2.3.3 Engineered Wood........................................................................................... 43 2.3.4 Bridges ........................................................................................................... 43 2.3.5 Concrete ......................................................................................................... 45 2.3.6 Aspects of Existing Codes Considered in this Work ..................................... 45 2.4 Previous Work on Reliability of FRP in Civil Infrastructure ............................... 46 2.4.1 FRP for Strengthening.................................................................................... 46 2.4.1.1 Limitations of Existing Studies ............................................................. 49 2.4.2 FRP for New Construction............................................................................. 50 2.4.2.1 General Design Standards ..................................................................... 51 2.5 Statistical Descriptors for Resistance Variables ................................................... 52 2.5.1 Concrete ......................................................................................................... 53 2.5.2 Reinforcing Steel............................................................................................ 55 2.5.3 Dimensions..................................................................................................... 56 2.5.3.1 Area of Steel.......................................................................................... 57 2.5.3.2 Slab Dimensions.................................................................................... 57 2.5.3.3 Beam Dimensions.................................................................................. 57 2.5.4 Modeling Uncertainty .................................................................................... 58 2.6 Description of Load Variables.............................................................................. 60 2.6.1 Dead Load ...................................................................................................... 61 2.6.2 Live and Impact Loads................................................................................... 61 2.7 Consideration of Continued Degradation ............................................................. 68 2.7.1 Modes of Reinforced Concrete Degradation.................................................. 68 2.7.2 Corrosion of Steel in Concrete ....................................................................... 69 vi 2.7.2.1 Carbonation Induced Corrosion ............................................................ 71 2.7.2.2 Chloride Induced Corrosion.................................................................. 72 2.7.2.3 Rates of Corrosion................................................................................. 72 2.7.3 Previous Work Modeling Corrosion Induced Degradation in Bridges.......... 74 2.7.4 Corrosion Models Used in this Report........................................................... 76 2.7.4.1 Major Assumptions for Corrosion Modeling ........................................ 76 2.7.4.2 Mathematical Models for Corrosion ..................................................... 78 2.8 Target Reliability Index........................................................................................ 81 2.8.1 Comparison to Other Acceptable Levels of Risk........................................... 82 2.8.2 Optimization of Cost Benefit ......................................................................... 83 2.8.3 Empirical Approaches.................................................................................... 83 2.8.4 Calibration to Safety Levels Implied by Existing Codes ............................... 86 2.8.4.1 Reliability Indices from Other LRFD Codes......................................... 87 2.8.5 Selection of Target β for this Work................................................................ 90 2.9 Discussion of Background Data ........................................................................... 92 Chapter 3. Characterization of Composite Properties for Reliability Analysis and Design..... 94 3.1 Introduction........................................................................................................... 94 3.2 Description of Data Sets ....................................................................................... 95 3.2.1 Testing Procedures ......................................................................................... 95 3.2.2 Wet Layup Composites .................................................................................. 95 3.3 Characterization of Random Variation ................................................................. 98 3.3.1 A Note on the Effect of Thickness ................................................................. 98 3.3.2 Basic Statistics ............................................................................................... 99 3.3.3 Statistical Distributions for Representing Composite Properties ................. 104 vii 3.3.3.1 Distributions ........................................................................................ 104 3.3.3.2 Distributions Fit to Wet Layup Composite Data ................................. 107 3.3.4 Best Fitting Distributions ............................................................................. 111 3.3.4.1 Strength ............................................................................................... 112 3.3.4.2 Modulus............................................................................................... 114 3.3.4.3 Thickness............................................................................................. 116 3.3.4.4 Summary of Distributions for Reliability Analysis............................. 118 3.3.5 Correlation between Variables ..................................................................... 119 3.4 Design Values for Composite Materials............................................................. 121 3.4.1 Current Approaches to Selection of Design Values..................................... 121 3.4.1.1 Reliability Implications of Current Design Approach......................... 123 3.4.2 Proposed Approach to Design Values.......................................................... 128 3.4.2.1 Accounting for Material Variability.................................................... 128 3.4.2.2 Use of the Mean as the Characteristic Value....................................... 130 3.4.2.3 Factors for Systematic Variation and Time Dependent Behavior....... 131 3.4.2.4 Promoting Reliability Based Design................................................... 131 3.5 Characterizing and Accounting for Systematic Differences between Laboratory Derived Design Values and In Situ Properties ................................................................ 133 3.5.1 Currently Used Factors ................................................................................ 133 3.5.2 Types of Systematic Variation ..................................................................... 136 3.5.3 Proposed Set of Application Factors............................................................ 137 3.5.4 Values of Factors for Wet Layup Composites ............................................. 139 3.5.4.1 Consideration of Thickness ................................................................. 140 3.5.4.2 Values for λpred .................................................................................... 140 viii 3.5.4.2.1 Predicted Value Based on Constitutive Properties .................... 140 3.5.4.2.2 Predicted Value Based on Manufacturer Data........................... 147 3.5.4.2.3 Predicted Value Based on Lamina or Laminate Level Tests ..... 149 3.5.4.3 Values for λlayers ................................................................................... 149 3.5.4.4 Values for λcure .................................................................................... 151 3.5.4.5 Values for λwork .................................................................................... 152 3.5.4.6 Summary of Factors for Systematic Variation of Wet Layup Composites. ........................................................................................................ 153 3.5.4.7 Assessment of Factor Accuracy .......................................................... 154 3.6 Time Dependent Degradation of FRP Properties............................................... 159 3.6.1 Current Approaches to Considering Time Dependent Behavior of FRP Properties................................................................................................................... 159 3.6.1.1 Environmental Exposure ..................................................................... 159 3.6.1.2 Sustained and Fatigue Loading ........................................................... 161 3.6.2 Proposed Method for Consideration of Time Dependent Degradation of FRP Properties................................................................................................................... 162 3.6.2.1 Factor for Environmental Degradation................................................ 162 3.6.2.1.1 Advantages of this Approach..................................................... 165 3.6.2.1.2 Limitations of Proposed Approach ............................................ 166 3.6.2.2 Stress Limitations for Sustained and Fatigue Loading........................ 167 3.6.2.2.1 Sustained Loading...................................................................... 167 3.6.2.2.2 Fatigue loading .......................................................................... 167 3.7 Summary............................................................................................................. 169 Chapter 4. Calibration of Resistance Factors for Flexural Strengthening of Bridge Girders. 171 ix 4.1 Introduction......................................................................................................... 171 4.2 Procedure for Calibration of Resistance Factors ................................................ 171 4.3 Summary of Previous Calibration Work ............................................................ 175 4.3.1 Load Factors for Strengthening Design ( Section C. 5) ................................. 175 4.3.2 Large Example Calibration without Corrosion ( Section C. 6)...................... 176 4.3.3 Example with Corrosion ( Section C. 8) ........................................................ 178 4.4 Range of Calibration........................................................................................... 178 4.4.1 Composite Materials .................................................................................... 179 4.4.1.1 Initial Properties .................................................................................. 179 4.4.1.2 States of FRP Degradation .................................................................. 180 4.4.2 Representative Members for Calibration ..................................................... 181 4.4.2.1 Typical Bridge Dimensions................................................................. 184 4.4.2.2 Selected Girders .................................................................................. 185 4.4.3 Time Periods Considered ............................................................................. 190 4.4.4 Cases of Continued Degradation.................................................................. 190 4.5 Design of Strengthening ..................................................................................... 191 4.5.1 Calculation of Design Load.......................................................................... 192 4.5.2 Calculation of Resistance ............................................................................. 197 4.5.2.1 Debonding Model................................................................................ 198 4.5.3 Computational Procedure............................................................................. 200 4.5.4 Summary of Designs .................................................................................... 201 4.6 Calculation of Reliability.................................................................................... 201 4.6.1 Description of Load Uncertainty.................................................................. 202 4.6.2 Description of Resistance Uncertainty......................................................... 205 x 4.6.3 Calculation Procedures................................................................................. 207 4.6.3.1 Simulation of Resistance ..................................................................... 207 4.6.3.1.1 Convergence .............................................................................. 208 4.7 Results ................................................................................................................ 209 4.7.1 Procedures Used to Analyze Reliability Results.......................................... 209 4.7.2 Effect of the Amount of Remaining Steel .................................................... 211 4.7.2.1 Significance ......................................................................................... 219 4.7.3 Effect of No Continuing Corrosion vs. Continuing Corrosion .................... 220 4.7.3.1 Significance ......................................................................................... 222 4.7.4 Effect of Different FRP Degradation Models .............................................. 223 4.7.4.1 Significance ......................................................................................... 225 4.7.5 Effect of Different Materials ........................................................................ 226 4.7.5.1 Significance ......................................................................................... 227 4.8 Extensions on the Large Calibration Example ................................................... 227 4.8.1 Effect of Changes in Modulus COV ............................................................ 227 4.8.1.1 Results ................................................................................................. 228 4.8.2 Effect of Different Bond Models ................................................................. 230 4.8.2.1 Results ................................................................................................. 231 4.9 Summary............................................................................................................. 232 Chapter 5. Recommended Design Procedure and Design Example....................................... 234 5.1 Proposed Design Procedure................................................................................ 234 5.1.1 Assess the Existing Structure ....................................................................... 235 5.1.2 Define the Objectives and Parameters for Strengthening ............................ 235 5.1.3 Determine Design Values for the Composite............................................... 236 xi 5.1.4 Select Appropriate Resistance Factors......................................................... 237 5.1.5 Calculate the Amount of FRP Needed to Meet the Design Objective ......... 239 5.1.6 Perform Final Checks on the Design............................................................ 240 5.1.7 Specify Appropriate Quality Control Measures to be Followed During Application of FRP.................................................................................................... 240 5.2 Design Example.................................................................................................. 240 5.2.1 Structural Assessment .................................................................................. 241 5.2.2 Objectives and Parameters for Strengthening .............................................. 241 5.2.3 Composite Design Values ............................................................................ 242 5.2.4 Selection of Resistance Factors.................................................................... 245 5.2.5 Calculating the Required Area of FRP......................................................... 249 5.2.6 Check the Stress in the FRP under Sustained Loads.................................... 254 5.3 Reliability Assessment of Design Example........................................................ 254 5.4 Summary............................................................................................................. 256 Chapter 6. Conclusions, Recommendations, and Areas for Further Study ............................ 257 6.1 Summary............................................................................................................. 257 6.2 Areas for Further Study ...................................................................................... 257 6.2.1 FRP Composite Material Properties and Design Factors............................. 258 6.2.1.1 Statistical Description of Properties .................................................... 258 6.2.1.2 Prefabricated Composites.................................................................... 259 6.2.1.3 Application Factors ............................................................................. 260 6.2.1.4 Degradation Models ............................................................................ 260 6.2.2 Limit States for Evaluation .......................................................................... 260 6.2.2.1 Flexure................................................................................................. 261 xii 6.2.2.2 Shear.................................................................................................... 261 6.2.2.3 Slabs .................................................................................................... 262 6.2.2.4 Serviceability....................................................................................... 262 6.2.2.5 Modeling Error .................................................................................... 262 6.2.2.6 Interaction of Limit States ................................................................... 262 6.2.3 Statistical Models of Load............................................................................ 263 6.2.4 Modeling Continued Structural Degradation ............................................... 264 6.2.5 Time Dependent Reliability......................................................................... 264 6.2.6 Selection of βT .............................................................................................. 265 6.2.7 Understanding the State of the Existing Structure ....................................... 265 6.3 Conclusion .......................................................................................................... 266 Appendix A. Live Load Statistics for Specified Design Life................................................. 267 A. 1 Introduction to Problem...................................................................................... 267 A. 2 Attempted Derivation of Extreme Value Distribution........................................ 267 A. 2.1 Basic Distribution of the Maximum ........................................................ 267 A. 2.2 Attempted Use of Distribution of the Maximum..................................... 268 A. 3 Different Methods Used to Assess Time Dependent Reliability........................ 272 A. 3.1 Definition of Trial Conditions ................................................................. 272 A. 3.2 Trial Calculation Techniques and Results ............................................... 273 A. 4 Conclusions ........................................................................................................ 276 Appendix B. Goodness of Fit Tests....................................................................................... 278 B. 1 Introduction......................................................................................................... 278 B. 2 Chi Squared Test ................................................................................................ 278 B. 3 EDF Tests ........................................................................................................... 279 xiii B. 3.1 Kolmogorov Smirnov Test .......................................................................... 280 B. 3.2 Anderson Darling Test................................................................................. 280 Appendix C. Preliminary Calibration Examples .................................................................... 282 C. 1 Introduction......................................................................................................... 282 C. 2 Sample Girder..................................................................................................... 282 C. 3 General Procedure for Strengthening Design ..................................................... 284 C. 4 Composite Material Properties for Calibration................................................... 285 C. 5 Load Factors for Use in Strengthening Design................................................... 287 C. 6 Large Example Calibration without Corrosion................................................... 290 C. 6.1 Description of Procedures and Variables ..................................................... 291 C. 6.1.1 Degraded Structure ............................................................................. 291 C. 6.1.2 FRP Properties .................................................................................... 291 C. 6.1.3 Degraded Properties............................................................................ 292 C. 6.1.4 Designs ............................................................................................... 293 C. 6.1.5 Reliability Analysis ............................................................................ 293 C. 6.1.6 Time Dependent Reliability ............................................................... 295 C. 6.1.7 Load Variables.................................................................................... 295 C. 6.1.8 Resistance Variables........................................................................... 297 C. 6.2 Results of Sample Calibration without Corrosion........................................ 298 C. 6.2.1 Effect of Reliability Calculation Method............................................ 298 C. 6.2.2 Effect of Different Amounts of Steel Loss ......................................... 302 C. 6.2.3 Effect of Time Span for Evaluation.................................................... 304 C. 6.2.4 Effect of Differences in Mean Value of FRP Properties .................... 308 C. 6.2.5 Effect of Changes in Modulus Coefficient of Variation..................... 312 xiv C. 6.2.6 Effect of Changes in Strength Coefficient of Variation...................... 315 C. 7 Effect of Resistance Variables Considered in Reliability Analysis.................... 316 C. 8 Example with Corrosion ..................................................................................... 318 C. 8.1 Design Philosophy ....................................................................................... 318 C. 8.2 Degraded Structure....................................................................................... 319 C. 8.3 Prediction of Remaining Steel...................................................................... 320 C. 8.4 FRP Properties ............................................................................................. 321 C. 8.5 Design of Strengthening............................................................................... 322 C. 8.6 Reliability Analysis...................................................................................... 322 C. 8.7 Random Variables........................................................................................ 322 C. 8.8 General vs. Pitting Corrosion....................................................................... 323 C. 8.9 Results of Sample Calibration with Corrosion............................................. 324 C. 9 Summary of Conclusions from Sample Calibrations.......................................... 326 Appendix D. Sectional Analysis ............................................................................................ 328 D. 1 Introduction......................................................................................................... 328 D. 2 RC Section without FRP..................................................................................... 328 D. 3 RC Section with Externally Bonded FRP........................................................... 330 Appendix E. Techniques Used in Reliability Assessment ................................................ 336 E. 1 Monte Carlo Simulation ..................................................................................... 336 E. 2 Generating Random Numbers from a Statistical Distribution............................ 339 E. 3 Implementation of FORM .................................................................................. 341 Appendix F. Java Programs.................................................................................................... 344 F. 1 Program for Design of Strengthening................................................................. 344 F. 1.1 Variables ...................................................................................................... 344 xv F. 1.2 Procedure...................................................................................................... 346 F. 1.3 Code ............................................................................................................. 347 F. 2 Program for Simulation and Evaluation of Resistance Statistics........................ 353 F. 2.1 Variables ...................................................................................................... 354 F. 2.2 Procedure...................................................................................................... 355 F. 2.3 Code ............................................................................................................. 356 Appendix G. Data from Bridge Survey ............................................................................. 366 G. 1 Summary of Dimensions Collected .................................................................... 366 Appendix H. Load Analysis in QConBridge ™ ...................................................................... 379 H. 1 Program Description........................................................................................... 379 H. 2 Input Details for Calibration Girders .................................................................. 381 References .............................................................................................................................. 385 xvi LIST OF FIGURES Figure 1 1 Components of Reliability Based Design for FRP Strengthening.......................... 17 Figure 2 1 Basic Structural Reliability Problem ...................................................................... 25 Figure 2 2 Graphical Representation of Probability of Failure................................................ 28 Figure 2 3 Interpretation of β in Terms of the Safety Margin ................................................. 31 Figure 2 4 Design Truck for HS 20 and HL 93 Load Models................................................. 64 Figure 2 5 Tuutti’s ( 1982) Model for Sequence of Steel Corrosion in Concrete.................... 70 Figure 2 6 Relation between Concrete Compressive Strength and Water Cement Ratio........ 80 Figure 3 1 Plot of Cumulative Distribution Functions for Set A1 Strength........................... 111 Figure 3 2 Changes in β with Additional Required Strengthening ........................................ 128 Figure 3 3 Ratio of Tested Strength to Predicted Strength vs. Fiber Volume Fraction for One Layer Samples ............................................................................................................... 144 Figure 3 4 Ratio of Tested Strength to Predicted Strength vs. Fiber Volume Fraction for One Layer Samples Without Set E1...................................................................................... 