ABSTRACT Title of Document: RELIABILITY-BASED DESIGN OF PIPING: Internal Pressure, Gravity, Earthquake, and Thermal Expansion Kleio Avrithi, Doctor of Philosophy, 2007 Directed By: Professor Bilal M. Ayyub Department of Civil and Environmental Engineering Although reliability theory has offered the means for reasonably accounting for the design uncertainties of structural components, limited effort has been made to estimate and control the probability of failure for mechanical components, such as piping. The ASME B&PV Code, Section III, used today for the design of safety piping in nuclear plants is based on the traditional Allowable Stress Design (ASD) method. This dissertation can be considered as a primary step towards the reliability- based design of nuclear safety piping. Design equations are developed according to the Load and Resistance Factor Design (LRFD) method. The loads addressed are the sustained weight, internal pressure, and dynamic loading (e.g., earthquake). The dissertation provides load combinations, and a database of statistical information on basic variables (strength of steel, geometry, and loads). Uncertainties associated with selected ultimate strength prediction models - burst or yielding due to internal pressure and the ultimate bending moment capacity- are quantified for piping. The procedure is based on evaluation of experimental results cited in literature. Partial load and resistance factors are computed for the load combinations and for selected values of the target reliability index, ȕ. Moreover, design examples demonstrate the procedure of the computations. A probabilistic-based method especially for Class 2 and 3 piping is proposed by considering only cycling moment loading (e.g., thermal expansion). Conclusions of the study and provided suggestions can be used for future research. RELIABILITY-BASED DESIGN OF PIPING: Internal Pressure, Gravity, Earthquake, and Thermal Expansion By Kleio Avrithi Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park, in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2007 Advisory Committee: Professor Bilal M. Ayyub, Chair/Advisor Professor Amde M. Amde Professor Donald B. Barker Research Professor Chung C. Fu Assistant Professor Ricardo A. Medina © Copyright by Kleio Avrithi 2007 ii Dedication With gratitude and love to my parents Zacharias P. Avrithis and Maria Xatzistergou-Avrithi iii Acknowledgements I would like to thank Professor Bilal M. Ayyub for the advising provided. I would also like to thank Professors Amde M. Amde, Donald B. Barker, Chung C. Fu, and Ricardo A. Medina for kindly serving as my advisory committee members. Help and software provided by Dr. Ibrahim A. Assakkaf as well as the recommendations of the ASME Special Working Group on Probabilistic Methods in Design and the ASME Steering Committee at the beginning of this work are recognized and appreciated. I would like to thank the University of Maryland library staff and especially Mr. Jim Miller (EPSL) for their invaluable help. Moreover, I am thankful to Dr. Mark Kaminskiy for letting me use a statistical program. I would like to thank Professor Antony E. Armenákas and my father for encouraging me to continue my studies in USA. The support and encouragement provided generously by my parents, sister, and brother-in-law are most appreciated. iv Table of Contents Dedication ..................................................................................................................... ii Acknowledgements...................................................................................................... iii Table of Contents......................................................................................................... iv List of Tables ............................................................................................................. viii List of Figures ............................................................................................................. xii CHAPTER 1: INTRODUCTION................................................................................. 1 1.1. Nuclear Energy and Nuclear Plants ............................................................... 1 1.2. Nuclear Plants ................................................................................................ 3 1.3. Nuclear Piping ............................................................................................... 4 1.4. Objective........................................................................................................ 7 1.5. Background.................................................................................................... 8 1.6. Organization................................................................................................. 11 CHAPTER 2: CURRENT PIPING DESIGN PRACTICES AND THE LRFD......... 13 2.1. Current Piping Design Practices .................................................................. 13 2.1.1. The ASME B&PV Code, Section III............................................... 13 2.1.2. Service Levels.................................................................................. 14 2.1.3. Stresses and Loads ........................................................................... 16 2.1.4. Equations in ASME B&PV Code.................................................... 22 2.1.4.1. Bending Primary Stresses ................................................. 23 2.1.4.2. Stress Indices .................................................................... 28 2.1.4.3. Internal Pressure ............................................................... 29 2.1.4.4. Other Codes for the Design of Piping............................... 32 2.2. Reliability-Based Design ............................................................................. 32 2.2.1. The Load and Resistance Factors Design Method........................... 33 2.2.1.1. Procedure for the Development of Load and Resistance Factor Design.................................................. 33 2.2.1.2. Data Space ........................................................................ 35 2.2.1.3. Performance Functions ..................................................... 36 2.2.1.4. Calculation of Safety Factors Applied to Mean Values of Variables........................................................... 39 2.2.2. Direct Reliability Design ................................................................. 43 2.3. Comparison of ASD and LRFD................................................................... 44 2.4. Advantages of LRFD................................................................................... 45 CHAPTER 3: BASIC RANDOM VARIABLES FOR PIPING................................. 47 3.1. Strength Variables........................................................................................ 48 3.1.1. Yield Strength of Steel..................................................................... 49 3.1.2. Ultimate Strength of Steel................................................................ 54 3.1.3. Comparison of LRFD Definition of Steel Strength with ASME Practice ................................................................................ 59 3.1.4. Geometrical properties..................................................................... 64 v3.1.4.1. External Diameter............................................................. 64 3.1.4.2. Thickness .......................................................................... 65 3.1.4.3. Section Modulus ............................................................... 66 3.1.4.4. Diameter to Thickness Ratio ............................................ 66 3.2. Load Variables............................................................................................. 67 3.2.1. Sustained Weight ............................................................................. 67 3.2.1.1. Self Weight ....................................................................... 68 3.2.1.2. Self Weight of Fittings and Components ......................... 68 3.2.1.3. Insulation .......................................................................... 69 3.2.1.4. Contents of Pipe................................................................ 70 3.2.1.5. Probabilistic Characteristics ............................................. 70 3.2.2. Internal Pressure .............................................................................. 71 3.2.3. Earthquake Loading......................................................................... 73 3.2.4. Mechanical Loading ........................................................................ 76 3.2.5. Thermal Loading.............................................................................. 79 3.2.6. LOCA Loading ................................................................................ 80 3.3. Generated Random Variables ...................................................................... 80 CHAPTER 4: STRENGTH MODELS UNCERTAINTY ......................................... 81 4.1. Definition of Bias......................................................................................... 81 4.2. Strength Model Uncertainty ........................................................................ 82 4.3. Piping Burst and Yielding Due to Internal Pressure.................................... 84 4.3.1. Strength Models............................................................................... 85 4.3.2. Total Bias Estimation for Yield and Burst Internal Pressure........... 88 4.3.3. Bias of Burst and Yield Pressure ..................................................... 95 4.4. Ultimate Moment Capacity of Piping.......................................................... 96 4.4.1. Strength Models for Failure Moment ............................... 96 4.4.2. Total Bias for Pure Bending ........................................... 101 4.4.3. Total Bias for Bending with Internal Pressure ............... 104 4.4.4. Bias of Flexural Strength for Straight Pipes ................... 105 4.5. Conclusions................................................................................................ 106 CHAPTER 5: LOAD COMBINATIONS AND PERFORMANCE FUNCTIONS FOR PIPING ........................................................................................................... 107 5.1. Performance Functions .............................................................................. 107 5.2. Load Combinations.................................................................................... 109 5.3. Normalization of Stresses with Respect to Sustained Weight................... 112 5.4. The Target Reliability Index, ȕ, for Piping................................................ 115 CHAPTER 6: CALCULATION OF PARTIAL SAFETY FACTORS.................... 120 6.1. Computations ............................................................................................. 120 6.2. Part I: Design for Internal Pressure............................................................ 123 6.2.1. Performance Functions and Probabilistic Characteristics of Variables ........................................................................................ 123 6.2.2. Partial Safety Factors ..................................................................... 126 6.2.3. Sensitivity Analysis ....................................................................... 136 6.2.4. Computational Example ................................................................ 138 vi 6.3. Part II: Design for Combined Loading ...................................................... 142 6.3.1. Design ............................................................................................ 142 6.3.1.1. Performance Function g2 ................................................ 143 6.3.1.2. Performance Function g3 ................................................ 145 6.3.2. Service Level A ............................................................................. 149 6.3.3. Service Level B.............................................................................. 153 6.3.3.1. Performance Function g6 ................................................ 153 6.3.3.2. Performance Function g7 ................................................ 158 6.3.3.3. Performance Function g8 ................................................ 163 6.3.3.4. Performance Function g9 ................................................ 166 6.3.4. Service Level C.............................................................................. 170 6.3.4.1. Performance Function g11 ............................................... 170 6.3.4.2. Performance Function g12 ............................................... 172 6.3.4.3. Performance Function g13 ............................................... 176 6.3.4.4. Performance Function g14 ............................................... 180 6.3.5. Service Level D ............................................................................. 184 6.3.5.1. Performance Function g16 ............................................... 184 6.3.5.2. Performance Function g17 ............................................... 187 6.3.5.3. Performance Function g18 ............................................... 191 6.3.5.4. Performance Function g19 ............................................... 197 6.4. Computational Example ............................................................................ 201 6.5. Conclusions................................................................................................ 205 CHAPTER 7: FATIGUE DESIGN OF PIPING ...................................................... 209 7.1. General Discussion .................................................................................... 209 7.2. ASME Practice for Class 2 and 3 Piping................................................... 213 7.3. Reliability-Based Fatigue Design .............................................................. 219 7.3.1. Performance Function and Equivalent Stress Range..................... 220 7.3.2. Strength and Loading Uncertainties .............................................. 224 7.3.3. Values for the Target Reliability Index, ȕ ..................................... 226 7.3.4. Discussion and Evaluation............................................................. 226 7.3.5. Computation Procedure ................................................................. 228 7.3.6. Combination of Secondary and Primary Stresses.......................... 229 7.3.6.1. Example 1 ....................................................................... 232 7.3.6.2. Example 2 ....................................................................... 233 7.3.7. Conclusions and Recommendations .............................................. 235 CHAPTER 8: CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH ........................................................................................................... 236 8.1. Summary.................................................................................................... 236 8.2. Conclusions................................................................................................ 237 8.3. Suggestions for Future Research ............................................................... 238 APPENDIX A: ASME B&PV CODE EQUATIONS.............................................. 240 APPENDIX B: STEEL USED IN B&PV CODE, PART III ................................... 246 vii APPENDIX C: PARTIAL MEAN RESISTANCE FACTORS AND ADJUSTED NOMINAL RESISTANCE FACTORS ................................................................... 252 C.1. Performance Functions g1, g5, g10, and g15 ................................................ 252 C.2. Performance Function g2 ........................................................................... 258 C.3. Performance Function g3 ........................................................................... 259 C.4. Performance Function g4 ........................................................................... 261 C.5. Performance Function g6 ........................................................................... 263 C.6. Performance Function g7 ........................................................................... 269 C.7. Performance Function g8 ........................................................................... 272 C.8. Performance Function g9 ........................................................................... 283 C.9. Performance Function g11 .......................................................................... 294 C.10. Performance Function g12 .......................................................................... 296 C.11. Performance Function g13 .......................................................................... 305 C.12. Performance Function g14 .......................................................................... 310 C.13. Performance Function g16 .......................................................................... 319 C.14. Performance Function g17 .......................................................................... 320 C.15. Performance Function g18 .......................................................................... 325 C.16. Performance Function g19 .......................................................................... 330 REFERENCES: ........................................................................................................ 339 viii List of Tables Table 1-1: Piping Systems in a Pressurized Water Reactor Plant (Rodabaugh, et al., 1987) ........................................................................................................... 6 Table 1-2: Load Combinations for Nuclear Plant Facilities and Components ........... 10 Table 2-1: Definition of Stresses Symbols Used in the Code..................................... 24 Table 2-2: Values of ȡ and Resultant Resistance Stress for Operation at Room Temperature .............................................................................................. 24 Table 2-3: Limit Theory Loading ............................................................................... 26 Table 2-4: Values of Additional Thickness A (ASME B&PV CODE, 2001) ............ 30 Table 2-5: Permissible Pressure for Class 1 Piping According to ASME B&PV Code, Section III.................................................................................................. 31 Table 2-6: Characteristics of the ASD and LRFD Methods ....................................... 45 Table 3-1: Data for the Yield Strength, Sy, of Carbon and Stainless Steel at Room Temperature .............................................................................................. 50 Table 3-2: Experimental Data for the Yield Strength of Carbon and Stainless Steel. 51 Table 3-3: Best Fit Curves for Yield Strength of Stainless Steel ............................... 52 Table 3-4: Mean Values and Bias of Yield Strength for Stainless Steel at Elevated Temperatures............................................................................................. 53 Table 3-5: Summary of Probabilistic Characteristics for Yield Strength ................... 54 Table 3-6: Data for the Ultimate Strength of Carbon and Stainless Steels for Nuclear Piping at Room Temperature .................................................................... 55 Table 3-7: Experimental Data for the Ultimate Strength of Carbon and Stainless Steel ................................................................................................................... 56 Table 3-8: Best Fit Curves for Ultimate Strength of Stainless Steel .......................... 57 Table 3-9: Mean Values and Bias of Ultimate Strength for Stainless Steel at Elevated Temperatures............................................................................................. 58 Table 3-10: Summary of Probabilistic Characteristics for Ultimate Strength............ 59 Table 3-11: Properties of Nominal Pipe Outside Diameter ........................................ 65 Table 3-12: Reported Probabilistic Characteristics for the Thickness of Pipes.......... 66 Table 3-13: Statistical Properties for the Thickness of Pipes ..................................... 66 Table 3-14: Self-Weight Tolerances of Steel Piping (Crocker, et al., 1967).............. 68 Table 3-15: Insulation Density, I (Helguero, 1983; Kannappan, 1986) ..................... 70 Table 3-16: Operating Pressure at Operating Temperature (Crocker, et al., 1967; Lamit, 1981).............................................................................................. 71 Table 3-17: Statistical Properties for Pressure Based on Literature Review .............. 72 Table 3-18: Statistics on Accidental Pressure............................................................. 72 Table 3-19: Proposed Probabilistic Characteristics for Internal Pressure for this Study ................................................................................................................... 73 Table 3-20: Probabilistic Characteristics for the Seismic Load.................................. 76 Table 3-21: Statistics on Safety Relief Valve (SRV) Discharge Loads, (Hwang, et al., 1983) ......................................................................................................... 78 Table 3-22: Statistics on Safety Relief Valve (SRV).................................................. 79 Table 4-1: Models Examined for Use in LRFD.......................................................... 86 Table 4-2: Models for Burst Pressure and Strain Hardening Steel............................. 87 ix Table 4-3a: Experimental Data Used for the Estimation of Total Bias for Burst of Pipes.......................................................................................................... 89 Table 4-3b: Experimental Data Used for the Estimation of Total Bias for Yielding of Pipes.......................................................................................................... 90 Table 4-4a: Probabilistic Characteristics of Total Bias for Burst Pressure ................ 91 Table 4-4b: Probabilistic Characteristics of Total Bias for Yield Pressure and Carbon Steel........................................................................................................... 91 Table 4-5: Models Examined and Parameters Calculated for the Predictionof Burst Pressure…………………………..………………………………….…...95 Table 4-6a: Probabilistic Characteristics of Bias of Burst Pressure for Carbon and Stainless Steel ........................................................................................... 95 Table 4-6b: Probabilistic Characteristics of Bias of Yield Pressure for Carbon Steel 96 Table 4-7a: Experimental Data Used for the Estimation of Total Bias for Bending of Pipes........................................................................................................ 100 Table 4-7b: Experimental Data Used for the Estimation of Total Bias for Bending of Pipes with Internal Pressure.................................................................... 101 Table 4-8a: Statistical Descriptive of Total Bias for Eq. (4-10) and Pure Bending, and Derivation of VM ..................................................................................... 103 Table 4-8b: Statistical Descriptive of Total Bias for Eq. (4-11) and Pure Bending, and Derivation of VM ..................................................................................... 103 Table 4-9a: Statistical Descriptive of Total Bias for Eq. (4-10) and Bending with Internal Pressure, and Derivation of VM.................................................. 105 Table 4-9b: Statistical Descriptive of Total Bias for Eq. (4-11) and Bending with Internal Pressure and Derivation of VM................................................... 105 Table 4-10: Bias for Bending Resistance and All Examined……………………….106 Table 5-1: Loads to Be Combined for Different Service Levels .............................. 110 Table 5-2: Recommended Performance Functions for Straight Pipes...................... 111 Table 5-3: Lifetime Limit State Probabilities for Nuclear Structures (for 40 years) 117 Table 5-4: Reliability Index ȕ and Corresponding Probability of Failure, Pf........... 118 Table 6-1: Probabilistic Characteristics of Variables Used for the Calculation of Partial Factors for g1, g5, g10, and g15 ...................................................... 126 Table 6-2: Probabilistic Properties of Variable XM................................................... 126 Table 6-3a: Recommended Adjusted Nominal Resistance Factor, ij, for Total Load Factor Ȗ=1.2 and Carbon Steel……………………………………….....135 Table 6-3b: Recommended Adjusted Nominal Resistance Factor, ij, for Total Load Factor Ȗ=1.2 and Stainless Steel……………………..……………..…..136 Table 6-4: Properties of Steel ...................................................................................139 Table 6-5a: Sample Computations for a Pipe Made of Carbon Steel ……………...140 Table 6-5b: Sample Computations for a Pipe Made of Stainless, Austenitic, Steel.141 Table 6-6: Parameters for the Calculations for g2.....................................................143 Table 6-7: Recommended Nominal Load and Resistance Factors for g2 ................ 145 Table 6-8: Parameters for the Calculations for g3.................................................... 146 Table 6-9: Ranges and Recommended Adjusted Nominal Resistance Factors for g3 ................................................................................................................ 148 Table 6-10: Parameters for the Calculations for g4.................................................. 150 xTable 6-11: Ranges and Recommended Adjusted Nominal Resistance Factors for g4 ................................................................................................................. 152 Table 6-12: Parameters for the Computations for g6 ................................................ 154 Table 6-13: Ranges and Recommended Values for Adjusted Nominal Resistance Factors for g6........................................................................................... 157 Table 6-14: Parameters for the Computations for g7 ................................................ 159 Table 6-15: Ranges and Recommended Nominal Adjusted Resistance Factors for g7 ................................................................................................................. 159 Table 6-16: Parameters for the Calculations for g8................................................... 164 Table 6-17: Recommended Values and Ranges for the Nominal Resistance Factor for g8 ............................................................................................................. 165 Table 6-18: Parameters for the Calculations for g9................................................... 167 Table 6-19: Recommended Values and Ranges for the Adjusted Nominal Resistance Factor for g9 ............................................................................................ 169 Table 6-20: Parameters for the Calculations for g11 ................................................. 171 Table 6-21: Recommended Values and Ranges for the Adjusted Nominal Resistance Factor for g11 ........................................................................................... 171 Table 6-22: Parameters for the Calculations for g12 ................................................. 172 Table 6-23: Recommended Values and Ranges for the Adjusted Nominal Resistance Factor for g12 ........................................................................................... 173 Table 6-24: Parameters for the Calculations for g13 ................................................. 177 Table 6-25: Ranges and Recommended Nominal Resistance Factors for g13 .......... 178 Table 6-26: Parameters for the Calculations for g14 ................................................. 181 Table 6-27: Recommended Values and Ranges for the Nominal Resistance Factor for g14............................................................................................................ 182 Table 6-28: Parameters for the Calculations for g16 ................................................. 185 Table 6-29: Recommended Values and Ranges for the Adjusted Nominal ............. 187 Table 6-30: Parameters for the Calculations for g17 ................................................. 188 Table 6-31: Recommended Values and Ranges for the Adjusted Nominal Resistance Factor for g17 ........................................................................................... 189 Table 6-32: Parameters for the Calculations for g18 ................................................. 192 Table 6-33: Ranges and Recommended Adjusted Nominal Resistance Factors for g18 ................................................................................................................. 195 Table 6-34: Parameters for the Calculations for g19 ................................................. 198 Table 6-35: Recommended Values and Ranges for the Adjusted Nominal Resistance Factor for g19 ........................................................................................... 199 Table 6-36: Data for Example Computations ........................................................... 202 Table 6-37: Nominal and Mean Normalized Stresses .............................................. 203 Table 6-38: Computations with the Developed LRFD Equations ............................ 204 Table 7-1: Design Transients and Cycles for the Reactor Coolant System (IAEA, 2003) ....................................................................................................... 211 Table 7-2: Values for the Reduction Factor, f........................................................... 215 Table 7-3: Variables for the Piping Fatigue Design ................................................. 226 Table 7-4: Values of Target Reliability Index for Fatigue Design and Different Structures ................................................................................................ 226 Table 7-5: Cycles and Stress Ranges for Example 1 Computations......................... 233 xi Table 7-6: Cycles and Stress Ranges for Example 2 Computations......................... 234 Table 7-7: Thermal Cycles and Stress Ranges ......................................................... 234 Table 7-8: Considered Probabilistic Characteristics of Variables in Eq. (7-21)....... 235 xii List of Figures Figure 1-1: Structure of the Dissertation .................................................................... 12 Figure 2-1: Piping Stresses and Loads........................................................................ 18 Figure 2-2: Failure Modes for Piping and their Cause ............................................... 19 Figure 2-3: (a) Constant Loads, (b) Pulse Loads and (c) Intermittent Loads (Hwang, et al., 1987) ............................................................................................... 21 Figure 2-4: Examples of Reversing and Nonreversing Dynamic Loads (ASME B&PV Code, 2001)............................................................................................... 22 Figure 2-5: Magnitude of Pressure in an Augmenting Scale for Different Service Levels for a Pipe’s Operation ................................................................... 31 Figure 2-6: Steps for the Development of Load and Resistance Factor Design......... 34 Figure 2-7: Space of Reduced Random Variables Showing the Reliability Index, ȕ, and the Most Probable Failure Point xƍ*.................................................... 43 Figure 2-8: Relationships among Nominal (LN, RN), Mean (ȝL, ȝR), and Factored Values (ȖLN, ijRN) for the Load and the Resistance .................................. 44 Figure 3-1: Histogram for the Yield Strength of Stainless Steel at Room Temperature Based on the Test Data in the Report of Simmons, et al. (1965), including all types of steel, except from steel TP321 and all types of specimens. ... 52 Figure 3-2: Histogram for the Ultimate Strength of Stainless Steel at Room Temperature Based on the Test Data in the Report of Simmons, et al. (1965) including all types of steel, except from steel TP321, and all types of specimens.............................................................................................. 57 Figure 3-3: Behavior of Type 316 Stainless Steel at Different Temperatures............ 60 Figure 3-4: Behavior of Carbon Steel SA106B at elevated temperatures, a) ultimate strength, b) yield strength (Simmons, 1955)............................................. 61 Figure 3-5: Physical Properties of Carbon Steel at Elevated Temperatures............... 61 Figure 3-6: Values of Yield Strength Used in LRFD and ASME Code for Carbon Steel........................................................................................................... 62 Figure 3-7: Values of Yield Strength Used in LRFD and ASME Code for Stainless Steel........................................................................................................... 62 Figure 3-8: Values of Ultimate Strength Used in LRFD and ASME Code for Carbon Steel SA 106B at Different Operating Temperatures ............................... 63 Figure 3-9: Values of Ultimate Strength Used in LRFD and ASME Code for Stainless Steel SA 312 Type 304 at Different Operating Temperatures.................. 63 Figure 4-1a: Total Bias of Burst Pressure for Model 1............................................... 92 Figure 4-1b: Total Bias of Burst Pressure for Model 2……….……………………..92 Figure 4-1c: Total Bias of Burst Pressure for Model 3………………………………92 Figure 4-1d: Total Bias of Burst Pressure for Model 4……………...………………92 Figure 4-1e: Total Bias of Burst Pressure for Model 5………………………………92 Figure 4-1f: Total Bias of Burst Pressure for Model 6……………………………....92 Figure 4-1g: Total Bias of Burst Pressure for Model 7………………………...……93 Figure 4-2a: Total Bias of Yield Pressure for Model 1 .............................................. 93 Figure 4-2b: Total Bias of Yield Pressure for Model 2………………………...……93 Figure 4-2c: Total Bias of Yield Pressure for Model 3………………………….…..93 xiii Figure 4-2d: Total Bias of Yield Pressure for Model 4………………………...……93 Figure 4-2e: Total Bias of Yield Pressure for Model 5………………………...……94 Figure 4-2f: Total Bias of Yield Pressure for Model 6……………………………....94 Figure 4-2g: Total Bias of Yield Pressure for Model 7………………………….…..94 Figure 4-3: Perfectly Plastic and Bilinear Approximations of Steel Behavior………97 Figure 4-4: Total Bias for the Bending Resistance According to Eq. (4-10) for Do/t>75 and Carbon Steel ....................................................................... 102 Figure 4-5: Total Bias for the Bending Resistance According to Eq. (4-10), (a) Stainless Steel for Do/t<50, (b) Carbon Steel for Do/t<65 ...................... 103 Figure 4-6: Total Bias for the Bending Resistance According to Eq. (4-11) for Do/t<50, (a) Stainless Steel, (b) Carbon Steel......................................... 104 Figure 4-7: Total Bias for the Bending Resistance with Internal Pressure for Stainless Steel and Do/t<50, (a) According to Eq. (4-10), (b) According to Eq. (4- 11) ........................................................................................................... 105 Figure 6-1a: Mean Partial Safety Factors φ′y for the Resistance, γ′Ρ for the Pressure, and γ′Μ for the Model Uncertainty Variable versus β for Design and Service Limit A, and Stainless Steel……………....................................128 Figure 6-1a: Mean Partial Safety Factors φ′y for the Resistance, γ′Ρ for the Pressure, and γ′Μ for the Model Uncertainty Variable versus β for Design and Service Limit A, and Carbon Steel………..……………………………128 Figure 6-2a: Mean Partial Safety Factors φ′y for the Resistance, γ′Ρ for the Pressure, and γ′Μ for the Model Uncertainty Variable versus β for Service Limit B and Stainless Steel………………………………………………………129 Figure 6-2b: Mean Partial Safety Factors φ′y for the Resistance, γ′Ρ for the Pressure, and γ′Μ for the Model Uncertainty Variable versus β for Service Limit B and Carbon Steel………………………………………………..........…129 Figure 6-3: Mean Partial Safety Factors φ′u for the Resistance, γ′Ρ for the Pressure, and γ′Μ for the Model Uncertainty Variable versus β for Service Limit C and Both Stainless and Carbon Steel. ..................................................... 130 Figure 6-4: Mean Partial Safety Factors φ′u for the Resistance, γ′Ρ for the Pressure, and γ′Μ for the Model Uncertainty Variable versus β for Service Limit D and Both Carbon and Stainless Steel. ..................................................... 130 Figure 6-5a: Adjusted Nominal Resistance Factor φ in Eq. (6-9) for γ=1.2, Various Operating Temperatures and Values of β, for Service Limit A and Stainless Steel…………………………………………………………..131 Figure 6-5b: Adjusted Nominal Resistance Factor φ in Eq. (6-9) for γ=1.2, Various Operating Temperatures and Values of β, for Service Limit A and Carbon Steel……………………………………………………………………..131 Figure 6-6a: Adjusted Nominal Resistance Factor φ in Eq. (6-9) for γ=1.2, Various Operating Temperatures and Values of β, for Service Limit B and Stainless Steel. ........................................................................................ 132 Figure 6-6b: Adjusted Nominal Resistance Factor φ in Eq. (6-9) for γ=1.2, Various Operating Temperatures and Values of β, for Service Limit B and Carbon Steel......................................................................................................... 132 xiv Figure 6-7a: Adjusted Nominal Resistance Factor ij in Eq. (6-9) for Ȗ=1.2, Various Operating Temperatures and Values of ȕ, for Service Limit C and Stainless Steel………………………………………………………......133 Figure 6-7b: Adjusted Nominal Resistance Factor ij in Eq. (6-9) for Ȗ=1.2, Various Operating Temperatures and Values of ȕ, for Service Limit C and Carbon Steel……………………………………………………………………133 Figure 6-8a: Adjusted Nominal Resistance Factor ij in Eq. (6-9) for Ȗ=1.2, Various Operating Temperatures and Values of ȕ, for Service Limit D and Stainless Steel………………………………………………………….134 Figure 6-8b: Adjusted Nominal Resistance Factor ij in Eq. (6-9) for Ȗ=1.2, Various Operating Temperatures and Values of ȕ, for Service Limit D and Carbon Steel……………………………………………………………………134 Figure 6-9a: Adjusted Nominal Resistance Factor for the Internal Pressure having Normal and Lognormal Distribution for Service Limit A and Stainless Steel……………………………………………………………………..137 Figure 6-9b: Adjusted Nominal Resistance Factor for the Internal Pressure having Normal and Lognormal Distribution for Service Limit A and Carbon Steel……………………………………………………………………..137 Figure 6-10a: Adjusted Nominal Resistance Factor for g2 and Stainless Steel…….144 Figure 6-10b: Adjusted Nominal Resistance Factor for g2 and Carbon Steel………144 Figure 6-11: Variation of Mean Partial Safety Factors with Respect to Normalized Stress due to Design Internal Pressure for Boundary Values of ȕ=2 and 5.5............................................................................................................ 147 Figure 6-12: Variation of Mean Partial Safety Factors with Respect to Normalized Stress due to Maximum Internal Pressure for Boundary Values of ȕ=2 and 5.5............................................................................................................ 151 Figure 6-13: Mean Partial Safety Factors with Respect to the Normalized Stress due to Internal Pressure for Preselected Value of ȕ=3.5 and fM=0.5 for g6. .. 155 Figure 6-14: Mean Partial Safety Factors with Respect to the Normalized Mechanical Load for Preselected Value of ȕ=3.5 and fPB=5 for g6. ........................... 156 Figure 6-15: Mean Partial Factors with Respect to the Normalized Stress due to OBE for Selected Values of ȕ=1.5 and 3.0 for g7............................................ 161 Figure 6-16: Variation with Temperature of the Recommended Nominal Resistance Factor for Carbon Steel and COV for Earthquake Loading.................... 162 Figure 6-17: Variation with Temperature of the Recommended Nominal Resistance Factor for Stainless Steel and COV for Earthquake Loading ................. 163 Figure 6-18: Mean Partial Factors with Respect to the Normalized Stress Pressure Coincident with Earthquake for Selected Value of ȕ=1.5, fM=0.5, and fO=2 for g9........................................................................................................ 168 Figure 6-19: Mean Partial Safety Factors for ȕ=1.5 and 3 for g16 ............................ 186 Figure 6-20: Variation of Mean Partial Safety Factors with Respect to the Normalized Pressure for fL=1 and ȕ=3 and 5.5........................................................... 193 Figure 6-21: Variation of Mean Partial Safety Factors with Respect to the Normalized Stress due to LOCA for fPB=5 and ȕ=3 and 5.5 ...................................... 194 Figure 6-22: (a) A Piping Segment with Anchored Ends and (b) Cross-Section Used for Sample Computations ....................................................................... 202 xv Figure7-1: Nomenclature for Cycling Loading with Mean Stress (ASME Criteria, 1969) ....................................................................................................... 212 Figure7-2: Girth Butt-Welded Straight Pipe............................................................. 215 Figure7-3: Fatigue Curves for Girth Butt-Welded Joints in Straight Pipes (Markl, et al., 1952), where the Nominal Stress Refers to the Stress Amplitude .... 218 Figure7-4: Allowable Fatigue Stress, SA, for Class 2 and 3 Piping, and Fatigue Test Stress versus Cycles to Failure for Girth Butt-Welded Pipe................... 219 Figure7-5: Probability Density Function (PDF) for the Bending Stresses (Ayyub, et. al., 2002) ................................................................................................. 223 Figure7-6: Flowchart for the Direct Reliability Design for Cyclic Bending Moments ................................................................................................................. 229 Figure7-7: Flowchart Using Direct Reliability-Based Design for the Combination of Thermal Expansion Stresses with Primary Stresses ............................... 232 1CHAPTER 1: INTRODUCTION This chapter provides the objective of the dissertation, the background and needs of research in the field, as well as the organization of the chapters. Initially, a brief discussion about nuclear energy, nuclear plants, and piping is provided. 1.1. Nuclear Energy and Nuclear Plants It is common knowledge and concern that although the consumption of energy is bound to increase the coming years, the resources of fossil energy (oil, gas, and coal) are limited and being exhausted in increasing rates. Research is turning towards the renewable energy (solar, geothermal, and biomass) and the improvement of technology (reactors and materials) for nuclear plants. Nuclear energy satisfies today 8% of the world’s energy needs and is an efficient source for the production of electricity. Power plants in U. S. produce 20% of country’s electricity, while in France the 80%. U.S. has the largest number of nuclear power plants (104) followed by France (59), Russia (32), Korea (26), U.K. (22), and India (21). More specifically, in U.S., as of April, 2005 there are 104 industrial power reactors, which are licensed by the U.S. Nuclear Regulatory Commission (NRC) to operate, producing in total 97,400 net megawatts (electric). From them, 69 reactors are categorized as Pressurized Water Reactors (PWR), producing almost two thirds of the total electric 2power and the rest 35 are Boiler Water Reactors (BWR). The plants are located at 64 sites and 31 states. From 1973 to 1987 there was a steady increase in the construction of nuclear plants in U.S. and from 1987 till now the number of reactor plants fluctuates insignificantly, with few of them shutting-down and others renewing their license. The last permission for a new nuclear power plant was issued in 1999. Nevertheless, the capacity of the plants nowadays is increased due to license extensions and upgrading of existing reactors. Moreover, several new plants are going to be built. Disadvantages of the use of nuclear energy are the production and disposal of radioactive waste and the high cost for the construction of nuclear reactors. Moreover, accidents in nuclear plants like in Chernobyl, near Kiev, Ukraine (1986), which is the worst accident ever with 31 deaths and radioactivity spread in the former Soviet Union and Europe, and the Three Mile Island accident near Harrisburg, Pennsylvania (1979), the worst in U.S. history, make the public opinion and governments skeptical about nuclear energy. On the other hand, an advantage of nuclear energy is that it does not contribute to global warming, since no carbon dioxide, CO2, is produced. Nevertheless, nuclear energy is a good source for production of electricity and with advanced security measures not as dangerous as the public thinks it is. Moreover, the energy crisis and the dependence of industrial countries on oil production of countries in Middle East, Africa or South America has made countries (USA, France, etc.) consider the construction of new power plants that will supplement their energy needs and their independence from foreigner economies. 3 The nuclear power industry has been developing and improving reactor technology for almost five decades. Generation I reactors were developed in 1950-60s and today still run only in England (GCR). Generation II reactors are in operation in all countries. About 85% of the world’s nuclear electricity is generated by reactors derived from designs originally developed for naval use. These and other second-generation nuclear power reactors have been considered to be reliable and safe, but Generation III reactors, the Advanced Reactors, have started to replace them. In Japan, Generation III reactors are running, while Generation IV reactors are in development. Generation IV technology is very promising, since not only can provide economical electric power, but also can generate hydrogen for other energy needs. 1.2. Nuclear Plants The dissertation deals with piping operating in Boiler Water Reactors (BWR) or Pressurized Water Reactors (PWR) nuclear plants, which are the types of nuclear plants that exist in U.S. today. Both BWR and PWR are characterized as Light Water Reactors (LWR), since they use light (common) water both as coolant and as neutron moderator. Water, therefore, is the main link in the process that converts the fission energy to electrical energy in such plants. More specifically, in BWR the water is heated by the nuclear fuel and boils to steam inside the reactor vessel and then is directed to the turbine for the production of electricity. In PWR the water is heated by the nuclear fuel and pressurized in order not to boil. Then it is pumped from the reactor pressure vessel to a steam generator and used to boil a separate supply of water, which turns into the steam that spins the turbine for the production of electricity. Both PWR and BWR reactors are based on the Rankine cycle. 4That is, through thermodynamic processes water is made to undergo changes involving energy transitions and subsequently returns to its original state. BWR are manufactured by General Electric, and PWR by the Babcock and Wilcox Company and the Westinghouse Electric Corporation that has also immersed the former Combustion Engineering Inc. 1.3. Nuclear Piping Nuclear piping consists of the pipes, their interconnection including in-line components such as pipe fittings and flanges, pipe supporting elements such as pumps and valves, the supports of the pipes but not the support structures like building frames, foundations, etc. that their design is also not included in the ASME B&PV Design Code (2001). This study deals with straight pipes and not fittings like elbows, tees, reducers, etc. that necessarily exist in all pipelines. Straight pipes constitute the majority of piping and unlike other components they do not have only a local influence. Although it can be claimed (Rodabaugh, 1978) that usually pipe failures do not occur on straight parts of piping, but mostly on other parts like elbows due to concentration of stresses, etc. straight pipes are the first step for developing a new design method, which can be then extended to include nuclear piping as defined above. Pipes in a structural context can be considered as secondary components in the power plant installation, whereas considering their functional role in the plant as passive components like vessels, electrical cables, and structures in opposition to the active components such as pumps, fans, relays, and transistors, whose functioning depends on 5an external input, such as actuation, mechanical movement, or supply of power and therefore influence in an active manner the system’s processes. In ASME B&PV Code, Division I, Part III safety piping is categorized in three classes. The level of importance, associated with the function of pipes as well as for other components, decreases from Class 1 to Class 3. In a nuclear power plant, approximately 66% of the total quantity of process piping is safety-related. From the safety related piping only 10% belongs to Class 1 (Stephenson, 1999). A brief description and the relevant Code’s sections for the piping classes are provided below. Class 1: Class 1 pipes are within the reactor coolant pressure boundary and they should prevent the release of fission products in the environment (Section NB). In Class 1 belong also the reactor vessel, other piping, pumps, and valves. Class 2: Class 2 pipes are important to the safety and designed for situations such as emergency core cooling (ECCS), accident mitigation containment heat removal, post-accident fission product removal and containment isolation (Section NC). Other Class 2 components are pressure vessels, other piping, pumps, valves, and storage tanks. Class 3: Class 3 pipes are part of the cooling water and auxiliary feed water systems, (Section ND). These pipes can also be designed according to the ASME B31.1 Code. Other Class 3 components are pressure vessels, other piping, pumps, valves, and storage tanks. Table 1-1 summarizes piping systems of a PWR plant and shows that the majority of piping is classified as Class 2 or 3 components. 6Table 1-1: Piping Systems in a Pressurized Water Reactor Plant (Rodabaugh, et al., 1987) System Class 1 Class 2 Class 3 Reactor Coolant 3 3 Residual Heat Removal 3 3 Safety Injection 3 3 Chemical and Volume Control 3 3 3 Primary Component Cooling Water 3 3 Spent Fuel Pool Cooling Cleanup 3 Reactor Makeup Water 3 Containment Spray 3 3 Steam Generator Blow-down 3 3 Sample 3 3 Service Water 3 Nitrogen Gas Service 3 3 Radioactive Gaseous Waste 3 3 Demineralized Water 3 Floor and equipment Drains 3 Main Steam 3 3 Condensate 3 Feed-water 3 3 Diesel Generator Air 3 Diesel Generator Fuel and Lube Oil 3 Diesel Generator Cooling Water 3 Post-accident Containment Combustible Gas Control 3 Reactor Coolant System Pressurized Relief Piping 3 3 Equipment Vent 3 Containment On-line Purge 3 Waste Process Liquid 3 Fire Protection 3 71.4. Objective Our society demands the safety and reliability of structures. When it comes to nuclear plants this demand is even more pronounced, considering the consequences that a failure can have in human lives and the environment. Although reliability theory has offered the means of reasonably accounting for the design uncertainties of structural components, little effort has been made to estimate and control the probability of failure for mechanical components. Mechanical components, such as piping in nuclear plants, although they are sheltered within buildings, they are themselves exposed to a variety of loading and it is due to them that failures in the past have caused the loss of human lives. A failure of a mechanical component can trigger many adverse events and finally lead to a disaster. This dissertation proposes a methodology for the reliability-based design of all classes of nuclear straight pipes by making use of the design equations and theories used for years in the ASME B&PV Code, Section III, NB, NC, and ND-3600. Design for fatigue addresses, under some limitations, only Class 2 and 3 piping. The design variables are determined and their uncertainties are estimated based on reported experimental data, literature review, and engineering judgment. Load combinations are proposed and the partial safety load and resistance factors are estimated for selected values of failure probabilities or else reliability indices. It is recognized that the piping engineering community and the writers of the Code (ASME, NRC) should contribute to the development of the LRFD with their experience. Moreover, as it is often mentioned, experimental data and advanced finite element analysis are necessary tools for the development of the LRFD, too. In this work, 8available data was collected and reasonable values were assumed in cases of unavailable or limited data. The LRFD method, among others, can provide a clear, simplified, and reasonable methodology for piping design. This dissertation is a primary step towards the probabilistic design of piping according to the LRFD method. 1.5. Background The need for quantifying the design uncertainties in a systematic and uniform way by using probabilistic methods such as the Load and Resistance Factor Design (LRFD), leaded many industries, mostly in the area of civil engineering, to use reliability-based codes. More specifically, in the United States LRFD is used for steel structures (AISC, 1986), concrete structures (ACI, 1977), timber structures (ASCE, 1992), offshore oil platforms (API, 1989), bridges (AASHTO, 1994), and Category I structures in nuclear power plants (AISC, 2003; ACI, 2001). Moreover, the pipeline industry is adopting the LRFD, too (API RP1111, 1999). LRFD Codes for civil structures are also used in Canada and the European Union (CISC, 1974; CEC, 1984). An effort to develop load and resistance factor equations for piping was made by Schwartz, et al. (1981) and Ravindra, et al. (1981), who presented a baseline and load combinations for the essential service water nuclear piping systems. More recently, Gupta, et al. (2003) performed an exploratory study for the use of LRFD for Class 1 piping, while Ayyub, et al. (2005) demonstrated the advantages and methodology of the LRFD for piping. Payne, et al. (1989) showed that an alternative probabilistic-based design for tubular members will facilitate decisions in design, which will properly balance economics, safety and uncertainties. 