145 Figure 3 5 Ratio of Tested Modulus to Predicted Modulus vs. Fiber Volume Fraction for One Layer Samples ............................................................................................................... 146 Figure 4 1 Basic Flowchart for Calibration Procedure ......................................................... 173 Figure 4 2 Histogram of Bridge Spans................................................................................... 184 Figure 4 3 Histogram of Number of Girders.......................................................................... 185 Figure 4 4 Histogram of Deck Width..................................................................................... 185 Figure 4 5 Plot of Convergence of Monte Carlo Results as a Function of the Number of Trials ............................................................................................................................... ....... 209 Figure 4 6 Example of Plots Used to Select Calibrated Resistance Factors .......................... 210 Figure 4 7 ψ vs. Strength COV for Girder 18, Corrosion Condition 4, SD, βT = 3.5, and φ = 0.85 ............................................................................................................................... 218 Figure A 1 PDF of Bias Factor for Maximum Load for Different Time Spans..................... 269 Figure A 2 CDF of Bias Factor for Maximum Load for Different Time Spans .................... 270 xvii Figure A 3 Comparison of Distributions for Mean Maximum 50 Year Load Bias Factor.... 271 Figure C 1 β vs. ψ for Material 1 Designs to Meet LRFD and LRFR Loads ........................ 290 Figure C 2 Monte Carlo Results for 20% Steel Loss, Strength COV = 0.25, Modulus COV = 0.05, 0 degradation, 75 year loads.............................................................................. 299 Figure C 3 Hybrid Results for 20% Steel Loss, Strength COV = 0.25, Modulus COV = 0.05, 0 degradation, 75 year loads .......................................................................................... 300 Figure C 4 Monte Carlo Results for 30% Steel Loss, Strength COV = 0.25, Modulus COV = 0.05, 0 degradation, 75 year loads ....................................................................... 301 Figure C 5 Hybrid Results for 30% Steel Loss, Strength COV = 0.25, Modulus COV = 0.05, 0 degradation, 75 year loads .......................................................................................... 301 Figure C 6 Hybrid Results for 20% Steel Loss, Strength COV = 0.15, Modulus COV = 0.15, no degradation, 75 year loads ........................................................................................ 305 Figure C 7 Hybrid Results for 20% Steel Loss, Strength COV = 0.15, Modulus COV = 0.15, 5 year exposure, 5 year loads ........................................................................................ 305 Figure C 8 Hybrid Results for 20% Steel Loss, Strength COV = 0.15, Modulus COV = 0.15, 50 year exposure, 50 year loads .................................................................................... 306 Figure C 9 Hybrid Results for 20% Steel Loss, Strength COV = 0.15, Modulus COV = 0.15, 5 year exposure, 5 year loads ........................................................................................ 307 Figure C 10 Hybrid Results for 30% Steel Loss, Strength COV = 0.25, Modulus COV = 0.05, 5 year exposure, 5 year loads ....................................................................................... 309 Figure C 11 Hybrid Results for 30% Steel Loss, Strength COV = 0.25, Modulus COV = 0.05, 50 year exposure, 50 year loads ................................................................................... 310 Figure C 12 Hybrid Results for 20% Steel Loss, Strength COV = 0.25, Modulus COV = 0.05, 5 year exposure, 5 year loads ....................................................................................... 313 Figure C 13 Hybrid Results for 20% Steel Loss, Strength COV = 0.25, Modulus COV = 0.15, 5 year exposure, 5 year loads ....................................................................................... 314 Figure C 14 Hybrid Results for 20% Steel Loss, Strength COV = 0.25, Modulus COV = 0.25, 5 year exposure, 5 year loads ....................................................................................... 314 Figure C 15 ψ as a function of Strength COV for 20% Steel Loss and Modulus COV = 15% ............................................................................................................................... ....... 315 Figure C 16 ψ as a function of Strength COV for 30% Steel Loss and Modulus COV = 15% ............................................................................................................................... ....... 316 xviii Figure C 17 Effect of Using Different Cases of Random Variables to Assess Reliability for Material 2 Designs......................................................................................................... 318 Figure C 18 Reliability Index vs. Composite Specific Resistance Factor for Material 1, φ = 0.90, ............................................................................................................................... 324 Figure D 1 Forces in a Rectangular Section at Ultimate ( Only Steel Reinforcement) .......... 329 Figure D 2 Forces in a Rectangular Section ( Steel and FRP Reinforcement) ....................... 331 Figure E 1 Flow Chart of Monte Carlo Simulation ............................................................... 337 Figure H 1 Example of Bridge Model for Girder 12 ( not to scale)....................................... 384 xix LIST OF TABLES Table 1 1 Questions to be Answered in LRFD Development.................................................. 18 Table 2 1 Probabilities of Failure and Corresponding β s ....................................................... 32 Table 2 2 Comparison of Live Load Factors for Inventory and Operating Levels.................. 44 Table 2 3 Distribution Properties for Slab Dimensions ........................................................... 57 Table 2 4 Distribution Properties for Beam Dimensions ......................................................... 58 Table 2 5 Comparison of HS 20 and HL 93 Load Models for Calculation of Maximum Positive Moment.............................................................................................................. 63 Table 2 6 Ratio of Mean Maximum Moments to HL 93 Moments ......................................... 65 Table 2 7 Causes of Deterioration of Concrete ( Bertolini et al., 2004) ................................. 68 Table 2 8 Rates of Corrosion Penetration of Steel in Concrete ( Bertolini et al., 2004)........... 73 Table 2 9 Rates of Corrosion Penetration Based on Concrete Cover and Exposure Condition74 Table 2 10 Approximate Relation between Concrete Strength and Water Cement Ratio....... 79 Table 2 11 Comparison of Common Risks and Structural Failure Probabilities .................... 82 Table 2 12 Target Failure Probabilities and Reliability Indices Based on CIRIA................... 84 Table 2 13 Target Failure Probabilities and Reliability Indices Based on Allen ( 1981) W= 0.1 ............................................................................................................................... ......... 85 Table 2 14 Target Reliability Levels and Corresponding Lifetime Probabilities of Failure from Nordic Report .................................................................................................................. 86 Table 2 15 Target Reliability Indices and Corresponding Annual Probabilities of Failure for Other Structural Design Codes ........................................................................................ 88 Table 2 16 Adjustments to Target Reliability for Canadian Bridge Evaluation ...................... 90 Table 3 1 Summary of Wet Layup Data Sets.......................................................................... 98 Table 3 2 Descriptive Statistics for Ultimate Tensile Strength.............................................. 100 Table 3 3 Descriptive Statistics for Longitudinal Modulus ................................................... 102 Table 3 4 Descriptive Statistics for Thickness....................................................................... 103 xx Table 3 5 Distribution Parameters for Ultimate Tensile Strength.......................................... 108 Table 3 6 Distribution Parameters for Longitudinal Modulus ............................................... 109 Table 3 7 Distribution Parameters for Composite Thickness ................................................ 110 Table 3 8 Chi Squared Goodness of Fit Results for Strength ............................................... 112 Table 3 9 Kolmogorov Smirnov Goodness of Fit Results for Strength, α= 0.10 .................. 113 Table 3 10 Anderson Darling Goodness of Fit Results for Strength, α= 0.25...................... 114 Table 3 11 Chi Squared Goodness of Fit Results for Modulus............................................. 115 Table 3 12 Kolmogorov Smirnov Goodness of Fit Results For Modulus, α= 0.10............... 115 Table 3 13 Anderson Darling Goodness of Fit Results for Modulus, α= 0.10 ...................... 116 Table 3 14 Chi Squared Goodness of Fit Results for Thickness........................................... 117 Table 3 15 Kolmogorov Smirnov Goodness of Fit Results for Thickness, α= 0.10.............. 117 Table 3 16 Anderson Darling Goodness of Fit Results for Thickness, α= 0.25 ................... 118 Table 3 17 Correlation Coefficients for Wet Layup Composites........................................... 121 Table 3 18 Different Ways of Specifying the Characteristic Value for FRP Strength ......... 123 Table 3 19 Properties of Model Composite .......................................................................... 125 Table 3 20 Reliability of Designs Using Different COVs for Strength ................................ 125 Table 3 21 Basic Description of System of Application Factors ........................................... 139 Table 3 22 Properties of Fibers and Matrices for Prediction of Strength and Modulus......... 141 Table 3 23 Mean and COV of Ratio of Tested Values to Values Predicted Using Properties of Fiber and Matrix for Strength........................................................................................ 142 Table 3 24 Mean and COV of Ratio of Tested Values to Values Predicted Using Properties of Fiber and Matrix for Modulus ....................................................................................... 142 Table 3 25 Manufacturer Properties for Sets E and F............................................................ 147 Table 3 26 Ratio of Tested Properties to Manufacturer Reported Properties........................ 148 Table 3 27 λlayers for Strength and Modulus........................................................................... 151 Table 3 28 Generalized λlayers for Design............................................................................. 151 xxi Table 3 29 Preliminary Values of Application Factors for Wet Layup Composites ............. 154 Table 3 30 Mean and COV of Ratio of Tested to Predicted Force per Unit Width............... 155 Table 3 31 Mean and COV of Ratio of Tested to Predicted Stiffness per Unit Width .......... 156 Table 3 32 Mean and COV of Ratio of Tested to Predicted Force per Unit Width............... 157 Table 3 33 Mean and COV of Ratio of Tested to Predicted Stiffness per Unit Width .......... 157 Table 3 34 Stress Limitations as Percentage of Ultimate Strength ....................................... 161 Table 3 35 Predictive Equations for Property Retention Based on an Arrhenius Rate Relation ( Abanilla, 2004)............................................................................................................. 165 Table 4 1 LRFR Load Factors for Design of Strengthening ( AASHTO, 2003) .................... 176 Table 4 2 Generalized Composite Properties Used for Calibration ....................................... 180 Table 4 3 Bridge Quantities Surveyed to Determine Common Values for Calibration......... 183 Table 4 4 Geometry of Representative Bridges for Calibration............................................. 187 Table 4 5 Comparison of Distribution of Span Lengths for Selected Bridges....................... 188 Table 4 6 Comparison of Distribution of Number of Girders for Selected Bridges .............. 188 Table 4 7 Comparison of Distribution of Deck Widths for Selected Bridges........................ 189 Table 4 8 Comparison of QConBridge ™ and CT BDS for Selected Girders........................ 193 Table 4 9 Load Components and LRFR Factored Load for Design ...................................... 196 Table 4 10 Distribution Parameters of Load for Reliability Analysis.................................... 204 Table 4 11 Statistical Distributions Used in Reliability Analysis.......................................... 206 Table 4 12 Baseline LRFR Steel Areas and Steel Areas for Each Corrosion Condition in mm2 ( in. 2) ............................................................................................................................... 213 Table 4 13 Summary of Resistance Factors for Different Target Reliabilities and Different Amounts of Relative Steel Loss .................................................................................... 217 Table 4 14 Example of Calibrated ψ for Girder 15, Corrosion Case 2, with FRP Degradation ............................................................................................................................... ....... 221 Table 4 15 Example of Calibrated ψ for Girder 3, Corrosion Case 5, with FRP Degradation ............................................................................................................................... ....... 222 xxii Table 4 16 Example of Calibrated ψ for Girder 5, Corrosion Case 2................................... 224 Table 4 17 Example of Calibrated ψ for Girder 14, Corrosion Case 4.................................. 225 Table 4 18 Comparison of Calibrated Resistance Factors with Changes in Strength and Modulus COVs for Girder 16, Corrosion Condition 4, βT = 3.5, φ = 0.85.................... 229 Table 4 19 Example of Calibrated ψ for Girder 4, β = 3.0, Corrosion Condition 1, φ = 0.9, AD ............................................................................................................................... ....... 232 Table 4 20 Example of Calibrated ψ for Girder 4, β = 3.0, Corrosion Condition 2, φ = 0.9, AD ............................................................................................................................... ....... 232 Table 5 1 Approximate Values for COVcharacteristic for Wet Layup Composites Based on Testing ............................................................................................................................... ....... 239 Table 5 2 Dimensions and Material Properties of Girder 15 ................................................ 241 Table 5 3 Results from Lamina Level Tests .......................................................................... 242 Table 5 4 Preliminary Values of Application Factors for Wet Layup Composites ............... 243 Table 5 5 Resistance Factors for Design Example................................................................. 248 Table 5 6 Final Design Quantities.......................................................................................... 254 Table 5 7 Statistical Distributions for Set A1 used in Reliability Analysis ........................... 255 Table A 1 Comparison of Estimated Bias Factors and Bias Factors from NCHRP Report 368 ............................................................................................................................... ....... 270 Table A 2 Basic Details of Strengthening Example............................................................... 273 Table A 3 Different Methods Used to Calculate Time Dependent Reliability...................... 274 Table A 4 Comparison of Reliabilities for Different Computation Techniques .................... 275 Table C 1 Bridge Deck Dimensions ...................................................................................... 283 Table C 2 Load Effects for Girder Design............................................................................. 283 Table C 3 Dimensions of Sample Girder ............................................................................... 284 Table C 4 Mean Property Values of Sample Composites...................................................... 288 Table C 5 Mean Property Values of Sample Composites...................................................... 292 Table C 6 Percent Retention of FRP Properties for Different Design Lives ......................... 293 xxiii Table C 7 Comparison of Two Different Reliability Procedures .......................................... 295 Table C 8 Statistical Description of Dead Loads................................................................... 296 Table C 9 Live Load Statistics for Different Design Lives ................................................... 296 Table C 10 Statistics of Total Load ....................................................................................... 297 Table C 11 FRP Rupture Strains at Different Design Lives .................................................. 311 Table C 12 Cases for Assessment of Resistance Variable Effect on Reliability ................... 317 Table C 13 Relation of Condition States for Bridge Management Systems to Structural Integrity of Bridge ......................................................................................................... 320 Table C 15 Remaining Steel Area for Various Design Lives ................................................ 321 Table C 16 Assumed Properties for Sample Composites ...................................................... 321 Table C 17 COV of Remaining Steel Area for Different Design Lives ................................ 325 Table E 1 Values of k for Different Two Sided Confidence Levels...................................... 338 Table F 1 Variables in Design Program................................................................................. 345 Table F 2 Variables in MCS Program.................................................................................... 354 Table G 1 Key to Bridge Dimensions in this Appendix ........................................................ 367 Table G 2 Key to Notes Column............................................................................................ 368 Table G 3 Data Collected in Bridge Survey .......................................................................... 369 Table H 1 Summary of Input Data for Girder Analysis......................................................... 382 xxiv xxv ABSTRACT Externally bonded fiber reinforced polymer ( FRP) composites are an increasingly adopted technology for the renewal of existing concrete structures. In order to encourage the further use of these materials, a design code is needed that considers the inherent material variability of the composite, as well as the variations introduced during field manufacture and environmental exposure while in service. Load and Resistance Factor Design ( LRFD) is a reliability based design methodology that provides an ideal framework for these considerations and is compatible with existing trends in civil engineering design codes. This investigation studies the application of LRFD to FRP strengthening schemes with an emphasis on wet layup, carbon fiber composites applied to reinforced concrete T beam bridge girders. Models to describe variation in the existing structural materials and the structural loading are drawn from the literature. Techniques for reliability analysis are discussed, and existing work on externally bonded FRP reliability is surveyed. Stochastic variation in the FRP is characterized based on tensile testing of several sets of field manufactured, wet layup composites. A general design procedure applicable to many different situations is proposed using a composite specific resistance factor to consider material variability, a set of Application Factors to account for deviations introduced through field manufacture, and an environment and service life specific factor for FRP degradation. Preliminary resistance factors for design of FRP strengthening are calibrated over a range of design scenarios. FRP degradation is considered based on existing durability models, and continued degradation of the structure due to general corrosion of the reinforcing steel is included. The girders used for calibration are selected as representative examples from a sample of California bridge plans. The reliability has been evaluated using simulation and xxv first order reliability methods. An example of the proposed design procedure, using the calibrated resistance factors, is provided. The results of this work bring to light the many variables affecting the reliability of strengthened members and the need for continuing research to better describe these variables. Two variables of particular significance, requiring extensive further study, are the state of the existing structure when strengthening is applied and the loads acting on the structure. xxvi Chapter 1. Introduction 1.1 Overview Externally bonded fiber reinforced polymer composites ( FRPs) are increasingly considered as a viable means of strengthening, retrofitting, and repairing existing reinforced concrete structures. In appropriate situations, these materials can offer significant advantages over more traditional techniques of adding new or replacing lost load carrying capacity. There is a pressing need for this type of technology as our country’s infrastructure ages. A prime example can be found in the U. S. bridge inventory; in 2004 the Federal Highway Administration deemed over a quarter of the nation’s bridges deficient based on data from 2002. Nearly fourteen percent of bridges were found to be structurally deficient, with an additional fourteen percent functionally obsolete ( FHWA, 2004). At the present time FRP strengthening is a technique seeing growing usage. In order to facilitate the continued growth of this technology and to provide for the long term safety of designs using FRPs, it is vital that a design code is developed for their use in strengthening. However, there are many challenges to be overcome in design code development such as the unique characteristics of FRPs, the incomplete database of material properties, and the somewhat limited understanding of the interaction between the FRP and the existing structure. 1.2 FRPs for Strengthening of Civil Structures 1.2.1 Fiber Reinforced Polymer Composites A composite is a material that is composed of two or more distinct phases. The constituent materials work together to produce properties that are more desirable than those of the individual materials. FRPs are composed of a fibrous reinforcing phase embedded in a polymeric matrix. Typical fiber types include carbon, glass, and aramid. Many different 1 polymers may be used for the matrix phase. In strengthening applications the resin system is typically a thermosetting polymer such an epoxy or vinylester. This combination of materials provides FRPs with a number of unique, and often advantageous, properties. FRPs are perhaps best known for their high specific strength and stiffness ( defined as the property divided by the material density). Unidirectional composites may have specific strengths nearly an order of magnitude greater than those of common metals, such as steel or aluminum ( Kaw, 1997). Other advantageous properties of composites include their enhanced fatigue resistance at the material level, resistance to corrosion, and tailorability. There are many different methods used to fabricate composite materials. Several, such as autoclave forming and resin transfer molding, are impractical for use in civil applications. The most common forms of FRP used for strengthening are wet layup systems, manufactured directly on the structure through a manual process, and prefabricated strips, which are often manufactured through pultrusion and then bonded to the structure with adhesives. Other special systems may be used to provide automated wrapping of columns or apply posttensioning ( International, 2001). 1.2.2 Strengthening and Repair of Civil Structures Structures designed by civil engineers are intended to have a long lifespan, and during that time there are many reasons why the structure may require strengthening or repair1. ( Täljsten, 2002; Ellingwood, 1996). The most significant of these reasons include: 1 It should be noted that strengthening generally implies adding capacity to a structure, while repair signifies returning a structure to its original capacity. This report treats these two applications of FRP to externally reinforce concrete structures interchangeably; however, the term strengthening is generally used. 2 1. Environmental Exposure  Civil structures are exposed to changing environmental conditions throughout their lifetimes. These factors can cause material degradation over time or impart significant damage during one extreme event. The impacts of environmental degradation will be especially felt in cases where regular maintenance is not performed. 2. Changing Usage  It is not uncommon for civil structures to outlive the purpose for which they were originally designed. Changes in tenancy or use may place different or larger load demands on the structure. 3. Changing Design Standards  Even if the use of the structure is not significantly changed, the standards the structure must meet may change over time. 4. Errors in Design or Construction  Civil structures may even require strengthening before they are ever used due to errors in the initial design or construction. Strengthening is not new to civil applications; however, in the past it generally meant placing more concrete, bonding steel plates, or applying some sort of post tensioning to the structure ( The Concrete Society, 2000). Now many types of strengthening can be accomplished with FRPs ( Täljsten, 2002; The Concrete Society, 2000). FRP strengthening can be applied to mitigate several failure modes. For flexural strengthening of beams, slabs, or girders, FRP plates can be applied to the tensile face of the concrete. Shear and torsional strengthening can be accomplished by placing FRP on the sides of beams. Columns are typically strengthened by wrapping the FRP around the column in the hoop direction, thus 3 increasing the confinement of the concrete core. This can be accomplished with wet lay up or prefabricated cylindrical jackets. 1.2.3 Advantages of FRPs for Strengthening The unique properties of FRPs result in many advantages from the perspective of strengthening designers ( The Concrete Society, 2000; Täljsten, 2002; International, 2001; ACI, 2002; Maruyama, 2001). FRPs do not suffer from corrosion as do steel plates, allowing the possibility of extended service lives or perhaps limiting required maintenance. Their high strength and stiffness to weight ratios mean that a smaller weight of FRP needs to be applied as compared to steel plate bonding. This low weight reduces transportation costs, significantly eases installation, even in tight spaces, and can eliminate the need for scaffolding, reducing traffic impact. The low weight also means that FRPs add only a small amount to the structure’s dead load. This allows more of the strengthening to be useful to the structure and also makes FRPs a repair option when significant additional weight could cause failure. Additionally, FRPs are typically applied in thin strips, resulting in very little change in the structural profile, an important feature on bridges or other structures that require clearances for vehicles or machinery. The way that FRPs are manufactured also provides useful properties. By designing the placement of the reinforcing fibers, properties such as strength and modulus can be controlled in different directions. This allows the strengthening to act only in the needed direction, preventing it from changing the structural behavior in unintended ways. Because they are made from long thin fibers, FRPs are very easy to handle. They can be made to wrap around curves and to accept the irregularities present in concrete surfaces. Furthermore, they can be manufactured in long lengths, eliminating the need for splices, and can be cut to length on site, eliminating sizing errors in the manufacturing stage. 