9 Table 1-2 summarizes load combinations for structural and mechanical components of nuclear plants based on literature review. More specifically, the ASCE, SMiRT-4 reference presents combinations of impulsive loads acting on a BWR-Mark I containment. Ravindra, et al. (1981) and Schwartz, et al. (1981) provide load combinations for the essential service water line (ESW) piping components, which are Class 2 components. Hwang, et al. (1987) developed practical probability-based load and resistance criteria for reinforced concrete containment and shear wall structures in nuclear plants. It can be noticed that in all these combinations, the loads involved are similar to the ones that are applicable to piping design such as the Operating Basis and Safe Shutdown Earthquake, the Safety Relief Discharge Load, the thermal loads, etc. Nevertheless, only the work of Ravindra, et al. (1981) and Schwartz, et al. (1981) refer exclusively to piping design. 10 Table 1-2: Load Combinations for Nuclear Plant Facilities and Components Load Combinations* Loads Reference SRVoPFLD 5.10.10.17.14.1  SRVoRoToPFLD 3.10.10.10.10.13.10.1  2 )( 2 )(25.10.10.10.10.10.10.1 SRVOBEoRoToPOFLD  LOCAIBASBASRVARATBPFLD 2 )/( 2 )(25.125.10.10.10.1  SRV A R A T A PFLD 0.10.10.125.10.10.10.1  LOCAIBASBASRVOBE ARATBPFLD 2 )/( 2 )( 2 )(1.1 0.10.11.10.10.10.1   } SRV A R A T A P o EFLD 0.10.10.11.11.10.10.10.1  2 )( 2 )(0.10.10.10.10.10.10.1 SRVSSEoRoToPFLD  2 )( 2 )(0.10.10.10.10.10.10.10.1 SRVSSE R R A R A T B PFLD  VSRRRARATAPSSEFLD 0.10.10.10.10.10.10.10.10.1  D=Dead L=Live F=Prestressing To=Operating Temperature Ro=Operating Reactions Po=Operating Pressure SRV=Safety/Relief Valve Eo=OBE Ess=SSE PB=SBA and IBA Pressure TA=Pipe Break Temperature Load RA=Pipe Break Temperature PA=LBA Pressure (including all pool hydrodynamic loadings) RR=Reactions and Jet Forces (Pipe Break) ASCE, SMiRT-4 (1977) Design: P+W Service Limit A/B: TRNG Service Limit B: P+W+OBE P+W+HYDTR Service Limit C: P+W+SSE P+W+OBE+HYDRT WWcW ȖdPPcP ȖRij t111 )(112 TRNGTH c TH ȖWWcW ȖdPPcP ȖRij t )(112 N HTR H c H ȖWWcW ȖdPPcP ȖRij t )( 1 112 OBEEo c oE ȖWWcW ȖdPPcP ȖRij t )( 1 )( 11 113 N HTR H c H ȖOBE Eo c E ȖWWcW ȖdPPcP ȖRij t )( 12 114 SSE SE c E ȖWWcW ȖdPPcP ȖRij t )( 1222 OBE oAE cEȖRij t )(2222 SSE SAE cEȖRij t )( 2 )( 23 22 N HTR AH c H ȖOBE oAE cEȖRij t )( 3 22 N HTR AH c H ȖRij t P=Design Pressure W=Weight OBE=Operating Basis Earthquake HYDTR=Hydrauli c Transient TRNG=Thermal Range SSE=Safe Shutdown Earthquake HTRN=Hydraulic Transient ijij=Resistance Factors Ȗij=Load Factors ci=Influence Coefficients that transform loads into moments Ravindra, et al. (1981) and Schwartz, et al. (1981) oRRȖoTLLȖDDȖ 1 )( 11 SRVSRVȖoRRȖoTLLȖDDȖ  oRRȖoTLLȖDDȖ  1 oRRȖoTt W tW Ȗor ss E ES ȖLLȖDDȖ 1 )( 1  a P p Ȗ Į R Į T ss E E ȖLLȖDDȖ  1 1 D=Dead L=Live To=Operating Temperature Ess=SSE PĮ=Loca Pressure Hwang, et al. (1987) * The notations are uniquely defined in “Loads” column of the table per respective cited references 11 1.6. Organization The dissertation consists of 8 chapters and 3 appendices. Chapter 1 presented the objective and the background of the work done in this area of study. Chapter 2 illustrates both the Allowable Stress Design (ASD) used in the Code † as well as the proposed Load and Resistance Factor Design (LRFD). It also describes how the partial safety factors applied to mean values of variables are calculated according to the LRFD. Chapter 3 summarizes the available probabilistic information for the resistance and load variables for piping. Chapter 4 presents strength models and calculates the bias of the burst or yielding resistance of pipes due to internal pressure and the bias for the ultimate bending capacity of pipes. Chapter 5 gives the developed load combinations and performance functions. Chapter 6 provides the specific probabilistic characteristics of the variables used in the calculation of the partial load and resistance factors, whereas also a summary of recommended adjusted resistance factors for a predefined set of partial load factors for each performance function and different reliability indices are presented. Chapter 7 introduces a probabilistic framework for the fatigue design of Class 2 and 3 nuclear piping. Finally, Chapter 8 presents the basic conclusions of this work and gives recommendations for future research. Appendix A provides a summary of the Code equations and Appendix B a summary of types and grades of steel used for the design of nuclear pipes operating today in nuclear plants. Appendix C shows analytically the calculated partial load and resistance factors applied to mean values of variables as well as the calculated adjusted nominal resistance factors for predefined sets of nominal load factors. Figure 1-1 presents the structure of the dissertation as described above. †: The word “Code” refers to the ASME B&PV Code (2001) 12 CHAPTER 1 INTRODUCTION: Nuclear Plants Nuclear Piping Background Objective CHAPTER 2 CURRENT PIPING DESIGN PRACTICES AND THE LRFD: Loads and Stresses Code Equations Reliability-Based Design Advantages of Load and Resistance Factor Design (LRFD) APPENDIX A SUMMARY OF ASME B&PV CODE EQUATIONS SECTION III, NB, NC, ND - 3600 CHAPTER 3 BASIC RANDOM VARIABLES FOR PIPING: Strength Loads CHAPTER 4 STRENGTH MODELS UNCERTAINTY: Bias Piping Burst and Yielding due to Internal Pressure Ultimate Moment Capacity APPENDIX B STEEL USED IN THE ASME B&PV CODE CHAPTER 5 LOAD COMBINATIONS AND PERFORMANCE FUNCTIONS: Performance Functions Normalization of Stresses Target Reliability Index CHAPTER 6 CALCULATION OF PARTIAL SAFETY FACTORS: Description of Calculations Part I: Design for Internal Pressure Part II: Design for Combined Loading Sensitivity Analysis Computational Examples Conclusions CHAPTER 7 FATIGUE DESIGN OF PIPES: ASME B&PV Code Practice, Section III Reliability Based Fatigue Design Combination of Primary and Secondary Stresses Computational Examples CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH APPENDIX C PARTIAL MEAN LOAD AND RESISTANCE FACTORS AND ADJUSTED NOMINAL RESISTANCE FACTOR Figure 1-1: Structure of the Dissertation 13 CHAPTER 2: CURRENT PIPING DESIGN PRACTICES AND THE LRFD This chapter describes initially the concepts used in the current Code and the methodology, namely the Allowable Stress Design (ASD) for piping. Then, the principles and methodology for a reliability-based design are presented. A comparison of the two methods, ASD and LRFD, moreover is provided. The steps for the calculation of mean safety factors according to the Load and Resistance Factor Design method are demonstrated and an overall illustration of the advantages of the LRFD for piping is given. 2.1. Current Piping Design Practices First, general information about the ASME B&PV Code, Section III, is provided and then a discussion about the service levels, the loads and stresses definitions is given. The equations and criteria used today for the design for moment loading and internal pressure are discussed, and the derived conclusions are used for the development of the performance functions for the LRFD. 2.1.1. The ASME B&PV Code, Section III The Allowable Stress Design also called Working Stress Design (WSD) is a reliability I method used for years in the ASME Codes. For this study the ASME B&PV 14 Code, Section III, Parts NC-3600 (Class 2 piping), and ND-3600 (Class 3 piping) with the included subsections referring to straight pipes are of interest. From Part NB-3600 (Class 1 piping) only equations similar to that of Classes 2 and 3 piping are addressed. The design of pipes in these sections is characterized as design by rule in opposition to the design by analysis (NB-3200) or by experiment. The ASME B&PV Code was first issued in 1963 covering only the design of vessels. It is just in the fourth edition of 1971 that the Code was expanded to cover the design of pumps, valves, and piping. New editions of the Code are provided every 3 years. Nevertheless, the last edition approved by the Nuclear Regulatory Commission (NRC) is that of 1992. In this study the equations provided in the edition of 2001 are summarized in Appendix A. The ASME B&PV Code is used or accepted in more than 80 countries around the world. A historical overview of the Code is provided by Ling (2000) and Canonico (2000). In what follows, an illustration of some of the basic concepts like the service levels used in the design of piping according to the ASME B&PV Code but also in the proposed new design is provided. Furthermore, a description of the loading and type of stresses is also presented. 2.1.2. Service Levels It is in the fourth edition of the ASME B&PV Code (1971) that the concept of normal, upset, emergency and faulted conditions of Class 1 components was introduced; and it is in the sixth edition of 1977 that the concept of Design and Service Limits was presented, covering moreover Class 2 and 3 components. The normal condition is referred thereafter in the Code as Service Limit A, the upset condition as Service Limit B, the emergency condition as Service Limit C, and the faulted condition as Service Limit 15 D. Other service levels except from the Service Limits A to D are the design and testing service levels. Nevertheless, in this study the testing service level is not addressed. The design and service limits define: a) the frequency of the occurrence of the loading, and b) the expected type of structural behavior. The most infrequent, namely accidental or faulted, loading and more severe damage are expected for Service Limit D. A definition for the different service levels is as follows: Design: The piping should remain functional under the design internal pressure and sustained weight. Design equations are applied to all classes of piping and moreover are used in order to define their geometry (thickness, external diameter, etc.). Then, if required, the piping is tested for the different service limits. Service Limit A: The piping must withstand the loading under normal operation of the plant. Moreover, the piping remains in elastic region under bending stress resulting from deadweight and peak pressure. Service Limit B: The loading in this service limit occurs often enough and also the effects of an Operating Basis Earthquake and larger pressure are taken into consideration. The piping must withstand these loadings without damage requiring repair. As it concerns bending loading, a hinge formation is not allowed. Service Limit C: For this service limit, loads as defined in Level B can also be taken into consideration. Nevertheless, this limit permits large deformations in areas of structural discontinuity, which may necessitate the removal of the pipe. Service Limit D: This service limit permits gross general deformation with some consequent loss of dimensional stability and damage requiring removal of the component 16 or support of service. The loads are expected once in the lifetime of the plant and are a result of faulted or accidental conditions. 2.1.3. Stresses and Loads The present Code equations for piping design are expressed in terms of stresses. There are three categories of stresses, as presented in the design by analysis part of the Code NB-3200, namely the primary, the secondary and peak stresses. Figure 2-1 summarizes the different stresses and the loads causing them, while Figure 2-2 presents the relevant failure modes. Of course, only some of these stresses will be encountered in this study in an effort to express the design in a probabilistic framework and achieve a consistent reliability (safety). Primary stresses are result of primary loads, which cause primary principal stresses, shear stresses or bending stresses and must satisfy the laws of equilibrium of external and internal forces and moments. A primary stress is not self-limiting. Therefore, as long as the load is applied, the stress is present and does not reduce with time or as deformation takes place. Primary stresses lead to gross deformations and finally to rupture. Secondary stresses are principal stresses, shear stresses or bending stresses that are developed due to constraints or displacements of the pipe. These displacements can be caused either by thermal expansion or by outwardly imposed restraint and anchor movements. Secondary stresses are self-limiting. Therefore, local yielding and minor distortions of the piping can relieve these stresses. Failure from one application of the stress is not expected. 17 Peak stresses are developed for example where there are stress concentrations at a structural discontinuity or thermal gradients through a pipe’s wall. The basic characteristic of peak stress is that it does not cause any noticeable distortion and is objectionable only as a possible source of a fatigue crack or brittle fracture. 18 Figure 2-1: Piping Stresses and Loads 19 Piping Failure Ductile Brittle Burst Fatigue Low Cycle Fatigue High Cycle Fatigue Primary Stresses (Pressure) Buckling Primary Stresses Do/t >75 Mashine Vibration Flow Induced Vibration Peak Stresses Secondary Stresses Primary (Earthquake, Pressure) Ratcheting Secondary Cycling Stresses Creep Secondary Stresses Time Dependent Behavior of Steel T > 900 o F Plastic Collapse Primary Stresses Figure 2-2: Failure Modes for Piping and their Cause Except from the above division that is provided with respect to the stresses that the loads produce, the primary loads can also be categorized according to their variation in time, as follows: Constant Loads: Such loads are the weight of pipes and other sustained equipment that do not vary significantly with time. 20 Pulse Loads: The accidental pressure can be considered as a pulse load. The normal operating pressure can not be considered constant with time, since it might not be always present. Moreover, due to earthquakes or mechanical loads its characteristics change and obtain peak values. Intermittent Loads: Such loads are dynamic like earthquake and mechanical loads (wave pulses), which vary in time and in magnitude as they take place. Figure 2-3 presents the loads as defined above. Repeating loads e.g. vibrations due to rotating equipment such as compressors, pumps, turbine drivers, etc. will not be addressed. Another division should be made among dynamic loads, since this division is considered in the Code, too. There are two types of dynamic loadings, namely reversing and not reversing. Reversing Dynamic Loads, as shown in Figure 2-1, are loads like earthquake, or reversing pressure pulses after an initial thrust force, which cycle about a mean value. Not Revering Dynamic Loads are those loads that do not cycle about a mean value. Such loads are the initial thrust force due to sudden opening or closure of valves, waterhammer, etc. Both types of loads we call in this study mechanical although in the ASME Code the term is used to express primary loads other than pressure. Figure 2-4 shows different categories of dynamic loads as illustrated in the ASME B&PV Code (2001). 21 Figure 2-3: (a) Constant Loads, (b) Pulse Loads and (c) Intermittent Loads (Hwang, et al., 1987) 22 Figure 2-4: Examples of Reversing and Nonreversing Dynamic Loads (ASME B&PV Code, 2001) 2.1.4. Equations in ASME B&PV Code The design equations in the Code are based on the Allowable (Working) Stress Design. Therefore, the resistance and the loads have nominal values RN and SN, respectively, which are a percentage of their mean values. In the case of piping under 23 consideration, the nominal resistance of steel can be considered to be about two standard deviations below the mean value, whereas the nominal values of loads can also be conservative values above their mean. As it will be shown in Chapter 3, for each design temperature the Code provides different nominal values for the yield, Sy, (Section II, Part D, Table Y-1) or ultimate, Su, (Section II, Part D, Table U) strength of steel. 2.1.4.1. Bending Primary Stresses The bending stresses in NC and ND-3600 of the Code are calculated for a linear elastic behavior of the material and a factor, ȡ, accounts for the nonlinearities as Eq. (2-1) shows. The safety factors in Eq. (2-2) are based on the judgment and past experience of the standard writers (ASME, NRC) in the analysis and structural behavior of piping, as well as on experimental results. ¦t n i iN f N A SS R ȡRȡ 1 (2-1) N N f S RS (2-2) where SNi are the nominal values of n stresses for elastic linear material given for bending as SNi=M/Z, where M is the developed moment and Z the elastic section modulus, or as PDo/4t for the effect of internal pressure. RA is an allowable stress equal to Sy, Sh or Sm depending on the piping class. Table 2-1 presents how the different allowable stresses are defined in the Code, including also the allowable stress for fatigue, SA. Table 2-2 presents the values of ȡ and the resultant resistance stress for operation at room temperature. 24 Table 2-1: Definition of Stresses Symbols Used in the Code Symbol Definition Value Use Class Sy Nominal Yield Strength at Loading Temperature Sy Section II, Part D Table Y-1 All Su Nominal Ultimate Strength at Loading Temperature Su Section II, Part D Table U All Sh Allowable Stress at Loading Temperature min{0.25Su, 0.667Sy) or min{0.25Su, 0.9Sy) if stainless at elevated temperatures Section II, Part D Table 1B 2, 3 Sm Allowable Stress at Loading Temperature min{0.333Su, 0.667Sy) or min{0.25Su, 0.9Sy) if stainless at elevated temperatures Section II, Part D Table 2A 1 Sc Allowable Stress at R.T. (cold) Equal to Sh for its value at room temperature Section II, Part D Table 1B 2, 3 SA Allowable Stress Range for Expansion Stresses SA=f (1.25Sc+0.25Sh) For f Table NC- 3611.2(e)-1 2, 3 Table 2-2: Values of ȡ and Resultant Resistance Stress for Operation at Room Temperature Class Service Level Equation in Appendix A ȡ Additional Limit Resultant Resistance Stress at Room Temperature Design A-4 1.5 Not Available min{0.5Su, Sy} B A-15 1.8 ”1.5Sy min{0.60Su, 1.2Sy) C A-22 2.25 ”1.8Sy min{0.74Su, 1.5Sy) 1 D A-32 3 ”2Sy min{Su, 2Sy) Design A-5 1.5 Not Available min{0.37Su, Sy} B A-16 1.8 ”1.5Sy min{0.46Su, 1.2Sy} C A-27 2.25 ”1.8Sy min{0.56Su, 1.5Sy) 2, 3 D A-38 3 ”2Sy min{0.75Su, 2Sy) As discussed by many researchers (Ayyub, et al., 2005; Mello, et al., 1974; Rodabaugh, et al., 1978) the resultant resistance stress reflects the accepted structural behavior of the piping. For example the 1.2Sy limit for Service Level B, can be considered as a conservative value for the shape factor, s, as explained below, and the 1.5 for Service Level C as a value representing the allowance for a formation of a plastic mechanism. Table 2-3 presents the limit theory loading for linear perfectly plastic behavior of the material. More specifically, it shows the load needed to cause a first yield at the exterior fiber of the pipe’s most critical cross-section, the load needed for the formation 25 of a single hinge and the one that causes a collapse mechanism for different loading and boundary conditions of pipes. In the table s is the shape factor and is equal to the ratio of the plastic, Eq. (2-3), to the elastic section modulus, Eq. (2-4), of pipe’s cross-section. For Do/t=10, s is 1.40, whereas as Do/tĺf, s equals 1.27. Thus, the margin between the yield moment, My and the limit moment, MP, which is the moment at first hinge, depends on the ratio of the external diameter to thickness, Do/t. Moreover, the margin between the yield moment and the plastic moment, which is the moment that causes a collapse mechanism, depends not only on the Do/t ratio but also on the indeterminacy of the pipe. In case 1 of Table 2-3, for example, the simply supported pipe, as other statically determined pipes, reaches the plastic moment as soon as the first hinge is formed. For Service Limit D the resultant resistance stress is much higher, since large deformations are allowed in order for strength hardening of the material to develop, too. o o D dDʌZ 32 44  (2-3) ttDdDZ ooP 2 33 )( 6 | (2-4a) or » » ¼ º « « ¬ ª ¸¸¹ · ¨¨© §  33 2 11 6 o o P D tDZ (2-4b) 26 Table 2-3: Limit Theory Loading Case Pipe First Yield Load, w First Hinge Load, wp Plastic Load, wc 1 L YM4 sw L PM 4 sw 2 L YM 3 16 sw L PM 3 16 sw125.1 3 2 8 L YM sw L PM 2 8 sw47.1 4 2 12 L YM sw L PM 2 12 sw33.1 5 2 34.9 L YM sw L PM 2 34.9 sw16.1 L=length as in figures, My=first yield moment, Mp=first hinge moment, w=applied load as in figures, s=shape factor The criterion used in the Code in order to express the effect of internal pressure and other accompanying primary loads is the Tresca. More specifically, given an element of straight, circular pipe subjected to three principal stresses, ıL>ıH>ıR, the Tresca criterion states that yielding is dependent on the maximum shear stress at yielding in a uniaxial tensile test and equals half the difference between the maximum and minimum principal stresses. Hence, 22 )( max yRL SııIJ  (2-5) where ıL=longitudinal stress, ıH=hoop stress, ıR=radial stress of pipe, and Sy=the yield strength of steel. Considering also conditions for thin wall cylinders and the radial stress, ıR, to be zero, Eq. (2-5) yields 27 yL Sı (2-6) If the straight pipe is subjected to internal pressure and bending and by neglecting the shear forces, or other axial forces for being relative small compared to the induced moment stresses, Eq. (2-6) gives: ySZ M t PD IJ  4 2 omax (2-7) where, PDo/4t is the longitudinal stress due to internal pressure P and M/Z is the term that represents the maximum longitudinal stress due to bending at the outer fiber of the most critical cross-section of the pipe. Moreover, M is considered as the resultant moment of three orthogonal components at any cross-section, and therefore the stress M/Z is not truly longitudinal but rather a conservative estimate of the longitudinal stress. The negligence of other forces (axial, shear) in the failure criterion is discussed by Antaki (1993), who actually states that the criterion is based on the assumption of large moments such as other forces are neglected. Nonetheless, it can be claimed that the negligence of shear stresses is not very important especially for slender beams-pipes, since shear stresses are normally much smaller than the moment stresses in piping systems. Moreover, the shear stresses are negligible at locations, where moment stresses are maximum. That is not the case though, when big concentrated loads arising from the attachment of heavy equipment on pipes are present, although in such cases the usual practice is to provide a support that sustains the increased load. For the combination of stresses also the Von-Mises criterion could have been used as other researchers have done (Rodabaugh, 1979). Under the same assumptions used for the Tresca criterion, the Von Mises criterion, which considers that yielding commences 28 when the energy, US, stored due to the change of pipe’s shape becomes equal to the yield strength, Sy, yields: 222 yHLHL Sıııı  (2-8) It is obvious that the Tresca criterion, Eq. (2-7), is much simpler than the Von Mises criterion, Eq. (2-8), which moreover involves the hoop stress ıǾ that is dependent on the boundary conditions of pipe. In the case of a thin wall circular, cylindrical tube subjected to axial force and internal pressure the maximum divergence of the two criteria is 15%, with the Tresca criterion being more conservative. This divergence is usually considered as not appreciable (Armenákas, 2006). The ASME B&PV Code, Section III uses the Tresca criterion with the assumption of large moments and so does the proposed LRFD method. 2.1.4.2. Stress Indices The equations in the ASME BVP Code have generally the following form: Sȡ Z MB t PD B d 2 o 1 2 (2-9) where, M is the resultant moment and P is the applicable pressure for the different service levels and accompanied loadings and ȡ a multiplier of the allowable stress. Moreover, S is the allowable stress of steel at the design temperature, t and Z are respectively the thickness and the elastic section modulus of pipe. It is noticeable that indices like B1 and B2 are also used. These indices for straight pipes are B1=0.5, and B2=1 that yield Eq. (2-7). Therefore, the indices have a meaning for pipe components other than straight pipes. More specifically, they are multipliers 29 used to obtain the stress of piping components such as elbows and reducers with respect to the stress of straight pipes. While Bi indices are multipliers for primary stresses there are also the Ci and i indices that are applicable to secondary and peak stresses, respectively, and account for the increased flexibility and intensification of stresses of components like elbows, tees, reducers, etc. Nevertheless, all these indices except B1 = 0.5, are equal to 1 for straight pipes and in this work their statistical properties are not examined. Discussion for these indices is provided among others by Kumar, et al. (2002), Yu, et al. (1999), Matzen, et al. (2002), Venkataramana, et al. (2004), Markl, et al. (1955). Stress indices are given in Table NB-3681(a)-1 of the Code. 2.1.4.3. Internal Pressure In the Code, in Sections NB-3600, NC-3600, and ND-3600 the design for internal pressure defines a) the minimum thickness of piping and b) the allowable pressure for each service level. a) Minimum thickness of piping The minimum thickness of piping is evaluated according to NB-3641, NC-3641, ND-3641, and here is given by Eq. (2-10). A yPS DP t Des oDes m  )(2 (2-10) where PDes is the design internal pressure, Do is the pipe’s outside diameter, S is the maximum allowable stress for pipes of Class 2 (Sm for pipes of Class1 and S E for pipes of Class 3) at the design temperature, y a coefficient having the value of 0.4 or y = d/(d+Do), when the ratio Do/tm is less than 6, d is the pipe’s inside diameter, E is a joint efficiency factor reducer of the allowable stress, depending on the type of longitudinal 30 joint used, and A an additional thickness that accounts for the reduction of the thickness during erection or during the functional life of piping (e.g., corrosion, erosion, etc.). Values of A as given in the Code are presented in Table 2-4. Table 2-4: Values of Additional Thickness A (ASME B&PV CODE, 2001) Type of Pipe A (in) Threaded steel and nonferrous pipe: ¾ in nominal (DN 20) and smaller 0.065 1 in nominal (DN25) and larger Depth of groove Grooved steel and nonferrous pipe Dept of groove plus 1/64 in Equation (2-10) is the Boardman model and constitutes an approximation of the hoop stress calculated by the Lamé model, considering moreover the maximum principal stress criterion of failure. In other ASME Codes (e.g., B31.1), the y in the Boardman model considers also the creep of steel in temperatures over 900 o F, but not in the ASME B&PV Code, since these pipes usually operate in temperatures lower than about 800 o F. b) Allowable pressure Figure 2-5 defines the terms of pressure used in the ASME B&PV Code for each service level, where different pressure loading can occur due for example to an acting earthquake, water hammer, a shock wave from a pipe that breaks, overpressure due to malfunctions of the system, etc. Moreover, although valves are responsible for keeping a constant pressure in piping, within each service level there is actually a variation of the pressure. Design pressure is the pressure that engineers estimate initially as the maximum pressure for the normal functioning of a pipe in order to calculate its schedule and minimum thickness. Should a higher value of thickness is selected than the minimum obtained using the design pressure, the resultant allowable pressure becomes greater than 31 the design pressure by a quantity, b, as Figure 2-5 shows, otherwise the two are equal. For Service Limits B, C, and D the ASME Code permits an augmentation of the allowable (maximum) pressure, PĮ, which reflects in return an augmentation of the allowable stress, S. Table 2-5 shows the permissible pressure for each service limit and the resultant permissible allowable stress for Class 1 pipes based on the definition of allowable stresses provided in Table 2-1. It can be seen for example that for Service Limit D the pipe’s stress can exceed the yield strength of steel for Class 1 pipes but also for Class 2 and 3 piping, since in that latter case the resultant allowable stress is equal to min{0.50fu, 1.33fy}. An extensive discussion for the calculation of hoop stress and the validity of different models is provided in Chapter 4. Figure 2-5: Magnitude of Pressure in an Augmenting Scale for Different Service Levels for a Pipe’s Operation Table 2-5: Permissible Pressure for Class 1 Piping According to ASME B&PV Code, Section III Service Level Symbol Permissible Pressure Sm* Resultant Allowable Stress Design PDes PĮ min{0.33Su, 0.67Sy} A PA PĮ min{0.33Su, 0.67Sy} B PB 1.1PĮ min{0.37Su, 0.73Sy} C PC 1.5PĮ min{0.5Su, Sy} D PD 2PĮ min{0.333Su, 0.667Sy} min{0.67Su, 1.33Sy} *Allowable stress according to ASME B&PV Code, Section II for Class 1 cold pipes 32 2.1.4.4. Other Codes for the Design of Piping Antaki (1999) presents a comparative study of Canadian and European Codes used for the design of piping. As it is the case of structural codes, the European countries (England, Germany, France, Finland and Norway are the countries that have developed their own codes for the design of nuclear piping) have unified their Codes under EN 13480-3. There is also the Japanese and Russian Codes but as far as the author knows, all these codes are using the ASD method, while some of them are more detailed about the load combinations than the ASME B&PV Code. 2.2. Reliability-Based Design Codes using reliability analysis can be divided in four levels (Madsen, et al., 1986). Level I codes are like the Allowable Stress Design used in the ASME B&PV Code that uses safety factors. In that context the proposed method, the Load and Resistance Factor Design can also be considered as Level I code, since it uses safety factors, although for their derivation level II reliability methods are utilized. Often the LRFD is referred as semi-probabilistic design method. For Level II codes a target reliability index defines the design criteria as for example does the direct reliability method. Level III and IV Codes are used in advanced research. More specifically, Level III codes use full reliability analysis and try to achieve an optimum reliability level or probability of failure, while Level IV codes use optimization in order to reduce the cost and increase the benefits of the design. 33 2.2.1. The Load and Resistance Factors Design Method The Load and Resistance Factor Design (LRFD) method is based on the requirement that a reduced by a factor ij nominal resistance is larger than the linear combination of magnified by factors Ȗi nominal loads, as Eq. (2-11) shows: ¦t n i inin LȖRij 1= (2-11) The factors are determined probabilistically and therefore correspond to a predefined level of safety and a predefined service life. Different load and strength factors are used for each type of load and the strength. This is a major difference from the ASD method, in which only one factor tries to account for all the uncertainties in the design. Usually, the higher the uncertainty associated with a load, the higher the corresponding load factor is; and the higher the uncertainty associated with strength, the lower the corresponding strength factor is. 2.2.1.1. Procedure for the Development of Load and Resistance Factor Design For the development of Load and Resistance Factor Design the following steps are followed that moreover are illustrated in Figure 2-6: x Definition of data space x Identification of limit states and derivation of performance functions x Characterization of model uncertainty and probabilistic characteristics of basic random variables x Selection of target reliability index, ȕ 34 x Computation of partial safety factors Information for the steps mentioned above is provided in the following sections or in separate chapters. More specifically, the definition of the data space is presented in Section 2.2.1.2, a general discussion about the performance functions is provided in Section 2.2.1.3, while specifically for the performance functions for piping as well as for the target reliability index, ȕ, details are given in Chapter 5. The probabilistic characteristics of the basic random variables are given in Chapter 3 and the characterization of the model uncertainty is illustrated in Chapter 4. In Section 2.2.1.4 the calculation of partial safety factors applied to mean values of variables is presented, whereas the procedure for obtaining partial safety factors applied to nominal values of variables is given in Chapter 6. Figure 2-6: Steps for the Development of Load and Resistance Factor Design 35 2.2.1.2. Data Space For the development of reliability-based codes, the data space should be first determined such as suitable limitations are applied. More specifically, the type of components, the material, the geometry, the structural modeling, and the considered loads need to be specified. For the reliability-based design of piping the following parameters are considered in this study: The pipes belong to all safety piping classes as defined in Section 1.3. Class 2 and 3 piping constitutes the majority of piping installed in nuclear plants. Class 3 piping is also present in other power plants like those that burn coal and their design can be made according to the ASME B&PV Code or ANSI B31. Chapter 7 that includes the design of piping for thermal expansion addresses only Class 2 and 3 piping. No equations are provided for the buckling of pipes or for the effects of external pressure (e.g., buried pipes). The working fluids in pipes are mainly water and steam. Pipes operate in different temperatures up to 800 o F for stainless steel, and 700 o F for carbon steel. In Chapter 6, calculations for carbon steel address 800 o F temperature, too. This is because statistical information was available for carbon steel and 800 o F. Similar limits in temperature exist also in the current design, which permits creep of steel not to be considered. The material of pipes is either stainless, austenitic steel or carbon steel. These types of steel show corrosion resistance and are intended to perform well under high temperatures. They are representative materials for nuclear piping, although the author recognizes the existence of other types of steel (e.g., duplex steel, low alloy stainless 36 steel, etc.), which are not addressed in this study. With the introduction of new technology reactors, new, improved types of steel may also get introduced. Although for Class 3 the usual material is carbon steel all calculations will address both materials. Different design equations are developed for carbon and stainless steel, since, as shown in Chapter 3, these steels have different statistical properties. The steel behavior is considered to be linear elastic–perfectly plastic and limit theory is considered. The pipes can be considered and analyzed as space frame structures. For Service Level D also strain hardening is judged to be considered, since for this service level accidental loads of large magnitude and large deformations are considered. Therefore, a larger allowable stress is permitted and the Tresca yield surface is expanded approximately 2.15 times. This value is the mean value of the ratio of the ultimate to yield strength with a coefficient of variation approximately equal to 0.19. Detailed discussion about Service Level C is provided in Section 5.1. 2.2.1.3. Performance Functions The performance functions or limit state equations are equations that express mathematically the relation between the load(s) and the resistance. Once the performance function becomes equal to zero, a limit state for the design is reached. The term limit state is used in the reliability-based design and shows that a component becomes unfit for its intended use under the specific loading conditions. Therefore, in a successful design the component should not reach the limit state. Limit states are divided in strength, serviceability or fatigue limit states. Serviceability limit states ensure the functionality of pipes (e.g., no yielding or acceptable deformations) while the strength ones characterize the ultimate capacity of components. 37 As mentioned previously, the performance functions include terms of strength, R, and loading, L. In order to avoid an undesirable limit state, the strength of the component should be greater than its loading. Equation (2-12) shows a general performance function as the difference of the strength and load, which can also be considered as demand minus offer. LRg  1 (2-12) where g1=performance function, R=strength (resistance) and L=load to which the structure is subjected. The failure in this case is defined in the region, where g1 is less than zero, or R is less than L, that is: LRg  or01 (2-13) As an alternative approach to that of Eq. (2-12), the performance function can also be written as: L Rg 2 (2-14) where now the failure is defined in the region where g2 is less than one, or R is less than L, that is: LRg  or12 (2-15) Equation (2-12) or (2-14) relate the strength and load and moreover, by considering them as random variables, the probability of failure of a component can be defined as: )(Prob)0.0(Prob 1 LRgPf   (2-16) 38 or )(Prob)0.1(Prob 2 LRgPf   (2-17) The probability of failure as defined by Eq. (2-16) is shown schematically in Figure 2-6. In the general case that the performance function is expressed according to Eq. (2-18) as: )...,,2,1()( nXXXgg X (2-18) in which X is a vector of basic random variables (X1, X2, ..., Xn), the probability of failure attains the form of the joint probability distribution function of Xi, as Eq. (2-19) shows. ³ ³ d0over 2121 ...)....,,,(... g nnXf dxdxdxxxxfP (2-19) The calculation of this integral is a difficult task and therefore usually a Monte Carlo simulation, preferably with variance–reduction techniques, is used. Alternatively, the probability of failure can be approximated by Eq. (2-20) (Ayyub, et al., 2003). )(1)( ȕȕPf ) ) (2-20) where ĭ(.)=cumulative probability distribution function of the standard normal distribution, and ȕ=reliability index. The probability of failure obtained by Eq. (2-20) is accurate when the performance function is linear and the implicated random variables normally distributed. Anyhow, in most practical cases the equation provides sufficient accuracy for Pf, (Ayyub, et al., 2003). 39 Load Effect (L) Strength (R) Density Function Origin 0 Random Value Area (for g < 0) = Failure probability Figure 2-6: Reliability Density Functions of Resistance (R) and Load (L) and Probability of Failure (Ayyub, et al., 2003) 2.2.1.4. Calculation of Safety Factors Applied to Mean Values of Variables The reliability methods used in order to quantify the partial safety factors applied to mean values of variables are the First Order Second Moment (FOSM) and the Advanced First–Order Second Moment (AFOSM) methods. Both are variations of the First-Order Reliability Method (FORM). More specifically, the FOSM method ignores the distribution of the random variables and considers only the moments (coefficient of variation and mean value). Therefore, it is accurate only when the random variables are normally distributed and the performance functions linear. Moreover, the performance function g(X) in Eq. (2-18) is linearized at the mean values of the Xi variables. This results in an invariance problem, since the converged reliability index is dependent on the specific form of the performance function (Novak, et al., 2002; Haldar, et al., 2000). The 40 FOSM method is usually used for preliminary design. Under the limitations of FOSM the target reliability index, ȕ, is given by Eq. (2-21) (Ayyub, et al., 2003). 22 LR LR ıı ȝȝ ȕ   (2-21) where ȝR=mean value of strength R, ȝL=mean value of the load effect L, ıR=standard deviation of strength R, and ıL=standard deviation of the load effect L. The reliability index according to this definition is commonly referred to as the Hasofer-Lind (1974) index. In this study, the AFOSM method will be utilized, which moreover takes into account any type of distribution for the random variables and even non-linear limit states, in order to evaluate the partial safety factors applied to mean variables (mean factors). In this study the variables are considered uncorrelated, although with transformations the method can be used also for correlated variables. If, generally, the performance function is expressed according to Eq. (2-18) in which X is a vector of basic random, uncorrelated variables (X1, X2, ..., Xn) for the strength and the loads, then the variables can be reduced in a coordinate system Xƍ having zero mean and unit standard deviation. Hence, the limit state is reached when g(Xƍ)=0, and therefore, failure occurs when g(Xƍ)<0. In a reduced coordinate system the reliability index, ȕ, is defined as the shortest distance from the origin to the failure surface at the most probable failure point, as Figure 2-7 shows. The AFOSM is beneficial not only because it considers variables with different distributions but also it has not the invariance problem described above. For the calculation of the mean partial safety factors a modified Rackwitz-Fiessler iterative algorithm is used (Ayyub, et al., 2003), as briefly described below: 41 1. Assume a design point ix and obtain ' ix , using the following equation: iX ı iX ȝix ix  ' (2-22) where ȕĮix  i ' , iX ȝ =mean value of the basic random variable, and iX ı =standard deviation of the basic random variable. Usually, the mean values of the basic random variables are used as initial values for the design points. The notations x and 'x are used for the design point in the regular coordinates and in the reduced coordinates, respectively. 2. Evaluate the equivalent normal distributions for the non-normal basic random variables at the design point using the following equations: NXı)(xXFxNXȝ ) 1 (2-23) and )(xXf )(xXFijN Xı ) 1 (2-24) where NXȝ mean of the equivalent normal distribution, N Xı standard deviation of the equivalent normal distribution, )(xFX original cumulative distribution function (CDF) of Xi evaluated at the design point, fX(x )=original probability density function (PDF) of Xi evaluated at the design point, )(˜)=cumulative density function of the standard normal distribution, and ij(˜)=probability density function of the standard normal distribution. 42 3. Compute the directional cosines ( iĮ , i=1, 2, ..., n), using the following equations: ¦ ¸¸¹ · ¨¨© § w w ¸¸¹ · ¨¨© § w w n i i i i X g X g Į 1 2 * ** ' ' (2-25) where, N X ii i ı X g X g ** ' ¸¸¹ · ¨¨© § w w ¸¸¹ · ¨¨© § w w (2-26) 4. With NX N Xi ii ıȝĮ and,, now known, the following equation can be solved for E: > @ 0)(...,),( 111  ȕıĮȝȕıĮȝg NXX N X N XX N X nnn (2-27) 5. Using the E obtained from step 4, a new design point can be obtained from the following equation: ȕıĮȝx NXi N Xi ii  for i=1, 2,…, n (2-28) 6. Repeat steps 1 to 5 until a convergence of E is achieved. Finally, the reliability index will be the shortest distance to the failure surface from the origin in the reduced coordinates system as explained above. 7. Calculate the mean partial safety factors, using the coordinates of the failure point. By noting R*, and Li* the values of resistance and loads at the design point on the failure boundary the partial factors are given as: Rȝ Rij (2-29a) 43 iL i i L P J (2-29b) where, ȝR, ȝLi are the mean values of resistance and loads, respectively. Figure 2-7: Space of Reduced Random Variables Showing the Reliability Index, ȕ, and the Most Probable Failure Point xƍ* 2.2.2. Direct Reliability Design This method is a level II reliability method and here is used in the fatigue design of piping presented in Chapter 7. For a given a performance function and the probabilistic characteristics of all implicated variables, the converged reliability index is computed and then compared to a predefined desirable reliability index, ȕ. For an adequate design the converged reliability index, ȕc, should be greater than the target reliability index. Hence, the convergence of the used AFOSM is achieved with respect to the value of the reliability index. 44 2.3. Comparison of ASD and LRFD Table 2-6 summarizes the characteristics of the two methods, such as their comparison will be facilitated. Figure 2-8, in addition, shows schematically the definition of symbols used in Table 2-6 for LRFD. The existing probability of failure is graphically presented by the intersection of the distributions of the load (L) and resistance (R) (dark grey). The use of factors for both the resistance and the load limit the probability of failure to a predefined quantity Pf . As the nominal values used in the ASD method are based only on the mean values of variables, it can be inferred that as the distribution or the coefficient of variation of the resistance or the load changes, the load and resistance partial factors of LRFD can follow these changes, whereas the deterministic safety factor, based only on mean values of variables, results in highly diverse and unknown probabilities of failure (Ang, et al., 1975; Rao, 1992). Figure 2-8: Relationships among Nominal (LN, RN), Mean (ȝL, ȝR), and Factored Values (ȖLN, ijRN) for the Load and the Resistance 45 Table 2-6: Characteristics of the ASD and LRFD Methods Allowable Stress Design (ASD) Load & Resistance Factor Design (LRFD) Format: ¦t k i Ni f N L S R 1 iNiN k i NN LȖRij ¦t 1 Safety factors: One: Sf applied to the nominal resistance RN Multiple: ijN for the nominal resistance RN and ȖNi for each nominal load or load effect LNi. Calculation of safety factors: 1. They are based only on nominal values of load and resistance (LN, RN) that correspond to percentile values of the mean. All uncertainties are considered through this one factor. 1. Although the method is not a full probabilistic method, it takes into consideration the types and the moments of the distribution of loads and resistance. Uncertainties are considered through multiple factors. 2. They are based on the experience of Code writers in the design, analysis and structural behavior of pipes under different loading conditions as well as on experimental results. Over the years and for particular structures, safety factors have been firmly established. 2. The resultant load and resistance factors are calculated for a reliability index, ȕ, which reflects the acceptable probability of failure and in consequence the acceptable risk for the service life of components. This attribute is significant, since values of ȕ can be used in reliability studies. Consistency: Through calibration the implemented values of the reliability index, ȕ, usually range significantly. The variation of the reliability index, ȕ, is small and controlled by the Code writers. 2.4. Advantages of LRFD LRFD offers important benefits when compared with ASD and more specifically: x It provides consistent reliability among classes of pipes and materials. In ASD the reliability index can vary for different materials and design temperatures and usually is unknown. 46 x It is consistent with the design of other industries (ACI, AISC, AASHTO, API) and other sections of the Code, such as the Section XI, where a probabilistic framework is used in order to program in-service inspections or decide for repairs. x It facilitates the risk analysis of systems in nuclear plants, since a known probability of failure can be assigned to critical pipes that may trigger a top event (e. g., break of a pipe) or be part of the events sequence, should for example the top event is the malfunction of a valve that leads to overpressure, etc. x It facilitates the understanding of the design and moreover it simplifies it. For example, the tables of allowable stresses will be substituted with factors applied to steel strength, which will be less in volume. x The understanding of the implied reliability indices (calibration) for the current piping design will help ameliorate the design by increasing the safety, where is needed, or by decreasing conservatisms that lead to higher costs. x The methodology can be used for many performance functions and for a reliability- based design for fatigue. x The LRFD format favors future changes as a result of gained information in prediction models, materials and load characterization. x The piping design according to the LRFD can be more analytical than the general equations proposed by the ASME B&PV Code in the sections of design by rule and namely, NB-3600, NC-3600, and ND-3600, facilitating hence the designers in selecting the appropriate equations for the piping design. 47 CHAPTER 3: BASIC RANDOM VARIABLES FOR PIPING Although the nominal or design values of dimensions, loads and strength of materials are used by the engineers as precised values, their actual values are random in nature. A reliability-based design necessitates that these uncertainties be quantified. Therefore, each load and the resistance are treated as random variables, where their distribution and moments –mean value and standard deviation or coefficient of variation- are determined by collecting and analyzing data from experimental results, published literature and experts’ opinion. The variables, whose probabilistic characteristics are obtained this way, are fundamental quantities in the design and are called basic. The basic random variables examined herein are: (a) the strength variables, namely the yield, Sy, and ultimate strength of steel, Su, (b) the geometrical characteristics such as the thickness, t, the outside diameter, Do, the section modulus Z, or ZP and the ratio of the external diameter to thickness, ș=Do/t, and (c) the load variables such as the sustained weight, the internal pressure P, other mechanical loads such as the valve relief surcharge, and the seismic loading. Information is also provided for accidental loads (thermal loading, LOCA). Section 3.1 presents the probabilistic characteristics of the strength basic variables and Section 3.2 under separate headings provides those for the considered loading. Nevertheless, variables concerning the reliability-based fatigue design of pipes are separately provided in Chapter 7. 48 3.1. Strength Variables Strength variables include the yield and ultimate strength of steel and the geometrical properties of pipes. The material mainly used for nuclear pipes are austenitic stainless steels. These are iron-based alloys with chromium, Cr, and nickel, Ni, as primary alloying elements and are specifically intended to operate at high-temperature, while they demonstrate also corrosion resistance. More common materials are AISI Type 304 complying with ASTM A312, A376, A358, A409, or A813, and Types 304L, 316, 316L, and 347. Carbon steels are also used for all classes of pipes and mainly the SA 106 Grade B and SA 333 Grade 6 steels. Carbon steels are the predominant materials mainly for oil refineries and chemical plants piping. The type of reactor specifies the material of the pipes in nuclear plants. Piping materials for Pressurized-Water Reactors (PWR), Boiling-Water Reactors (BWR), Sodium-cooled Fast and Thermal Reactors (SGR), are as provided in the previous paragraph. Other less used materials for PWR reactors are ferritic steel ASTM 516 clad with Type 308L austenitic stainless steel. A detailed table of steels used for nuclear pipes is given in Appendix B. In Section 3.1.1 the statistical properties of the yield strength are discussed, and in 3.1.2 these of ultimate strength. Pipes in nuclear plants are designed to operate at room temperature or in elevated temperatures, therefore properties and probabilistic characteristics of steel should be considered for different operating temperatures. Stainless pipes are permitted to operate up to 800 o F, whereas carbon pipes up to 700 o F. For higher temperatures steel is susceptible to time dependent behavior and continuous 49 deformation (namely creep), which is beyond the scope of this work. Section 3.1.3 presents statistical information about the geometrical properties of pipes. 3.1.1. Yield Strength of Steel The yield strength is specified by the offset method of 0.2 per cent. Tables in this chapter present the collected data. More specifically, Table 3-1 presents the minimum and maximum values of the yield strength of steel based on the reviewed literature. Table 3-2 presents statistical data from cited experiments. The nominal yield strength is the specified minimum yield strength value (SMYS) given in the ASME B&PV Code, Part II and also presented in Appendix B. The bias given in the following tables is defined as the ratio of the mean strength of steel at operating temperature to its nominal strength at room temperature. Ware (1995) estimated the margins in the ASME Code for nuclear piping stainless steels, Types 304 (cast and wrought) and 316. He gave the best fit curves shown in Table 3-3 for different temperatures, x (oF), for the yield strength, Sy, (ksi). He assumed a normal distribution for the yield strength in order to estimate the confidence level that the specified ASME Code minimum yield strength (SMYS) has with respect to the experimental data. He concludes that the yield strength value on the best-fit curve could be used as the mean of the yield strength distribution for a given temperature and the ASME Code value (SMYS), as the 97% lower confidence limit. Table 3-3 shows also best fit curves obtained by Sikka, et al. (1977). Moreover, Table 3-4 provides information about the bias of steel at elevated temperatures. Table 3-5 summarizes the statistical properties of yield strength. Properties of carbon steel in elevated temperatures are based on the work of Simmons, et al. (1955). Table 3-5 also shows the recommended 50 values used for the calculation of the partial safety factors. It can be noticed that the bias was slightly lowered no more than 3% from the average value, in order to consider the fact that usually steel not fulfilling the requirements of a grade is classified as steel of the immediate lower grade. Table 3-1: Data for the Yield Strength, Sy, of Carbon and Stainless Steel at Room Temperature Steel Type Min (ksi) Max (ksi) Reference Carbon SA-106B 35 NA Davis (1996) Carbon Steels 30 40 GP Courseware (1982) SA-106B 28.9 39.5 Simmons, et al. (1955) Stainless AISI, TP 304 42 NA Lynch (1989) AISI, TP 304-L 39 NA Lynch (1989) Stainless Steels 40 50 GP Courseware (1982) AISI, TP 316 42 NA Benjamin (1983) AISI, TP 316-L 39 NA Benjamin (1983) AISI, TP 304 29.73 110.23 Cardarelli (1999) AISI, TP 304L 24.66 44.96 AISI, TP 304LN 29.73 NA AISI, TP 304N 34.81 NA AISI, TP 347 29.73 44.96 AISI, TP 304 NA 35 Macdonald, et al. (1989), annealed sheet and strip AISI, TP 304L NA 38 AISI, TP 316 NA 40 AISI, TP 316L NA 32 AISI, Type 316LN NA 38 AISI, Type 321 NA 35 AISI, TP 347 NA 40 NA=Not Available 51 Table 3-2: Experimental Data for the Yield Strength of Carbon and Stainless Steel Steel Mean (ksi) Bias COV Number of Specimens Reference Carbon steel SA106-GR B 45.6 1.30 NA NA Scott, et al. (1994) SA106-GRB 43.65 1.25 NA NA ANL (Chopra, et al. 1996) SA106-GRB 43.51 1.24 NA NA Terrell (Chopra, et al. 1996) SA106-GRB 35.08 1.00 0.10 6 Simmons, et al. (1955) SA 106-GR B 36 1.02 NA NA Wesley (1993) SA 106-GR B 41.93 1.20 NA 3 Marschall, et al. (1993) SA333-6 43.8 1.25 NA NA Higuchi (1991) SA333-6 55.55 1.59 NA NA A-515, Grade 60 39.1 1.22 NA 2 Brust, et al. (1994) Low strength NA 1.10 0.07 NA Assakaff (1998) High strength NA 1.20 0.09 NA Stainless steel TP 304 38 1.27 NA NA Stoner, et al. (1991) SA312-TP 304 37 1.23 NA NA Wesley (1993) SA 376-TP 304 36.05 1.20 NA 2 SA 358-TP 304 42.77 1.43 0.027 3 TP 316L 37.5 1.50 NA 2 TP 316L 37.71 1.51 NA NA Touboul, et al. (1999) AISI 316 39.16 1.31 NA NA Prost, et al. (1983) TP 304 35.67 1.19 NA NA Spaeder, et al. (1974) TP 304 Pipe 37.7 1.26 0.17 14 ASTM DS5S2 (1969)† TP 304 L 33.8 1.35 0.1065 13 TP 316 38 1.27 0.168 14 TP 316L 33.7 1.35 0.19 6 TP 347 38.2 1.27 0.136 18 SA312, TP316, 316H 42 1.40 NA NA Wesley, et al. (1990) NA=Not Available, † Stainless steel type 321 is reported also in the study but not considered here, since as commented the data are unreasonable. 