4 1.2.4 Disadvantages of FRPs for Strengthening Despite their numerous advantages FRPs are not without drawbacks ( The Concrete Society, 2000; Täljsten, 2002; International, 2001). Unidirectional FRP materials are characterized by linear elastic behavior up to failure; this lack of yielding can result in less ductile structures unless this behavior is specifically considered at the design stage. These materials are very susceptible to damage from impact, fire, or vandalism, and as such need to be protected. Though FRPs do not exhibit corrosion, they are not immune to environmental impacts and do suffer degradation due to moisture, temperature, and UV rays. This disadvantage is of particular importance because there is currently little long term information on the durability of composites in exposed environments. The initially high material cost of FRPs is also a drawback to many engineers, however, due to the cost advantages in transportation and installation offered by composites, the cost of a whole strengthening project can be comparable or even less than the same project strengthened with steel plates. 1.3 Design Code for FRP Strengthening 1.3.1 Need for a Design Code Other limits to the use of composites in strengthening are related to the unique aspects of civil design ( Ellingwood, 2003). Composite materials were initially developed and used in the mechanical and aerospace fields, fields that are significantly different from civil engineering. The typical mechanical or aerospace part will be mass produced at the end of engineering design, making it economically feasible to conduct testing throughout the design stage and to specifically tailor materials for a particular project. Furthermore, the design requirements, such as load demands, are clearly defined and the manufacturing processes used in these fields allow for very tight control of finished properties. In contrast, each civil design is a unique project that is usually designed and built just once. Due to cost, size, and time 5 constraints, routine civil designs are rarely tested before construction, and when testing does occur it is usually performed on a scale model or only a critical portion of the design. In place of testing civil design is based on knowledge of material properties, analysis, and prior experience, which are often actualized in codes of practice. For example, the International Building Code is a model code that is based on recognized standards and specifications developed by individual organizations with expertise in different aspects of construction, such as the American Institute of Steel Construction ( AISC) or the American Concrete Institute ( ACI). Bridge design is usually based on the specifications of the American Association of State Highway and Transportation Officials ( AASHTO). However, civil design is usually characterized by substantial uncertainty in load demands, especially those due to natural phenomena, and material properties that cannot be as tightly controlled. These uncertainties result in conservative specification of loads and material strengths in design codes. When governments adopt design codes they become part of local, state, and federal law, exposing civil engineers to liability concerns for designs that do not meet the standard. In addition to their legal implications, design codes also serve as a set of minimum technical requirements for acceptable design and provide a pathway for research findings to make their way into practice ( Ellingwood, 2000b). Thus, most design in civil engineering is based on codes of practice, and, without a comprehensive specification for FRP, it is unlikely that this promising new material will gain widespread acceptance and utilization. This is especially true because design with composite materials is not a typical component of the undergraduate civil engineering education. The lack of design code and designer experience are the most significant obstacles limiting the present use of composites in civil infrastructure. 6 1.3.2 Uncertainty in Structural Design The goal of the structural engineer is to achieve structural safety in the face of numerous uncertainties. Nearly every variable considered in design is uncertain to varying degrees. Loads can be highly variable, especially when natural effects such as wind and earthquakes are considered. Materials have inherent variability and may suffer degradation when they are put in service. The models describing structural behavior are just that, models, and the uncertainty in their results is usually unknown. Even the service life of the design is an uncertain quantity. The result of uncertainty is risk, which is often defined as the product of the probability of failure and the costs associated with failure ( Ellingwood, 1994). Since the design variables are uncertain, there is a risk that the structure will fail due to overloading, when the loads exceed those for which the structure was designed, or that the structure will be understrength due weak materials or incorrect dimensions. Though it is impossible to completely eliminate risk, good engineering design can hold the risk to acceptable levels by accounting for the uncertainty inherent in design. 1.3.3 Design Philosophies as the Basis for Design Codes Currently there are two main philosophies behind civil design: Working or Allowable Stress Design and probabilistic based limit states design. Other approaches, such as Ultimate Strength Design or Load Factor Design, fall somewhere between these two approaches. In the United States probabilistic limit states design is typically implemented in the Load and Resistance Factor Design ( LRFD) format. Other parts of the world, such as Europe and Canada, also have design codes with a probabilistic basis; however, the implementation differs from the LRFD format ( Ellingwood, 1996). 7 1.3.3.1 Working Stress Design Working Stress Design has served as the basis for structural calculations since the late 19th century when calculations first started to be used for design ( Ellingwood, 2000a). In Working Stress Design the stresses in members due to service loads are elastically computed and compared to a specified allowable stress divided by a factor of safety. The basic checking equation used for Working Stress Design is shown in Eq. 1 1 wherein f is the elastically computed stress in the structure, F is the allowable stress, and FS is the factor of safety. FS f ≤ F Eq. 1 1 The factors of safety used in Working Stress Design are based on past experience and engineering judgment, not specific consideration of the uncertainties involved in design. As experience has been gained over time, factors of safety have generally been decreased ( Ellingwood, 1994). In this design format there is only one factor to account for all the uncertainties that may be encountered in loads and resistance. This neglects the fact that different types of load may have different degrees of variation, resulting in a range of structural reliabilities. This variation in structural reliability is one of the key drawbacks to Working Stress Design. 1.3.3.2 Load and Resistance Factor Design Load and Resistance Factor Design is a relatively new development in civil design. The theoretical basis for LRFD, structural reliability theory, was developed during the period from the late 1940s to the mid 1960s, at which point interest grew in incorporating the reliability research into standards for design ( Ellingwood, 1994). The first LRFD specification was adopted in 1986 by AISC with the first LRFD edition of the AISC Manual of Steel Construction ( Salmon and Johnson, 1996). 8 The LRFD approach to design is distinct from Working or Allowable Stress Design in two ways. First, it is based on a philosophy of defining pertinent limit states. A structure is said to reach a limit state when it fails to reach a level of performance for which it was designed. Limit states are typically divided into two categories: strength and serviceability. Strength limit states relate to the structure’s ability to carry load and include limits such as the plastic capacity of a ductile member, fracture of brittle materials, and instability or buckling. Service limit states are primarily related to the comfort of occupants and include excessive deflection, vibration, and/ or cracking ( Salmon and Johnson, 1996). Including strength and serviceability, AASHTO defines four different kinds of limit states in the AAHSTO LRFD Bridge Design Specifications ( AASHTO, 2004). Fatigue and fracture provisions are considered separately from the strength provisions and are intended to prevent failure through cyclic loading. The extreme event limit state specifically considers rare events such as earthquakes, floods, or collisions, which can be considered as statistically insignificant loads. In WSD structures are evaluated at typical service conditions; in LRFD structures are evaluated in the ways they are likely to fail by considering the applicable limit states. LRFD is also different from Working Stress Design in that it is based on probabilistic analysis of the uncertainties present in design. The factors in LRFD based specifications are specifically calibrated such that the probability of reaching a particular limit state is acceptably small. This probability is most often measured in terms of the reliability index, β. In the development of a LRFD code, a target value of β is set, and design factors for load and resistance are selected such that a wide range of designs will be close to this target, usually with a bit of conservatism. The reliability index and the methods of structural reliability theory used to calibrate design factors are discussed further in Chapter 2 of this report. 9 The basic design equation in LRFD is shown in Eq. 1 2 where φ is the resistance factor, usually specific to material and failure mode and sometimes to a particular limit state, Rn is the nominal resistance, γ i is the load factor specific to load i, and Qi is the load effect due to load i. ≥ Σ i n i i φR γ Q Eq. 1 2 In the LRFD format different types of loads such as dead, live, wind, snow, earthquake, etc. each have their own load factor. Different types of loads are given different load factors depending on their coefficient of variation. These factors were calibrated for buildings in the late seventies and are intended to be applicable for all design materials ( Ellingwood, 1994; Galambos et al., 1982; Ellingwood et al., 1982). Load factors for bridge design were calibrated in the nineties for use in the AASHTO LRFD Bridge Design Specifications ( Nowak, 1999). The variations in capacity caused by material variability, geometric uncertainty, and modeling error are accounted for by the resistance factor φ. Resistance factors generally depend on the material being used and the limit state being checked. 1.3.3.3 Advantages of LRFD From the viewpoint of the designer LRFD is still a deterministic format with no explicit reliability calculations required. However, the probabilistic basis of LRFD is much more complex than the empirical basis of Working Stress Design. There are many advantages to the LRFD format ( Ellingwood, 2000a; Salmon and Johnson, 1996). 1. Designs created with LRFD have much more uniform reliabilities than those created with Working Stress Design. 2. The random nature of materials and loads is handled in a rational and analytical manner; the factors are derived based on statistics not just experience. Since 10 the factors have an statistical basis it is much clearer when factors should be changed based on new research or technology. 3. Since LRFD is a limit states approach, structures are evaluated in the ways they are likely to fail. This can provide for better evaluation of serviceability limit states. It also makes the relationship between behavior and design easier to comprehend. 4. Since separate factors are used for load and resistance, research on one or the other can be conducted independently, and changes can be made to either side as new information is gained. 5. With load factors common to all materials LRFD simplifies the design process. 6. For unusual load cases or new materials LRFD provides a framework with which to approach design code development. These advantages, plus the fact that LRFD is the design philosophy that most civil design is moving towards, make LRFD the design philosophy of choice for development of a code for FRP strengthening. 1.3.4 Current Design Guidelines for FRP Strengthening There are several current guidelines for the use of FRP to strengthen reinforced concrete structures: 1. Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures, published by the American Concrete Institute, 2002 11 2. Externally Bonded FRP Reinforcement for RC Structures published by the International Federation for Structural Concrete ( fib), 2001 3. Design Guidance for Strengthening Concrete Structures Using Fibre Composite Materials from The Concrete Society, 2000 4. Strengthening Reinforced Concrete Structures with Externally Bonded Fibre Reinforced Polymers from ISIS Canada, 2001 ( Neale, 2001) 5. FRP Strengthening of Existing Concrete Structures, Design Guidelines by Björn Täljsten, 2002 6. Recommendations for Upgrading of Concrete Structures with Use of Continuous Fiber Sheets, from the Japanese Society of Civil Engineers, 2001 ( Maruyama, 2001) All of these guidelines draw together a large body of research into a document that is easily understood by designers, and as such they are a valuable advance in the use of FRPs to renew existing concrete structures. However, these documents also share many limitations. The guidelines are all quite similar in their design approach, and at first glance all appear very similar to LRFD. All use a limit states approach to defining design checking equations. Design procedures and equations are given for basic strength limit states such as flexure or shear; however, the accuracy of these approaches is questionable in some cases, particularly shear. Debonding is discussed in all; however, the level of detail varies significantly. The approach to serviceability limit states also varies from guide to guide. All of the guidelines rely on the load factors already developed in relevant specifications for new design, and many use familiar resistance factors or partial material factors from probabilistic design codes for other materials. However, these design guidelines are not true probabilistic codes. No 12 calibration procedure was used to specifically derive the resistance factors in order to achieve a target reliability; in fact, no such target was even set. Thus, while these guidelines are a significant advance in the use of FRPs for the strengthening of concrete structures, there is still work to be done to develop a code in the preferred LRFD format. 1.4 Problem Statement and Research Objectives 1.4.1 Problem Description Externally bonded fiber reinforced polymer composites have shown promising performance as a means of strengthening existing reinforced concrete structures. As the infrastructure of our country continues to become deficient due to ageing, environmental attack, and growing usage, the development and implementation of repair strategies, such as the external bonding of FRPs, can serve a vital role in economically promoting the safety of engineered structures. However, due in large part to a lack of design code and designer experience, this technology is currently under utilized. Design guidelines are already available for the use of bonded FRPs as a strengthening measure. However, these guidelines are based on a deterministic format for design. Given the high level of material variability that can be exhibited by FRP systems, deterministic design is likely to produce an unacceptably large range of project reliabilities. The framework provided by LRFD is an ideal method for considering the inherent material uncertainty, as well as the time dependent material behavior, to produce designs with an acceptable level of structural reliability. Therefore, a design code in the LRFD format must be developed to promote the usage of FRP for strengthening. Development of a LRFD based design procedure requires extensive data: 1. Statistical data characterizing the load and resistance variables 13 2. Methods for defining nominal or design values of load and resistance 3. Definition of applicable limit states and models for structural behavior at these limits 4. A range of application defining the cases for which the code is valid 5. A target reliability index 6. A method by which reliabilities can be calculated during the calibration process With regard to FRP strengthening, some of this information is already available. For example, the statistical variation of traditional materials, such as steel and concrete, has been examined by previous researchers. Stochastic load models have been developed and used for both building and bridge structures. Researchers have developed many different expressions for modeling the behavior of structures strengthened with FRP. Structural reliability theory has reached the stage where there are a number of mature techniques for evaluation of structural reliability. Despite this foundation of available information, there are still significant gaps in knowledge as well as significant challenges to the development of a probabilistic code for the design of FRP strengthening. For example, there is not an adequate existing database of FRP properties as used in civil infrastructure projects. Given the variation that is introduced during field manufacture and application of FRPs this is a significant shortcoming. The multitude of possible fiber/ matrix combinations creates an additional level of complexity. Time dependent material properties have not been considered in the development of codes for other materials; however, this is an important concern for FRPs exposed to severe environments. Furthermore, as the existing load descriptions are intended for new designs, they contain many conservative assumptions and may be too demanding for renewal of existing structures. Many of the 14 existing equations for modeling the effect of FRPs on structural behavior are too involved for routine design use. Selection of an appropriate reliability target is also a challenge in that there is no direct basis for comparison. 1.4.2 Research Objectives The intent of this research is to answer many of the questions regarding probabilistic design of FRP strengthening  at least in a preliminary fashion  and to use these answers to develop a LRFD approach to strengthening design. LRFD can be thought of as a general philosophy for approaching design; in order to tailor this philosophy to composite materials and to strengthening rather than new design, several sub objectives have been identified and will be addressed in this report: Given the variety of composites available for strengthening, the proposed procedure must have the flexibility to accommodate this variety. The initial as well as time dependent properties of FRPs must be explicitly considered in order to ensure acceptable reliability over the proposed life of the strengthening. This procedure must recognize that the addition of extra capacity during strengthening can be much more costly than at the design stage. This requires special consideration of the assessment of initial accuracy of the target reliability level and the statistical models used for loading. Many structures will require strengthening due to deterioration, and the application of strengthening does not necessarily halt the deterioration process. Therefore, continued degradation of the structure and its effect on the time dependent reliability should be considered. 15 The design methodology must be compatible with existing codes for new design. Given the current state of knowledge regarding FRPs and their use for strengthening, it is certain that advances will occur in the future. Therefore, as much as possible, the procedure should lend itself to incorporating new research results into the specifications without requiring full reliability based recalibration of the design code. However, because the design factors developed in this research must be considered as preliminary and likely to require recalibration in the future, the underlying data and procedures should be documented as clearly as possible to aid future work. To guide further development of LRFD for FRP strengthening critical gaps in the existing research should be identified and suggestions made as to how to fill the gaps in the most advantageous manner. 1.4.3 Research Approach As discussed in Section 1.4.2 LRFD is simply a way of approaching structural design. In order to develop an effective LRFD procedure this general approach must be tailored to the circumstances in question. With regard to FRP strengthening of existing structures there are three main components to be considered: the FRP material, the structure to be strengthened, and the requirements of reliability based design. These three components are considered schematically in Figure 1 1. 16 FRP Reliability Structure Theory Figure 1 1 Components of Reliability Based Design for FRP Strengthening The three components identified in Figure 1 1 are shown with significant amounts of overlap. This represents the need to consider the problem as a whole, rather than as three independent components. Each region on this diagram can be used to identify an aspect of developing LRFD for FRP strengthening. The significant questions associated with each portion of the diagram are given in Table 1 1. 17 Table 1 1 Questions to be Answered in LRFD Study Region Questions to be Answered Structural Reliability What is the reliability basis of a LRFD code? What reliability methods are available for use? FRP / Structural Reliability How is the FRP characterized statistically? How is this statistical characterization related to the design value? FRP What types of FRP is the code applicable to? FRP / Structure How is the FRP applied to the structure? ( What limit states?) How is the effect of strengthening modeled? Structure What kind of structures is the code applicable to? Structure / Structural Reliability How can the existing structure be described statistically? How are loads on the structure described? FRP / Structure / Structural Reliability How should the design equation be formatted? ( Where should the factors go?) What is an appropriate reliability target? What reliability method can accommodate the materials and limit states? What is the range of applicability of the design code? The questions given in Table 1 1 have driven this research project, and their answers form the basis of this report. For purposes of the present work, several of these questions can be answered immediately, including the type of FRP, the type of structure, and the limit states considered. Though the design procedure, in particular the definition of material values for design, has been developed with an eye toward accommodating the full range of FRP materials, the specific examples given herein are for carbon fiber reinforced, wet layup composites. This focus was driven by the frequent use of carbon composites for strengthening and by the availability of data and material for assessing the variability of wet layup composites. Again, while the general format of design presented in this report is applicable to all types of structures, the specific example considered in this report is the class of T beam bridge superstructures. This choice was largely motivated by the sponsor of this research, the 18 California Department of Transportation. Furthermore, the limit state considered in this example is the flexural capacity of the girders. This choice was made based on the availability of load and resistance models and will be discussed further later in the report. Answers to the remaining questions shown in Table 1 1 are developed throughout the remainder of this report. It should be noted that, as research progressed on this project, it became abundantly clear that there are many topics where the existing state of knowledge is simply insufficient to provide for LRFD development without the use of extensive assumptions. The areas of uncertainty are diverse and most merit significant amounts of further study. The topic of strengthening of box girders is a pertinent one but is outside the scope of this investigation as defined by the funding agency. 1.4.4 Outline of the Report This report follows the progressive development of a LRFD based design procedure for FRP strengthening. Chapter 2 is devoted to developing the background for the project including structural reliability methods, existing statistical data, and the model chosen for continuing structural degradation. Chapter 3 develops the statistical description and defines the design values for the FRP material. In Chapter 4 a full sample calibration is conducted and preliminary factors for design are given. The complete design procedure is summarized and a design example is provided in Chapter 5. Finally, Chapter 6 concludes this report with a summary of the main accomplishments of this work and a detailed discussion of the areas remaining for further study. Numerous appendices supplement the text by providing more detail on issues raised, descriptions of some of the procedures used herein, and additional tabulated data. 19 The background information for this research is presented in Chapter 2. This chapter starts with a discussion of structural reliability methods, including basic topics such as the safety margin and reliability index. Computational methods for determining the reliability index are briefly reviewed as well as special topics, such as system reliability. The chapter then gives an overview of the prior implementation of LRFD for design using other materials, such as steel, wood, and concrete. Previous work on applying reliability based design to FRP strengthening of concrete structures is discussed. The chapter also summarizes the available statistical data and degradation models used throughout the remainder of the report. The chapter concludes by discussing the selection of the target reliability index used to calibrate resistance factors. In Chapter 3, the results of several sets of material test data are analyzed to determine appropriate statistical descriptors for FRP, including the distribution type, correlation between variables, and ranges for the mean and coefficient of variation of material properties. A value for use in design is specified as well as modification factors intended to account for the differences between laboratory tested properties and field manufactured properties. The effect of environmental degradation on FRP properties is also considered. A large example calibration is the topic of Chapter 4. This chapter thoroughly describes the assumptions and calculation methods used to derive design factors. Specific data for the example, such as the range of material properties and geometric quantities and the models used for design, are also described. Preliminary factors for the design of flexural strengthening of T beam bridge girders, considering the possibility of continued degradation, are presented. Chapter 5 summarizes the results of this project. It describes the proposed design procedure in detail and provides a full design example using the FRP material design values 20 and preliminary design factors derived in this project. It is shown that the design procedure and calibrated factors are able to create designs meeting the target reliability; however, a thorough verification of the design factors is still required in the future. The final chapter in this report, Chapter 6, is devoted to an inventory of further work needed to fully develop a LRFD specification for FRP strengthening. This work primarily involves improving and expanding available data for load models, resistance modeling and FRP properties, as well as research to improve the understanding of the state of the existing structure before the FRP is applied. 21 Chapter 2. Background for Structural Reliability, LRFD and Design Uncertainty 2.1 Introduction The goal of this chapter is to develop a firm foundation for the development of reliability based design for FRP strengthening. Topics covered herein include structural reliability methods, previous implementation of LRFD for other materials, prior work on the reliability of FRPs in civil applications, statistical descriptors for resistance and load variables, procedures for modeling the continued deterioration of structures through corrosion of the reinforcement, and selection of a target reliability index. For each specific topic, this chapter is organized to first provide some general background and to then discuss specific data, models, procedures, and assumptions that are utilized throughout the remainder of this report. 