52 Table 3-3: Best Fit Curves for Yield Strength of Stainless Steel Stainless Steel Best Fit Curve (Mean Value) x=70 oF (ksi) COV Reference TP 304, Wrought 9.42) 2 10(23.8 2 ) 4 10(10.1 3 ) 8 10(12.5        xxxy R2=0.81 37.66 0.108 Ware (1995) TP 304, Cast 8.41) 2 10(423.7 2 ) 4 10(04.1 3 ) 8 10(12.5        xxxy R2=0.877 37.09 0.102 Type 316 8.42) 2 10(55.6 2 ) 5 10(81.6 3 ) 8 10(44.2        xxxy R2=0.862 38.54 0.118 TP 304 09.42) 2 10(61.6 2 ) 5 10(62.7 3 ) 8 10(26.3        xxxy R2=0.866 37.82 0.069 Sikka, et al. (1977) TP 316 76.42) 2 10(98.6 2 ) 5 10(76.7 3 ) 8 10(88.2        xxxy R2=0.841 38.24 0.100 tube specimens R2=Coefficient of multiple determination, TP=Type 15 20 25 30 35 40 45 50 55 60 65 Sy (ksi) 0 10 20 30 40 50 60 F re q u e n c y Lognormal Normal Weibull Observations=169 Mean=37.47 ksi COV=0.178 Max=57.7 ksi Min=21.5 ksi Figure 3-1: Histogram for the Yield Strength of Stainless Steel at Room Temperature Based on the Test Data in the Report of Simmons, et al. (1965) Including All Types of Steel Except from Steel TP321 and All Types of Specimens T ab le 3 -4 : M ea n V al u es a n d B ia s o f Y ie ld S tr en g th f o r S ta in le ss S te el a t E le v at ed T em p er at u re s 3 0 4 3 0 4 L 3 1 6 3 1 6 L 3 4 7 T em p . o F M ea n (k si ) B ia s M ea n (k si ) B ia s M ea n (k si ) B ia s M ea n (k si ) B ia s M ea n (k si ) B ia s T em p . o F 7 5 3 7 .7 1 .2 6 3 3 .8 1 .3 5 3 8 1 .2 7 3 3 .7 1 .3 5 3 8 .2 1 .2 7 7 5 1 0 0 3 6 .1 9 1 .2 1 3 2 .7 8 1 .3 1 3 6 .8 6 1 .2 3 3 2 .6 9 1 .3 1 3 7 .4 3 1 .2 5 1 0 0 1 5 0 * 3 3 .9 3 1 .1 3 3 0 .7 6 1 .2 3 3 4 .7 7 1 .1 6 3 0 .6 7 1 .2 3 3 6 .2 9 1 .2 1 1 5 0 * 2 0 0 3 1 .6 7 1 .0 6 2 8 .7 3 1 .1 5 3 2 .6 8 1 .0 9 2 8 .6 5 1 .1 5 3 5 .1 4 1 .1 7 2 0 0 3 0 0 2 8 .2 7 0 .9 4 2 6 .0 3 1 .0 4 2 9 .6 4 0 .9 9 2 5 .6 1 1 .0 2 3 2 .4 7 1 .0 8 3 0 0 4 0 0 2 6 .0 1 0 .8 7 2 3 .6 6 0 .9 5 2 6 .6 0 0 .8 9 2 3 .5 9 0 .9 4 3 0 .5 6 1 .0 2 4 0 0 5 0 0 2 4 .5 0 0 .8 2 2 1 .9 7 0 .8 8 2 5 .0 8 0 .8 4 2 1 .5 7 0 .8 6 2 8 .6 5 0 .9 6 5 0 0 6 0 0 2 3 .3 7 0 .7 8 2 0 .9 6 0 .8 4 2 3 .9 4 0 .8 0 2 0 .5 6 0 .8 2 2 7 .5 0 0 .9 2 6 0 0 7 0 0 2 2 .2 4 0 .7 4 2 0 .2 8 0 .8 1 2 2 .8 0 .7 6 1 9 .5 5 0 .7 8 2 6 .3 6 0 .8 8 7 0 0 8 0 0 2 1 .4 9 0 .7 2 1 9 .6 0 0 .7 8 2 2 .4 2 0 .7 5 1 8 .5 3 0 .7 4 2 5 .9 8 0 .8 7 8 0 0 9 0 0 2 0 .3 6 0 .6 8 1 8 .9 3 0 .7 6 2 2 .0 4 0 .7 3 1 7 .8 6 0 .7 1 2 5 .5 9 0 .8 5 9 0 0 1 0 0 0 1 9 .6 0 0 .6 5 1 7 .9 1 0 .7 2 2 1 .6 6 0 .7 2 1 6 .8 5 0 .6 7 2 5 .5 9 0 .8 5 1 0 0 0 1 1 0 0 1 9 .2 3 0 .6 4 1 6 .9 0 0 .6 8 2 0 .9 0 0 .7 0 1 5 .8 4 0 .6 3 2 5 .2 1 0 .8 4 1 1 0 0 1 2 0 0 1 9 .2 3 0 .6 4 1 5 .2 1 0 .6 1 2 0 .5 2 0 .6 8 1 4 .1 5 0 .5 7 2 4 .8 3 0 .8 3 1 2 0 0 * In te rp o la te d V al u e 53 54 Table 3-5: Summary of Probabilistic Characteristics for Yield Strength Bias COV Steel Temperature (oF) Min Max Avg Rec. Min Max Avg Rec. Distribution Room Temperature 1.00 1.59 1.16 1.13 0.07 0.10 0.09 0.08 200 0.75 1.01 0.95 0.93 NA NA 0.10 0.08 400 0.66 1.03 0.90 0.87 NA NA 0.15 0.13 600 0.67 0.89 0.77 0.75 NA NA 0.13 0.13 C ar b o n 800 0.77 0.86 0.82 0.70 NA NA 0.06 0.13 Lognormal Room Temperature 1.06 1.50 1.30 1.26 0.03 0.19 0.14 0.15 200 1.06 1.17 1.12 1.10 0.11 0.13 0.12 0.15 400 0.87 1.02 0.93 0.90 0.04 0.26 0.14 0.15 600 0.78 0.92 0.83 0.80 0.08 0.27 0.18 0.15S ta in le ss 800 0.72 0.87 0.77 0.75 0.07 0.22 0.17 0.15 Lognormal Rec.=Recommended value, Avg.=Average value, Min=Minimum value, Max=Maximum value, N.A.=Not Available 3.1.2. Ultimate Strength of Steel Data for the ultimate strength, Su, of carbon and stainless steels are provided in Tables 3-6 and 3-7. Table 3-8 gives the best fit curves for different temperatures, x (oF), for the ultimate steel strength, Su (ksi), predicted by Ware (1995) and Sikka, et al. (1977). For carbon steel SA106-GrB Stevenson, et al. (1999) suggest a lognormal distribution for the ultimate strength, while Hill, et al. (2000) propose an average value of 67.2 ksi with standard deviation 4.05 ksi for operation at room temperature. Table 3-9 shows the bias of the ultimate strength of stainless steel at elevated temperatures, while Table 3-10 summarizes the probabilistic properties and recommended values to be used for the calculation of the partial safety factors. For the recommended bias of ultimate strength evaluation was based on same criteria as for the yield strength. The properties of ultimate strength at elevated temperatures are based on the work of Simmons, et al. (1965), and Simmons, et al. (1955) for stainless and carbon steel, respectively. 55 Table 3-6: Data for the Ultimate Strength of Carbon and Stainless Steels for Nuclear Piping at Room Temperature Steel Type Min Su (ksi) Max Su (ksi) Reference Carbon Steel SA-106B 60 NA Davis (1996) Carbon Steels 55 65 GP Courseware (1982) SA 106 B 60 NA Rajdeep Metals, Mumbai SA 106 B 59.7 72 Simmons, et al. (1955) Stainless Steel Stainless Steels 78 100 GP Courseware (1982) AISI, Type 316 84 NA Benjamin (1983) AISI, Type 316-L 81 NA AISI, Type 347 85 NA Davis (2000) Type 304 74.7 NA Ukrainian Industrial Energetic Type 304-L 70.34 NA Company Type 316 74.7 NA Type 316-L 70.34 NA A 312 TP 304 75 NA Rajdeep Metals, Mumbai A 312 TP 304L 70 NA AISI, Type 304 74.69 150.11 Cardarelli (1999) AISI, Type 304L 65.27 89.92 AISI, Type 304LN 74.69 NA AISI, Type 304N 79.77 NA AISI, Type 347 74.69 89.92 AISI, Type 304 NA 85 Macdonald, et al. (1989) AISI, Type 304L NA 75 annealed sheet and strip AISI, Type 316 NA 90 AISI, Type 316L NA 75 AISI, Type 316LN NA 85 AISI, Type 321 NA 90 AISI, Type 347 NA 95 AISI, Type 304 84 NA Lynch (1989) AISI, Type 304-L 81 NA NA=Not Available 56 Table 3-7: Experimental Data for the Ultimate Strength of Carbon and Stainless Steel Steel Mean (ksi) Bias COV Specimens Reference Carbon Steel SA106-GR B 75.4 1.26 NA NA Scott, et al. (1994) SA106-GR B 68.03 1.13 0.06 6 Simmons, et al. (1955) SA 106-GR B 68 1.13 NA NA Wesley (1993) SA 106-GR B 75.24 1.25 0.035 3 Marschall, et al. (1993) SA106-GRB 82.96 1.38 NA NA ANL (Chopra, et al. 1996) SA106-GRB 75.85 1.26 NA NA Terrell (Chopra, et al. 1996) SA333-6 70.92 1.18 NA NA Higuchi (1991) SA333-6 79.62 1.33 NA NA Higuchi (1995) Low strength NA 1.05 0.06 NA Assakkaf (1998) SA516 GR70 84 1.20 NA NA Wesley, et al. (1990) Stainless Steel SA 376-Type 304 87.65 1.17 NA 2 Marschall, et al. SA 358-Type 304 105.23 1.40 0.034 3 (1993) A-515, Grade 60 63.9 1.07 NA 2 Brust, et al. (1994) Type 316L 86.6 1.24 NA 2 Type 316L 88.33 1.26 NA NA Touboul, et al. (1999) AISI 316 86.3 1.15 NA NA Prost, et al. (1983) Type 304 91 1.21 NA NA Stoner, et al. (1991) SA312-Type 304 86 1.15 NA NA Wesley (1993) 304 Pipe 84.0 1.12 0.063 14 304 L 79.2 1.13 0.034 14 316 83.3 1.11 0.077 14 316L 78.9 1.13 0.037 9 347 87.0 1.16 0.057 18 ASTM DS5S2 (1969)† NA=Not Available, † Stainless steel type 321 is reported also in the study but not considered here, since as commented the data are unreasonable. 57 Table 3-8: Best Fit Curves for Ultimate Strength of Stainless Steel Steel Best Fit Curve (Mean Value) Value for x=70oF (ksi) COV Reference TP 304, Wrought 95) 1 10(46.1 2 ) 4 10(38.2 3 ) 7 10(27.1        xxxy R 2 =0.871 85.90 NA TP 304, Cast 1.93) 1 10(56.1 2 ) 4 10(56.2 3 ) 7 10(43.1        xxxy R 2 =0.831 83.38 NA TP 316 3.69) 1 10(31.2 2 ) 4 10(46.5 3 ) 7 10(75.4 4 ) 10 10(28.1         x xxxy R 2 =0.957 82.65 NA Ware (1995) TP 304 44.95291.1 2 ) 4 10(90.1 3 ) 8 10(55.9      xxxy R 2 =0.969 79.56 0.035 TP 316 58.83) 2 10(35.6 2 ) 5 10(45.9 3 ) 8 10(39.5        xxxy R 2 =0.954 79.58 0.05 Sikka, et al. (1977) tube specimens R2=Coefficient of multiple determination, TP=Type 55 60 65 70 75 80 85 90 95 100 105 110 Su (ksi) 0 10 20 30 40 50 60 70 80 F re q u e n c y Lognormal Normal Observations=170 Mean=83.55 ksi COV=0.063 Max=103.9 ksi Min=63.5 ksi Figure 3-2: Histogram for the Ultimate Strength of Stainless Steel at Room Temperature Based on the Test Data in the Report of Simmons, et al. (1965) Including All Types of Steel, Except from Steel TP321, and All Types of Specimens T ab le 3 -9 : M ea n V al u es a n d B ia s o f U lt im at e S tr en g th f o r S ta in le ss S te el a t E le v at ed T em p er at u re s 3 0 4 3 0 4 L 3 1 6 3 1 6 L 3 4 7 T em p . o F M ea n (k si ) B ia s M ea n (k si ) B ia s M ea n (k si ) B ia s M ea n (k si ) B ia s M ea n (k si ) B ia s T em p . o F 7 5 8 4 1 .1 2 7 9 .2 1 .1 3 8 3 .3 1 .1 1 7 8 .9 1 .1 3 8 7 1 .1 6 7 5 1 0 0 8 1 .4 8 1 .0 9 7 6 .8 2 1 .1 0 8 0 .8 1 .0 8 7 6 .5 3 1 .0 9 8 4 .3 9 1 .1 3 1 0 0 1 5 0 * 7 8 1 .0 4 7 2 .7 8 1 .0 4 7 8 .3 0 1 .0 4 7 2 .9 8 1 .0 4 8 0 .0 4 1 .0 7 1 5 0 * 2 0 0 7 3 .9 2 0 .9 9 6 8 .7 3 0 .9 8 7 5 .8 0 1 .0 1 6 9 .4 3 0 .9 9 7 5 .6 9 1 .0 1 2 0 0 3 0 0 6 9 .7 2 0 .9 3 6 2 .5 7 0 .8 9 7 4 .1 4 0 .9 9 6 5 .4 9 0 .9 3 6 9 .6 0 .9 3 3 0 0 4 0 0 6 7 .2 0 0 .9 0 6 0 .1 9 0 .8 6 7 2 .4 7 0 .9 7 6 3 .9 1 0 .9 1 6 5 .2 5 0 .8 7 4 0 0 5 0 0 6 6 .3 6 0 .8 9 5 9 .4 0 .8 5 7 2 .4 7 0 .9 7 6 3 .1 2 0 .9 0 6 3 .5 1 0 .8 5 5 0 0 6 0 0 6 6 .3 6 0 .8 9 5 8 .6 1 0 .8 4 7 3 .3 0 .9 8 6 3 .1 2 0 .9 0 6 2 .6 4 0 .8 3 6 0 0 7 0 0 6 6 .3 6 0 .8 9 5 7 .8 2 0 .8 3 7 2 .4 7 0 .9 7 6 3 .1 2 0 .9 0 6 1 .7 7 0 .8 2 7 0 0 8 0 0 6 5 .5 2 0 .8 7 5 7 .0 2 0 .8 2 7 1 .6 4 0 .9 6 6 2 .3 3 0 .8 9 6 1 .7 7 0 .8 2 8 0 0 9 0 0 6 3 .8 4 0 .8 5 5 5 .4 4 0 .7 9 6 9 .1 4 0 .9 2 5 9 .9 6 0 .8 6 6 1 .7 7 0 .8 2 9 0 0 1 0 0 0 5 9 .6 4 0 .8 0 5 2 .2 7 0 .7 5 6 4 .9 7 0 .8 7 5 6 .8 1 0 .8 1 6 0 .9 0 0 .8 1 1 0 0 0 1 1 0 0 5 3 .7 6 0 .7 2 4 7 .5 2 0 .6 8 5 9 .1 4 0 .8 0 5 2 .0 7 0 .7 4 5 8 .2 9 0 .7 8 1 1 0 0 1 2 0 0 4 3 .6 8 0 .5 8 4 1 .9 8 0 .6 0 5 1 .6 5 0 .6 9 4 6 .5 5 0 .6 7 5 3 .9 4 1 2 0 0 * In te rp o la te d V al u e 58 59 Table 3-10: Summary of Probabilistic Characteristics for Ultimate Strength Bias COV Steel Temperature (oF) Min Max Avg. Rec. Min Max Avg. Rec. Distribution Room Temperature 1.05 1.38 1.19 1.15 0.035 0.06 0.05 0.06 200 0.91 1.15 1.07 1.05 NA NA 0.08 0.06 400 1.02 1.36 1.20 1.17 NA NA 0.10 0.10 600 0.95 1.30 1.13 1.10 NA NA 0.10 0.10 C ar b o n 800 0.80 0.99 0.91 0.88 NA NA 0.10 0.10 Lognormal Room Temperature 1.07 1.40 1.15 1.13 0.034 0.077 0.06 0.06 200 0.98 1.01 1.00 0.98 0.04 0.05 0.05 0.06 400 0.86 0.97 0.90 0.88 0.03 0.06 0.05 0.06 600 0.83 0.98 0.89 0.88 0.03 0.06 0.05 0.06S ta in le ss 800 0.82 0.96 0.87 0.85 0.03 0.06 0.05 0.06 Lognormal Rec.=Recommended value for calculations, Avg.=Average value, Min=Minimum value, Max=Maximum value, NA=Not Available 3.1.3. Comparison of LRFD Definition of Steel Strength with ASME Practice Initially, in Figures 3-3 and 3-4 the yield and ultimate strength of steel at different operating temperatures is shown for stainless and carbon steel, respectively. In addition, Figure 3-5 provides physical properties of carbon steel at different temperatures. In all cases the curves represent the mean values of properties. Figures 3-6 to 3-9 show (for carbon steel SA 106B and austenitic steel SA 312 Type 312) the steel properties used in LRFD, namely the mean values of the steel strength and the nominal values that are the specified minimum at room temperature for any operating temperature. These values are given in Tables 1A and 2A in Section II, Part D of the ASME B&PV Code. The mean and nominal values (Sy, Su) for the LRFD are connected with a line. The bias, calculated in previous sections, is the ratio of these values for the steel under consideration. Figures 3-6 to 3-9 moreover present the nominal (minimum) strength of steel for different operating temperatures (Sƍy, Sƍu) that is used in the ASME B&PV Code. More 60 specifically, these values are given in Table U1 for the ultimate strength of steel and in Y-1 for the yield strength of steel of the ASME Code, Section II, Part D. The figures also show the allowable stresses of the Code for pipes of Class 1, Sm, and S for pipes of Classes 2 and 3. Figure 3-3: Behavior of Type 316 Stainless Steel at Different Temperatures (Cullen, et al., 1969) 61 (a) (b) Figure 3-4: Behavior of Carbon Steel SA106B at Elevated Temperatures, a) Ultimate Strength, b) Yield Strength (Simmons, 1955) Figure 3-5: Physical Properties of Carbon Steel at Elevated Temperatures (Timoshenko, 1930) 62 Carbon Steel (SA 106B) 0 5 10 15 20 25 30 35 40 45 0 100 200 300 400 500 600 700 800 Temperature (oF) Y ie ld S tr e n g th M ean Code Nominal (S'y) Code Allowable (S) Code Allowable (Sm) Nominal LRFD (Sy) Figure 3-6: Values of Yield Strength Used in LRFD and ASME Code for Carbon Steel SA 106B at Different Operating Temperatures Stainless Steel (SA 312 Type 304) 0 5 10 15 20 25 30 35 40 0 100 200 300 400 500 600 700 800 900 Temperature (oF) Y ie ld S tr e n g th M ean Code Nominal (S'y) Code A llowable (S) Code A llowable (Sm) Nominal LRFD (Sy) Figure 3-7: Values of Yield Strength Used in LRFD and ASME Code for Stainless Steel SA 312 Type 304 at Different Operating Temperatures 63 Carbon Steel (SA 106B) 5 15 25 35 45 55 65 75 0 100 200 300 400 500 600 700 800 Temperature (oF) U lt im a te S tr e n g th M ean Code Nominal (S'u) Code Allowable (S) Code Allowable (Sm) Nominal LRFD (Su) Figure 3-8: Values of Ultimate Strength Used in LRFD and ASME Code for Carbon Steel SA 106B at Different Operating Temperatures Stainless Steel (SA 312 Type 304) 0 10 20 30 40 50 60 70 80 90 0 100 200 300 400 500 600 700 800 900 Temperature (oF) U lt im a te S tr e n g th M ean Code Nominal (S'u) Code A llowable (S) Code A llowable (Sm) Nominal LRFD (Su) Figure 3-9: Values of Ultimate Strength Used in LRFD and ASME Code for Stainless Steel SA 312 Type 304 at Different Operating Temperatures Considering Figures 3-6 to 3-9 some inferences can be derived for the criteria used in the ASME B&PV Code that certainly can be assured by examining the properties of more types of steel, which will be included in the LRFD. Thus, for example, by 64 comparing the nominal yield strength cited in the Code (Sƍy) with the allowable stresses (S for Classes 2 and 3 and Sm for Class 1), it appears that the Code is more conservative for pipes operating in temperatures less than 200 o F. Therefore, calibration of the Code for use in LRFD will result in lower values of ȕ for pipes operating at high temperatures. That is though not the case for the ultimate strength, as Figures 3-8 and 3-9 show. 3.1.4. Geometrical properties Statistical data on pipe outside diameter, Do, pipe thickness, t, and the ratio ș = Do/t is provided in this section. These variables are investigated under separate headings. 3.1.4.1. External Diameter In a nuclear power plant pipes of different diameters are used (Zhao, 1994; Crocker, et al., 1967). Dimensional standards (ASME B36.10M and ANSI/ASME B36.19M) provide the diameter, thickness and weight for all piping schedules for both welded and seamless wrought steel as well as stainless steel piping. Diameter tolerances are quite tight and vary with pipe size, while they are governed by the requirements of ASME B16.9 (2003). For example for a 4 in pipe the tolerance is 1/16 in, while for 24 in pipe the tolerance is 1/8. For straight pipes these diameter tolerances are for the entire pipe, but for fittings, these only apply to the ends. The basic idea is that it must be possible to create good welds between pipes and fittings. In typical probabilistic risk assessment studies conducted for existing plants the statistical characteristics for diameter are considered to follow normal distribution with a value of twice standard deviation = r tolerance values. Table 3-11 provides the outside diameter 65 variations for various pipe sizes. Based on the assumption that tolerances fall within the 4ı area, the coefficient of variation for the outside diameter in Table 3-11 is estimated. Table 3-11: Properties of Nominal Pipe Outside Diameter Nominal Diameter Diameter Variation (in) Mean COV Distribution Do +1/64, -1/32 Do 0.012/Do in4in2 dd oD +1/32, -1/32 Do 0.016/Do in8in5 dd oD +1/16, -1/32 Do 0.023/Do in20in10 d oD +3/32, -1/32 Do 0.031/Do in20toD +1/8, -1/32 Do 0.039/Do Normal 3.1.4.2. Thickness The pipe thickness usually is indicated by a schedule number. The higher the schedule number, the thicker the pipe is. Its value can be approximated by the following equation: SPnumberSchedule /1000 (3-1) where P=the steam pressure (lb/in2); and S=the working stress of pipe material (usually taken as 10 to 15 percent of the ultimate strength of steel). The minimum thickness for fittings and straight pipes is the nominal thickness minus 12.5%. There is no maximum thickness for fittings, but modern fabrication techniques allow manufacturers to create products that do not exceed nominal values by much except, perhaps at points like the intrados of elbows, (Ayyub, et al., 2005). There is an average weight tolerance for straight pipes that, in effect, limits the average thickness tolerances to 5%. In typical probabilistic risk assessment studies conducted for existing plants the statistical characteristics for thickness are considered to follow normal distribution with a value of twice standard deviation = r tolerance values. A minimum 66 coefficient of variation for the thickness of 0.03 can be used. Table 3-12 summarizes the probabilistic characteristics used for the reliability–based design for onshore and offshore pipelines and dented pipes, and Table 3-13 the estimated coefficient of variation (COV) to be used in this study. Table 3-12: Reported Probabilistic Characteristics for the Thickness of Pipes Distribution Mean COV Reference Normal Nominal (mm) 0.25 / Mean Zimmerman, et al. (1998) Normal NA 0.025 Sotberg, et al. (1994) Normal NA 0.02 Bai, et al. (1997) Normal 0.925*(Nominal) 0.03 Stewart, et al. (2002) NA=Not Available Table 3-13: Statistical Properties for the Thickness of Pipes Nominal Pipe Size Nominal Thickness Thickness Variation Mean COV Distribution Do t -12.5% t 0.035/t in4in2 dd oD t -12.5% t 0.035/t in8in5 dd oD t -12.5% t 0.035/t in20in10 d oD t -12.5% t 0.035/t in20toD t -12.5% t 0.035/t Normal 3.1.4.3. Section Modulus The elastic, Z, and plastic, ZP, section modulus of a pipe were given by Eqs. (2-3) and (2-4), respectively. Monte Carlo simulation was used in order to assess the coefficient of variation and bias for this variable. As a result a lognormal distribution is recommended with a coefficient of variation equal to 0.05 and bias 1.0. 3.1.4.4. Diameter to Thickness Ratio The diameter to thickness ratio ș=Do/t constitutes an indicator for the following: 67 x The susceptibility of pipes to buckling when the latter are subjected to bending loading. Relative information is provided in Chapter 4. x The shape factor defined as the ratio of the plastic section modulus to the elastic one, which defines the margin between the first yield moment and the first hinge moment. x The characterization of a pipe as thin or thick is related with the ratio of the internal diameter, d, to the pipe thickness, t, where d=Do–2t. Thin pipes are considered those having a ratio d/t greater than 20. ASME Code Criteria are based on the assumption of thin pipes, as explained in a previous chapter. Taking into consideration the tables for standard commercial pipe sizes, it can be noticed that the ratio for pipes made of carbon steel varies between 6 to 100, while for stainless steel has an upper limit of 40. In order to assess the probabilistic characteristics of ș,Ȃonte Carlo simulation was performed and the following probabilistic characteristics were derived: Coefficient of variation equal to 0.037, while the distribution can be considered either normal or lognormal. 3.2. Load Variables This section gives information and the probabilistic characteristics for the loads impacting nuclear piping. 3.2.1. Sustained Weight The sustained weight includes the own weight of the pipe, the weight of the attachments or components mounted on it, the insulation, and the weight of the pipe’s 68 contents. A description of these loads and the probabilistic characteristics for the sustained weight are provided in the following sections. 3.2.1.1. Self Weight For the computation of the weight of the pipe, the following equation is proposed (King, 1967). )(68.10)/(, tDtFftlbpipeofweight o  (3-2) where F=relative weight factor, t=wall thickness (in), and Do=outside diameter (in). The pipe weight calculation in Eq. (3-2) is based on low–carbon steel properties, which weighs 0.2833 lb/in 3 and is extended to other materials through the factor F, which takes the values: 1.02 for austenitic stainless steel, 0.98 for wrought iron, 1.00 for carbon steel, and 0.95 for ferritic stainless steel. Normally, the weight of piping can be found in piping catalogs from vendors, where the same weight for carbon and austenitic stainless steel is used. The weight per foot of steel pipe is subject to tolerances as illustrated in Table 3-14. Table 3-14: Self-Weight Tolerances of Steel Piping (Crocker, et al., 1967) Specification Size Tolerance ASTM A376 and A312 12 in and under +6.5% -3.5% ASTM A106 Sch. 10-120 Sch. 140-160 +6.5% -3.5% +10% -3.5% ASTM A53 Std wt and XS wt +5% -5% 3.2.1.2. Self Weight of Fittings and Components Except from the weight of the pipe itself, the sustained weight should include all the fittings and components that are part of the pipe run, including valves, meters and 69 other special equipment. Because of the relative large diameters and wall thickness of the nuclear pipes the accompanying equipment is usually of greater weight than the one mounted on a conventional pipeline, for example valves with motorized or hydraulic actuators (Lamit, 1981). 3.2.1.3. Insulation All nuclear pipes connecting the nuclear vessel with the steam generator, turbines and condenser require high quality insulation in order to protect surrounding equipment and instrumentation. For these pipes the insulation is never permanently bonded; it can be easily removed and stored during the regular inspection of the pipes. Block insulation, metal insulation and blanket insulation have been used in such cases. Most protective insulation must be clad with stainless steel or aluminum sheets. For pipes that inspection is not necessary, common commercial and industrial piping insulation materials are used that demonstrate resistance in high temperatures and can reduce heat losses. Such materials are provided in ASTM specifications. The weight of pipe insulation, W, is given by the following relation (Helguero, 1983): )(0218.0)/( KDKIftlbW o  (3-3) where K=insulation thickness, Do=outside diameter of pipe (in), I=insulation density (lb/ft 3 ). Table 3-15 gives values for I for some insulation materials. 70 Table 3-15: Insulation Density, I (Helguero, 1983; Kannappan, 1986) Material Density, I (lb/ft3) Calcium Silicate 11, 12.25 * 85% Magnesium 10 to 11 Thermobestos 11.53 Kalo 19 to 21 High Temperature 24 Super -X 25 Poly-Urethane 2.3, 2.0 * Mineral Wool 8 Fiber Glass 3.25 * Foam Glass 8.50 * Polystyrene 2.00 * *Values provided by Kannappan, 1986 3.2.1.4. Contents of Pipe The weight of pipe’s contents (lb/ft) is given by the following equation (Crocker, et al., 1967): 2)2(3405.0 tDGpipeofcontentsofweight o  (3-4) where G=specific gravity of contents, t=pipe thickness (in), and Do=outside diameter of pipe (in). Different coolants are used for different types of reactors. The density of these coolant materials varies considerably. For example gas cooled reactors have coolant densities below 0.1 lb/ft 3 and water cooled reactors may have coolant densities over 60 lb/ft 3 . 3.2.1.5. Probabilistic Characteristics The uniformly distributed weight, including the insulation and fluid weights, as well as the concentrated weight due to valves, flanges, etc. may vary by r 20% from the as-analyzed weight (Mikitka, et al., 1988). Typical probabilistic risk assessment studies 71 conducted for existing plants consider the statistical characteristics for dead weight to follow normal distribution with a value of twice standard deviation =r tolerance values, (Ayyub, et al., 2005). Hwang, et al. (1983) estimate that the coefficient of variation for sustained weight of pipes can be considered as low as 0.05, but since often there are unexpected loads on pipes like cable trays, etc., a value of 0.10 will be used in this study. This coefficient of variation is used also for the own weight of building structures (Ellingwood, 1981). 3.2.2. Internal Pressure Within each service level there is actually a variation of pressure- indicated although valves are responsible for keeping a constant pressure within the pipes. Table 3-16 gives representative average operating pressure (Service Limit A) and the corresponding operating temperature for different reactors and piping systems (primary and secondary piping). Nevertheless, pressure information for PWR and BWR are of interest for this study. Table 3-16: Operating Pressure at Operating Temperature (Crocker, et al., 1967; Lamit, 1981) Reactor Type Primary Pressure Average (psig) Steam Pressure (psig) Pressurized-water (PWR) 2,500 at 514 o F 453 at 460 o F Boiling-water (BWR) 1,000 at 546 o F 500 at 825 o F Sodium-cooled Graphite- moderated Thermal Reactor atmospheric 800 at 825 o F Sodium cooled Fast-breeder Reactor atmospheric 900 at 780 o F Gas-cooled Reactor Systems 360 1,450 at 1,000 o F Organic-moderated Reactor- Systems 120 450 at 550 o F 72 The probabilistic characteristics in existed studies for internal pressure are tabulated in Tables 3-17 for the operating pressure, while in Table 3-18 characteristics for the accidental pressure are provided, which can be related to the Service Level D pressure. Table 3-19 gives the suggested probabilistic characteristics for the different pressure types encountered in the design of piping in this study. Table 3-17: Statistical Properties for Pressure Based on Literature Review Pressure Type Mean COV Distribution Reference Design DESP 0.04 Normal Bishop, et al. (1993) Pmax 1.03 ( AP ) NA Weibull Stancampiano, et al. (1976) Operating AP 0.05 Normal Stewart, et al. (2002) Pmax 1.07 DESP 0.02 Extreme I Sotberg, et al. (1994) Operating AP 0.10 Lognormal Saigal (2005) NA=Not Available Table 3-18: Statistics on Accidental Pressure Occurrence Intensity Rate per year Duration Mean/Design COV Distribution Reference NA NA 0.90 0.12 Normal Hwang, et al. (1983) 1.7(10 -3 ) 20 min 0.80 0.20 Type I Ellingwood, et al. (1996) NA=Not Available More, specifically the pressure for each service level in this study is considered to have different probabilistic characteristics. This arises from the fact that the different loading conditions for each service level can generate pressure loading (e.g., for accidental, Service Level D, or emergency conditions of Service Level C the COV is expected to be higher and the bias less than that for other service levels. Nevertheless, in 73 this study the COV for pressure is between 0.10 and 0.20 for all service limits as Table 3- 19 shows. Extreme I of the largest values distribution is used for the faulted or accidental conditions (Service Limits C and D) as well as for the maximum (peak) pressure under normal operating conditions. In Table 3-19, PO and PS are the pressures coincident with the Operating Basis Earthquake (OBE) and the Safe Shutdown Earthquake (SSE), respectively. These pressures are considered to have the same probabilistic characteristics, since they are generated from the same physical load, namely earthquake. Nevertheless, PS is expected to attain higher values than the PO pressure. Table 3-19: Proposed Probabilistic Characteristics for Internal Pressure for this Study Pressure COV Bias Distribution Design, PDes 0.10 1.0 Normal PA 0.10 1.0 Normal PB 0.13 0.95 Normal PC 0.15 0.85 Extreme I PD 0.20 0.80 Extreme I Pmax 0.13 0.90 Extreme I PO 0.13 0.95 Lognormal PS 0.13 0.95 Lognormal 3.2.3. Earthquake Loading Seismic loading can be classified as a reversing dynamic loading as shown in Figure 2-1. For Service Limits B and C the pipes should be designed to withstand an Operating Basis Earthquake (OBE), while for Service Limit D pipes should also be designed for the Safe Shutdown Earthquake (SSE). More specifically, a definition for these earthquakes according to the Article N-1000 of the ASME BPV Code, (2001) is provided herein: 74 The Operating Basis Earthquake (OBE), Eo, is that earthquake which, considering the regional and local geology and seismology and specific characteristics of local subsurface material could be reasonably be expected to affect the plant site during the operating life of the plant. This earthquake is typically expected to occur once in one hundred years (Rodabaugh, 1984). The maximum vibratory ground motion of the OBE should be at least one-half the maximum vibratory ground motion of the SSE, unless a lower OBE can be justified on the basis of probability calculations. The Safe Shutdown Earthquake (SSE), ES, is that earthquake, which is based upon an evaluation of the maximum earthquake potential considering the regional and local geology and seismology and specific characteristics of local subsurface material. It is the earthquake which produces the maximum vibratory ground motion for which the pipes should remain functional. This earthquake is typically expected to occur once in one thousand years (Rodabaugh, 1984). The Code requires the moment responses caused by a postulated earthquake loading at the nuclear plant foundation. The seismic loading is transmitted to the piping system through the pipe support connections and the structure that supports it. For the calculation of the maximum response of piping under earthquake loading, many uncertainties arise, and namely: ƒ The intensity of the peak ground motion at the plant site. The usual approach is to perform a seismic hazard analysis, where a probability distribution of effective peak ground acceleration at the site is determined by considering all possible 75 magnitudes of earthquake, epicenter distances and depths possible for the site under consideration, (Cornell, 1968; Ellingwood, 1994). This way a cumulative distribution function GA(x) is generated, showing the annual probability of exceeding a specified ground acceleration, x, as a function of x. The peak ground acceleration, A, is supposed to have a Type II distribution of largest values. ƒ The pipe response, which implicates different uncertainties by itself, such as: a) The method of analysis. The most precise method but cost prohibitive is to analyze with a time history the piping system together with the other components and the building. As an alternative, the pipes are decoupled from the structure and input load functions are considered at the supports. Some pipes are within rigid containments, and in such case the supports vibration can be considered in phase, but in generally the supports may not be in phase and this way two types of stresses are generated that are considered in the ASME B&PV Code (2001): the primary stresses due to the inertia of mass, and secondary stresses due to relative movement of the supports. Primary stresses are considered in this study. In usual practice the response of piping is obtained by using the Response Spectrum Method. b) The damping of the pipe. Over the different eras of piping design different response spectra were utilized and values of critical damping between 0.5% and 5% were considered, with the damping for the SSE to be larger than the one considered for the OBE. When the Response Spectrum Method is considered, the variability in the modal frequency of piping systems should 76 also be considered. Discussion about the criteria for seismic piping design is provided in detail by Stephenson (1995, 2003). For the development of LRFD for buildings (Ellingwood, et al., 1980) and other structures (e.g., dams) the coefficient of variation of the seismic load and its distribution were determined. As in civil engineering structures, where the largest uncertainty comes from the peak ground acceleration since all the other uncertainties mentioned above, (method of analysis, damping, modal frequency) have a smaller coefficient of variation, for pipes also the acceleration determines the characteristics of the seismic load. Table 3- 20 summarizes the probabilistic characteristics of seismic loads used in different studies for the development of load and resistance safety factors. For piping also the distribution of the seismic load can be considered as Extreme II of largest values with a coefficient of variation varying between 0.40 and 0.90. Table 3-20: Probabilistic Characteristics for the Seismic Load Occurrence Intensity Rate per year Duration Mean COV Distribution Reference 0.02 30sec Site-dep. 0.85 Type II Ellingwood (1995) 0.05 30sec (0.08Es) 0.90 Type II Ellingwood, et al. (1996) lnFA(0.05)/year 10-20sec NA 0.85 Type II Hwang & Ellingwood, et al. (1987) 2/year 60sec NA 0.70 Type I Casciati (1983) 1 to 4/year 60sec NA 0.35- 0.70 Type I or II Casciati, et al. (1982) NA=Not Available 3.2.4. Mechanical Loading In this study the term “mechanical” loads is used to describe loads such as water hammer or pressure surges. Water hammer refers to shocks sounding like hammer blows produced by a rapid change of fluid flow velocity in a closed pipeline. It can happen due 77 to rapid closure of the valve, where the fluid stops suddenly. The kinetic energy is then converted into pressure energy. The pressure rise causes in turn elastic waves that travel upstream and downstream from the point of origin. These elastic waves cause increases or decreases in pressure that are called water hammer surge or transient pressure. In most cases water hammer occurs during a plant startup or during return of an isolated plant system into service or when safety valves are actuated. The discharge piping is more susceptible to transient hydrodynamic loads, since the opening time of safety valves is very fast and can induce large hydrodynamic loads on the downstream piping, especially when loop seals are present. The pressure rise, P, for instantaneous valve closure is given in the following equation (AWWA Manual, 2004). g VWaP 144 (3-5) where W is the weight of fluid (lb/ft3), V is the velocity of flow (fps), g=32.2 fps/sec, and Į is the magnitude of the surge wave velocity, which is independent of the length of the pipe and for steel is equal to: fps ))100/(1( 4660 tD a i (3-6) where, Di is the inside diameter of the pipe and t its thickness. The pressure rise exert a force to the pipe which is equal to the pressure times the cross-sectional area of the pipe. The water hammer mechanism can be also generally described by Joukowsky’s Law as follows: uĮȡp ' ' (3-7) 78 where ǻp=the dynamic pressure change resulting from the change of the flow velocity in the pipe by an amount ǻu, ȡ=the density of the fluid, Į=the speed of a pressure wave in the fluid flowing in the pipe. Table 3-21 presents the statistics on Safety Relief Valve (SRV) discharge loads as presented in the consensus estimation studies of Hwang, et al. (1983), and Table 3-22 shows the statistical properties of Safety Relief Valve (SRV) discharge loads, presented in other studies. SRV loads occur mostly in BWR plants. McGeorge (1974) refers that in most cases of steam flow in main steam lines, the fluctuation of pressure following a valve closure is of the order of 10-20% of the normal operating pressure. Table 3-21: Statistics on Safety Relief Valve (SRV) Discharge Loads, (Hwang, et al., 1983) Load Case Property A B C D Design Value (psi d) 13.49 17.40 16.46 28.23 Design Value (bar d) 0.93 1.20 1.14 1.95 Predicted Value (bar d) 0.60 0.79 0.74 1.11 Mean/Design Value 0.65 0.66 0.65 0.57 Variance 0.00357 0.00407 0.00363 0.0154 Standard Deviation 0.0597 0.0638 0.0602 0.124 COV 0.10 0.08 0.08 0.11 Number of Occurrence in 40 years 271 1313 NA 1620 Occurrence Rate per year 6.775 32.825 NA 40.5 A: First actuation of one or two valves (100oF suppression pool) B: First actuation of three or more adjacent valves (100oF suppression pool) C: First actuation of an ADS valve (120oF suppression pool) D: Subsequent actuation of a single valve (120oF suppression pool) 79 Table 3-22: Statistics on Safety Relief Valve (SRV) Occurrence Intensity Rate per year Duration Mean COV Distribution Reference N/A 1sec 0.8PSRV 0.14 Normal Ellingwood, et al. (1996) N/A=Not Available Mechanical loads such as water hammer except from the overpressure listed in the above tables result also in out-of balance forces developed at areas of piping such as elbows, tees, etc. The time varying force develops a dynamic excitation of the piping system and the development of moments, which are considered for the piping design for Service Limits B and C. 3.2.5. Thermal Loading Thermal loading is considered in the Code for Class 2 and 3 piping only as a cause for fatigue, while for Class 1 pipes both for fatigue and ratcheting. More specifically, for Classes 2 and 3 the range of expansion moment loading resulting from a temperature difference (expansion or contraction) at locations where the pipe is restraint to move are taken into consideration. More discussion is provided in Chapter 7. Nevertheless, as a result of accidental temperature, which is not a repeated load, moments can be developed, too. Probabilistic characteristics of accidental temperature are suggested for containments by Hwang, et al. (1983), namely a coefficient of variation equal to 0.12 and bias (ratio of mean to design value) equal to 0.90. The temperature differences, ǻȉ, produce also axial forces, F=E Į ǻȉ, where E is the modulus of elasticity and, Į, the thermal expansion coefficient. These forces were not considered herein. 80 3.2.6. LOCA Loading Loss Of Coolant Accident (LOCA), as water hammer, leads to an increase in the pressure boundary and dynamic loading on the piping. LOCA loading is characterized as small or large depending on what pipe breaks and the effects of the pipe’s failure. Its cause is that low pressure piping gets over-pressurized. This type of loading is accidental and therefore is considered only for Service Limit D conditions. Moreover, it is mostly usual for Class 1 piping. 3.3. Generated Random Variables The variables used in the performance functions are stresses. The basic loading random variables on the other hand, as presented previously, are different types of loadings. Using Monte Carlo simulation in order to approximate the characteristics of stresses, it was inferred that the loading has the greatest impact such as the stresses can also be approximated to have the same probabilistic characteristics as the loads themselves. 81 CHAPTER 4: STRENGTH MODELS UNCERTAINTY In this chapter initially a brief discussion about the bias and the factors applied to mean and nominal values of variables is provided. Then, the procedure for the estimation of the uncertainty introduced by the design strength models for yield or burst of pipes due to internal pressure and ultimate moment capacity is discussed. Moreover, the bias of the resistance for the previously mentioned failure modes is assessed. 4.1. Definition of Bias The determined in the previous chapter probabilistic characteristics of the design variables are used for the calculation of partial safety factors applicable to the mean value of variables, R, Li, as Eq. (4-1) shows. For convenience these factors are called mean partial factors. Since engineers use the nominal values, Rn, Lni, in structural design, the partial load and resistance factors, ij, Ȗi applicable to the mean values of variables are converted to factors applicable to nominal values, ijn, Ȗni, by using the ratio shown in Eq. (4-2) also called bias, a statistical term usually used to express deviation from reality. These factors are called nominal factors. Hence, Eq. (4-1) yields Eq. (4-3) by using Eq. (4-2). ¦t k i ii LȖRij 1= (4-1) 82 valueNominal mean valueTrue Bias b (4-2) ¦t k i inLiinR LbȖRbij 1= or ¦t k i inninn LȖRij 1= (4-3) where, bR=R/Rn and bLi=Li/Lni are the bias for the resistance and loads, respectively. The calculation of bias for the resistance of piping for burst or yielding due to internal pressure and ultimate bending capacity is calculated according to Eq. (4-2). The true mean value in that equation is obtained as: )valuemeanSimulated(valuemeanTrue MX (4-4) where the simulated mean value is the mean value obtained from a variety of pipes strengths and dimensions, considering, moreover, the probabilistic characteristics of the basic random variables given in Chapter 3, and using Monte Carlo simulation. XM is a variable, which introduces the uncertainty of the strength model used to obtain the nominal value of the resistance. 4.2. Strength Model Uncertainty Detailed description for the determination of the uncertainty generated by the use of strength models in order to obtain the resistance of structural elements is encountered and discussed for different structures (Galambos, 1973; MacGregor, 1976; Ellingwood, et 83 al., 1980; Assakkaf, 1998). The strength model uncertainty or strength model bias is the result of simplifications and assumptions made for the derivation of the model. Initially, by comparing experimental results with the model predictions and using representative mean values of variables, the total bias shown in Eq. (4-5) is computed. VME XXX mean valuetiveRepresenta valuealExperiment BiasTotal (4-5) The mean of total bias, ȝȉǺ, can be therefore expressed as: VMETB ȝȝȝȝ (4-6) and the coefficient of variation, VTB, (considering XE, XM, XV as uncorrelated variables and neglecting terms higher than second order), as: 222 MVETB VVVV  (4-7) The variables XE, XV, and XM in Eq. (4-5) are defined as: mean valueTrue valuealExperiment EX (4-8a) which is a variable with mean value, ȝE equal to 1 and coefficient of variation, VE, that includes uncertainties arising from erroneous readings, inaccuracies of the gages and small errors in the setting of the experiments. Ellingwood, et al. (1980) proposes a value between 0.02-0.04 for VE. mean valuetiveRepresenta mean valueSimulated VX (4-8b) 84 which is a variable with mean value ȝV=1.0 and coefficient of variation, VV, which considers the uncertainties introduced by the basic variables such as the dimensions, and the strength of steel, and mean valueSimulated mean valueTrue MX (4-8c) which is the variable that as mentioned previously represents the uncertainty of the strength model alone. From the previous discussion and Eq. (4-6), it is evident that it has a mean value equal to that of total bias and coefficient of variation given as: 222 EVTBM VVVV  (4-9) In what follows, the total bias (mean and coefficient of variation) for strength models predicting the burst and yielding pressure of piping and their ultimate moment capacity is calculated. Moreover, the bias of the resistance, bR, according to Eq. (4-2), for each case is estimated. 4.3. Piping Burst and Yielding Due to Internal Pressure Piping burst is a brittle failure highly undesirable in all engineering designs. Yielding starts at the inner surface of the pipe and progress towards the outside surface. As the internal pressure in the pipe arises to a bursting point, the generated tension by the circumferential or hoop stress expands the walls of the pipe, creates thinning of the wall and finally causes a split along a longitudinal line, which usually is not an existing longitudinal joint. As yielding precedes the burst of the pipe, it is usual demand in the design codes that pipes should resist yielding and therefore permanent deformations 85 under normal operating conditions. The following sections describe criteria, models and assumptions for the design against burst and yielding of piping due to internal pressure. 4.3.1. Strength Models The models analytically examined here are the ones usually used in the ASME codes. They are based on the maximum principal stress criterion, on the Tresca criterion or are arbitrary approximations like the Barlow model. The models are simple, facilitating hence hand calculations, since their use goes back almost six decades and are based on elastic prediction of strength. In other words, the yield strength, Sy, is substituted by the ultimate strength, Su, in order to predict the burst pressure. Table 4-1 presents these models. It also shows two models, 6 and 7, based on elastic perfectly plastic behavior of the material, using the Tresca and the Von Mises Criterion, respectively. From these two criteria the Tresca is more conservative but favorites the calculations, since unlike the Von Mises criterion it does not consider the longitudinal stress, fL, and therefore is invariant for the different boundary conditions of the pipe (Szabó, 1972; Benham, et al., 1996). Nevertheless, the most common case is to consider pipes in closed end conditions. There are also models that consider the strain hardening of the material, when burst is examined. In such cases the material is selected to follow a law, usually the power law stress-strain relationship, where the strengthening coefficient, n, is calculated as a ratio of the ultimate strain. Such models are presented in Table 4-2. Burrows, et al. (1954) describe a great variety of models for the design of pipes for bursting. They group the models that give similar results and suggest the Boardman 86 model, also used in the ASME B&PV Code, Section III (2001), for the prediction of burst pressure of pipes, operating at room as well as at elevated temperatures. Table 4-1: Models Examined for Use in LRFD Model Description 1 Lamé ¸ ¹ ·¨ © §  ¸ ¹ ·¨ © § 1 15.0 2 t D t D t D Pff o oo H Maximum principal stress criterion 2 Boardman t tyD Pf oH 2 2 y=0.4 and for Do/t<6, y=d/(Do+d) Approximation of model 1 3 Thin Theory t dPf 2 + Tresca criterion with fR = 0 based on internal diameter 4 Average Diameter t D Pf m 2 + Pseudo-average hoop stress based on the average diameter 5 Barlow t D Pf o 2 + Pseudo-average hoop stress based on the outer diameter 6 d oD Pf ln 1 Tresca criterion for perfectly plastic behavior of the material 7 d oD Pf ln 1 2 3 Von Mises criterion for perfectly plastic behavior of the material, considering end caps and fL = 0.5fH Do=external diameter, d=internal diameter, f=equivalent stress, fH=hoop stress, fR=radial stress, fL=longitudinal stress, P=internal pressure, t=thickness 87 Table 4-2: Models for Burst Pressure and Strain Hardening Steel Criterion Strain Hardening Material Reference T re sc a uS mD t nu P 2 2 1 with corresponding ultimate strain 2 n uİ Based on pure power-law curve Zhu, et al. (2005); Steward, et al. (1994) Closed-End Cylinders u S mD t nuP 4 1 3 1  where the strengthening coefficient n is 596.0 1239.0 ¸ ¸ ¹ · ¨ ¨ © §  yS uSn with corresponding ultimate strain 3 n uİ Zhu, et al. (2005); Steward, et al. (1994); Cooper (1957); Svensson (1958) Closed-End Cylinders n oD uSt uP ¸¹ ·¨ © § ¸¸¹ · ¨¨© § ¸ ¹ ·¨ © § 3 12 3 2 with corresponding ultimate strain 2 n uİ Open-End Cylinders ¸¸¹ · ¨¨© § ¸ ¹ ·¨ © § oD uSt uP 2 3 2 with corresponding ultimate strain 3 2 n uİ Gerdeen (1976) V o n M is es Closed-End Cylinders ¸¸¹ · ¨¨© § ¸¸¹ · ¨¨© § ¸ ¹ ·¨ © §   toD tySuS uP 075.23 4 Material follows the Ramberg-Osgood power law Kirkemo (2001) Dm=average diameter, Do=external diameter, n=strength hardening coefficient, Pu=burst pressure, Sy=yield strength of steel, Su=ultimate strength of steel, t=thickness of steel, İu=ultimate strain 88 4.3.2. Total Bias Estimation for Yield and Burst Internal Pressure The nominal value of yield or burst pressure can be estimated, as mentioned previously by using a variety of models, when the hoop stress, fǾ or equivalent stress, f is substituted by the either the yield strength of steel, Sy, or the ultimate strength of steel, Su. The test data and the predicted by the models pressure are given in Tables 4-3a, and 4-3b for burst and yielding of pipes, respectively. Criteria for the selection of tests are that the pipes are straight, without defects and remote from discontinuities. The data include different categories of pipes, both thin ( 20/ dtd ) and thick pipes. It should be nonetheless mentioned that in some cases (e.g. yielding of stainless or carbon steel pipes) data is either not available or very limited and uniform in order to pursue the statistical analysis described in Section 4.2. In such cases the trend of the results is only considered and assumptions are made in order to evaluate the bias for the resistance of pipes, except for the yielding of pipes made of stainless steel, where no data at all are available. The coefficient of variation VE, as described in Section 4.2, is the same for all the models and assumed to be 0.02. VV is estimated, using Monte Carlo simulation and the statistical properties of the basic random variables presented in Chapter 3. VTB is the coefficient of variation of the total bias derived from the experimental results and lastly VM is calculated, using Eq. (4-9) with respect to the measured, experimental burst or yielding pressure. The mean value and coefficient of variation of the total bias are summarized in Tables 4-4a and 4-4b for burst and yielding pressure, respectively, while a distinction between thick and thin pipes is also mad T ab le 4 -3 a: E x p er im en ta l D at a U se d f o r th e E st im at io n o f T o ta l B ia s fo r B u rs t o f P ip es M o d el d (m m ) d/ t f y† (M P a) f u† (M P a) P b u rs t (M P a) 1 * 2 * 3 * 4 * 5 * 6 * 7 * R ef er en ce 8 0 .9 8 0 .9 7 1 .3 7 1 .3 8 1 .6 1 1 4 .7 1 1 4 .7 8 0 .9 8 0 .9 2 0 .2 2 0 .2 8 .1 8 .1 8 .2 9 .2 9 .2 2 0 .2 2 0 .2 2 3 8 5 1 2 3 9 1 6 4 2 4 7 4 2 9 4 .2 1 0 0 .6 9 7 .6 7 3 .5 7 6 5 7 .9 6 1 .8 3 6 .7 6 3 6 .7 6 8 4 .8 9 8 4 .8 9 8 4 .3 7 7 6 .1 1 7 6 .1 1 6 0 .3 6 6 0 .3 6 3 6 .5 0 3 6 .5 0 8 4 .0 7 8 4 .0 7 8 3 .5 5 7 5 .3 7 7 5 .3 7 5 9 .9 3 5 9 .9 3 3 8 .6 7 3 8 .6 7 9 6 .5 2 9 6 .5 2 9 5 .8 3 8 5 .2 2 8 5 .2 2 6 3 .4 9 6 3 .4 9 3 6 .8 4 3 6 .8 4 8 5 .9 1 8 5 .9 1 8 5 .3 7 7 6 .8 5 7 6 .8 5 6 0 .4 9 6 0 .4 9 3 5 .1 9 3 5 .1 9 7 7 .4 1 7 7 .4 1 7 6 .9 7 6 9 .9 7 6 9 .9 7 5 7 .7 7 5 7 .7 7 3 6 .8 7 3 6 .8 7 8 6 .2 6 8 6 .2 6 8 5 .7 1 7 7 .1 0 7 7 .1 0 6 0 .5 4 6 0 .5 4 4 2 .5 7 4 2 .5 7 9 9 .6 1 9 9 .6 1 9 8 .9 7 8 9 .0 2 8 9 .0 2 6 9 .9 1 6 9 .9 1 W el li n g er , et a l. ( 1 9 7 1 ) (C ) 3 5 .7 3 4 .9 4 8 .4 4 7 .6 6 1 .1 6 0 .3 7 3 .8 7 3 2 9 .8 2 1 .8 4 0 .3 2 9 .8 5 0 .9 3 7 .7 6 1 .5 4 5 .6 2 3 9 5 5 7 4 1 .4 1 5 5 .8 6 2 9 .5 3 9 2 3 .5 3 3 .4 1 8 .9 2 6 .7 3 6 .1 9 4 8 .7 4 2 6 .9 4 3 6 .1 9 2 1 .4 5 2 8 .7 8 1 7 .8 2 2 3 .8 8 3 5 .9 9 4 8 .4 1 2 6 .8 2 3 5 .9 9 2 1 .3 8 2 8 .6 5 1 7 .7 7 2 3 .7 9 3 7 .4 5 5 1 .0 7 2 7 .6 2 3 7 .4 5 2 1 .8 8 2 9 .5 6 1 8 .1 1 2 4 .4 2 3 6 .2 3 4 8 .8 3 2 6 .9 5 3 6 .2 3 2 1 .4 6 2 8 .7 9 1 7 .8 2 2 3 .8 9 3 5 .0 9 4 6 .7 8 2 6 .3 1 3 5 .0 9 2 1 .0 5 2 8 .0 7 1 7 .5 4 2 3 .3 9 3 6 .2 4 4 8 .8 6 2 6 .9 6 3 6 .2 4 2 1 .4 6 2 8 .8 0 1 7 .8 3 2 3 .9 0 4 1 .8 5 5 6 .4 2 3 1 .1 3 4 1 .8 5 2 4 .7 8 3 3 .2 6 2 0 .5 8 2 7 .5 9 P re to ri u s, e t al . (1 9 9 6 ) (S ) 5 7 3 .6 5 7 2 .3 5 7 2 .3 3 1 .5 3 0 .3 3 0 .3 6 3 5 5 5 6 7 3 3 6 6 7 4 1 .7 6 3 7 .8 6 4 0 .7 9 4 5 .0 4 4 2 .6 0 4 2 .6 0 4 4 .8 1 4 2 .3 8 4 2 .3 8 4 6 .5 2 4 4 .0 5 4 4 .0 5 4 5 .0 8 4 2 .6 5 4 2 .6 5 4 3 .7 4 4 1 .3 3 4 1 .3 3 4 5 .1 0 4 2 .6 6 4 2 .6 6 5 2 .0 8 4 9 .2 6 4 9 .2 6 M ax ey ( 1 9 8 6 ) (C ) 8 7 1 .1 1 5 2 .4 3 8 .7 1 5 .6 4 7 6 6 3 5 6 1 0 7 3 3 2 7 .9 3 8 6 .6 3 0 .7 0 8 8 .2 5 3 0 .5 6 8 7 .5 2 3 1 .5 1 9 7 .2 4 3 0 .7 2 8 8 .5 7 2 9 .9 6 8 3 .5 3 3 0 .7 2 8 8 .6 8 3 5 .4 8 1 0 2 .4 S te w ar t, e t al . (1 9 9 4 ) (C ) 3 8 4 3 7 8 2 8 .4 2 9 .5 2 9 0 7 9 3 5 4 9 9 3 5 3 6 .5 5 9 .6 3 7 .2 5 6 1 .1 8 3 7 .0 4 6 0 .8 5 3 8 .6 0 6 3 .3 2 3 7 .2 9 6 1 .2 5 3 6 .0 7 5 9 .3 1 3 7 .3 0 6 1 .2 7 4 3 .0 8 7 0 .7 5 R o y er , et a l. ( 1 9 7 4 ) (C ) 4 9 2 .1 5 4 9 3 .7 3 5 9 6 .9 5 9 5 .3 3 5 9 0 .5 5 6 2 .1 6 9 .2 9 4 .0 8 3 .4 6 2 .0 4 7 6 4 1 3 5 1 5 .2 4 4 4 6 1 0 5 3 6 .9 6 2 6 .6 5 7 8 .2 1 8 .7 3 1 6 .9 7 1 1 .7 7 1 5 .1 1 7 .9 5 1 9 .3 3 1 5 .3 0 1 1 .3 0 1 4 .8 4 1 8 .3 5 1 9 .2 7 1 5 .2 6 1 1 .2 8 1 4 .8 1 1 8 .3 0 1 9 .6 5 1 5 .5 2 1 1 .4 2 1 5 .0 2 1 8 .6 5 1 9 .3 3 1 5 .3 0 1 1 .3 0 1 4 .8 5 1 8 .3 6 1 9 .0 3 1 5 .0 9 1 1 .1 9 1 4 .6 7 1 8 .0 7 1 9 .3 4 1 5 .3 0 1 1 .3 0 1 4 .8 5 1 8 .3 6 2 2 .3 3 1 7 .6 7 1 3 .0 5 1 7 .1 4 2 1 .2 0 N ak ai , et a l. ( 1 9 8 2 ) (C ) * : R ef er t o T ab le 4 -1 , † : R ep re se n ta ti v e m ea n v al u e fo r st ee l an d n o t m ea su re d , (C ): C ar b o n S te el , (S ): S ta in le ss S te el 89 T ab le 4 -3 b : E x p er im en ta l D at a U se d f o r th e E st im at io n o f T o ta l B ia s fo r Y ie ld in g o f P ip es M o d el d (m m ) d/ t f y† (M P a) P y ie ld (M P a) 1 * 2 * 3 * 4 * 5 * 6 * 7 * R ef er en ce 4 9 2 .1 5 4 9 3 .7 3 5 9 6 .9 5 9 5 .3 3 5 9 0 .5 5 6 2 .1 6 9 .2 9 4 .0 8 3 .4 6 2 .0 4 7 6 4 1 3 5 1 5 .2 4 4 4 1 5 .1 1 4 .0 2 1 0 .2 1 2 .9 4 1 5 .5 9 1 5 .0 8 1 1 .7 7 8 .6 9 1 2 .2 0 1 4 .0 9 1 5 .0 4 1 1 .7 4 8 .6 8 1 2 .1 8 1 4 .0 5 1 5 .3 3 1 1 .9 4 8 .7 9 1 2 .3 5 1 4 .3 2 1 5 .0 9 1 1 .7 7 8 .6 9 1 2 .2 1 1 4 .1 0 1 4 .8 5 1 1 .6 0 8 .6 0 1 2 .0 6 1 3 .8 8 1 5 .0 9 1 1 .7 7 8 .7 0 1 2 .2 1 1 4 .1 0 1 7 .4 2 1 3 .5 9 1 0 .0 4 1 4 .1 0 1 6 .2 8 N ak ai , et a l. (1 9 8 2 ) (C ) 7 1 .3 8 0 .9 8 .1 2 0 .2 2 3 8 5 8 .9 2 5 .5 5 1 .6 7 2 2 .3 8 5 1 .1 7 2 2 .2 2 5 8 .7 5 2 3 .5 4 5 2 .2 9 2 2 .4 3 4 7 .1 2 2 1 .4 2 5 2 .5 1 2 2 .4 4 6 0 .6 3 2 5 .9 1 W el li n g er , et a l. ( 1 9 7 1 ) (C ) * : R ef er t o T ab le 4 -1 , † : R ep re se n ta ti v e m ea n v al u e fo r st ee l an d n o t m ea su re d , (C ): C ar b o n s te el 90 91 Table 4-4a: Probabilistic Characteristics of Total Bias for Burst Pressure Ratio internal diameter-thickness, (d/t) Any 20d >20 Model* Mean (ȝȉǺ) Coef. Var. (VTB) Mean (ȝȉǺ) Coef. Var. (VTB) Mean (ȝȉǺ) Coef. Var. (VTB) 1 1.050 0.091 1.066 0.090 1.046 0.092 2 1.056 0.091 1.076 0.090 1.051 0.093 3 1.002 0.090 0.951 0.075 1.015 0.090 4 1.047 0.090 1.055 0.088 1.045 0.092 5 1.092 0.100 1.159 0.100 1.075 0.096 6 1.046 0.090 1.052 0.087 1.044 0.092 7 0.906 0.090 0.911 0.087 0.905 0.092 *: Refer to Table 4-1 Table 4-4b: Probabilistic Characteristics of Total Bias for Yield Pressure and Carbon Steel Ratio internal diameter-thickness, (d/t) Any 20d >20 Model* Mean (ȝȉǺ) Coef. Var. (VTB) Mean (ȝȉǺ) Coef. Var. (VTB) Mean (ȝȉǺ) Coef. Var. (VTB) 1 1.116 0.060 NA NA 1.112 0.065 2 1.121 0.060 NA NA 1.116 0.065 3 1.077 0.067 NA NA 1.090 0.065 4 1.114 0.059 NA NA 1.111 0.065 5 1.150 0.072 NA NA 1.133 0.067 6 1.113 0.059 NA NA 1.111 0.064 7 0.964 0.059 NA NA 0.962 0.064 *: Refer to Table 4-1, NA=Not Available, since only one test is available Figures 4-1a to 4-1g show the histograms of the estimated total bias for all seven models for burst and Figures 4-2a to 4-2g those for yielding of pipes. Normal and lognormal distributions are moreover fitted in the histograms. 