2.2 Structural Reliability Methods 2.2.1 Uncertainty and Risk Uncertainty exists in nearly everything humans do, and the design of structures is no exception. There are several different types of uncertainty that contribute to the uncertain performance of structural systems. Uncertainty is traditionally divided into two categories, aleatory, referring to inherent or “ natural” variability, and epistemic, describing errors that are related to a lack of complete knowledge and may be reduced with more information ( Melchers, 1999). Haldar and Mahadevan ( 2000) recognize two broader categories of uncertainty, cognitive or qualitative uncertainty, which is related to vagueness caused by trying to represent reality in abstract form, and noncognitive or quantitative uncertainty. Melchers ( 1999) gives a thorough breakdown of the different types of uncertainty that affect the design and performance of structures: 22 Phenomenological uncertainty is encountered whenever the structure is of a form that causes uncertainty in its eventual behavior or performance. This type of uncertainty usually affects novel structures, where designers do not have the benefit of previous experience to aid in the design process. This type of uncertainty is cognitive and cannot be included in a reliability analysis in a quantitative way. Decision uncertainty is caused by the difficulty in determining whether or not an event has occurred. In terms of structures this type of uncertainty refers to whether or not a limit state has been violated. Decision uncertainty may be of particular importance with regard to serviceability limit states where it is difficult to draw a clear line distinguishing acceptable and unacceptable performance; however, it would be very difficult to model for analysis purposes. Modeling uncertainty arises from the need to model natural phenomena with mathematical equations. This is an epistemic error that can typically be reduced through further research. If a relation between model predictions and actual test results can be found, modeling error can be accounted for in reliability analysis. Prediction uncertainty arises from the need of designers to anticipate future conditions, such as the material properties achieved during construction and the loading the structure will be subject to during its lifetime. This type of uncertainty affects how variables are modeled and how the reliability calculations are carried out, but is not explicitly accounted for in the calculations themselves. Physical uncertainty is the inherent randomness that exists in material properties and in the loads applied to a structure. This type of uncertainty may, perhaps, be reduced for materials through the use of better quality control, but it cannot be eliminated. This type of uncertainty makes up the bulk of the uncertainty considered in modern reliability analysis. 23 Statistical uncertainty arises from the need to estimate statistical descriptors such as the mean, variance, and probability distribution from limited sets of data. This type of uncertainty may be considered in analysis by allowing statistics such as mean and variance to be random variables or by performing multiple analyses with different values of the parameters. Human error is the final category of uncertainty. Human errors may be divided into routine variations in performance and gross errors. Gross human errors are likely the most common cause of structural failures; however, they are generally not included in reliability calculations because they are very hard to model in a quantitative fashion. This is an important reason why calculated reliabilities do not directly correspond to actual observed failure rates, which are generally much higher ( Melchers, 2001). Correctly modeling boundary conditions such as in scour in order to predict performance is also crucial. The presence of uncertainty creates risk. Risk is defined in some contexts as simply the probability that failure may occur. In other contexts risk is associated with the severity of the failure, and is calculated as the probability of failure times the cost of failure ( Ellingwood, 1994). 2.2.2 Evaluation of Structural Reliability 2.2.2.1 Effect of Uncertainty A primary goal of structural design is to create a structure such that the resistance capacity of the structure is greater than the load demands placed on it. This task is complicated by uncertainties because, in their presence, resistance and load cannot be described by deterministic quantities. Instead, uncertainties result in a range of possible resistance and load values and introduce the possibility that the applied load will exceed the capacity of the structure. The most basic situation, where only one resistance variable acts against just one load variable, is shown in Figure 2 1. In this figure the uncertainty in the resistance, R, and the load effect, S, is represented by probability density functions fR( r) and 24 fS( s). Following customary notation, random variables will be represented herein by capital letters, and individual realizations of a random variable will be represented with lower case letters. The following discussion on structural reliability is drawn from a number of sources ( Melchers, 1999; Madsen et al., 1986; Haldar and Mahadevan, 2000; Conte, 2004). μS μR r, s fR( r) , fS ( s) Sn Rn Figure 2 1 Basic Structural Reliability Problem 2.2.2.2 Deterministic Safety Factors In Figure 2 1, μ S and μR are the mean of the load and resistance, respectively, and Sn and Rn are the nominal load and capacity used for design. The nominal value is determined according to the design procedure in use. It may be a specified as a percentile of test results, calculated through the use of empirical or analytical equations, or prescribed in the code governing the design. In traditional deterministic design a safety factor would be calculated as the ratio of the nominal resistance to the nominal load, or a central safety factor could be calculated as the ratio of the mean of resistance to the mean of load. However, since they do not take into account the full distributions of load and resistance, safety factors calculated in this manner are unable to give an accurate assessment of the design safety. 25 The structural element whose load and resistance curves are shown in Figure 2 1 will fail if the load effect exceeds the resistance of the member. This can be seen to occur in the region of overlap between these two curves. The area of this region is not the probability of failure, as it is commonly mistaken to be. However, the size of this region can be considered to qualitatively indicate the probability of failure. By considering Figure 2 1 it can be seen that the probability of failure will change as the relative position of the two distributions changes ( i. e. the mean values are changed causing a shift along the axis of one or both of the curves), as the amount of spread in one or both of the distributions changes ( increased spread will increase the area of overlap), and as the shape of the distributions changes ( for example from a Normal distribution to a Lognormal distribution). The traditional approach to design seeks to provide acceptable designs by shifting the position of the distributions through the use of safety factors. These design approaches do not, however, consider the shape or spread of the distribution. A more rational approach to design is to consider all three issues in selection of design criteria. 2.2.2.3 Basic Reliability Problem The basic problem of structural reliability is to use statistical knowledge of uncertainties to compute the probability of structural failure. In actuality, computing the reliability of an entire structure is a very difficult task that will be briefly discussed in Section 2.2.2.7. The present discussion is pertinent to computing the reliability of a particular member with respect to a particular limit state. Given a random resistance, R, and a random load demand, S, the probability of structural failure can be expressed in many ways, as shown by Melchers ( 1999): p P( R S) f = ≤ Eq. 2 1a 26 p = P( R − S ≤ 0) f Eq. 2 1b ⎟⎠ ⎞ ⎜⎝ = ⎛ ≤ 1 S p R f Eq. 2 1c p = P( ln R − ln S ≤ 0) f Eq. 2 1d p = P[ G( R, S) ≤ 0] f Eq. 2 1e Eq. 2 1e shows the most general form. In this statement G( ) is referred to as the limit state function. This function is used to denote the boundary between safe ( or acceptable) structural behavior and unsafe ( or unacceptable) behavior. When the value of G( ) is less than zero, the limit state is said to have been violated and the structure is in the unsafe zone. Thus, the probability of failure can be expressed as the probability that the limit state has been violated. Failure in this sense means that the structure fails to meet some criteria of performance, not necessarily that the member has indeed failed; for example, serviceability limit states may be evaluated where “ failure” is defined as excessive deflection, even though the member can still support load. Furthermore, the limit state may be expressed in terms of many structural variables, not just the load and the resistance. For example, for the case of a bar element in pure tension, rather than expressing the limit state in terms of only R and S, the resistance could be expressed as a product of the area, A, and yield stress, σ y , of the member as shown in Eq. 2 2. In the following discussion of structural reliability methods, just two variables, R and S, will be considered for simplicity and clarity. G A S A S y y ( , σ , ) = σ − Eq. 2 2 27 In the most general case, load and resistance may be correlated random variables, and in order to evaluate the probability of limit state violation the joint probability density function ( PDF), fRS( r, s), is required. Figure 2 2 shows contour lines of a general joint PDF. The volume contained in an area of dimensions Δr, Δs underneath the joint PDF represents the probability that the random variable R takes a value between r and Δr while at the same time the random variable S takes a value between s and Δs. The limit state function is shown by the dotted line. R S Limit State Surface r = s Failure Domain Safe Domain fRS( r, s) Figure 2 2 Graphical Representation of Probability of Failure The area of the joint PDF shaded in grey represents the values for R and S where the limit state function is violated. In order to calculate the probability of failure, integration must be used to compute the total volume of probability under the joint PDF. This integral is expressed in Eq. 2 3. In the case where R and S are independent random variables, this equation can be expressed in terms of the marginal distributions of R and S as shown in Eq. 2 4 and Eq. 2 5. In these equations FR( ) and FS( ) are the cumulative density functions ( CDF) 28 of R and S, respectively. The lower bound of integration on these equations is zero since resistance cannot be negative. ∫∫ ≤ = () 0 ( , ) G f RS p f r s drds Eq. 2 3 ∫ ∞ = 0 p F ( s) f ( s) ds f R S Eq. 2 4 ∫[ ] ∞ = − 0 p 1 F ( r) f ( r) dr f S R Eq. 2 5 Eq. 2 3 may be generalized to account for as many random variables as are present in the problem by using the joint PDF for all variables. In general, obtaining a joint PDF for all variables is very difficult, and, even if one is available, analytical solution of these integrals is often impossible. Therefore other techniques must be used to estimate the reliability. However, there are some special cases that allow for direct solution. 2.2.2.4 The Reliability Index The reliability index is denoted with the symbol β. It is often used as a substitute for the probability of failure. This substitution is typically used because the difference between the probability of failure calculated using reliability methods and that actually witnessed in the field makes it desirable to avoid stating an explicit probability of failure. The causes of this difference are discussed in Section 2.2.2.9. β may be used to compare different structures and can be used as the target in reliability based design without mentioning a specific probability of failure. The meaning of β can be best interpreted through consideration of one of the few cases where the probability integrals of Section 2.2.2.3 can be computed analytically. If R and S are 29 independent, Normally distributed variables, the linear safety margin, Z= R S, is also a Normally distributed variable. Given the means, μR and μ S , and variances, σR 2 and σ S 2, of resistance and load, respectively, the mean and variance of Z can be expressed as shown in Eq. 2 6 and Eq. 2 7, respectively, following basic formulas for linear combinations of Normally distributed variables. Z R S μ = μ − μ Eq. 2 6 2 2 2 Z R S σ = σ + σ Eq. 2 7 By computing the standard Normal variate of Z, Eq. 2 1b can now be expressed in terms of Φ, the cumulative distribution function of the standard Normal distribution, as shown in Eq. 2 8. This function is available in tabulated form in many statistics texts and can also be accessed using the NORMSDIST function in Microsoft Excel. ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − − Φ = ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − = − ≤ = ≤ = Φ 2 2 0 0 ( ) ( 0) ( 0) R S R S Z Z f p P R S P Z σ σ μ μ σ μ Eq. 2 8 Cornell ( 1969) defined the reliability index, or safety index, as shown in Eq. 2 9, as the mean of the safety margin divided by the standard deviation of the safety margin. This formulation of the reliability index is referred to as the Cornell reliability index. In this formulation β can be interpreted as the distance between the mean of the safety margin and the point of failure, measured in terms of the standard deviation of the safety margin. This is shown in Figure 2 3. Z Z σ μ β = Eq. 2 9 30 Cornell’s definition of β holds even if the safety margin is a function of more than two variables, as long as the failure surface is linear. The definition of β has been extended to cover other cases, see Melchers ( 1999) and Madsen et al.( 1986). However, for all definitions, the basic concept of β remains the same; it is a measure of the distance between the most likely state of the structure ( mean) and the most likely failure point, in terms of the variation. β σ μ Failure Domain Safe Domain PDF of Safety Margin, Z 0 Figure 2 3 Interpretation of β in Terms of the Safety Margin Though it is only exact in the case of Normally distributed variables and a linear limit state function, β is often approximately related to the probability of failure through Eq. 2 10, which is a generalization of Eq. 2 8. Table 2 1 shows values of β corresponding to several probabilities of failure based on this approximate equation. = Φ(− β ) f p Eq. 2 10 31 Table 2 1 Probabilities of Failure and Corresponding β s pf β 0.5 0 0.1 1.28 0.01 2.33 0.001 3.09 0.0001 3.72 0.00001 4.27 2.2.2.5 Methods of Computing the Reliability Index In some cases the probability integral in Eq. 2 3 may be solved using numerical integration. However, this integral is an n fold integral where n represents the number of design variables, and thus the complexity rapidly increases as variables are added to the problem. Several other methods have been developed to calculate the probability of failure, or the reliability index directly. The following brief summary of some of the most popular techniques is based on Melchers ( 1999), Madsen et al. ( 1986), Haldar and Mahadevan ( 2000), and Conte ( 2004). In Section 2.2.2.10 the specific methods used for this report and the reasons why they were selected will be discussed. 2.2.2.5.1 First Order, Second Moment Reliability Index This formulation of the reliability index is an extension of the Cornell reliability index. βFOSM is still defined as the mean of the safety margin divided by the standard deviation of the safety margin and still requires only the mean and standard deviation of the design variables for calculation. However, to allow for non linear limit state functions, the mean and standard deviation of the safety margin are computed by linearizing the safety margin using the linear terms of a Taylor series expansion. The value of βFOSM depends on the point that is chosen for linearization of the limit state function. A common choice is the point where each random 32 variable takes on its mean value, resulting in the mean value, first order, second moment reliability index. Though this method is very simple, it has several drawbacks. The most significant drawback is that the value of βFOSM is not invariant with respect to the limit state formulation; two mechanically equivalent formulations of the same limit state can produce different values for the reliability index. The ambiguity of βFOSM caused by the formulation of the limit state function can be removed by choosing a point on the limit state surface as the expansion point. The point used for expansion is called the design or checking point. When all variables and the limit state function are transferred to standard Normal space the design point is found through a minimization procedure as the point on the limit state surface with the shortest distance to the origin. The reliability index found in this manner is sometimes referred to as βHL, the Hasofer Lind reliability index ( Hasofer and Lind, 1974). This measure of reliability is invariant with respect to the limit state formulation, but it uses only second moment information about the variables and is not comparable because it does not depend on the curvature of the limit state function at the design point. The generalized reliability index introduced by Ditlevsen makes use of a weighting function to overcome the lack of comparability ( Madsen et al., 1986). 2.2.2.5.2 First and Second Order Reliability Methods ( FORM and SORM) The first order reliability method is very similar to the first order, second moment method. The limit state function is still linearized using a first order approximation about the design point. The significant difference between these two methods is that the first order, second moment reliability index only uses second moment information about the design variables. FORM uses knowledge of the full distribution of the design variables. The distributions of the design variables are transformed into standard Normal space using 33 appropriate techniques, such as the Normal tail transformation for independent variables or the Rosenblatt transformation for dependent variables ( Melchers, 1999). βFO can then be found using the same optimization techniques used in calculation of βFOSM. To find the probability of failure, Eq. 2 10 may be used to give a first order approximation. SORM makes use of non linear expansions of the limit state function. Determining the probability content of these non linear surfaces is complicated, and approximate techniques must be used to estimate pf. The accuracy of both of these techniques depends on the ability of the approximating surface to represent the true limit state. 2.2.2.5.3 Monte Carlo Simulation ( MCS) Monte Carlo Simulation is a technique with applications to many different problems ( Rubinstein, 1981). In general the Monte Carlo technique involves using random sampling to generate a large set of artificial data that may be analyzed. In application to structural reliability, a random value is generated for each design variable based on the statistical distribution of that variable, and the random values are used to check the limit state equation. If the limit state function is less than zero, the structure is considered to have “ failed”. This process is repeated a large number of times, and the probability of failure is estimated as the number of failed samples divided by the total number of simulations. This approach is very robust and can be applied to almost any limit state formulation. However, the accuracy of this approach depends on the number of simulations, and for small failure probabilities the required computing time can be very demanding. With additional knowledge about the failure region, variance reduction techniques, such as importance sampling, can be used to concentrate the simulations in the region of interest and reduce the necessary number of simulations. 34 2.2.2.5.4 Other Techniques In some cases a hybrid approach combining simulation with FORM or SORM is used. This may be the case when convergence cannot be reached using all the variables in FORM. This approach was taken by Plevris et al. ( 1995). The statistical description of resistance of a strengthened member was determined using Monte Carlo Simulation, and then FORM was used with distributions for load to compute the structural reliability. Response surface techniques can be used when the limit state function is not in an explicit functional form, such as when structural behavior is modeled using finite element analysis. This technique is based on evaluation of the implicit limit state at a limited number of points, and then fitting a function to these points. This functional form can then be directly used with first order second moment methods or FORM/ SORM. 2.2.2.6 Levels of Reliability Methods The structural reliability methods described above have been grouped into different levels defined by the types of information they use and the types of calculation they imply ( Madsen et al., 1974; Melchers, 1999). These levels are now briefly described in order to illuminate the relation between reliability computation techniques and the LRFD format. Level 1 methods are the simplest techniques, and are used to provide safety in the design of structures. These techniques rely on just one value to describe each design parameter. ASD, LRFD and other code level techniques are examples of Level 1 methods. Level 2 methods make use of two values describing each design parameter ( most often the mean and variance) along with a description of the correlation between parameters to calculate a reliability index. These methods are more approximate in their estimate of the probability of failure than methods such as FORM or simulation. The first order, secondmoment reliability index is an example of this level. 35 Level 3 methods are those methods that seek the best possible estimate of the probability of failure by making use of full distributions to describe the design variables. Examples of this level include FORM/ SORM and Monte Carlo Simulation. Level 4 methods combine consideration of the structure from the previous levels with economic data and seek to provide a cost benefit analysis. These methods can provide a rational basis for decision making. The goal of the study is to develop factors for a LRFD design procedure for FRP strengthening. The design format itself is an example of Level 1 reliability methods; however, the design factors will be calibrated based on Level 2 techniques for computing the reliability. 2.2.2.7 Component vs. System Reliability The first order second moment, FORM, and SORM methods described above are all capable of predicting the probability of failure for a particular member with respect to a particular mode of failure. In reliability analysis this is referred to as component reliability. However, most structural elements are susceptible to failure in several different modes, for example a beam must resist both flexural and shear loads. Furthermore, nearly all structures are composed of many members. The probability of failure of a single element in one of several different mechanisms or the probability of failure of an entire structure are both problems of system reliability. Computation of system reliability is significantly more difficult than computation of component reliability. Monte Carlo Simulation is an option if a model of the complete system can be developed. Or the reliability of a system can be computed based on the reliability of individual components, often relying on simplifying approximations such as assuming the system is a series system or a parallel system. However, the resulting methods remain too complicated for use in calibration of a reliability based design procedure. Design factors are therefore based on the reliability of a single member 36 with respect to a particular limit state. This would directly correspond to the weakest link analogy when considering the reliability of the complete structure; however, due to the indeterminate nature of most structures, the reliability of the structure as a whole is usually much higher than the reliability of individual components. This fact must be kept in mind when selecting a target reliability for code calibration. More information on system reliability may be found in Madsen et al. ( 1986) and Melchers ( 1999). 2.2.2.8 Time dependent Reliability When the statistical description of load or resistance ( as for a deteriorating structure) changes with time, these changes must be considered through the use of time dependent reliability techniques. The most rigorous approaches involve the use of stochastic processes, and the probability of failure is determined as the first passage probability, i. e. the probability that the limit state will be violated for the first time during the time period in question. Direct solution of these theoretical formulations is often intractable except for the simplest cases, and thus numerical or simulation based approaches are necessary ( Melchers, 1999). Researchers studying degradation of concrete structures have formulated the reliability problem as seen in Eq. 2 11. In this equation pf represents the probability of failure, R( ti) is the time dependent resistance, and Si are independent random loading events during the life of the structure. This equation has been solved using simulation for deterministic loading increments ( Stewart 1998), as well as stochastic load processes ( Mori 1993). 1 [ ( ) ( ) ... ( ) ] f 1 1 2 2 n n p = − P R t > S ∩ R t > S ∩ ∩ R t > S Eq. 2 11 While the form of Eq. 2 11 is far easier to handle than dealing directly with outcrossing rates ( the probability of leaving the safe domain from a certain point weighted with the probability of being at that point), this equation is still quite demanding from a calibration 37 standpoint. The commonly used approach in development of reliability based codes is referred to as the time integrated approach ( Melchers, 1999). In this method, the complete lifetime ( or at least the lifetime of interest) of the structure is considered to be a single unit. All random variables must be related to this period of time by choosing appropriate statistical descriptions. This approach has been used in the calibration of previous reliability based codes. For example, the service life considered in development of the AASHTO LRFD Bridge Design Specifications is 75 years, and thus the code is calibrated based on an estimated distribution of the 75 year maximum loading ( Nowak, 1999). In this case, deterioration of the structure was not considered, so the resist
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Title  Development of resistance factors for LRFD design for FRP strengthening of reinforced concrete bridges 
Subject  TG340.A83 2006 compu/d; Concrete bridgesMaintenance and repair.; Fiberreinforced concrete.; Load factor design.; T989.F772 CD 
Description  Title from PDF title page.; "May 2006."; Includes bibliographical references.; Final report.; Electronic text (PDF: xxvi, 378 p. : col. ill.).; Submitted to the California Department of Transportation, Engineering Service Center, under contract no. 
Creator  Atadero, Rebecca. 
Publisher  Dept. of Structural Engineering, University of California, San Diego 
Contributors  Karbhari, Vistasp M.; Structural Systems Research Project.; University of California, San Diego. Dept. of Structural Engineering.; California. Dept. of Transportation. Engineering Service Center. 
Type  Text 
Language  eng 
Relation  Also available online.; http://svctwww1.dot.ca.gov/hq/esc/earthquake_engineering/Research_Reports/vendor/uc_san_diego/200613/2006.13%20Report%20Final.pdf; http://worldcat.org/oclc/213416790/viewonline 
DateIssued  [2006] 
FormatExtent  1 CDROM : col. ; 4 3/4 in. 
RelationRequires  System requirements: Adobe Acrobat Reader; CDROM drive. 
RelationIs Part Of  Report / Structural Systems Research Project ; no. SSRP06/13; Report (Structural Systems Research Project) ; no. SSRP06/13. 