92 0.889 0.967 1.045 1.123 1.201 1.279 Total bias 0 2 4 6 8 10 12 F re q u e n c y Normal Lognormal 1 0.893 0.972 1.051 1.130 1.209 1.288 Total bias 0 2 4 6 8 10 12 F re q u e n c y 2 Normal Lognormal Figure 4-1a: Total Bias of Burst Pressure Figure 4-1b: Total Bias of Burst Pressure for Model 1 for Model 2 0.859 0.931 1.002 1.073 1.144 1.216 Total bias 0 1 2 3 4 5 6 7 8 9 10 F re q u e n c y 3 NormalLognormal 0.888 0.965 1.043 1.121 1.198 1.276 Total bias 0 2 4 6 8 10 12 F re q u e n c y 4 Normal Lognormal Figure 4-1c: Total Bias of Burst Pressure Figure 4-1d: Total Bias of Burst Pressure for Model 3 for Model 4 0.916 1.000 1.084 1.168 1.252 1.336 Total bias 0 1 2 3 4 5 6 7 8 9 10 F re q u e n c y 5 Normal Lognormal 0.887 0.965 1.042 1.120 1.197 1.275 Total bias 0 2 4 6 8 10 12 F re q u e n c y Normal Lognormal 6 Figure 4-1e: Total Bias of Burst Pressure Figure 4-1f: Total Bias of Burst Pressure for Model 5 for Model 6 93 0.769 0.836 0.903 0.970 1.037 1.104 Total bias 0 2 4 6 8 10 12 F re q u e n c y 7 Lognormal Normal Figure 4-1g: Total Bias of Burst Pressure for Model 7 1.001 1.049 1.096 1.144 1.191 Total bias 0 1 2 3 4 F re q u e n c y 1 Normal Lognormal 1.004 1.052 1.099 1.147 1.195 Total bias 0 1 2 3 4 5 F re q u e n c y 2 Lognormal Normal Figure 4-2a: Total Bias of Yield Pressure Figure 4-2b: Total Bias of Yield Pressure for Model 1 and Carbon Steel for Model 2 and Carbon Steel 0.985 1.032 1.080 1.127 1.174 Total bias 0 1 2 3 F re q u e n c y 3 Lognormal Normal 1.001 1.048 1.096 1.144 1.191 Total bias 0 1 2 3 4 F re q u e n c y 4 Lognormal Normal Figure 4-2c: Total Bias of Yield Pressure Figure 4-2d: Total Bias of Yield Pressure for Model 3 and Carbon Steel for Model 4 and Carbon Steel 94 1.017 1.075 1.133 1.192 1.250 Total bias 0 1 2 3 F re q u e n c y 5 Lognormal Normal 1.001 1.048 1.096 1.144 1.191 Total bias 0 1 2 3 4 F re q u e n c y 6 Lognormal Normal Figure 4-2e: Total Bias of Yield Pressure Figure 4-2f: Total Bias of Yield Pressure for Model 5 and Carbon Steel for Model 6 and Carbon Steel 0.867 0.908 0.949 0.990 1.032 Total bias 0 1 2 3 4 F re q u e n c y 7 Lognormal Normal Figure 4-2g: Total Bias of Yield Pressure for Model 7 and Carbon Steel Table 4-5 shows that all models, except from model 7, give similar results, when burst is considered. More specifically, model 7 gives a value of total bias less than one, and therefore is not recommended to be used for the prediction of burst or yield pressure. Model 3 based on the thin theory yields a mean value equal to unity, when burst is considered. As shown in Table 4-4a, the model is not conservative for thick pipes and therefore it is also unsuitable since it underestimates the burst pressure for thick pipes. 95 Table 4-5: Models Examined and Parameters Calculated for the Prediction of Burst Pressure Mean Coefficient of Variation Model* (ȝȉǺ) (VTB) (VV) (VM) (VE) 1 1.050 0.091 0.072 0.052 0.02 2 1.056 0.091 0.072 0.052 0.02 3 1.002 0.090 0.073 0.049 0.02 4 1.047 0.090 0.072 0.050 0.02 5 1.092 0.100 0.071 0.068 0.02 6 1.046 0.090 0.072 0.050 0.02 7 0.906 0.090 0.072 0.050 0.02 *: Refer to Table 4-1 4.3.3. Bias of Burst and Yield Pressure Tables 4-6a and 4-6b give the calculated bias of burst and yield pressure for the models discussed herein, respectively. For carbon steel and yielding, a column is moreover added showing an assumed coefficient of variation for the model uncertainty variable, XM, since it could not be derived according to the procedure of Section 4.2 due to a uniform and small sample that gives an unrealistic small coefficient of variation. Nonetheless, as mentioned earlier, no estimation is provided for the yielding of pipes made of stainless steel, since for these pipes there is no information at all. Table 4-6a: Probabilistic Characteristics of Bias of Burst Pressure for Carbon and Stainless Steel Stainless Steel Carbon Steel Model* Mean Coef. of Variation Mean Coef. of Variation 1 1.184 0.091 1.205 0.091 2 1.190 0.091 1.211 0.091 3 1.130 0.094 1.150 0.094 4 1.180 0.091 1.201 0.091 5 1.230 0.101 1.252 0.101 6 1.179 0.091 1.200 0.091 7 1.021 0.091 1.040 0.091 *: Refer to Table 4-1 96 Table 4-6b: Probabilistic Characteristics of Bias of Yield Pressure for Carbon Steel Carbon Steel Model* Assumed Coefficient of Variation for XM Mean Coefficient of Variation 1 0.05 1.257 0.104 2 0.05 1.263 0.105 3 0.06 1.214 0.110 4 0.05 1.255 0.106 5 0.06 1.295 0.109 6 0.05 1.200 0.104 7 0.05 1.086 0.105 *: Refer to Table 4-1 4.4. Ultimate Moment Capacity of Piping A failure due to excessive bending is of principal interest for the design of piping. Moreover, the equations in the ASME Code, as discussed in Chapter 2, are based on the assumption that pipes are subjected to large bending moments compared to other loading e.g., shear or axial forces. Bending moments result from a variety of loading such as own weight, seismic, other reversing or not reversing dynamic loadings and thermal loads, when free expansion is restraint. A failure due to excessive bending can be visualized as local buckling, ovalization of the cross-section or collapse. The following sections describe design strength models for the ultimate moment capacity and calculate the total bias and bias for the ultimate flexural strength of straight pipes, operating at room temperature. 4.4.1. Strength Models for Failure Moment Nuclear pipes, as explained in Chapter 3, are principally made of carbon (e.g., SA 106 Gr. B) or stainless austenitic steel (e.g., Types 304, 316, and 347) that show corrosion resistance and good performance under high temperatures. These materials are ductile and their behavior can be considered as linear elastic–perfectly plastic, as Figure 97 4-3 shows. Given this fact, limit theory is considered for the analysis of pipes. More specifically, as load increases a hinge is formatted and then a redistribution of moments occurs until the number of hinges exceeds by one the degree of indeterminacy of the pipe and hence a collapse mechanism is formed. For limit theory small deformations of pipes are considered and the failure moment equals: pyPf ZSMM (4-10) where Sy is the yield strength and Zp is the plastic section modulus. Figure 4-3: Perfectly Plastic and Bilinear Approximations of Steel Behavior Sherman (1976) examined if the above theory is applicable to straight pipes by conducting experiments for simply supported, fixed ends and cantilever pipes of different ratios of external diameter to thickness, Do/t. He concluded that pipes actually develop a collapse mechanism, whereas pipes of Do/t approximately equal to 100 fail due to buckling, without even being able to form a hinge first. 98 If the material is also considered work-hardening, the failure moment has the following upper bound: puuf ZSMM (4-11) where Su is the ultimate strength of the material. Should now a bilinear approximation of work hardening material is considered, as shown in Figure 4-3, the failure moment, Mf, is given as: ZSSZSM yfpyf )(  (4-12) where Zp is the plastic and Z is the elastic cross-section modulus. In Eq. (4-12) Sf is the stress that corresponds to a predefined failure strain, ef, different from the ultimate, eu, as shown in Figure 4-3. If the stress–strain curve is not available, Sf can be considered equal to the ultimate strength, Su (Belke, 1983). In the following sections the total bias for excessive bending is estimated according to Eq. (4-5). The experimental results of bending tests on straight pipes without and with internal pressure are considered and are based on bibliographical data (Rodabaugh, 1978; Belke, 1983). Rodabaugh (1978) summarizes experimental data from other researchers (Schroeder, et al., 1974; Ellyin, et al., 1976, 1977; Franzen, et al., 1972; Gerber, 1974; Sherman, 1974; Jirsa, 1972; and Sorenson, 1970), and graphically calculates the experimental failure moment as that, which causes a deformation equal to two times the elastic deformation. As explained by Ayyub, et al. (2005), there are also other ways to define the failure moment of a pipe. Belke (2003) does not refer to the way experimentally the collapse failure moment is estimated. Simply supported, clamped and cantilevered pipes are examined. The models evaluated are these of Eqs. (4-10) and (4- 99 11) because are more suitable, due to their simplicity, to be used in an LRFD format. The data used for the analysis are summarized in Tables 4-7a and 4-7b for bending without and with internal pressure, respectively. The mean value of steel shown in these tables is again the representative mean of the material, and where not available, a mean of the reported data was considered. As in the case of burst and yielding of pipes, data for bending are also limited and in some cases not available. 100 Table 4-7a: Experimental Data Used for the Estimation of Total Bias for Bending of Pipes Do (mm) Do/t fy† (MPa) fu† (MPa) Mp(Eq. 4-10) (kN m) Mu (Eq. 4-11) (kN m) Mexp (kN m) Reference 33.7 33.7 33.7 33.7 32.3 32.3 32.3 32.3 29.8 29.8 29.8 29.8 12 12 12 12 17 17 17 17 33 33 33 33 264 600 0.71 0.71 0.71 0.71 0.46 0.46 0.46 0.46 0.20 0.20 0.20 0.20 1.61 1.61 1.61 1.61 1.06 1.06 1.06 1.06 0.45 0.45 0.45 0.45 1.24 1.19 1.31 1.25 0.68 0.71 0.66 0.62 0.28 0.27 0.27 0.24 Belke (1984) (S) 27.13 27.13 14.8 14.8 259.94 579.17 0.34 0.34 0.68 0.68 0.28 0.40 Franzen, et al. (1972), (S) 105.66 81.03 106.43 160.78 106.43 12.3 25.3 25.1 24.1 25.1 259.94 579.17 21.02 5.08 11.53 41.24 11.53 46.89 11.19 25.65 91.97 25.65 29.38 4.52 11.30 63.27 11.30 Gerber (1974) (S) 80.01 90.68 21 51 194.44 4.29 2.71 4.18 2.82 Schroeder, et al. (1974), (C) 108.20 108.20 105.66 108.20 108.20 100.84 111.25 111.25 111.25 18 18 12.3 18 18 7.5 36.5 36.5 36.5 277.52 488.16 17.51 17.51 22.48 17.51 17.51 28.81 9.94 9.94 9.94 30.72 30.72 39.50 30.72 30.72 50.64 17.42 17.42 17.42 23.73 22.60 29.38 24.86 22.56 39.55 15.82 18.08 19.21 Gerber (1974) (C) 67.56 21.3 592.27 7.80 8.59 Ellyin, et al. (1976) (C) 117.60 146.56 30.7 46.2 277.52 488.16 13.78 18.08 24.29 31.86 14.69 18.98 Ellyin, et al. (1977) (C) 258.32 265.94 273.05 17.4 33.6 107.5 277.52 488.16 244.73 146.43 51.64 430.48 257.61 90.73 259.87 135.58 39.55 Sherman (1974) (C) 273.05 273.05 406.4 508 46.1 30.7 61.5 78.4 365.43 154.34 226.76 385.74 692.38 158.18 192.08 361.56 576.23 Jirsa, et al. (1972) (C) 101 Table 4-7a (Continued) Do (mm) Do/t fy† (MPa) fu† (MPa) Mp (Eq. 4- 10) (kN m) Mu (Eq. 4-11) (kN m) Mexp (kN m) Reference 32.56 500.13 32.08 32.08 273.05 32.05 32.03 32.03 32.05 32.08 39.4 63.1 94 99 108 101 115 112 103 100.2 586.06 360.60 586.06 309.58 586.06 0.45 692.38 0.23 0.23 57.51 0.23 0.11 0.23 0.23 0.19 0.46 796.56 0.15 0.17 39.55 0.14 0.11 0.12 0.14 0.17 Sorenson (1970) (C) (C): Carbon steel, (S): Stainless steel, †: Representative mean value for steel and not measured Table 4-7b: Experimental Data Used for the Estimation of Total Bias for Bending of Pipes with Internal Pressure Do (mm) Do/t fy† (MPa) fu† (MPa) P (MPa) Mp (Eq. 4-10) (kN m) Mu (Eq. 4- 11) (kN m) Mexp (kN m) Reference 33.7 41.26 194.44 5.18 3.40 3.16 Schroeder, et al. (1974), (C) 27.13 27.13 27.13 27.13 27.13 27.13 27.13 27.13 14.83 14.83 14.83 14.83 14.83 14.83 14.83 14.83 277.52 488.16 13.48 20.68 27.58 20.64 27.58 15.15 18.97 24.13 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.24 0.21 0.16 0.34 0.32 0.33 0.28 0.21 Franzen, et al. (1974), (S) (C): Carbon steel, (S): Stainless steel, †: Representative mean value for steel and not measured 4.4.2. Total Bias for Pure Bending From the examined experimental results it is evident that the failure moment (ultimate bending resistance) is dependent on the ratio of the external diameter to the thickness of the pipe, Do/t. More specifically, for the model of Eq. (4-10), pipes with Do/t>75 are susceptible to failure due to buckling and ovalization of the cross-section and 102 therefore are expected not to reach the limit pressure, MP, as shown in Figure 4-4, for carbon steel. For Do/t<50 the failure moment either exceeds the moment calculated by Eq. (4-10) or approaches it. Computation results are shown in Figure 4-5 for both stainless and carbon steel. In addition, in Table 4-8a the statistical properties of the total bias for Eq. (4-10) are given, where moreover the coefficient of variation for the model, VM, is derived. 0.684 0.779 0.873 0.968 Total bias 0 1 2 3 4 5 6 7 8 F re q u e n c y Lognormal Normal Figure 4-4: Total Bias for the Bending Resistance According to Eq. (4-10) for Do/t>75 and Carbon Steel 103 0.896 1.134 1.371 1.608 1.845 Total Bias 0 1 2 3 4 5 6 7 F re q u e n c y Normal Lognormal 0.847 1.120 1.393 1.666 1.939 Total bias 0 2 4 6 8 10 12 F re q u e n c y Lognormal Normal (a) (b) Figure 4-5: Total Bias for the Bending Resistance According to Eq. (4-10), (a) Stainless Steel for Do/t<50, (b) Carbon Steel for Do/t<65 Table 4-8a: Statistical Descriptive of Total Bias for Eq. (4-10) and Pure Bending, and Derivation of VM Properties Coefficient of Variation Do/t Steel Mean (ȝȉǺ) (VTB) (VV) (VM) (VE) <50 Stainless 1.373 0.207 0.16 0.13 0.02 65d Carbon 1.216 0.242 0.10 0.22 0.02 t75 Carbon 0.780 0.126 0.10 0.07 0.02 The model of Eq. (4-11) gives always values of total bias less than one and as expected is not conservative. Figure 4-6 shows histograms of the total bias with fitted distributions for Do/t<50 for both stainless and carbon steel. Table 4-8b shows further the statistical properties of the model and the derivation of the coefficient of variation, VM. Table 4-8b: Statistical Descriptive of Total Bias for Eq. (4-11) and Pure Bending, and Derivation of VM Properties Coefficient of Variation Do/t Steel Mean (ȝȉǺ) (VTB) (VV) (VM) (VE) <50 Stainless 0.608 0.202 0.08 0.18 0.02 <50 Carbon 0.766 0.223 0.08 0.21 0.02 104 0.402 0.505 0.607 0.709 0.812 Total bias 0 1 2 3 4 5 6 7 F re q u e n c y Lognormal Normal 0.526 0.670 0.814 0.958 1.102 Total bias 0 1 2 3 4 5 6 7 F re q u e n c y Lognormal Normal (a) (b) Figure 4-6: Total Bias for the Bending Resistance According to Eq. (4-11) for Do/t<50, (a) Stainless Steel, (b) Carbon Steel 4.4.3. Total Bias for Bending with Internal Pressure As presented by Rodabaugh, et al. (1978) and as expected by theory, internal pressure reduces the moment capacity of straight pipes. Experimental results were evaluated for data summarized by Rodabaugh, et al. (1978) and the total bias was estimated. Figure 4-7 shows histograms of calculated total bias for stainless steel for both models of Eqs. (4-10) and (4-11). Moreover, Tables 4-9a and 4-9b give the statistical properties for the total bias and the derivation of the coefficient of variation for the models of Eqs. (4-10) and (4-11), respectively. 105 0.519 0.717 0.914 1.112 Total bias 0 1 2 3 4 5 Fr eq ue nc y Lognormal Normal 0.233 0.322 0.410 0.499 Total bias 0 1 2 3 4 5 Fr eq ue nc y Lognormal Normal (a) (b) Figure 4-7: Total Bias for the Bending Resistance with Internal Pressure for Stainless Steel and Do/t<50, (a) According to Eq. (4-10), (b) According to Eq. (4-11) Table 4-9a: Statistical Descriptive of Total Bias for Eq. (4-10) and Bending with Internal Pressure, and Derivation of VM Properties Coefficient of variation Do/t Steel Mean (μΤΒ) (VTB) (VV) (VM) (VE) <50 Stainless 0.857 0.248 0.16 0.19 0.02 Table 4-9b: Statistical Descriptive of Total Bias for Eq. (4-11) and Bending with Internal Pressure and Derivation of VM Properties Coefficient of variation Do/t Steel Mean (μΤΒ) (VTB) (VV) (VM) (VE) <50 Stainless 0.385 0.248 0.16 0.19 0.02 4.4.4. Bias of Flexural Strength for Straight Pipes Considering the probabilistic properties of the variables in Chapter 3 and the derived model uncertainty, the bias for the bending resistance is estimated according to Eqs. (4-2) and (4-4). Results are shown comprehensively in Table 4-10 for all the examined cases. 106 Table 4-10: Bias for Bending Resistance and All Examined Cases Bias (Eq. 4-10) Bias (Eq. 4-11) Steel Do/t Mean COV Mean COV Bending Stainless <50 1.71 0.21 0.68 0.20 Carbon d65 1.37 0.24 0.87 0.23 t75 0.88 0.12 NA NA Bending with internal pressure Stainless <50 1.07 0.25 0.43 0.21 NA: Not Available 4.5. Conclusions A definitive step for the development of Load and Resistance Factor Design (LRFD) for nuclear pipes is the quantification of the uncertainties introduced by the design strength models. The methodology for that quantification was described, and in addition, models for the burst or yielding due to internal pressure and for the flexural resistance of pipes were evaluated. Further the bias of the piping resistance was estimated for these cases. The collected data for the analysis are limited and moreover the provided information concerns only straight pipes operating at room temperature. Therefore, results and assumptions can be further expanded by the acquisition of additional data. Moreover, in this section the bias was estimated only from experimental data, whereas the use of advanced finite element analysis of pipes can also provide useful data for the bias of the resistance. 107 CHAPTER 5: LOAD COMBINATIONS AND PERFORMANCE FUNCTIONS FOR PIPING This chapter presents load combinations and the resultant performance functions necessary for the LRFD for piping, as explained in Chapter 2. The partial load and resistance factors corresponding to the performance functions are summarized in Chapter 6, while analytical calculations are provided in Appendix C. Performance functions for fatigue are given separately in Chapter 7. 5.1. Performance Functions The performance functions for the LRFD are based on the criteria and format used in the ASME B&PV Code, Section III. This is judged necessary, since a propable transition- even in the form of a code case - from the current Allowable Stress Design to the LRFD, should provide the familiar formats to the designers such as it will be easier for them to understand the philosophy of the LRFD. In consequence, the Boardman model is used for the design for internal pressure, and the Tresca criterion with the assumption of large moments for the combination of primary loads. As it has already been mentioned, LRFD favorites future changes in models and characteristics in variables. Therefore the amelioration of the current design criteria will be easier accomplished by using the LRFD rather than the ASD. 108 The performance functions are given in the following section based on combination of loads that coincide in time. The Design and Service Levels A and B equations have the steel yield strength as the resistance variable. Service Level D has the ultimate steel strength, since larger displacements are expected in order for strength hardening to develop and having in consequence an expansion of the Tresca yield surface. For the development of the performance functions for Service Level C there is actually a challenge. By examining the allowable stresses of the Code given in Table 2-2 of Section 2.1.4.1 for the combination of primary loads, it can be inferred that the yield strength of steel is exceeded more than an approximated value of the shape factor. Hence, the use of the ultimate strength seems reasonable such as calibration of the current Code equations will not lead in extremely low values for the target reliability index, ȕ, which can be unusual for a structural design. Nevertheless, in design examples with arbitrary chosen load intensities the use of the yield strength of steel permitted reasonable safety margins, but since calibration of a great range of real piping design calculations is necessary to confirm this inference, the ultimate strength is selected as the resistance variable for Service Level C performance functions. Should the yield strength of steel is used in the performance functions of Service Level C, a lower value of the target reliability index is to be expected compared to that for performance functions of Service Level B. In Table 5-2 of the following section, it can be seen that the performance functions g9, g14, and g19 include a factor E that is applied to the sum of dynamic loads. Since no analytical coputations were made in order to evaluate its value, for the 109 calculation of the partial load and resistance factors E is considered as constant equal to one. Nevertheless, its mean value is expected to be less than one and its physical meaning is that by summing the individual stress quantities of dynamic loading may result in conservative values as by combining the dynamic loading before the calculation of stresses. 5.2. Load Combinations In this chapter primary loads/stresses are considered as described in Chapter 3, (secondary stresses cause fatigue or ratcheting) and are summarized below: x The sustained weight x Internal Pressure x Earthquake load (OBE, SSE) x Mechanical loading (e.g. sudden valve closure, water hammer) x Accidental loading (LOCA) Loads that coincide in time are combined and moreover the Turkstra’s Rule is considered, which simply states that when combining loads, it is logical to assume that only one load can attain its maximum value for the specified design life of the component, whereas the other loads attain an arbitrary-point-in-time value. Other rules for combining loads are the Borges model (Borges, et al., 1971) and the combination of loads by considering them as Poisson pulse processes (Wen, 1990). Table 5-1 presents the loads to be combined. Also secondary loads (thermal) are shown, although as mentioned earlier these loads are considered only in fatigue evaluations. The performance functions arising from these load combinations are provided in Table 5-2. 110 Table 5-1: Loads to Be Combined for Different Service Levels Service Limit Weight Pressure Mechanical* OBE SSE LOCA Thermal 3 3 Design 3 3 3 3 3 3 3A 3 3 3 3 3 3 3 3 3 3 B & C 3 3 3 3 3 3 3 3 3 3 3 3 D 3 3 3 *Dynamic loadings other than earthquake 111 Table 5-2: Recommended Performance Functions for Straight Pipes Service Level Performance Function Definition of Variables Design HSySg  1 ASySg  2 PDesSASySg  3 t ytoDDes P H S 2 )2(  Z AM AS t D Des P PDesS 4 o A (Operating Condition) max 4 PSASySg  t DP PS 4 max o Z AM AS B (Upset Loading Condition) HSySg  5 MSASPBSySg  6 OSASySg  7 OSPOSASySg  8 )( 9 O SMSEASPO SySg  t DP PBS B 4 o t DP POS O 4 o t ytoDB P H S 2 )2(  PZ OM OS PZ AM AS P Z MM MS C (Emergency Loading Condition) HSuSg  10 OSASuSg  11 OSASPOSuSg  12 MSASPCSSg u  13 )( 14 OSMSEASPOSuSg  t D O P POS 4 o t D C P PCS 4 o t ytoDC P H S 2 )2(  PZ AM AS PZ OM OS 112 Table 5-2: (Continued) Service Level Performance Function Definition of Variables D (Faulted Loading Condition) HSSg u  15 SAu SSSg  16 SSASSPSuSg  17 LSPDSASuSg  18 PSSLSSSEASuSg  )(19 t D D P PDS 4 o t ytoDD P H S 2 )2(  PZ SM SS PZ AM A S PZ L M LS t D S P PSS 4 o Nomenclature Sy=Nominal Yield Strength of Steel, Su=Nominal Ultimate Strength of Steel, SH =Hoop Stress, Si, i=PDes, Pmax, PC , PD, PO, PS= Stress due to Normal Operating Pressure, Maximum Operating Pressure, Pressure Coincident with Service Limit C Conditions, Pressure Coincident with Service D Conditions, Pressure coincident with OBE, Pressure coincident with SSE SL. SM = Stresses Due to Mechanical Loading, SL=stress due to LOCA loadind, y = 0.4, Z = Elastic Section Modulus, ZP = Plastic Section Modulus, SO= Stress Due to OBE, SS = Stress Due to SSE, M = Moment, E=Factor for Combining Dynamic Loading 5.3. Normalization of Stresses with Respect to Sustained Weight For the calculation of the partial load and resistance factors the performance functions listed in Table 5-2, except from the performance functions g1, g5, g10, and g15, are normalized with respect to the stress due to sustained weight. All the stresses noted as Si in that table, therefore become fi as follows: SA i i ȝ S f (5-1) 113 where Si is the stress due the load of origin, i, and ȝSA the mean stress due to sustained weight. The normalized stress fi becomes a new variable with the following characteristics: AS iS if ȝ ȝ ȝ (5-2a) ii fCOVSCOV ofof (5-2b) ii fS ofPDFofPDF (5-2c) where ȝfi, ȝSi, ȝSA are the mean values of variables fi, Si, and SA, respectively. The acronyms COV and PDF stand for Coefficient Of Variation and Probability Density Function, respectively. This procedure is necessary in order to be able to consider different magnitude of loading for the pipes under consideration. The sustained weight was selected to be used for the normalization of other stresses, for uniformity reasons, since it is present in all the performance functions. Of course, other variables could have been used in order to normalize the stresses. The proposed LRFD equations intend to serve as an alternative to the equations of the ASME Code, Section III and namely the design by rule part, and therefore should take into consideration a representative range of the loading magnitude for piping. This makes necessary the normalization of stresses. Below, in addition, some explanation is provided in order to demonstrate the great variability of loading that piping is subjected to. Hence: 114 x Nuclear pipes come with different sizes, especially those complying with the design rules of Classes 2 and 3, which constitute the majority of piping, and are part of different sections of the nuclear plants. x The earthquake loading not only varies between different territories in U.S. (west- east), but significantly varies within close regions where special tectonic characteristics define the subsoil’s behavior to seismic excitations. Even in the same plant the pipes can be differently impacted, depending on the floor they are mounted on, the characteristics of the building structure, and their own properties (flexibility, mass). x Pressure differs highly depending on the size of the pipe and the operating temperature. The produced longitudinal stresses attain diverse values and if they are to be compared with the stresses due to sustained weight, diverse values are obtained for light or heavy pipes pressurized at different levels. x Generally, values of loading being accepted as Service Limit B for some pipes, for others can be loading characterized as Service Limit C, etc. This fact may also lead to small and large load ratios for all service levels. x The sustained weight can vary significantly even for the same schedule pipes depending on the sustained weight (superimposed valves, etc.). This method of analysis was also considered for the development of LRFD equations for other structures, buildings, ships, etc., too. In order to obtain values for the ratio of Eq. (5-2a) calculations for simple pipes-beams were considered and the ranges were extended such as to include feasible quantities for the variables under consideration. 