Transcript  STRUCTURAL SYSTEMS RESEARCH PROJECT Report No. SSRP– 06/ 13 DEVELOPMENT OF RESISTANCE FACTORS FOR LRFD DESIGN FOR FRP STRENGTHENING OF REINFORCED CONCRETE BRIDGES by REBECCA ATADERO VISTASP M. KARBHARI Final Report Submitted to the California Department of Transportation Under Contract No. 59A0401. May 2006 Department of Structural Engineering University of California, San Diego La Jolla, California 92093 0085 University of California, San Diego Department of Structural Engineering Structural Systems Research Project Report No. SSRP– 06/ 15 DRAFT Development of Resistance Factors for LRFD Design for FRP Strengthening of Reinforced Concrete Bridges by Rebecca Atadero Graduate Student Researcher Vistasp M. Karbhari Professor of Structural Engineering Final Report Submitted to the California Department of Transportation Under Contract No. 59A0401. Department of Structural Engineering University of California, San Diego La Jolla, California 92093 0085 May 2006 Technical Report Documentation Page 1. Report No. FHWA/ CA/ ES 2006/ 11 2. Government Accession No. 3. Recipient’s Catalog No. 4. Title and Subtitle Development of Load and Resistance Factor Design for FRP Strengthening of Reinforced Concrete Structures 5. Report Date 5/ 26/ 2006 6. Performing Organization Code 7. Author( s) Rebecca A. Atadero and Vistasp M. Karbhari 8. Performing Organization Report No. UCSD / SSRP 06/ 13 9. Performing Organization Name and Address Department of Structural Engineering School of Engineering 10. Work Unit No. ( TRAIS) University of California, San Diego La Jolla, California 92093 0085 11. Contract or Grant No. 59A0401 12. Sponsoring Agency Name and Address California Department of Transportation 13. Type of Report and Period Covered Final Report – 5/ 26/ 2006 Engineering Service Center 1801 30th St., West Building MS 9 Sacramento, California 95807 14. Sponsoring Agency Code 15. Supplementary Notes Prepared in cooperation with the State of California Department of Transportation. This report is the first of multiple reports. 16. Abstract Externally bonded fiber reinforced polymer ( FRP) composites are an increasingly adopted technology for the renewal of existing concrete structures. In order to encourage the further use of these materials, a design code is needed that considers the inherent material variability of the composite, as well as the variations introduced during field manufacture and environmental exposure while in service. Load and Resistance Factor Design ( LRFD) is a reliability based design methodology that provides an ideal framework for these considerations and is compatible with existing trends in civil engineering design codes. This investigation studies the application of LRFD to FRP strengthening schemes, with an emphasis on wet layup, carbon fiber composites applied to reinforced concrete T beam bridge girders. Models to describe variation in the existing structural materials and the structural loading are drawn from the literature. Techniques for reliability analysis are discussed and existing work on externally bonded FRP reliability is surveyed. Stochastic variation in the FRP is characterized based on tensile testing of several sets of field manufactured wet layup composites. A general design procedure applicable to many different situations is proposed using a composite specific resistance factor to consider material variability, a set of Application Factors to account for deviations introduced through field manufacture, and an environment and service life specific factor for FRP degradation. Preliminary resistance factors for design of FRP strengthening are calibrated over a range of design scenarios. FRP degradation is considered based on existing durability models, and continued degradation of the structure due to general corrosion of the reinforcing steel is included. The girders used for calibration are selected as representative examples from a sample of California bridge plans. The reliability has been evaluated using simulation and first order reliability methods. An example of the proposed design procedure, using the calibrated resistance factors, is provided. The results of this work bring to light the many variables affecting the reliability of strengthened members, and the need for continuing research to better describe these variables. Two variables of particular significance, requiring extensive further study, are the state of the existing structure when strengthening is applied and the loads acting on the structure. 17. Key Words FRP, LRFD, Design of Strengthening 18. Distribution Statement No restrictions. This document is available to the public through the National Technical Information Service, Springfield, Virginia 22161. 19. Security Classification ( of this report) Unclassified 20. Security Classification ( of this page) Unclassified 21. No. of Pages ~ 420 22. Price Form DOT F 1700.7 ( 8 72) Reproduction of completed page authorized DISCLAIMER The opinions expressed in this report are those of the authors and do not represent positions of the California Department of Transportation. iii TABLE OF CONTENTS Disclaimer ............................................................................................................................... . iii Table of Contents ...................................................................................................................... iv List of Figures ........................................................................................................................ xvii List of Tables......................................................................................................................... .. xx Abstract ............................................................................................................................... .. xxv Chapter 1. Introduction .............................................................................................................. 1 1.1 Overview ................................................................................................................ 1 1.2 FRPs for Strengthening of Civil Structures ............................................................ 1 1.2.1 Fiber Reinforced Polymer Composites ............................................................ 1 1.2.2 Strengthening and Repair of Civil Structures................................................... 2 1.2.3 Advantages of FRPs for Strengthening............................................................ 4 1.2.4 Disadvantages of FRPs for Strengthening ....................................................... 5 1.3 Design Code for FRP Strengthening ...................................................................... 5 1.3.1 Need for a Design Code ................................................................................... 5 1.3.2 Uncertainty in Structural Design...................................................................... 7 1.3.3 Design Philosophies as the Basis for Design Codes ........................................ 7 1.3.3.1 Working Stress Design............................................................................ 8 1.3.3.2 Load and Resistance Factor Design ........................................................ 8 1.3.3.3 Advantages of LRFD............................................................................. 10 1.3.4 Current Design Guidelines for FRP Strengthening........................................ 11 1.4 Problem Statement and Research Objectives ....................................................... 13 1.4.1 Problem Description....................................................................................... 13 1.4.2 Research Objectives ....................................................................................... 15 iv 1.4.3 Research Approach ........................................................................................ 16 1.4.4 Outline of the Report...................................................................................... 19 Chapter 2. Background for Structural Reliability, LRFD and Design Uncertainty.................. 22 2.1 Introduction........................................................................................................... 22 2.2 Structural Reliability Methods.............................................................................. 22 2.2.1 Uncertainty and Risk...................................................................................... 22 2.2.2 Evaluation of Structural Reliability................................................................ 24 2.2.2.1 Effect of Uncertainty ............................................................................. 24 2.2.2.2 Deterministic Safety Factors ................................................................. 25 2.2.2.3 Basic Reliability Problem...................................................................... 26 2.2.2.4 The Reliability Index............................................................................. 29 2.2.2.5 Methods of Computing the Reliability Index........................................ 32 2.2.2.5.1 First Order, Second Moment Reliability Index ........................... 32 2.2.2.5.2 First and Second Order Reliability Methods ( FORM and SORM) …………….................................................................................................... 33 2.2.2.5.3 Monte Carlo Simulation ( MCS)................................................... 34 2.2.2.5.4 Other Techniques......................................................................... 35 2.2.2.6 Levels of Reliability Methods ............................................................... 35 2.2.2.7 Component vs. System Reliability ........................................................ 36 2.2.2.8 Time dependent Reliability ................................................................... 37 2.2.2.9 Limitations of Reliability Methods ....................................................... 38 2.2.2.10 Reliability Methods Used for This Report .......................................... 39 2.3 Previous Development of LRFD .......................................................................... 41 2.3.1 Steel................................................................................................................ 41 v 2.3.2 Loads.............................................................................................................. 42 2.3.3 Engineered Wood........................................................................................... 43 2.3.4 Bridges ........................................................................................................... 43 2.3.5 Concrete ......................................................................................................... 45 2.3.6 Aspects of Existing Codes Considered in this Work ..................................... 45 2.4 Previous Work on Reliability of FRP in Civil Infrastructure ............................... 46 2.4.1 FRP for Strengthening.................................................................................... 46 2.4.1.1 Limitations of Existing Studies ............................................................. 49 2.4.2 FRP for New Construction............................................................................. 50 2.4.2.1 General Design Standards ..................................................................... 51 2.5 Statistical Descriptors for Resistance Variables ................................................... 52 2.5.1 Concrete ......................................................................................................... 53 2.5.2 Reinforcing Steel............................................................................................ 55 2.5.3 Dimensions..................................................................................................... 56 2.5.3.1 Area of Steel.......................................................................................... 57 2.5.3.2 Slab Dimensions.................................................................................... 57 2.5.3.3 Beam Dimensions.................................................................................. 57 2.5.4 Modeling Uncertainty .................................................................................... 58 2.6 Description of Load Variables.............................................................................. 60 2.6.1 Dead Load ...................................................................................................... 61 2.6.2 Live and Impact Loads................................................................................... 61 2.7 Consideration of Continued Degradation ............................................................. 68 2.7.1 Modes of Reinforced Concrete Degradation.................................................. 68 2.7.2 Corrosion of Steel in Concrete ....................................................................... 69 vi 2.7.2.1 Carbonation Induced Corrosion ............................................................ 71 2.7.2.2 Chloride Induced Corrosion.................................................................. 72 2.7.2.3 Rates of Corrosion................................................................................. 72 2.7.3 Previous Work Modeling Corrosion Induced Degradation in Bridges.......... 74 2.7.4 Corrosion Models Used in this Report........................................................... 76 2.7.4.1 Major Assumptions for Corrosion Modeling ........................................ 76 2.7.4.2 Mathematical Models for Corrosion ..................................................... 78 2.8 Target Reliability Index........................................................................................ 81 2.8.1 Comparison to Other Acceptable Levels of Risk........................................... 82 2.8.2 Optimization of Cost Benefit ......................................................................... 83 2.8.3 Empirical Approaches.................................................................................... 83 2.8.4 Calibration to Safety Levels Implied by Existing Codes ............................... 86 2.8.4.1 Reliability Indices from Other LRFD Codes......................................... 87 2.8.5 Selection of Target β for this Work................................................................ 90 2.9 Discussion of Background Data ........................................................................... 92 Chapter 3. Characterization of Composite Properties for Reliability Analysis and Design..... 94 3.1 Introduction........................................................................................................... 94 3.2 Description of Data Sets ....................................................................................... 95 3.2.1 Testing Procedures ......................................................................................... 95 3.2.2 Wet Layup Composites .................................................................................. 95 3.3 Characterization of Random Variation ................................................................. 98 3.3.1 A Note on the Effect of Thickness ................................................................. 98 3.3.2 Basic Statistics ............................................................................................... 99 3.3.3 Statistical Distributions for Representing Composite Properties ................. 104 vii 3.3.3.1 Distributions ........................................................................................ 104 3.3.3.2 Distributions Fit to Wet Layup Composite Data ................................. 107 3.3.4 Best Fitting Distributions ............................................................................. 111 3.3.4.1 Strength ............................................................................................... 112 3.3.4.2 Modulus............................................................................................... 114 3.3.4.3 Thickness............................................................................................. 116 3.3.4.4 Summary of Distributions for Reliability Analysis............................. 118 3.3.5 Correlation between Variables ..................................................................... 119 3.4 Design Values for Composite Materials............................................................. 121 3.4.1 Current Approaches to Selection of Design Values..................................... 121 3.4.1.1 Reliability Implications of Current Design Approach......................... 123 3.4.2 Proposed Approach to Design Values.......................................................... 128 3.4.2.1 Accounting for Material Variability.................................................... 128 3.4.2.2 Use of the Mean as the Characteristic Value....................................... 130 3.4.2.3 Factors for Systematic Variation and Time Dependent Behavior....... 131 3.4.2.4 Promoting Reliability Based Design................................................... 131 3.5 Characterizing and Accounting for Systematic Differences between Laboratory Derived Design Values and In Situ Properties ................................................................ 133 3.5.1 Currently Used Factors ................................................................................ 133 3.5.2 Types of Systematic Variation ..................................................................... 136 3.5.3 Proposed Set of Application Factors............................................................ 137 3.5.4 Values of Factors for Wet Layup Composites ............................................. 139 3.5.4.1 Consideration of Thickness ................................................................. 140 3.5.4.2 Values for λpred .................................................................................... 140 viii 3.5.4.2.1 Predicted Value Based on Constitutive Properties .................... 140 3.5.4.2.2 Predicted Value Based on Manufacturer Data........................... 147 3.5.4.2.3 Predicted Value Based on Lamina or Laminate Level Tests ..... 149 3.5.4.3 Values for λlayers ................................................................................... 149 3.5.4.4 Values for λcure .................................................................................... 151 3.5.4.5 Values for λwork .................................................................................... 152 3.5.4.6 Summary of Factors for Systematic Variation of Wet Layup Composites. ........................................................................................................ 153 3.5.4.7 Assessment of Factor Accuracy .......................................................... 154 3.6 Time Dependent Degradation of FRP Properties............................................... 159 3.6.1 Current Approaches to Considering Time Dependent Behavior of FRP Properties................................................................................................................... 159 3.6.1.1 Environmental Exposure ..................................................................... 159 3.6.1.2 Sustained and Fatigue Loading ........................................................... 161 3.6.2 Proposed Method for Consideration of Time Dependent Degradation of FRP Properties................................................................................................................... 162 3.6.2.1 Factor for Environmental Degradation................................................ 162 3.6.2.1.1 Advantages of this Approach..................................................... 165 3.6.2.1.2 Limitations of Proposed Approach ............................................ 166 3.6.2.2 Stress Limitations for Sustained and Fatigue Loading........................ 167 3.6.2.2.1 Sustained Loading...................................................................... 167 3.6.2.2.2 Fatigue loading .......................................................................... 167 3.7 Summary............................................................................................................. 169 Chapter 4. Calibration of Resistance Factors for Flexural Strengthening of Bridge Girders. 171 ix 4.1 Introduction......................................................................................................... 171 4.2 Procedure for Calibration of Resistance Factors ................................................ 171 4.3 Summary of Previous Calibration Work ............................................................ 175 4.3.1 Load Factors for Strengthening Design ( Section C. 5) ................................. 175 4.3.2 Large Example Calibration without Corrosion ( Section C. 6)...................... 176 4.3.3 Example with Corrosion ( Section C. 8) ........................................................ 178 4.4 Range of Calibration........................................................................................... 178 4.4.1 Composite Materials .................................................................................... 179 4.4.1.1 Initial Properties .................................................................................. 179 4.4.1.2 States of FRP Degradation .................................................................. 180 4.4.2 Representative Members for Calibration ..................................................... 181 4.4.2.1 Typical Bridge Dimensions................................................................. 184 4.4.2.2 Selected Girders .................................................................................. 185 4.4.3 Time Periods Considered ............................................................................. 190 4.4.4 Cases of Continued Degradation.................................................................. 190 4.5 Design of Strengthening ..................................................................................... 191 4.5.1 Calculation of Design Load.......................................................................... 192 4.5.2 Calculation of Resistance ............................................................................. 197 4.5.2.1 Debonding Model................................................................................ 198 4.5.3 Computational Procedure............................................................................. 200 4.5.4 Summary of Designs .................................................................................... 201 4.6 Calculation of Reliability.................................................................................... 201 4.6.1 Description of Load Uncertainty.................................................................. 202 4.6.2 Description of Resistance Uncertainty......................................................... 205 x 4.6.3 Calculation Procedures................................................................................. 207 4.6.3.1 Simulation of Resistance ..................................................................... 207 4.6.3.1.1 Convergence .............................................................................. 208 4.7 Results ................................................................................................................ 209 4.7.1 Procedures Used to Analyze Reliability Results.......................................... 209 4.7.2 Effect of the Amount of Remaining Steel .................................................... 211 4.7.2.1 Significance ......................................................................................... 219 4.7.3 Effect of No Continuing Corrosion vs. Continuing Corrosion .................... 220 4.7.3.1 Significance ......................................................................................... 222 4.7.4 Effect of Different FRP Degradation Models .............................................. 223 4.7.4.1 Significance ......................................................................................... 225 4.7.5 Effect of Different Materials ........................................................................ 226 4.7.5.1 Significance ......................................................................................... 227 4.8 Extensions on the Large Calibration Example ................................................... 227 4.8.1 Effect of Changes in Modulus COV ............................................................ 227 4.8.1.1 Results ................................................................................................. 228 4.8.2 Effect of Different Bond Models ................................................................. 230 4.8.2.1 Results ................................................................................................. 231 4.9 Summary............................................................................................................. 232 Chapter 5. Recommended Design Procedure and Design Example....................................... 234 5.1 Proposed Design Procedure................................................................................ 234 5.1.1 Assess the Existing Structure ....................................................................... 235 5.1.2 Define the Objectives and Parameters for Strengthening ............................ 235 5.1.3 Determine Design Values for the Composite............................................... 236 xi 5.1.4 Select Appropriate Resistance Factors......................................................... 237 5.1.5 Calculate the Amount of FRP Needed to Meet the Design Objective ......... 239 5.1.6 Perform Final Checks on the Design............................................................ 240 5.1.7 Specify Appropriate Quality Control Measures to be Followed During Application of FRP.................................................................................................... 240 5.2 Design Example.................................................................................................. 240 5.2.1 Structural Assessment .................................................................................. 241 5.2.2 Objectives and Parameters for Strengthening .............................................. 241 5.2.3 Composite Design Values ............................................................................ 242 5.2.4 Selection of Resistance Factors.................................................................... 245 5.2.5 Calculating the Required Area of FRP......................................................... 249 5.2.6 Check the Stress in the FRP under Sustained Loads.................................... 254 5.3 Reliability Assessment of Design Example........................................................ 254 5.4 Summary............................................................................................................. 256 Chapter 6. Conclusions, Recommendations, and Areas for Further Study ............................ 257 6.1 Summary............................................................................................................. 257 6.2 Areas for Further Study ...................................................................................... 257 6.2.1 FRP Composite Material Properties and Design Factors............................. 258 6.2.1.1 Statistical Description of Properties .................................................... 258 6.2.1.2 Prefabricated Composites.................................................................... 259 6.2.1.3 Application Factors ............................................................................. 260 6.2.1.4 Degradation Models ............................................................................ 260 6.2.2 Limit States for Evaluation .......................................................................... 260 6.2.2.1 Flexure................................................................................................. 261 xii 6.2.2.2 Shear.................................................................................................... 261 6.2.2.3 Slabs .................................................................................................... 262 6.2.2.4 Serviceability....................................................................................... 262 6.2.2.5 Modeling Error .................................................................................... 262 6.2.2.6 Interaction of Limit States ................................................................... 262 6.2.3 Statistical Models of Load............................................................................ 263 6.2.4 Modeling Continued Structural Degradation ............................................... 264 6.2.5 Time Dependent Reliability......................................................................... 264 6.2.6 Selection of βT .............................................................................................. 265 6.2.7 Understanding the State of the Existing Structure ....................................... 265 6.3 Conclusion .......................................................................................................... 266 Appendix A. Live Load Statistics for Specified Design Life................................................. 267 A. 1 Introduction to Problem...................................................................................... 267 A. 2 Attempted Derivation of Extreme Value Distribution........................................ 267 A. 2.1 Basic Distribution of the Maximum ........................................................ 267 A. 2.2 Attempted Use of Distribution of the Maximum..................................... 268 A. 3 Different Methods Used to Assess Time Dependent Reliability........................ 272 A. 3.1 Definition of Trial Conditions ................................................................. 272 A. 3.2 Trial Calculation Techniques and Results ............................................... 273 A. 4 Conclusions ........................................................................................................ 276 Appendix B. Goodness of Fit Tests....................................................................................... 278 B. 1 Introduction......................................................................................................... 278 B. 2 Chi Squared Test ................................................................................................ 278 B. 3 EDF Tests ........................................................................................................... 279 xiii B. 3.1 Kolmogorov Smirnov Test .......................................................................... 280 B. 3.2 Anderson Darling Test................................................................................. 280 Appendix C. Preliminary Calibration Examples .................................................................... 282 C. 1 Introduction......................................................................................................... 282 C. 2 Sample Girder..................................................................................................... 282 C. 3 General Procedure for Strengthening Design ..................................................... 284 C. 4 Composite Material Properties for Calibration................................................... 285 C. 5 Load Factors for Use in Strengthening Design................................................... 287 C. 6 Large Example Calibration without Corrosion................................................... 290 C. 6.1 Description of Procedures and Variables ..................................................... 291 C. 6.1.1 Degraded Structure ............................................................................. 291 C. 6.1.2 FRP Properties .................................................................................... 291 C. 6.1.3 Degraded Properties............................................................................ 292 C. 6.1.4 Designs ............................................................................................... 293 C. 6.1.5 Reliability Analysis ............................................................................ 293 C. 6.1.6 Time Dependent Reliability ............................................................... 295 C. 6.1.7 Load Variables.................................................................................... 295 C. 6.1.8 Resistance Variables........................................................................... 297 C. 6.2 Results of Sample Calibration without Corrosion........................................ 298 C. 6.2.1 Effect of Reliability Calculation Method............................................ 298 C. 6.2.2 Effect of Different Amounts of Steel Loss ......................................... 302 C. 6.2.3 Effect of Time Span for Evaluation.................................................... 304 C. 6.2.4 Effect of Differences in Mean Value of FRP Properties .................... 308 C. 6.2.5 Effect of Changes in Modulus Coefficient of Variation..................... 312 xiv C. 6.2.6 Effect of Changes in Strength Coefficient of Variation...................... 315 C. 7 Effect of Resistance Variables Considered in Reliability Analysis.................... 316 C. 8 Example with Corrosion ..................................................................................... 318 C. 8.1 Design Philosophy ....................................................................................... 318 C. 8.2 Degraded Structure....................................................................................... 319 C. 8.3 Prediction of Remaining Steel...................................................................... 320 C. 8.4 FRP Properties ............................................................................................. 321 C. 8.5 Design of Strengthening............................................................................... 322 C. 8.6 Reliability Analysis...................................................................................... 322 C. 8.7 Random Variables........................................................................................ 322 C. 8.8 General vs. Pitting Corrosion....................................................................... 323 C. 8.9 Results of Sample Calibration with Corrosion............................................. 324 C. 9 Summary of Conclusions from Sample Calibrations.......................................... 326 Appendix D. Sectional Analysis ............................................................................................ 328 D. 1 Introduction......................................................................................................... 328 D. 2 RC Section without FRP..................................................................................... 328 D. 3 RC Section with Externally Bonded FRP........................................................... 330 Appendix E. Techniques Used in Reliability Assessment ................................................ 336 E. 1 Monte Carlo Simulation ..................................................................................... 336 E. 2 Generating Random Numbers from a Statistical Distribution............................ 339 E. 3 Implementation of FORM .................................................................................. 341 Appendix F. Java Programs.................................................................................................... 344 F. 1 Program for Design of Strengthening................................................................. 344 F. 1.1 Variables ...................................................................................................... 344 xv F. 1.2 Procedure...................................................................................................... 346 F. 1.3 Code ............................................................................................................. 347 F. 2 Program for Simulation and Evaluation of Resistance Statistics........................ 353 F. 2.1 Variables ...................................................................................................... 354 F. 2.2 Procedure...................................................................................................... 355 F. 2.3 Code ............................................................................................................. 356 Appendix G. Data from Bridge Survey ............................................................................. 366 G. 1 Summary of Dimensions Collected .................................................................... 366 Appendix H. Load Analysis in QConBridge ™ ...................................................................... 379 H. 1 Program Description........................................................................................... 379 H. 2 Input Details for Calibration Girders .................................................................. 381 References .............................................................................................................................. 385 xvi LIST OF FIGURES Figure 1 1 Components of Reliability Based Design for FRP Strengthening.......................... 17 Figure 2 1 Basic Structural Reliability Problem ...................................................................... 25 Figure 2 2 Graphical Representation of Probability of Failure................................................ 28 Figure 2 3 Interpretation of β in Terms of the Safety Margin ................................................. 31 Figure 2 4 Design Truck for HS 20 and HL 93 Load Models................................................. 64 Figure 2 5 Tuutti’s ( 1982) Model for Sequence of Steel Corrosion in Concrete.................... 70 Figure 2 6 Relation between Concrete Compressive Strength and Water Cement Ratio........ 80 Figure 3 1 Plot of Cumulative Distribution Functions for Set A1 Strength........................... 111 Figure 3 2 Changes in β with Additional Required Strengthening ........................................ 128 Figure 3 3 Ratio of Tested Strength to Predicted Strength vs. Fiber Volume Fraction for One Layer Samples ............................................................................................................... 144 Figure 3 4 Ratio of Tested Strength to Predicted Strength vs. Fiber Volume Fraction for One Layer Samples Without Set E1...................................................................................... 145 Figure 3 5 Ratio of Tested Modulus to Predicted Modulus vs. Fiber Volume Fraction for One Layer Samples ............................................................................................................... 146 Figure 4 1 Basic Flowchart for Calibration Procedure ......................................................... 173 Figure 4 2 Histogram of Bridge Spans................................................................................... 184 Figure 4 3 Histogram of Number of Girders.......................................................................... 185 Figure 4 4 Histogram of Deck Width..................................................................................... 185 Figure 4 5 Plot of Convergence of Monte Carlo Results as a Function of the Number of Trials ............................................................................................................................... ....... 209 Figure 4 6 Example of Plots Used to Select Calibrated Resistance Factors .......................... 210 Figure 4 7 ψ vs. Strength COV for Girder 18, Corrosion Condition 4, SD, βT = 3.5, and φ = 0.85 ............................................................................................................................... 218 Figure A 1 PDF of Bias Factor for Maximum Load for Different Time Spans..................... 269 Figure A 2 CDF of Bias Factor for Maximum Load for Different Time Spans .................... 270 xvii Figure A 3 Comparison of Distributions for Mean Maximum 50 Year Load Bias Factor.... 271 Figure C 1 β vs. ψ for Material 1 Designs to Meet LRFD and LRFR Loads ........................ 290 Figure C 2 Monte Carlo Results for 20% Steel Loss, Strength COV = 0.25, Modulus COV = 0.05, 0 degradation, 75 year loads.............................................................................. 299 Figure C 3 Hybrid Results for 20% Steel Loss, Strength COV = 0.25, Modulus COV = 0.05, 0 degradation, 75 year loads .......................................................................................... 300 Figure C 4 Monte Carlo Results for 30% Steel Loss, Strength COV = 0.25, Modulus COV = 0.05, 0 degradation, 75 year loads ....................................................................... 301 Figure C 5 Hybrid Results for 30% Steel Loss, Strength COV = 0.25, Modulus COV = 0.05, 0 degradation, 75 year loads .......................................................................................... 301 Figure C 6 Hybrid Results for 20% Steel Loss, Strength COV = 0.15, Modulus COV = 0.15, no degradation, 75 year loads ........................................................................................ 305 Figure C 7 Hybrid Results for 20% Steel Loss, Strength COV = 0.15, Modulus COV = 0.15, 5 year exposure, 5 year loads ........................................................................................ 305 Figure C 8 Hybrid Results for 20% Steel Loss, Strength COV = 0.15, Modulus COV = 0.15, 50 year exposure, 50 year loads .................................................................................... 306 Figure C 9 Hybrid Results for 20% Steel Loss, Strength COV = 0.15, Modulus COV = 0.15, 5 year exposure, 5 year loads ........................................................................................ 307 Figure C 10 Hybrid Results for 30% Steel Loss, Strength COV = 0.25, Modulus COV = 0.05, 5 year exposure, 5 year loads ....................................................................................... 309 Figure C 11 Hybrid Results for 30% Steel Loss, Strength COV = 0.25, Modulus COV = 0.05, 50 year exposure, 50 year loads ................................................................................... 310 Figure C 12 Hybrid Results for 20% Steel Loss, Strength COV = 0.25, Modulus COV = 0.05, 5 year exposure, 5 year loads ....................................................................................... 313 Figure C 13 Hybrid Results for 20% Steel Loss, Strength COV = 0.25, Modulus COV = 0.15, 5 year exposure, 5 year loads ....................................................................................... 314 Figure C 14 Hybrid Results for 20% Steel Loss, Strength COV = 0.25, Modulus COV = 0.25, 5 year exposure, 5 year loads ....................................................................................... 314 Figure C 15 ψ as a function of Strength COV for 20% Steel Loss and Modulus COV = 15% ............................................................................................................................... ....... 315 Figure C 16 ψ as a function of Strength COV for 30% Steel Loss and Modulus COV = 15% ............................................................................................................................... ....... 316 xviii Figure C 17 Effect of Using Different Cases of Random Variables to Assess Reliability for Material 2 Designs......................................................................................................... 318 Figure C 18 Reliability Index vs. Composite Specific Resistance Factor for Material 1, φ = 0.90, ............................................................................................................................... 324 Figure D 1 Forces in a Rectangular Section at Ultimate ( Only Steel Reinforcement) .......... 329 Figure D 2 Forces in a Rectangular Section ( Steel and FRP Reinforcement) ....................... 331 Figure E 1 Flow Chart of Monte Carlo Simulation ............................................................... 337 Figure H 1 Example of Bridge Model for Girder 12 ( not to scale)....................................... 384 xix LIST OF TABLES Table 1 1 Questions to be Answered in LRFD Development.................................................. 18 Table 2 1 Probabilities of Failure and Corresponding β s ....................................................... 32 Table 2 2 Comparison of Live Load Factors for Inventory and Operating Levels.................. 44 Table 2 3 Distribution Properties for Slab Dimensions ........................................................... 57 Table 2 4 Distribution Properties for Beam Dimensions ......................................................... 58 Table 2 5 Comparison of HS 20 and HL 93 Load Models for Calculation of Maximum Positive Moment.............................................................................................................. 63 Table 2 6 Ratio of Mean Maximum Moments to HL 93 Moments ......................................... 65 Table 2 7 Causes of Deterioration of Concrete ( Bertolini et al., 2004) ................................. 68 Table 2 8 Rates of Corrosion Penetration of Steel in Concrete ( Bertolini et al., 2004)........... 73 Table 2 9 Rates of Corrosion Penetration Based on Concrete Cover and Exposure Condition74 Table 2 10 Approximate Relation between Concrete Strength and Water Cement Ratio....... 79 Table 2 11 Comparison of Common Risks and Structural Failure Probabilities .................... 82 Table 2 12 Target Failure Probabilities and Reliability Indices Based on CIRIA................... 84 Table 2 13 Target Failure Probabilities and Reliability Indices Based on Allen ( 1981) W= 0.1 ............................................................................................................................... ......... 85 Table 2 14 Target Reliability Levels and Corresponding Lifetime Probabilities of Failure from Nordic Report .................................................................................................................. 86 Table 2 15 Target Reliability Indices and Corresponding Annual Probabilities of Failure for Other Structural Design Codes ........................................................................................ 88 Table 2 16 Adjustments to Target Reliability for Canadian Bridge Evaluation ...................... 90 Table 3 1 Summary of Wet Layup Data Sets.......................................................................... 98 Table 3 2 Descriptive Statistics for Ultimate Tensile Strength.............................................. 100 Table 3 3 Descriptive Statistics for Longitudinal Modulus ................................................... 102 Table 3 4 Descriptive Statistics for Thickness....................................................................... 103 xx Table 3 5 Distribution Parameters for Ultimate Tensile Strength.......................................... 108 Table 3 6 Distribution Parameters for Longitudinal Modulus ............................................... 109 Table 3 7 Distribution Parameters for Composite Thickness ................................................ 110 Table 3 8 Chi Squared Goodness of Fit Results for Strength ............................................... 112 Table 3 9 Kolmogorov Smirnov Goodness of Fit Results for Strength, α= 0.10 .................. 113 Table 3 10 Anderson Darling Goodness of Fit Results for Strength, α= 0.25...................... 114 Table 3 11 Chi Squared Goodness of Fit Results for Modulus............................................. 115 Table 3 12 Kolmogorov Smirnov Goodness of Fit Results For Modulus, α= 0.10............... 115 Table 3 13 Anderson Darling Goodness of Fit Results for Modulus, α= 0.10 ...................... 116 Table 3 14 Chi Squared Goodness of Fit Results for Thickness........................................... 117 Table 3 15 Kolmogorov Smirnov Goodness of Fit Results for Thickness, α= 0.10.............. 117 Table 3 16 Anderson Darling Goodness of Fit Results for Thickness, α= 0.25 ................... 118 Table 3 17 Correlation Coefficients for Wet Layup Composites........................................... 121 Table 3 18 Different Ways of Specifying the Characteristic Value for FRP Strength ......... 123 Table 3 19 Properties of Model Composite .......................................................................... 125 Table 3 20 Reliability of Designs Using Different COVs for Strength ................................ 125 Table 3 21 Basic Description of System of Application Factors ........................................... 139 Table 3 22 Properties of Fibers and Matrices for Prediction of Strength and Modulus......... 141 Table 3 23 Mean and COV of Ratio of Tested Values to Values Predicted Using Properties of Fiber and Matrix for Strength........................................................................................ 142 Table 3 24 Mean and COV of Ratio of Tested Values to Values Predicted Using Properties of Fiber and Matrix for Modulus ....................................................................................... 142 Table 3 25 Manufacturer Properties for Sets E and F............................................................ 147 Table 3 26 Ratio of Tested Properties to Manufacturer Reported Properties........................ 148 Table 3 27 λlayers for Strength and Modulus........................................................................... 151 Table 3 28 Generalized λlayers for Design............................................................................. 151 xxi Table 3 29 Preliminary Values of Application Factors for Wet Layup Composites ............. 154 Table 3 30 Mean and COV of Ratio of Tested to Predicted Force per Unit Width............... 155 Table 3 31 Mean and COV of Ratio of Tested to Predicted Stiffness per Unit Width .......... 156 Table 3 32 Mean and COV of Ratio of Tested to Predicted Force per Unit Width............... 157 Table 3 33 Mean and COV of Ratio of Tested to Predicted Stiffness per Unit Width .......... 157 Table 3 34 Stress Limitations as Percentage of Ultimate Strength ....................................... 161 Table 3 35 Predictive Equations for Property Retention Based on an Arrhenius Rate Relation ( Abanilla, 2004)............................................................................................................. 165 Table 4 1 LRFR Load Factors for Design of Strengthening ( AASHTO, 2003) .................... 176 Table 4 2 Generalized Composite Properties Used for Calibration ....................................... 180 Table 4 3 Bridge Quantities Surveyed to Determine Common Values for Calibration......... 183 Table 4 4 Geometry of Representative Bridges for Calibration............................................. 187 Table 4 5 Comparison of Distribution of Span Lengths for Selected Bridges....................... 188 Table 4 6 Comparison of Distribution of Number of Girders for Selected Bridges .............. 188 Table 4 7 Comparison of Distribution of Deck Widths for Selected Bridges........................ 189 Table 4 8 Comparison of QConBridge ™ and CT BDS for Selected Girders........................ 193 Table 4 9 Load Components and LRFR Factored Load for Design ...................................... 196 Table 4 10 Distribution Parameters of Load for Reliability Analysis.................................... 204 Table 4 11 Statistical Distributions Used in Reliability Analysis.......................................... 206 Table 4 12 Baseline LRFR Steel Areas and Steel Areas for Each Corrosion Condition in mm2 ( in. 2) ............................................................................................................................... 213 Table 4 13 Summary of Resistance Factors for Different Target Reliabilities and Different Amounts of Relative Steel Loss .................................................................................... 217 Table 4 14 Example of Calibrated ψ for Girder 15, Corrosion Case 2, with FRP Degradation ............................................................................................................................... ....... 221 Table 4 15 Example of Calibrated ψ for Girder 3, Corrosion Case 5, with FRP Degradation ............................................................................................................................... ....... 222 xxii Table 4 16 Example of Calibrated ψ for Girder 5, Corrosion Case 2................................... 224 Table 4 17 Example of Calibrated ψ for Girder 14, Corrosion Case 4.................................. 225 Table 4 18 Comparison of Calibrated Resistance Factors with Changes in Strength and Modulus COVs for Girder 16, Corrosion Condition 4, βT = 3.5, φ = 0.85.................... 229 Table 4 19 Example of Calibrated ψ for Girder 4, β = 3.0, Corrosion Condition 1, φ = 0.9, AD ............................................................................................................................... ....... 232 Table 4 20 Example of Calibrated ψ for Girder 4, β = 3.0, Corrosion Condition 2, φ = 0.9, AD ............................................................................................................................... ....... 232 Table 5 1 Approximate Values for COVcharacteristic for Wet Layup Composites Based on Testing ............................................................................................................................... ....... 239 Table 5 2 Dimensions and Material Properties of Girder 15 ................................................ 241 Table 5 3 Results from Lamina Level Tests .......................................................................... 242 Table 5 4 Preliminary Values of Application Factors for Wet Layup Composites ............... 243 Table 5 5 Resistance Factors for Design Example................................................................. 248 Table 5 6 Final Design Quantities.......................................................................................... 254 Table 5 7 Statistical Distributions for Set A1 used in Reliability Analysis ........................... 255 Table A 1 Comparison of Estimated Bias Factors and Bias Factors from NCHRP Report 368 ............................................................................................................................... ....... 270 Table A 2 Basic Details of Strengthening Example............................................................... 273 Table A 3 Different Methods Used to Calculate Time Dependent Reliability...................... 274 Table A 4 Comparison of Reliabilities for Different Computation Techniques .................... 275 Table C 1 Bridge Deck Dimensions ...................................................................................... 283 Table C 2 Load Effects for Girder Design............................................................................. 283 Table C 3 Dimensions of Sample Girder ............................................................................... 284 Table C 4 Mean Property Values of Sample Composites...................................................... 288 Table C 5 Mean Property Values of Sample Composites...................................................... 292 Table C 6 Percent Retention of FRP Properties for Different Design Lives ......................... 293 xxiii Table C 7 Comparison of Two Different Reliability Procedures .......................................... 295 Table C 8 Statistical Description of Dead Loads................................................................... 296 Table C 9 Live Load Statistics for Different Design Lives ................................................... 296 Table C 10 Statistics of Total Load ....................................................................................... 297 Table C 11 FRP Rupture Strains at Different Design Lives .................................................. 311 Table C 12 Cases for Assessment of Resistance Variable Effect on Reliability ................... 317 Table C 13 Relation of Condition States for Bridge Management Systems to Structural Integrity of Bridge ......................................................................................................... 320 Table C 15 Remaining Steel Area for Various Design Lives ................................................ 321 Table C 16 Assumed Properties for Sample Composites ...................................................... 321 Table C 17 COV of Remaining Steel Area for Different Design Lives ................................ 325 Table E 1 Values of k for Different Two Sided Confidence Levels...................................... 338 Table F 1 Variables in Design Program................................................................................. 345 Table F 2 Variables in MCS Program.................................................................................... 354 Table G 1 Key to Bridge Dimensions in this Appendix ........................................................ 367 Table G 2 Key to Notes Column............................................................................................ 368 Table G 3 Data Collected in Bridge Survey .......................................................................... 369 Table H 1 Summary of Input Data for Girder Analysis......................................................... 382 xxiv xxv ABSTRACT Externally bonded fiber reinforced polymer ( FRP) composites are an increasingly adopted technology for the renewal of existing concrete structures. In order to encourage the further use of these materials, a design code is needed that considers the inherent material variability of the composite, as well as the variations introduced during field manufacture and environmental exposure while in service. Load and Resistance Factor Design ( LRFD) is a reliability based design methodology that provides an ideal framework for these considerations and is compatible with existing trends in civil engineering design codes. This investigation studies the application of LRFD to FRP strengthening schemes with an emphasis on wet layup, carbon fiber composites applied to reinforced concrete T beam bridge girders. Models to describe variation in the existing structural materials and the structural loading are drawn from the literature. Techniques for reliability analysis are discussed, and existing work on externally bonded FRP reliability is surveyed. Stochastic variation in the FRP is characterized based on tensile testing of several sets of field manufactured, wet layup composites. A general design procedure applicable to many different situations is proposed using a composite specific resistance factor to consider material variability, a set of Application Factors to account for deviations introduced through field manufacture, and an environment and service life specific factor for FRP degradation. Preliminary resistance factors for design of FRP strengthening are calibrated over a range of design scenarios. FRP degradation is considered based on existing durability models, and continued degradation of the structure due to general corrosion of the reinforcing steel is included. The girders used for calibration are selected as representative examples from a sample of California bridge plans. The reliability has been evaluated using simulation and xxv first order reliability methods. An example of the proposed design procedure, using the calibrated resistance factors, is provided. The results of this work bring to light the many variables affecting the reliability of strengthened members and the need for continuing research to better describe these variables. Two variables of particular significance, requiring extensive further study, are the state of the existing structure when strengthening is applied and the loads acting on the structure. xxvi Chapter 1. Introduction 1.1 Overview Externally bonded fiber reinforced polymer composites ( FRPs) are increasingly considered as a viable means of strengthening, retrofitting, and repairing existing reinforced concrete structures. In appropriate situations, these materials can offer significant advantages over more traditional techniques of adding new or replacing lost load carrying capacity. There is a pressing need for this type of technology as our country’s infrastructure ages. A prime example can be found in the U. S. bridge inventory; in 2004 the Federal Highway Administration deemed over a quarter of the nation’s bridges deficient based on data from 2002. Nearly fourteen percent of bridges were found to be structurally deficient, with an additional fourteen percent functionally obsolete ( FHWA, 2004). At the present time FRP strengthening is a technique seeing growing usage. In order to facilitate the continued growth of this technology and to provide for the long term safety of designs using FRPs, it is vital that a design code is developed for their use in strengthening. However, there are many challenges to be overcome in design code development such as the unique characteristics of FRPs, the incomplete database of material properties, and the somewhat limited understanding of the interaction between the FRP and the existing structure. 1.2 FRPs for Strengthening of Civil Structures 1.2.1 Fiber Reinforced Polymer Composites A composite is a material that is composed of two or more distinct phases. The constituent materials work together to produce properties that are more desirable than those of the individual materials. FRPs are composed of a fibrous reinforcing phase embedded in a polymeric matrix. Typical fiber types include carbon, glass, and aramid. Many different 1 polymers may be used for the matrix phase. In strengthening applications the resin system is typically a thermosetting polymer such an epoxy or vinylester. This combination of materials provides FRPs with a number of unique, and often advantageous, properties. FRPs are perhaps best known for their high specific strength and stiffness ( defined as the property divided by the material density). Unidirectional composites may have specific strengths nearly an order of magnitude greater than those of common metals, such as steel or aluminum ( Kaw, 1997). Other advantageous properties of composites include their enhanced fatigue resistance at the material level, resistance to corrosion, and tailorability. There are many different methods used to fabricate composite materials. Several, such as autoclave forming and resin transfer molding, are impractical for use in civil applications. The most common forms of FRP used for strengthening are wet layup systems, manufactured directly on the structure through a manual process, and prefabricated strips, which are often manufactured through pultrusion and then bonded to the structure with adhesives. Other special systems may be used to provide automated wrapping of columns or apply posttensioning ( International, 2001). 1.2.2 Strengthening and Repair of Civil Structures Structures designed by civil engineers are intended to have a long lifespan, and during that time there are many reasons why the structure may require strengthening or repair1. ( Täljsten, 2002; Ellingwood, 1996). The most significant of these reasons include: 1 It should be noted that strengthening generally implies adding capacity to a structure, while repair signifies returning a structure to its original capacity. This report treats these two applications of FRP to externally reinforce concrete structures interchangeably; however, the term strengthening is generally used. 2 1. Environmental Exposure  Civil structures are exposed to changing environmental conditions throughout their lifetimes. These factors can cause material degradation over time or impart significant damage during one extreme event. The impacts of environmental degradation will be especially felt in cases where regular maintenance is not performed. 2. Changing Usage  It is not uncommon for civil structures to outlive the purpose for which they were originally designed. Changes in tenancy or use may place different or larger load demands on the structure. 3. Changing Design Standards  Even if the use of the structure is not significantly changed, the standards the structure must meet may change over time. 4. Errors in Design or Construction  Civil structures may even require strengthening before they are ever used due to errors in the initial design or construction. Strengthening is not new to civil applications; however, in the past it generally meant placing more concrete, bonding steel plates, or applying some sort of post tensioning to the structure ( The Concrete Society, 2000). Now many types of strengthening can be accomplished with FRPs ( Täljsten, 2002; The Concrete Society, 2000). FRP strengthening can be applied to mitigate several failure modes. For flexural strengthening of beams, slabs, or girders, FRP plates can be applied to the tensile face of the concrete. Shear and torsional strengthening can be accomplished by placing FRP on the sides of beams. Columns are typically strengthened by wrapping the FRP around the column in the hoop direction, thus 3 increasing the confinement of the concrete core. This can be accomplished with wet lay up or prefabricated cylindrical jackets. 1.2.3 Advantages of FRPs for Strengthening The unique properties of FRPs result in many advantages from the perspective of strengthening designers ( The Concrete Society, 2000; Täljsten, 2002; International, 2001; ACI, 2002; Maruyama, 2001). FRPs do not suffer from corrosion as do steel plates, allowing the possibility of extended service lives or perhaps limiting required maintenance. Their high strength and stiffness to weight ratios mean that a smaller weight of FRP needs to be applied as compared to steel plate bonding. This low weight reduces transportation costs, significantly eases installation, even in tight spaces, and can eliminate the need for scaffolding, reducing traffic impact. The low weight also means that FRPs add only a small amount to the structure’s dead load. This allows more of the strengthening to be useful to the structure and also makes FRPs a repair option when significant additional weight could cause failure. Additionally, FRPs are typically applied in thin strips, resulting in very little change in the structural profile, an important feature on bridges or other structures that require clearances for vehicles or machinery. The way that FRPs are manufactured also provides useful properties. By designing the placement of the reinforcing fibers, properties such as strength and modulus can be controlled in different directions. This allows the strengthening to act only in the needed direction, preventing it from changing the structural behavior in unintended ways. Because they are made from long thin fibers, FRPs are very easy to handle. They can be made to wrap around curves and to accept the irregularities present in concrete surfaces. Furthermore, they can be manufactured in long lengths, eliminating the need for splices, and can be cut to length on site, eliminating sizing errors in the manufacturing stage. 4 1.2.4 Disadvantages of FRPs for Strengthening Despite their numerous advantages FRPs are not without drawbacks ( The Concrete Society, 2000; Täljsten, 2002; International, 2001). Unidirectional FRP materials are characterized by linear elastic behavior up to failure; this lack of yielding can result in less ductile structures unless this behavior is specifically considered at the design stage. These materials are very susceptible to damage from impact, fire, or vandalism, and as such need to be protected. Though FRPs do not exhibit corrosion, they are not immune to environmental impacts and do suffer degradation due to moisture, temperature, and UV rays. This disadvantage is of particular importance because there is currently little long term information on the durability of composites in exposed environments. The initially high material cost of FRPs is also a drawback to many engineers, however, due to the cost advantages in transportation and installation offered by composites, the cost of a whole strengthening project can be comparable or even less than the same project strengthened with steel plates. 1.3 Design Code for FRP Strengthening 1.3.1 Need for a Design Code Other limits to the use of composites in strengthening are related to the unique aspects of civil design ( Ellingwood, 2003). Composite materials were initially developed and used in the mechanical and aerospace fields, fields that are significantly different from civil engineering. The typical mechanical or aerospace part will be mass produced at the end of engineering design, making it economically feasible to conduct testing throughout the design stage and to specifically tailor materials for a particular project. Furthermore, the design requirements, such as load demands, are clearly defined and the manufacturing processes used in these fields allow for very tight control of finished properties. In contrast, each civil design is a unique project that is usually designed and built just once. Due to cost, size, and time 5 constraints, routine civil designs are rarely tested before construction, and when testing does occur it is usually performed on a scale model or only a critical portion of the design. In place of testing civil design is based on knowledge of material properties, analysis, and prior experience, which are often actualized in codes of practice. For example, the International Building Code is a model code that is based on recognized standards and specifications developed by individual organizations with expertise in different aspects of construction, such as the American Institute of Steel Construction ( AISC) or the American Concrete Institute ( ACI). Bridge design is usually based on the specifications of the American Association of State Highway and Transportation Officials ( AASHTO). However, civil design is usually characterized by substantial uncertainty in load demands, especially those due to natural phenomena, and material properties that cannot be as tightly controlled. These uncertainties result in conservative specification of loads and material strengths in design codes. When governments adopt design codes they become part of local, state, and federal law, exposing civil engineers to liability concerns for designs that do not meet the standard. In addition to their legal implications, design codes also serve as a set of minimum technical requirements for acceptable design and provide a pathway for research findings to make their way into practice ( Ellingwood, 2000b). Thus, most design in civil engineering is based on codes of practice, and, without a comprehensive specification for FRP, it is unlikely that this promising new material will gain widespread acceptance and utilization. This is especially true because design with composite materials is not a typical component of the undergraduate civil engineering education. The lack of design code and designer experience are the most significant obstacles limiting the present use of composites in civil infrastructure. 6 1.3.2 Uncertainty in Structural Design The goal of the structural engineer is to achieve structural safety in the face of numerous uncertainties. Nearly every variable considered in design is uncertain to varying degrees. Loads can be highly variable, especially when natural effects such as wind and earthquakes are considered. Materials have inherent variability and may suffer degradation when they are put in service. The models describing structural behavior are just that, models, and the uncertainty in their results is usually unknown. Even the service life of the design is an uncertain quantity. The result of uncertainty is risk, which is often defined as the product of the probability of failure and the costs associated with failure ( Ellingwood, 1994). Since the design variables are uncertain, there is a risk that the structure will fail due to overloading, when the loads exceed those for which the structure was designed, or that the structure will be understrength due weak materials or incorrect dimensions. Though it is impossible to completely eliminate risk, good engineering design can hold the risk to acceptable levels by accounting for the uncertainty inherent in design. 1.3.3 Design Philosophies as the Basis for Design Codes Currently there are two main philosophies behind civil design: Working or Allowable Stress Design and probabilistic based limit states design. Other approaches, such as Ultimate Strength Design or Load Factor Design, fall somewhere between these two approaches. In the United States probabilistic limit states design is typically implemented in the Load and Resistance Factor Design ( LRFD) format. Other parts of the world, such as Europe and Canada, also have design codes with a probabilistic basis; however, the implementation differs from the LRFD format ( Ellingwood, 1996). 7 1.3.3.1 Working Stress Design Working Stress Design has served as the basis for structural calculations since the late 19th century when calculations first started to be used for design ( Ellingwood, 2000a). In Working Stress Design the stresses in members due to service loads are elastically computed and compared to a specified allowable stress divided by a factor of safety. The basic checking equation used for Working Stress Design is shown in Eq. 1 1 wherein f is the elastically computed stress in the structure, F is the allowable stress, and FS is the factor of safety. FS f ≤ F Eq. 1 1 The factors of safety used in Working Stress Design are based on past experience and engineering judgment, not specific consideration of the uncertainties involved in design. As experience has been gained over time, factors of safety have generally been decreased ( Ellingwood, 1994). In this design format there is only one factor to account for all the uncertainties that may be encountered in loads and resistance. This neglects the fact that different types of load may have different degrees of variation, resulting in a range of structural reliabilities. This variation in structural reliability is one of the key drawbacks to Working Stress Design. 1.3.3.2 Load and Resistance Factor Design Load and Resistance Factor Design is a relatively new development in civil design. The theoretical basis for LRFD, structural reliability theory, was developed during the period from the late 1940s to the mid 1960s, at which point interest grew in incorporating the reliability research into standards for design ( Ellingwood, 1994). The first LRFD specification was adopted in 1986 by AISC with the first LRFD edition of the AISC Manual of Steel Construction ( Salmon and Johnson, 1996). 8 The LRFD approach to design is distinct from Working or Allowable Stress Design in two ways. First, it is based on a philosophy of defining pertinent limit states. A structure is said to reach a limit state when it fails to reach a level of performance for which it was designed. Limit states are typically divided into two categories: strength and serviceability. Strength limit states relate to the structure’s ability to carry load and include limits such as the plastic capacity of a ductile member, fracture of brittle materials, and instability or buckling. Service limit states are primarily related to the comfort of occupants and include excessive deflection, vibration, and/ or cracking ( Salmon and Johnson, 1996). Including strength and serviceability, AASHTO defines four different kinds of limit states in the AAHSTO LRFD Bridge Design Specifications ( AASHTO, 2004). Fatigue and fracture provisions are considered separately from the strength provisions and are intended to prevent failure through cyclic loading. The extreme event limit state specifically considers rare events such as earthquakes, floods, or collisions, which can be considered as statistically insignificant loads. In WSD structures are evaluated at typical service conditions; in LRFD structures are evaluated in the ways they are likely to fail by considering the applicable limit states. LRFD is also different from Working Stress Design in that it is based on probabilistic analysis of the uncertainties present in design. The factors in LRFD based specifications are specifically calibrated such that the probability of reaching a particular limit state is acceptably small. This probability is most often measured in terms of the reliability index, β. In the development of a LRFD code, a target value of β is set, and design factors for load and resistance are selected such that a wide range of designs will be close to this target, usually with a bit of conservatism. The reliability index and the methods of structural reliability theory used to calibrate design factors are discussed further in Chapter 2 of this report. 9 The basic design equation in LRFD is shown in Eq. 1 2 where φ is the resistance factor, usually specific to material and failure mode and sometimes to a particular limit state, Rn is the nominal resistance, γ i is the load factor specific to load i, and Qi is the load effect due to load i. ≥ Σ i n i i φR γ Q Eq. 1 2 In the LRFD format different types of loads such as dead, live, wind, snow, earthquake, etc. each have their own load factor. Different types of loads are given different load factors depending on their coefficient of variation. These factors were calibrated for buildings in the late seventies and are intended to be applicable for all design materials ( Ellingwood, 1994; Galambos et al., 1982; Ellingwood et al., 1982). Load factors for bridge design were calibrated in the nineties for use in the AASHTO LRFD Bridge Design Specifications ( Nowak, 1999). The variations in capacity caused by material variability, geometric uncertainty, and modeling error are accounted for by the resistance factor φ. Resistance factors generally depend on the material being used and the limit state being checked. 1.3.3.3 Advantages of LRFD From the viewpoint of the designer LRFD is still a deterministic format with no explicit reliability calculations required. However, the probabilistic basis of LRFD is much more complex than the empirical basis of Working Stress Design. There are many advantages to the LRFD format ( Ellingwood, 2000a; Salmon and Johnson, 1996). 1. Designs created with LRFD have much more uniform reliabilities than those created with Working Stress Design. 2. The random nature of materials and loads is handled in a rational and analytical manner; the factors are derived based on statistics not just experience. Since 10 the factors have an statistical basis it is much clearer when factors should be changed based on new research or technology. 3. Since LRFD is a limit states approach, structures are evaluated in the ways they are likely to fail. This can provide for better evaluation of serviceability limit states. It also makes the relationship between behavior and design easier to comprehend. 4. Since separate factors are used for load and resistance, research on one or the other can be conducted independently, and changes can be made to either side as new information is gained. 5. With load factors common to all materials LRFD simplifies the design process. 6. For unusual load cases or new materials LRFD provides a framework with which to approach design code development. These advantages, plus the fact that LRFD is the design philosophy that most civil design is moving towards, make LRFD the design philosophy of choice for development of a code for FRP strengthening. 1.3.4 Current Design Guidelines for FRP Strengthening There are several current guidelines for the use of FRP to strengthen reinforced concrete structures: 1. Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures, published by the American Concrete Institute, 2002 11 2. Externally Bonded FRP Reinforcement for RC Structures published by the International Federation for Structural Concrete ( fib), 2001 3. Design Guidance for Strengthening Concrete Structures Using Fibre Composite Materials from The Concrete Society, 2000 4. Strengthening Reinforced Concrete Structures with Externally Bonded Fibre Reinforced Polymers from ISIS Canada, 2001 ( Neale, 2001) 5. FRP Strengthening of Existing Concrete Structures, Design Guidelines by Björn Täljsten, 2002 6. Recommendations for Upgrading of Concrete Structures with Use of Continuous Fiber Sheets, from the Japanese Society of Civil Engineers, 2001 ( Maruyama, 2001) All of these guidelines draw together a large body of research into a document that is easily understood by designers, and as such they are a valuable advance in the use of FRPs to renew existing concrete structures. However, these documents also share many limitations. The guidelines are all quite similar in their design approach, and at first glance all appear very similar to LRFD. All use a limit states approach to defining design checking equations. Design procedures and equations are given for basic strength limit states such as flexure or shear; however, the accuracy of these approaches is questionable in some cases, particularly shear. Debonding is discussed in all; however, the level of detail varies significantly. The approach to serviceability limit states also varies from guide to guide. All of the guidelines rely on the load factors already developed in relevant specifications for new design, and many use familiar resistance factors or partial material factors from probabilistic design codes for other materials. However, these design guidelines are not true probabilistic codes. No 12 calibration procedure was used to specifically derive the resistance factors in order to achieve a target reliability; in fact, no such target was even set. Thus, while these guidelines are a significant advance in the use of FRPs for the strengthening of concrete structures, there is still work to be done to develop a code in the preferred LRFD format. 1.4 Problem Statement and Research Objectives 1.4.1 Problem Description Externally bonded fiber reinforced polymer composites have shown promising performance as a means of strengthening existing reinforced concrete structures. As the infrastructure of our country continues to become deficient due to ageing, environmental attack, and growing usage, the development and implementation of repair strategies, such as the external bonding of FRPs, can serve a vital role in economically promoting the safety of engineered structures. However, due in large part to a lack of design code and designer experience, this technology is currently under utilized. Design guidelines are already available for the use of bonded FRPs as a strengthening measure. However, these guidelines are based on a deterministic format for design. Given the high level of material variability that can be exhibited by FRP systems, deterministic design is likely to produce an unacceptably large range of project reliabilities. The framework provided by LRFD is an ideal method for considering the inherent material uncertainty, as well as the time dependent material behavior, to produce designs with an acceptable level of structural reliability. Therefore, a design code in the LRFD format must be developed to promote the usage of FRP for strengthening. Development of a LRFD based design procedure requires extensive data: 1. Statistical data characterizing the load and resistance variables 13 2. Methods for defining nominal or design values of load and resistance 3. Definition of applicable limit states and models for structural behavior at these limits 4. A range of application defining the cases for which the code is valid 5. A target reliability index 6. A method by which reliabilities can be calculated during the calibration process With regard to FRP strengthening, some of this information is already available. For example, the statistical variation of traditional materials, such as steel and concrete, has been examined by previous researchers. Stochastic load models have been developed and used for both building and bridge structures. Researchers have developed many different expressions for modeling the behavior of structures strengthened with FRP. Structural reliability theory has reached the stage where there are a number of mature techniques for evaluation of structural reliability. Despite this foundation of available information, there are still significant gaps in knowledge as well as significant challenges to the development of a probabilistic code for the design of FRP strengthening. For example, there is not an adequate existing database of FRP properties as used in civil infrastructure projects. Given the variation that is introduced during field manufacture and application of FRPs this is a significant shortcoming. The multitude of possible fiber/ matrix combinations creates an additional level of complexity. Time dependent material properties have not been considered in the development of codes for other materials; however, this is an important concern for FRPs exposed to severe environments. Furthermore, as the existing load descriptions are intended for new designs, they contain many conservative assumptions and may be too demanding for renewal of existing structures. Many of the 14 existing equations for modeling the effect of FRPs on structural behavior are too involved for routine design use. Selection of an appropriate reliability target is also a challenge in that there is no direct basis for comparison. 1.4.2 Research Objectives The intent of this research is to answer many of the questions regarding probabilistic design of FRP strengthening  at least in a preliminary fashion  and to use these answers to develop a LRFD approach to strengthening design. LRFD can be thought of as a general philosophy for approaching design; in order to tailor this philosophy to composite materials and to strengthening rather than new design, several sub objectives have been identified and will be addressed in this report: Given the variety of composites available for strengthening, the proposed procedure must have the flexibility to accommodate this variety. The initial as well as time dependent properties of FRPs must be explicitly considered in order to ensure acceptable reliability over the proposed life of the strengthening. This procedure must recognize that the addition of extra capacity during strengthening can be much more costly than at the design stage. This requires special consideration of the assessment of initial accuracy of the target reliability level and the statistical models used for loading. Many structures will require strengthening due to deterioration, and the application of strengthening does not necessarily halt the deterioration process. Therefore, continued degradation of the structure and its effect on the time dependent reliability should be considered. 15 The design methodology must be compatible with existing codes for new design. Given the current state of knowledge regarding FRPs and their use for strengthening, it is certain that advances will occur in the future. Therefore, as much as possible, the procedure should lend itself to incorporating new research results into the specifications without requiring full reliability based recalibration of the design code. However, because the design factors developed in this research must be considered as preliminary and likely to require recalibration in the future, the underlying data and procedures should be documented as clearly as possible to aid future work. To guide further development of LRFD for FRP strengthening critical gaps in the existing research should be identified and suggestions made as to how to fill the gaps in the most advantageous manner. 1.4.3 Research Approach As discussed in Section 1.4.2 LRFD is simply a way of approaching structural design. In order to develop an effective LRFD procedure this general approach must be tailored to the circumstances in question. With regard to FRP strengthening of existing structures there are three main components to be considered: the FRP material, the structure to be strengthened, and the requirements of reliability based design. These three components are considered schematically in Figure 1 1. 16 FRP Reliability Structure Theory Figure 1 1 Components of Reliability Based Design for FRP Strengthening The three components identified in Figure 1 1 are shown with significant amounts of overlap. This represents the need to consider the problem as a whole, rather than as three independent components. Each region on this diagram can be used to identify an aspect of developing LRFD for FRP strengthening. The significant questions associated with each portion of the diagram are given in Table 1 1. 17 Table 1 1 Questions to be Answered in LRFD Study Region Questions to be Answered Structural Reliability What is the reliability basis of a LRFD code? What reliability methods are available for use? FRP / Structural Reliability How is the FRP characterized statistically? How is this statistical characterization related to the design value? FRP What types of FRP is the code applicable to? FRP / Structure How is the FRP applied to the structure? ( What limit states?) How is the effect of strengthening modeled? Structure What kind of structures is the code applicable to? Structure / Structural Reliability How can the existing structure be described statistically? How are loads on the structure described? FRP / Structure / Structural Reliability How should the design equation be formatted? ( Where should the factors go?) What is an appropriate reliability target? What reliability method can accommodate the materials and limit states? What is the range of applicability of the design code? The questions given in Table 1 1 have driven this research project, and their answers form the basis of this report. For purposes of the present work, several of these questions can be answered immediately, including the type of FRP, the type of structure, and the limit states considered. Though the design procedure, in particular the definition of material values for design, has been developed with an eye toward accommodating the full range of FRP materials, the specific examples given herein are for carbon fiber reinforced, wet layup composites. This focus was driven by the frequent use of carbon composites for strengthening and by the availability of data and material for assessing the variability of wet layup composites. Again, while the general format of design presented in this report is applicable to all types of structures, the specific example considered in this report is the class of T beam bridge superstructures. This choice was largely motivated by the sponsor of this research, the 18 California Department of Transportation. Furthermore, the limit state considered in this example is the flexural capacity of the girders. This choice was made based on the availability of load and resistance models and will be discussed further later in the report. Answers to the remaining questions shown in Table 1 1 are developed throughout the remainder of this report. It should be noted that, as research progressed on this project, it became abundantly clear that there are many topics where the existing state of knowledge is simply insufficient to provide for LRFD development without the use of extensive assumptions. The areas of uncertainty are diverse and most merit significant amounts of further study. The topic of strengthening of box girders is a pertinent one but is outside the scope of this investigation as defined by the funding agency. 1.4.4 Outline of the Report This report follows the progressive development of a LRFD based design procedure for FRP strengthening. Chapter 2 is devoted to developing the background for the project including structural reliability methods, existing statistical data, and the model chosen for continuing structural degradation. Chapter 3 develops the statistical description and defines the design values for the FRP material. In Chapter 4 a full sample calibration is conducted and preliminary factors for design are given. The complete design procedure is summarized and a design example is provided in Chapter 5. Finally, Chapter 6 concludes this report with a summary of the main accomplishments of this work and a detailed discussion of the areas remaining for further study. Numerous appendices supplement the text by providing more detail on issues raised, descriptions of some of the procedures used herein, and additional tabulated data. 19 The background information for this research is presented in Chapter 2. This chapter starts with a discussion of structural reliability methods, including basic topics such as the safety margin and reliability index. Computational methods for determining the reliability index are briefly reviewed as well as special topics, such as system reliability. The chapter then gives an overview of the prior implementation of LRFD for design using other materials, such as steel, wood, and concrete. Previous work on applying reliability based design to FRP strengthening of concrete structures is discussed. The chapter also summarizes the available statistical data and degradation models used throughout the remainder of the report. The chapter concludes by discussing the selection of the target reliability index used to calibrate resistance factors. In Chapter 3, the results of several sets of material test data are analyzed to determine appropriate statistical descriptors for FRP, including the distribution type, correlation between variables, and ranges for the mean and coefficient of variation of material properties. A value for use in design is specified as well as modification factors intended to account for the differences between laboratory tested properties and field manufactured properties. The effect of environmental degradation on FRP properties is also considered. A large example calibration is the topic of Chapter 4. This chapter thoroughly describes the assumptions and calculation methods used to derive design factors. Specific data for the example, such as the range of material properties and geometric quantities and the models used for design, are also described. Preliminary factors for the design of flexural strengthening of T beam bridge girders, considering the possibility of continued degradation, are presented. Chapter 5 summarizes the results of this project. It describes the proposed design procedure in detail and provides a full design example using the FRP material design values 20 and preliminary design factors derived in this project. It is shown that the design procedure and calibrated factors are able to create designs meeting the target reliability; however, a thorough verification of the design factors is still required in the future. The final chapter in this report, Chapter 6, is devoted to an inventory of further work needed to fully develop a LRFD specification for FRP strengthening. This work primarily involves improving and expanding available data for load models, resistance modeling and FRP properties, as well as research to improve the understanding of the state of the existing structure before the FRP is applied. 21 Chapter 2. Background for Structural Reliability, LRFD and Design Uncertainty 2.1 Introduction The goal of this chapter is to develop a firm foundation for the development of reliability based design for FRP strengthening. Topics covered herein include structural reliability methods, previous implementation of LRFD for other materials, prior work on the reliability of FRPs in civil applications, statistical descriptors for resistance and load variables, procedures for modeling the continued deterioration of structures through corrosion of the reinforcement, and selection of a target reliability index. For each specific topic, this chapter is organized to first provide some general background and to then discuss specific data, models, procedures, and assumptions that are utilized throughout the remainder of this report. 2.2 Structural Reliability Methods 2.2.1 Uncertainty and Risk Uncertainty exists in nearly everything humans do, and the design of structures is no exception. There are several different types of uncertainty that contribute to the uncertain performance of structural systems. Uncertainty is traditionally divided into two categories, aleatory, referring to inherent or “ natural” variability, and epistemic, describing errors that are related to a lack of complete knowledge and may be reduced with more information ( Melchers, 1999). Haldar and Mahadevan ( 2000) recognize two broader categories of uncertainty, cognitive or qualitative uncertainty, which is related to vagueness caused by trying to represent reality in abstract form, and noncognitive or quantitative uncertainty. Melchers ( 1999) gives a thorough breakdown of the different types of uncertainty that affect the design and performance of structures: 22 Phenomenological uncertainty is encountered whenever the structure is of a form that causes uncertainty in its eventual behavior or performance. This type of uncertainty usually affects novel structures, where designers do not have the benefit of previous experience to aid in the design process. This type of uncertainty is cognitive and cannot be included in a reliability analysis in a quantitative way. Decision uncertainty is caused by the difficulty in determining whether or not an event has occurred. In terms of structures this type of uncertainty refers to whether or not a limit state has been violated. Decision uncertainty may be of particular importance with regard to serviceability limit states where it is difficult to draw a clear line distinguishing acceptable and unacceptable performance; however, it would be very difficult to model for analysis purposes. Modeling uncertainty arises from the need to model natural phenomena with mathematical equations. This is an epistemic error that can typically be reduced through further research. If a relation between model predictions and actual test results can be found, modeling error can be accounted for in reliability analysis. Prediction uncertainty arises from the need of designers to anticipate future conditions, such as the material properties achieved during construction and the loading the structure will be subject to during its lifetime. This type of uncertainty affects how variables are modeled and how the reliability calculations are carried out, but is not explicitly accounted for in the calculations themselves. Physical uncertainty is the inherent randomness that exists in material properties and in the loads applied to a structure. This type of uncertainty may, perhaps, be reduced for materials through the use of better quality control, but it cannot be eliminated. This type of uncertainty makes up the bulk of the uncertainty considered in modern reliability analysis. 23 Statistical uncertainty arises from the need to estimate statistical descriptors such as the mean, variance, and probability distribution from limited sets of data. This type of uncertainty may be considered in analysis by allowing statistics such as mean and variance to be random variables or by performing multiple analyses with different values of the parameters. Human error is the final category of uncertainty. Human errors may be divided into routine variations in performance and gross errors. Gross human errors are likely the most common cause of structural failures; however, they are generally not included in reliability calculations because they are very hard to model in a quantitative fashion. This is an important reason why calculated reliabilities do not directly correspond to actual observed failure rates, which are generally much higher ( Melchers, 2001). Correctly modeling boundary conditions such as in scour in order to predict performance is also crucial. The presence of uncertainty creates risk. Risk is defined in some contexts as simply the probability that failure may occur. In other contexts risk is associated with the severity of the failure, and is calculated as the probability of failure times the cost of failure ( Ellingwood, 1994). 2.2.2 Evaluation of Structural Reliability 2.2.2.1 Effect of Uncertainty A primary goal of structural design is to create a structure such that the resistance capacity of the structure is greater than the load demands placed on it. This task is complicated by uncertainties because, in their presence, resistance and load cannot be described by deterministic quantities. Instead, uncertainties result in a range of possible resistance and load values and introduce the possibility that the applied load will exceed the capacity of the structure. The most basic situation, where only one resistance variable acts against just one load variable, is shown in Figure 2 1. In this figure the uncertainty in the resistance, R, and the load effect, S, is represented by probability density functions fR( r) and 24 fS( s). Following customary notation, random variables will be represented herein by capital letters, and individual realizations of a random variable will be represented with lower case letters. The following discussion on structural reliability is drawn from a number of sources ( Melchers, 1999; Madsen et al., 1986; Haldar and Mahadevan, 2000; Conte, 2004). μS μR r, s fR( r) , fS ( s) Sn Rn Figure 2 1 Basic Structural Reliability Problem 2.2.2.2 Deterministic Safety Factors In Figure 2 1, μ S and μR are the mean of the load and resistance, respectively, and Sn and Rn are the nominal load and capacity used for design. The nominal value is determined according to the design procedure in use. It may be a specified as a percentile of test results, calculated through the use of empirical or analytical equations, or prescribed in the code governing the design. In traditional deterministic design a safety factor would be calculated as the ratio of the nominal resistance to the nominal load, or a central safety factor could be calculated as the ratio of the mean of resistance to the mean of load. However, since they do not take into account the full distributions of load and resistance, safety factors calculated in this manner are unable to give an accurate assessment of the design safety. 25 The structural element whose load and resistance curves are shown in Figure 2 1 will fail if the load effect exceeds the resistance of the member. This can be seen to occur in the region of overlap between these two curves. The area of this region is not the probability of failure, as it is commonly mistaken to be. However, the size of this region can be considered to qualitatively indicate the probability of failure. By considering Figure 2 1 it can be seen that the probability of failure will change as the relative position of the two distributions changes ( i. e. the mean values are changed causing a shift along the axis of one or both of the curves), as the amount of spread in one or both of the distributions changes ( increased spread will increase the area of overlap), and as the shape of the distributions changes ( for example from a Normal distribution to a Lognormal distribution). The traditional approach to design seeks to provide acceptable designs by shifting the position of the distributions through the use of safety factors. These design approaches do not, however, consider the shape or spread of the distribution. A more rational approach to design is to consider all three issues in selection of design criteria. 2.2.2.3 Basic Reliability Problem The basic problem of structural reliability is to use statistical knowledge of uncertainties to compute the probability of structural failure. In actuality, computing the reliability of an entire structure is a very difficult task that will be briefly discussed in Section 2.2.2.7. The present discussion is pertinent to computing the reliability of a particular member with respect to a particular limit state. Given a random resistance, R, and a random load demand, S, the probability of structural failure can be expressed in many ways, as shown by Melchers ( 1999): p P( R S) f = ≤ Eq. 2 1a 26 p = P( R − S ≤ 0) f Eq. 2 1b ⎟⎠ ⎞ ⎜⎝ = ⎛ ≤ 1 S p R f Eq. 2 1c p = P( ln R − ln S ≤ 0) f Eq. 2 1d p = P[ G( R, S) ≤ 0] f Eq. 2 1e Eq. 2 1e shows the most general form. In this statement G( ) is referred to as the limit state function. This function is used to denote the boundary between safe ( or acceptable) structural behavior and unsafe ( or unacceptable) behavior. When the value of G( ) is less than zero, the limit state is said to have been violated and the structure is in the unsafe zone. Thus, the probability of failure can be expressed as the probability that the limit state has been violated. Failure in this sense means that the structure fails to meet some criteria of performance, not necessarily that the member has indeed failed; for example, serviceability limit states may be evaluated where “ failure” is defined as excessive deflection, even though the member can still support load. Furthermore, the limit state may be expressed in terms of many structural variables, not just the load and the resistance. For example, for the case of a bar element in pure tension, rather than expressing the limit state in terms of only R and S, the resistance could be expressed as a product of the area, A, and yield stress, σ y , of the member as shown in Eq. 2 2. In the following discussion of structural reliability methods, just two variables, R and S, will be considered for simplicity and clarity. G A S A S y y ( , σ , ) = σ − Eq. 2 2 27 In the most general case, load and resistance may be correlated random variables, and in order to evaluate the probability of limit state violation the joint probability density function ( PDF), fRS( r, s), is required. Figure 2 2 shows contour lines of a general joint PDF. The volume contained in an area of dimensions Δr, Δs underneath the joint PDF represents the probability that the random variable R takes a value between r and Δr while at the same time the random variable S takes a value between s and Δs. The limit state function is shown by the dotted line. R S Limit State Surface r = s Failure Domain Safe Domain fRS( r, s) Figure 2 2 Graphical Representation of Probability of Failure The area of the joint PDF shaded in grey represents the values for R and S where the limit state function is violated. In order to calculate the probability of failure, integration must be used to compute the total volume of probability under the joint PDF. This integral is expressed in Eq. 2 3. In the case where R and S are independent random variables, this equation can be expressed in terms of the marginal distributions of R and S as shown in Eq. 2 4 and Eq. 2 5. In these equations FR( ) and FS( ) are the cumulative density functions ( CDF) 28 of R and S, respectively. The lower bound of integration on these equations is zero since resistance cannot be negative. ∫∫ ≤ = () 0 ( , ) G f RS p f r s drds Eq. 2 3 ∫ ∞ = 0 p F ( s) f ( s) ds f R S Eq. 2 4 ∫[ ] ∞ = − 0 p 1 F ( r) f ( r) dr f S R Eq. 2 5 Eq. 2 3 may be generalized to account for as many random variables as are present in the problem by using the joint PDF for all variables. In general, obtaining a joint PDF for all variables is very difficult, and, even if one is available, analytical solution of these integrals is often impossible. Therefore other techniques must be used to estimate the reliability. However, there are some special cases that allow for direct solution. 2.2.2.4 The Reliability Index The reliability index is denoted with the symbol β. It is often used as a substitute for the probability of failure. This substitution is typically used because the difference between the probability of failure calculated using reliability methods and that actually witnessed in the field makes it desirable to avoid stating an explicit probability of failure. The causes of this difference are discussed in Section 2.2.2.9. β may be used to compare different structures and can be used as the target in reliability based design without mentioning a specific probability of failure. The meaning of β can be best interpreted through consideration of one of the few cases where the probability integrals of Section 2.2.2.3 can be computed analytically. If R and S are 29 independent, Normally distributed variables, the linear safety margin, Z= R S, is also a Normally distributed variable. Given the means, μR and μ S , and variances, σR 2 and σ S 2, of resistance and load, respectively, the mean and variance of Z can be expressed as shown in Eq. 2 6 and Eq. 2 7, respectively, following basic formulas for linear combinations of Normally distributed variables. Z R S μ = μ − μ Eq. 2 6 2 2 2 Z R S σ = σ + σ Eq. 2 7 By computing the standard Normal variate of Z, Eq. 2 1b can now be expressed in terms of Φ, the cumulative distribution function of the standard Normal distribution, as shown in Eq. 2 8. This function is available in tabulated form in many statistics texts and can also be accessed using the NORMSDIST function in Microsoft Excel. ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − − Φ = ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ − = − ≤ = ≤ = Φ 2 2 0 0 ( ) ( 0) ( 0) R S R S Z Z f p P R S P Z σ σ μ μ σ μ Eq. 2 8 Cornell ( 1969) defined the reliability index, or safety index, as shown in Eq. 2 9, as the mean of the safety margin divided by the standard deviation of the safety margin. This formulation of the reliability index is referred to as the Cornell reliability index. In this formulation β can be interpreted as the distance between the mean of the safety margin and the point of failure, measured in terms of the standard deviation of the safety margin. This is shown in Figure 2 3. Z Z σ μ β = Eq. 2 9 30 Cornell’s definition of β holds even if the safety margin is a function of more than two variables, as long as the failure surface is linear. The definition of β has been extended to cover other cases, see Melchers ( 1999) and Madsen et al.( 1986). However, for all definitions, the basic concept of β remains the same; it is a measure of the distance between the most likely state of the structure ( mean) and the most likely failure point, in terms of the variation. β σ μ Failure Domain Safe Domain PDF of Safety Margin, Z 0 Figure 2 3 Interpretation of β in Terms of the Safety Margin Though it is only exact in the case of Normally distributed variables and a linear limit state function, β is often approximately related to the probability of failure through Eq. 2 10, which is a generalization of Eq. 2 8. Table 2 1 shows values of β corresponding to several probabilities of failure based on this approximate equation. = Φ(− β ) f p Eq. 2 10 31 Table 2 1 Probabilities of Failure and Corresponding β s pf β 0.5 0 0.1 1.28 0.01 2.33 0.001 3.09 0.0001 3.72 0.00001 4.27 2.2.2.5 Methods of Computing the Reliability Index In some cases the probability integral in Eq. 2 3 may be solved using numerical integration. However, this integral is an n fold integral where n represents the number of design variables, and thus the complexity rapidly increases as variables are added to the problem. Several other methods have been developed to calculate the probability of failure, or the reliability index directly. The following brief summary of some of the most popular techniques is based on Melchers ( 1999), Madsen et al. ( 1986), Haldar and Mahadevan ( 2000), and Conte ( 2004). In Section 2.2.2.10 the specific methods used for this report and the reasons why they were selected will be discussed. 2.2.2.5.1 First Order, Second Moment Reliability Index This formulation of the reliability index is an extension of the Cornell reliability index. βFOSM is still defined as the mean of the safety margin divided by the standard deviation of the safety margin and still requires only the mean and standard deviation of the design variables for calculation. However, to allow for non linear limit state functions, the mean and standard deviation of the safety margin are computed by linearizing the safety margin using the linear terms of a Taylor series expansion. The value of βFOSM depends on the point that is chosen for linearization of the limit state function. A common choice is the point where each random 32 variable takes on its mean value, resulting in the mean value, first order, second moment reliability index. Though this method is very simple, it has several drawbacks. The most significant drawback is that the value of βFOSM is not invariant with respect to the limit state formulation; two mechanically equivalent formulations of the same limit state can produce different values for the reliability index. The ambiguity of βFOSM caused by the formulation of the limit state function can be removed by choosing a point on the limit state surface as the expansion point. The point used for expansion is called the design or checking point. When all variables and the limit state function are transferred to standard Normal space the design point is found through a minimization procedure as the point on the limit state surface with the shortest distance to the origin. The reliability index found in this manner is sometimes referred to as βHL, the Hasofer Lind reliability index ( Hasofer and Lind, 1974). This measure of reliability is invariant with respect to the limit state formulation, but it uses only second moment information about the variables and is not comparable because it does not depend on the curvature of the limit state function at the design point. The generalized reliability index introduced by Ditlevsen makes use of a weighting function to overcome the lack of comparability ( Madsen et al., 1986). 2.2.2.5.2 First and Second Order Reliability Methods ( FORM and SORM) The first order reliability method is very similar to the first order, second moment method. The limit state function is still linearized using a first order approximation about the design point. The significant difference between these two methods is that the first order, second moment reliability index only uses second moment information about the design variables. FORM uses knowledge of the full distribution of the design variables. The distributions of the design variables are transformed into standard Normal space using 33 appropriate techniques, such as the Normal tail transformation for independent variables or the Rosenblatt transformation for dependent variables ( Melchers, 1999). βFO can then be found using the same optimization techniques used in calculation of βFOSM. To find the probability of failure, Eq. 2 10 may be used to give a first order approximation. SORM makes use of non linear expansions of the limit state function. Determining the probability content of these non linear surfaces is complicated, and approximate techniques must be used to estimate pf. The accuracy of both of these techniques depends on the ability of the approximating surface to represent the true limit state. 2.2.2.5.3 Monte Carlo Simulation ( MCS) Monte Carlo Simulation is a technique with applications to many different problems ( Rubinstein, 1981). In general the Monte Carlo technique involves using random sampling to generate a large set of artificial data that may be analyzed. In application to structural reliability, a random value is generated for each design variable based on the statistical distribution of that variable, and the random values are used to check the limit state equation. If the limit state function is less than zero, the structure is considered to have “ failed”. This process is repeated a large number of times, and the probability of failure is estimated as the number of failed samples divided by the total number of simulations. This approach is very robust and can be applied to almost any limit state formulation. However, the accuracy of this approach depends on the number of simulations, and for small failure probabilities the required computing time can be very demanding. With additional knowledge about the failure region, variance reduction techniques, such as importance sampling, can be used to concentrate the simulations in the region of interest and reduce the necessary number of simulations. 34 2.2.2.5.4 Other Techniques In some cases a hybrid approach combining simulation with FORM or SORM is used. This may be the case when convergence cannot be reached using all the variables in FORM. This approach was taken by Plevris et al. ( 1995). The statistical description of resistance of a strengthened member was determined using Monte Carlo Simulation, and then FORM was used with distributions for load to compute the structural reliability. Response surface techniques can be used when the limit state function is not in an explicit functional form, such as when structural behavior is modeled using finite element analysis. This technique is based on evaluation of the implicit limit state at a limited number of points, and then fitting a function to these points. This functional form can then be directly used with first order second moment methods or FORM/ SORM. 2.2.2.6 Levels of Reliability Methods The structural reliability methods described above have been grouped into different levels defined by the types of information they use and the types of calculation they imply ( Madsen et al., 1974; Melchers, 1999). These levels are now briefly described in order to illuminate the relation between reliability computation techniques and the LRFD format. Level 1 methods are the simplest techniques, and are used to provide safety in the design of structures. These techniques rely on just one value to describe each design parameter. ASD, LRFD and other code level techniques are examples of Level 1 methods. Level 2 methods make use of two values describing each design parameter ( most often the mean and variance) along with a description of the correlation between parameters to calculate a reliability index. These methods are more approximate in their estimate of the probability of failure than methods such as FORM or simulation. The first order, secondmoment reliability index is an example of this level. 35 Level 3 methods are those methods that seek the best possible estimate of the probability of failure by making use of full distributions to describe the design variables. Examples of this level include FORM/ SORM and Monte Carlo Simulation. Level 4 methods combine consideration of the structure from the previous levels with economic data and seek to provide a cost benefit analysis. These methods can provide a rational basis for decision making. The goal of the study is to develop factors for a LRFD design procedure for FRP strengthening. The design format itself is an example of Level 1 reliability methods; however, the design factors will be calibrated based on Level 2 techniques for computing the reliability. 2.2.2.7 Component vs. System Reliability The first order second moment, FORM, and SORM methods described above are all capable of predicting the probability of failure for a particular member with respect to a particular mode of failure. In reliability analysis this is referred to as component reliability. However, most structural elements are susceptible to failure in several different modes, for example a beam must resist both flexural and shear loads. Furthermore, nearly all structures are composed of many members. The probability of failure of a single element in one of several different mechanisms or the probability of failure of an entire structure are both problems of system reliability. Computation of system reliability is significantly more difficult than computation of component reliability. Monte Carlo Simulation is an option if a model of the complete system can be developed. Or the reliability of a system can be computed based on the reliability of individual components, often relying on simplifying approximations such as assuming the system is a series system or a parallel system. However, the resulting methods remain too complicated for use in calibration of a reliability based design procedure. Design factors are therefore based on the reliability of a single member 36 with respect to a particular limit state. This would directly correspond to the weakest link analogy when considering the reliability of the complete structure; however, due to the indeterminate nature of most structures, the reliability of the structure as a whole is usually much higher than the reliability of individual components. This fact must be kept in mind when selecting a target reliability for code calibration. More information on system reliability may be found in Madsen et al. ( 1986) and Melchers ( 1999). 2.2.2.8 Time dependent Reliability When the statistical description of load or resistance ( as for a deteriorating structure) changes with time, these changes must be considered through the use of time dependent reliability techniques. The most rigorous approaches involve the use of stochastic processes, and the probability of failure is determined as the first passage probability, i. e. the probability that the limit state will be violated for the first time during the time period in question. Direct solution of these theoretical formulations is often intractable except for the simplest cases, and thus numerical or simulation based approaches are necessary ( Melchers, 1999). Researchers studying degradation of concrete structures have formulated the reliability problem as seen in Eq. 2 11. In this equation pf represents the probability of failure, R( ti) is the time dependent resistance, and Si are independent random loading events during the life of the structure. This equation has been solved using simulation for deterministic loading increments ( Stewart 1998), as well as stochastic load processes ( Mori 1993). 1 [ ( ) ( ) ... ( ) ] f 1 1 2 2 n n p = − P R t > S ∩ R t > S ∩ ∩ R t > S Eq. 2 11 While the form of Eq. 2 11 is far easier to handle than dealing directly with outcrossing rates ( the probability of leaving the safe domain from a certain point weighted with the probability of being at that point), this equation is still quite demanding from a calibration 37 standpoint. The commonly used approach in development of reliability based codes is referred to as the time integrated approach ( Melchers, 1999). In this method, the complete lifetime ( or at least the lifetime of interest) of the structure is considered to be a single unit. All random variables must be related to this period of time by choosing appropriate statistical descriptions. This approach has been used in the calibration of previous reliability based codes. For example, the service life considered in development of the AASHTO LRFD Bridge Design Specifications is 75 years, and thus the code is calibrated based on an estimated distribution of the 75 year maximum loading ( Nowak, 1999). In this case, deterioration of the structure was not considered, so the resist 



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