115 These ranges and the summary of the probabilistic characteristics of normalized variables are given in Chapter 6. 5.4. The Target Reliability Index, ȕ, for Piping The use of a probabilistic based approach for the design of nuclear components necessitates that a target probability of failure be established before the design proceeds. This probability refers to the design life of the structure, which in the case of nuclear plants and components is 40 years, with a possibility of extension for another 20 years. The target probability of failure sometimes may also be expressed as probability of failure per year or for example for pipelines as a probability of failure per year and kilometer. Although the actual probability of failure can also be estimated, using Monte Carlo simulation or by collecting historical data on relevant, reported failures of pipes, when selecting the reliability index and in consequence assigning the probability of failure to pipes, the following factors should be considered: a. The consequences that a failure can cause e.g., deaths, injuries, economic losses, environmental damage, etc. and the cost for increasing the reliability of piping. Accordingly, a cost-benefit analysis can be performed for finding an optimal design. Nevertheless, this method of analysis necessitates the use of higher level reliability methods. b. The probability of occurrence of the loading event. Loading such as weight and pressure are associated with smaller probabilities of failure compared to loads with smaller probability of occurrence, such as earthquake loading or a shock wave from a pipe that breaks. Consequently, in case of similar performance functions, higher 116 reliability indices are expected for Service Limits A and B, where the loading is operational or occasional as compared to Service Limits C and D, where in the case of the latter, the occurrence of loading might as well not be within the service life of the component. c. The probabilities of failure, which are implicit in the current design. This procedure, known also as code calibration, quantifies the probabilities of failure for the design of piping according to the ASME B&PV Code, Section III. The results from calibration are expecting to give a range of probabilities of failures. This way, the probability of failure actually becomes a variable, where a histogram can be created and a corresponding value -often in the upper percentile of the distribution- can be considered as the target reliability index for future designs for piping according to LRFD. Therefore, LRFD will become more consistent as compared to Allowable Stress Design and the probability of failure for the component will not only be more consistent but also known. d. Assignment of an initial probability of failure for the whole plant. Schwartz, et al. (1981) suggest that the acceptable probability of failure or ȕ for certain categories of pipes can be assigned by considering an initial acceptable probability of failure for the whole plant that by using risk analysis methods can break down to a suggested value of ȕ for the pipes under consideration in a system’s framework. As an example for suggested probabilities of failure for nuclear structures of Category I, such as containments, etc. are given for different load combinations in Table 5-3 by Schuëller, et al. (1992). 117 Table 5-3: Lifetime Limit State Probabilities for Nuclear Structures (for 40 years) Load Combination Limit State Probability ȕ D+L @.).(11 SLPȕ )  (5-3) where P(L.S.)=the limit state probability. A pilot project was successful in calculating probability of piping failures based upon deterministic Code rules (Barnes, et al., 2000). Phase I investigations found that pipe cross-section failure probabilities are generally below 10-6 per year, when credit is taken for the probability of the seismic event occurring. Phase II analysis used the example from Phase I modified to account for cyclic stresses in hot piping. In this instance, the cumulative leak probabilities were computed to be quite high, as high as 3x10-3 per year. The double-ended pipe break probabilities were several orders of magnitude lower. In this study, calculations will address a range of reliability indices that are judged suitable for each performance function, understanding that in order to estimate the target reliability index for each performance function the above criteria should be considered. 118 The Code calibration, which is the procedure of determining the target (desired) reliability index for each load combination based on inconsistent probabilities of failure that exist in the current Code and stochastic analysis of loads (Schwartz, et al., 1981) is out of the scope of this work. Nonetheless, only in some example calculations presented in Chapter 6, calibration is attempted in order, for the specific examples, to quantify for illustration purposes the implied reliability indices of the ASME Code design equations. Table 5-4 presents values of ȕ and the corresponding probability of failure, Pf, addressing the design life of piping calculated according to Eq. (2-20). Table 5-4: Reliability Index ȕ and Corresponding Probability of Failure, Pf Reliability Index, ȕ Probability of failure, Pf 1.5 6.68E-02 2.0 2.28E-02 2.5 6.21E-03 3 1.35E-03 3.5 2.33E-04 4.5 3.40E-06 5.5 1.90E-08 6 9.87E-10 7 1.82E-12 8 2.5 there is a slight difference for carbon steel. The higher difference is noticed for ȕ=4.5 and carbon steel to be 3.24% with an impact on ȕ of 0.02. Nevertheless, the use of either distribution is recommended, since there is no significant impact on the value of the target reliability index, ȕ. 137 Stainless Steel 0 0.2 0.4 0.6 0.8 1 1.2 1 1.5 2 2.5 3 3.5 4 4.5 5 Reliability Index, ȕ A d ju s te d R e s is ta n c e F a c to r, ij ij(lognormal) ij(normal) Figure 6-9a: Adjusted Nominal Resistance Factor for the Internal Pressure having Normal and Lognormal Distribution for Service Limit A and Stainless Steel Carbon Steel 0 0.2 0.4 0.6 0.8 1 1.2 1 1.5 2 2.5 3 3.5 4 4.5 5 Reliability Index, ȕ A d ju s te d R e s is ta n c e F a c to r, ij ij(lognormal) ij(normal) Figure 6-9b: Adjusted Nominal Resistance Factor for the Internal Pressure having Normal and Lognormal Distribution, for Service Limit A and Carbon Steel 138 6.2.4. Computational Example This section provides a computational example, where for two pipes made of stainless and carbon steel the required minimum thickness is first estimated. Then, the maximum pressure for all service limits is computed according to the ASME B&PV Code, Section III. Calibration is performed, that is the partial safety factors and the corresponding reliability index, are calculated such as the LRFD equations will give the same results as the ones obtained from the Code equations. Nevertheless, in order to derive the implied reliability indices for the design for internal pressure according to the Code, a large number of calculations are needed, which as previously mentioned are above the scope of this work. Hence, a Class 1 pipe made of carbon steel A106 B and a pipe made of austenitic steel Type 304 are designed for operation at room temperature and operation at 400oF. The pipe’s outside diameter is Do=12.75 in and the design pressure, PDes, assumed to be 800psi for all cases. The thickness of the pipe and the allowable (maximum) pressure for all service limits are to be estimated. Table 6-4 shows the mechanical properties of the considered materials such as the nominal yield and ultimate strength of steel, namely the minimum specified yield or ultimate strength of steel, as well as the allowable stresses, all given in the ASME B&PV Code, II, Table 2-a. Table 6-5 presents the computations. In the LRFD column the derived reliability index, ȕ, and the partial factors are provided, in order to obtain the same thickness and maximum pressure as the ASME Code equations. The additional thickness, A, which is added and accounts for corrosion- erosion effects, is not considered in the calculations since no new recommendations are provided for its value. 139 Table 6-4: Properties of Steel Steel Sy ksi (MPa) Su ksi (MPa) SRT ksi (MPa) S400 ksi (MPa) Carbon: A106B 35 (241.32) 60 (413.69) 20 (137.90) 20 (137.90) Stainless: Type 304 30 (206.85) 75 (517.12) 20 (137.90) 18.7 (128.93) 140 Table 6-5a: Sample Computations for a Pipe Made of Carbon Steel ASME LRFD For Design at Room Temperature Design: (6.37mm)in25.0 )800*4.020000(2 )75.12(800 )(2min   yPS oDDesPt Service Limit A: Selecting t=0.28in (7.14mm) Pa=PA=897.39 psi (6.19MPa) Service Limit B: PB=1.1 Pa=987.13 psi (6.81MPa) Service Limit C: PC=1.5 Pa=1346.09 psi (9.28MPa) Service Limit D: PD=2 Pa=1794.78 psi (12.38MPa) t tyoDPȖSij 2 2 *†  (a) Design: For ij=0.68, Ȗ=1.2 and solving (a) for t, yields tmin=0.25 in (6.37mm) (ȕ=4.42) Service Limit A: For ij=0.68, Ȗ=1.2 and by selecting t=0.28 in (7.14mm) and solving (a) for P, Pa=PA=897.39 psi (6.19MPa) (ȕ=4.42) Service Limit B: For ij=0.75, Ȗ=1.2 and solving (a) for P produces P=PB=987.13 psi (6.81MPa) (ȕ=3.58) Service Limit C: For ij=0.60, Ȗ=1.2 and solving (a) for P produces P=PC=1346.09 psi (9.28MPa) (ȕ=4.54) Service Limit D: For ij=0.80, Ȗ=1.2 and solving (a) for P produces P=PD=1794.78 psi (12.38MPa) (ȕ=3.08) For Design Temperature 400oF Design: (6.37mm)in25.0 )800*4.020000(2 )75.12(800 )(2min   yPS oDDesPt Selecting t=0.28in (7.14mm) Service Limit A: Pa=897.39 psi (6.19MPa) Service Limit B: PB=1.1 Pa=987.13 psi (6.81MPa) Service Limit C: PC=1.5 Pa=1346.09 psi (9.28MPa) Service Limit D: PD=2 Pa=1794.78 psi (12.38MPa) Design: For ij=0.68, Ȗ=1.2 and solving (a) for t produces tmin=0.25in (6.37mm) (ȕ=1.78) Service Limit A: Using t=0.28 in (7.14mm) and for ij=0.68, Ȗ=1.2 , produces P=897.39 psi (6.19MPa) (ȕ=1.78) Service Limit B: For ij=0.75, Ȗ=1.2 and solving (a) for P produces PB=987.13 psi (6.81MPa) (ȕ=1.37) Service Limit C: For ij=0.60, Ȗ=1.2 and solving (a) for P produces PC=1346.09 psi (9.28MPa) (ȕ=4.37) Service Limit D: For ij=0.80, Ȗ=1.2 and solving (a) for P produces P=PD=1794.78 psi (12.38MPa) (ȕ=3.01) †S*=Sy or Su depending on the Service Limit 141 Table 6-5b: Sample Computations for a Pipe Made of Stainless, Austenitic Steel ASME LRFD For Design at Room Temperature Design: (6.37mm)in25.0 )800*4.020000(2 )75.12(800 )(2min   yPS oDDesPt Service Limit A: Selecting t=0.28in (7.14 mm) Pa=PA=897.39psi (6.19 MPa) Service Limit B: PB=1.1 Pa=987.13psi (6.81 MPa) Service Limit C: PC=1.5 Pa=1346.09psi (9.28 MPa) Service Limit D: PD=2 Pa=1794.78psi (12.38MPa) t tyoDPȖSij 2 2 *†  (a) Design: For ij=0.80, Ȗ=1.2 and solving (a) for t, it is tmin=0.25 in (6.37mm) (ȕ=2.80) Service Limit A: Selecting here also t= 0.28in (7.14mm) and for ij=0.80, Ȗ=1.2 and solving (a) for Pa=PA=897.39 psi (6.19MPa)(ȕ=2.80) Service Limit B: For ij=0.88, Ȗ=1.2 and solving (a) for P, produces P=PB=987.13 psi (6.81MPa) (ȕ=2.37) Service Limit C: For ij=0.48, Ȗ=1.2 and solving (a) for P, produces P=PC=1346.09 psi (9.28 MPa), (ȕ=5.40) Service Limit D: For ij=0.64, Ȗ=1.2 and solving (a) for P, produces PD=1567.67 psi (12.38MPa) (ȕ=3.82) For Design Temperature 400oF Design: (6.8mm)in27.0 )800*4.018700(2 )75.12(800 )(2min   yPS oDDesPt Service Limit A: Selecting t=0.28in (7.14mm) Pa=839.06 psi (5.79MPa) Service Limit B: PB=1.1 Pa=922.67psi (6.36MPa) Service Limit C: PC=1.5 Pa=1258.59psi (8.68MPa) Service Limit D: PD=2 Pa=1678.21psi (11.57MPa) Design: For ij=0.75, Ȗ=1.2 Solving (a) for t, produces t=0.27in (6.8mm) (ȕ=1.32) Service Limit A: Selecting here also t=0.28in (7.14mm) and solving (a) for P=PA=839.06psi (5.79MPa) (ȕ=1.32) Service Limit B: For ij=0.82, Ȗ=1.2 Solving (a) for P, produces PB=922.67psi (6.36MPa) (ȕ=0.99) Service Limit C: For ij =0.45, Ȗ=1.2 and solving (a) for P, produces PC=1258.59psi (8.68MPa) (ȕ=4.64) Service Limit D: For ij=0.60, Ȗ=1.2 and solving (a) for P, produces PD=1678.21 psi (11.57MPa) (ȕ=3.16) †*S= Sy or Su depending on the Service Limit 142 6.3. Part II: Design for Combined Loading In this part, the partial safety factors for the performance functions combining primary stresses such as pressure, earthquake, and moments generated by alternating or not mechanical loads (e.g., sudden valve closure, water-hammer) are evaluated. The following analysis may address pipes with a ratio of external diameter to thickness, ș, approximately less than 75. Since, as explained in Chapter 4, limit theory and the development of plastic hinges are possible for pipes with ș about less than 75. In Table NB-3681(a)-1 of the Code, where the values for the primary indices B1, B2 are proposed, for the calculation of bending stresses, the ratio ș is limited to values less than 50. Nevertheless, for values of ș greater than approximately 75 the failure mode is buckling and in this study performance functions especially for buckling are not provided. Part II is separated in five subsections where the partial safety factors for the performance functions of the five service levels (Design, A, B, C, and D) are presented, respectively. A simple design example, using the derived LRFD equations, is provided at the end of the chapter together with the computation conclusions. 6.3.1. Design The partial safety factors for the performance functions g2 and g3 in Table 5-2 are evaluated herein. The following sections summarize the considered probabilistic characteristics of variables and the results of the computations. Stresses for these performance functions are evaluated considering the elastic section modulus of the pipe’s cross-section. 143 6.3.1.1. Performance Function g2 For this performance function (P.F.) the partial safety factors are calculated for ȕ=6, 7, 8, therefore for a low probability of failure, in order to show that a failure of a pipe due to its sustained weight should be impossible. The normalized performance function is given by Eq. (6-10). 0 2  Afyfg (6-10) where, fy is the normalized yield strength of steel and fA the normalized stress due to sustained weight, all with respect to the stress due to sustained weight. Table 6-6 gives the parameters for the calculation of safety factors and Table C-6 presents the calculated mean partial safety factors (Iƍy, Ȗƍǹ for the yield strength and sustained weight, respectively). Table C-7 gives the calculated adjusted nominal resistance factor, Iy, for a load factor Ȗǹ=1.2 for all cases. Figure 6-10, in addition, shows the computed, adjusted nominal resistance factor for different operating temperatures and the considered values of the reliability index, ȕ. Table 6-7 gives recommended (rounded) values for the nominal resistance factors. Table 6-6: Parameters for the Calculations for g2 Parameter Range Recommended Value Distribution ȕ na 6, 7 ,8 na Mean NA 1 COV 0.05 to 0.10 0.10 fA Bias 1.0-1.05 1.0 Normal Mean NA NA COV Table 3-5 Table 3-5 fy Bias Table 3-5 Table 3-5 Lognormal NA=Not Available, na=not applicable 144 Stainless Steel, P.F.:g 2 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0 100 200 300 400 500 600 700 800 900 Temperature ( o F) ij y ȕ=6 ȕ=7 ȕ=8 Figure 6-10a: Adjusted Nominal Resistance Factor for g2 and Stainless Steel Carbon Steel, P.F.:g 2 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0 100 200 300 400 500 600 700 800 900 Temperature ( o F) ij y ȕ=6 ȕ=7 ȕ=8 Figure 6-10b: Adjusted Nominal Resistance Factor for g2 and Carbon Steel 145 Table 6-7: Recommended Nominal Load and Resistance Factors for g2 Carbon Steel Stainless Steel Temperature T (oF) ȕ Iy Ȗǹ Iy Ȗǹ Room Temperature 0.68 1.2 0.53 1.2 200 0.55 1.2 0.45 1.2 400 0.40 1.2 0.38 1.2 t600 6 0.35 1.2 0.33 1.2 Room Temperature 0.60 1.2 0.45 1.2 200 0.50 1.2 0.40 1.2 400 0.35 1.2 0.32 1.2 t600 7 0.30 1.2 0.30 1.2 Room Temperature 0.55 1.2 0.38 1.2 200 0.45 1.2 0.33 1.2 400 0.30 1.2 0.27 1.2 t600 8 0.25 1.2 0.23 1.2 6.3.1.2. Performance Function g3 For this performance function the partial safety factors are calculated for various values of the target reliability index, ȕ. As the previous performance function, this one is also used for designing the pipe, which then can be checked for other service levels, too. The normalized performance function is given by Eq. (6-11). The parameters of the calculations are given in Table 6-8. Table C-8 presents the mean partial safety factors. Figure 6-11 shows, for selected values of ȕ, how the mean factors (ijƍy, Ȗƍǹ, ȖƍPDes for the yield strength, sustained weight, and design internal pressure, respectively) vary with respect to the normalized stress due to the design pressure. DesPfAfyfg  3 (6-11) 146 Table 6-8: Parameters for the Calculations for g3 Parameter Statistical Properties Range Recommended Value Distribution ȕ na na 2, 3, 3.5, 4.5, 5.5 na Mean NA 1 COV 0.05 to 0.10 0.10 fA Bias 1 to 1.05 1 Normal Mean 0.5 to 50* 0.5, 1, 5, 10, 50 COV 0.04 to 0.10 0.10 fPDes Bias NA 1 Lognormal Mean NA NA COV Table 3-5 Table 3-5 fy Bias Table 3-5 Table 3-5 Lognormal NA=Not Available, na=not applicable, *The value of 50 is shown, whereas the same results are obtained for values greater than 50 The adjusted nominal resistance factors are evaluated for representative values of the normalized pressure with respect to the sustained weight stress, and for nominal factors Ȗǹ=1.1 and ȖPDes=1.2 for the sustained weight and design internal pressure, respectively. The adjusted nominal resistance factors are shown in Table C-9. Table 6-9 presents the recommended nominal values for the adjusted resistance factor by grouping the results of Table C-9. 147 g 3: Carbon Steel for T <=200oF (ȕ =2) 0.85 0.88 0.91 0.94 0.97 1 1.03 1.06 1.09 1.12 1.15 1.18 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f PDes P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'PDes 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f PDes P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'PDes g 3 : Carbon Steel for T<= 200oF (ȕ =5.5) g 3 : Carbon Steel for T> 200ȠF (ȕ =2) 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f PDes P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'PDes g 3 : Carbon Steel for T> 200oF (ȕ= 5.5) 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f PDes P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'PDes g 3 : Stainless Steel for any T (ȕ =2) 0.7 0.74 0.78 0.82 0.86 0.9 0.94 0.98 1.02 1.06 1.1 1.14 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f PDes P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'PDes g 3 : Stainless Steel for any T (ȕ= 5.5) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f PDes P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'PDes Figure 6-11: Variation of Mean Partial Safety Factors with Respect to Normalized Stress due to Design Internal Pressure for Boundary Values of ȕ=2 and 5.5 148 Table 6-9: Ranges and Recommended (1) Adjusted Nominal Resistance Factors for g3 ijy for Ȗǹ=1.1 & ȖPDes=1.2 ȕ CARBON STEEL STAINLESS STEEL Room Temperature Room Temperature 2 0.95 (0.94 to 0.97) (2) 0.95 (0.93 to 0.96) (2) 3 0.93 (0.93 to 0.94) 0.87 (0.86 to 0.88) 3.5 0.88 (0.87 to 0.89) 0.80 (0.79 to 0.81) 4.5 0.78 (0.77 to 0.80) 0.68 (0.67 to 0.69) 5.5 0.70 (0.68 to 0.73) 0.57 (0.56 to 0.58) 200oF 200oF 2 0.86 (0.85 to 0.87) 0.91 (0.89 to 0.92) 3 0.77 (0.76 to 0.78) 0.76 (0.75 to 0.77) 3.5 0.73 (0.72 to 0.74) 0.70 (0.69 to 0.71) 4.5 0.65 (0.63 to 0.67) 0.59 (0.59 to 0.60) 5.5 0.58 (0.56 to 0.60) 0.50 (0.49 to 0.51) 400oF 400oF 2 0.74 (0.73 to 0.75) 0.74 (0.73 to 0.75) 3 0.63 (0.63 to 0.64) 0.62 (0.61 to 0.63) 3.5 0.59 (0.58 to 0.60) 0.58 (0.57 to 0.58) 4.5 0.50 (0.50 to 0.51) 0.48 (0.48 to 0.49) 5.5 0.43 (0.43 to 0.44) 0.41 (0.40 to 0.42) 600oF 600oF 2 0.64 (0.63 to 0.65) 0.66 (0.64 to 0.67) 3 0.54 (0.54 to 0.55) 0.55 (0.55 to 0.56) 3.5 0.50 (0.50 to 0.51) 0.50 (0.50 to 0.51) 4.5 0.43 (0.43 to 0.44) 0.43 (0.43 to 0.44) 5.5 0.37 (0.37 to 0.38) 0.36 (0.36 to 0.37) 800oF 800oF 2 0.62 (0.62 to 0.63) 0.62 (0.60 to 0.63) 3 0.51 (0.51 to 0.52) 0.52 (0.51 to 0.53) 3.5 0.47 (0.47 to 0.48) 0.47 (0.47 to 0.48) 4.5 0.41 (0.40 to 0.41) 0.40 (0.40 to 0.41) 5.5 0.35 (0.34 to 0.36) 0.34 (0.33 to 0.35) (1) Recommended values are in bold face, (2) Ȗǹ=1.0 and ȖPDes=1.1 149 6.3.2. Service Level A In this section only the partial safety factors for the performance function g4 are evaluated. The probabilistic characteristics for the computations are provided in Table 6- 10. The normalized performance function is given by Eq. (6-12), where fy is the normalized yield strength, fA the normalized stress due to sustained weight, and fPmax the normalized stress due to peak pressure. For Service Level (Limit) A stresses are evaluated considering the elastic section modulus of the pipe’s cross-section. Table C-10 presents the calculated mean partial safety factors for different values of ȕ and properties of steel. Table C-11 gives the calculated nominal resistance factors, considering a given set of load partial safety factors, and Table 6-11 the recommended adjusted nominal resistance factors derived by summarizing and grouping in categories the results of Table C-11. In Figure 6-12 some cases are plotted, showing how the mean partial safety factors (ijƍy, Ȗƍǹ, ȖƍPmax for the yield strength, sustained weight, and peak internal pressure, respectively) vary with respect to the normalized maximum pressure. 0 max 4  PfAfyfg (6-12) 150 Table 6-10: Parameters for the Calculations for g4 Parameter Statistical Properties Range Recommended Value Distribution ȕ na na 2, 3, 3.5, 4.5, 5.5 na Mean NA 1 COV 0.05 to 0.10 0.10 fA Bias 1 to 1.05 1 Normal Mean 0.5 to 50* 0.5, 1, 3, 5, 10, 50 COV 0.10 to 0.14 0.13 fPmax Bias NA 0.95 Extr. I (Largest) Mean NA NA COV Table 3-5 Table 3-5 fy Bias Table 3-5 Table 3-5 Lognormal NA=Not Available, na=not applicable, *The value of 50 is shown, whereas the same results are obtained also for values greater than 50. 151 g 4 : Carbon Steel for T <=200oF (ȕ =2) 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f Pmax P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'Pmax 0.65 0.85 1.05 1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f Pmax P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'Pmax g 4 : Carbon Steel for T<= 200oF (ȕ =5.5) g 4 : Carbon Steel for T >200ȠF (ȕ =2) 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f Pmax P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'Pmax g 4 : Carbon Steel for T> 200oF (ȕ= 5.5) 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f Pmax P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'Pmax g 4 : Stainless Steel for any T (ȕ =2) 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f Pmax P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'Pmax 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f Pmax P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'Pmax g 4 : Stainless Steel for any T (ȕ =5.5) Figure 6-12: Variation of Mean Partial Safety Factors with Respect to Normalized Stress due to Maximum Internal Pressure for Boundary Values of ȕ=2 and 5.5 152 Table 6-11: Ranges and Recommended (1) Adjusted Nominal Resistance Factors for g4 ijy for Ȗǹ=1.1 & ȖPmax=1.2 ȕ CARBON STEEL STAINLESS STEEL Room Temperature Room Temperature 2 0.96 (2) (0.95 to 0.97) 0.96 (2) (0.94 to 0.98) 3 For fPmax”1, 0.93 (0.93 to 0.94) For fPmax>1, 0.87 (0.85 to 0.89) 0.86 (0.84 to 0.88) 3.5 For fPmax”1, 0.88 (0.87 to 0.89) For fPmax>1, 0.77 (0.76 to 0.79) For fPmax”1, 0.81 For fPmax>1, 0.76 (0.75 to 0.77) 4.5 For fPmax”1, 0.72 (0.74 to 0.78) For fPmax>1, 0.62 (0.61 to 0.64) For fPmax”1, 0.67 (0.67 to 0.68) For fPmax>1, 0.59 (0.58 to 0.61) 200oF 200oF 2 0.87 (0.86 to 0.88) 0.92 (0.90 to 0.94) 3 For fP”1, 0.77 For fPmaxt1, 0.72 (0.70 to 0.73) 0.76 (0.74 to 0.78) 3.5 For fPmax”1, 0.72 (0.71 to 0.73) For fPmax>1, 0.65 (0.63 to 0.67) For fPmax”5, 0.69 (0.67 to 0.71) For fPmax>5, 0.66 (0.65 to 0.67) 4.5 For 3”fPmax”5 0.54 (0.53 to 0.64) For fPmax<3 0.62 (0.61 to 0.64) For fPmax>5, 0.51 (0.50 to 0.53) For fP”1, 0.59 For fPmax>1, 0.53 (0.51 to 0.55) 400oF 400oF 2 0.75 (0.74 to 0.76) 0.75 (0.74 to 0.77) 3 0.62 (0.61 to 0.64) 0.62 (0.60 to 0.63) 3.5 0.56 (0.54 to 0.59) For fPmax”1, 0.58 For fPmaxt1, 0.54 (0.53 to 0.56) 4.5 For fPmax”1, 0.50 (0.50 to 0.51) For fPmax>1, 0.44 (0.42 to 0.46) For fPmax”1, 0.48 (0.48 to 0.49) For fPmax>1, 0.43 (0.41 to 0.45) 600oF 600oF 2 0.65 (0.64 to 0.66) 0.67 (0.66 to 0.68) 3 0.54 (0.52 to 0.56) 0.55 (0.54 to 0.56) 3.5 For fPmax”1, 0.51 For fPmax>1, 0.47 (0.46 to 0.49) 0.49 (0.47 to 0.51) 4.5 For fPmax”1, 0.43 (0.43 to 0.44) For fPmax>1, 0.37 (0.36 to 0.39) For fPmax”1, 0.43 For fPmax>1, 0.38 (0.37 to 0.40) 800oF 800oF 2 0.60 (0.60 to 0.61) 0.62 (0.61 to 0.64) 3 0.50 (0.49 to 0.52) 0.51 (0.50 to 0.52) 3.5 0.45 (0.43 to 0.48) 0.46 (0.44 to 0.48) 4.5 For fPmax”1, 0.40 (0.40 to 0.41) For fPmax>1, 0.35 (0.34 to 0.37) For fPmax”1, 0.40 For fPmax>1, 0.35 (0.34 to 0.36) (1) Recommended values are in bold face, (2) Ȗǹ=1.0 and ȖDes=1.1 153 6.3.3. Service Level B For Service Level B the partial safety factors are calculated for four performance functions, namely g6, g7 g8, and g9 in Table 5-2, under separate headings. Stresses are evaluated using the plastic section modulus of the pipe’s cross-section. 6.3.3.1. Performance Function g6 The normalized performance function is given by Eq. (6-13), where fy is the normalized yield strength, fA the normalized stress due to sustained weight, fPB the normalized pressure stress for Service Level B, and fM the normalized stress due to dynamic mechanical loading (e.g., generation of moments due to sudden valve closure, etc.). MfPBfAfyfg  6 (6-13) Table 6-12 provides the probabilistic characteristics for the variables under consideration and recommended values for the computation of the mean partial safety factors. Table C-12 provides the calculated mean safety factors for all cases examined. Table C-13 provides the adjusted nominal resistance factors for different operating temperatures for piping and a predefined set of load factors. A summary of the results and recommended values for the adjusted nominal resistance factors are provided in Table 6-13. Figure 6-13 presents how the mean partial loads (ijƍy, Ȗƍǹ, ȖƍPB, ȖƍM, for the yield strength, sustained weight, pressure under Service Level B conditions, and mechanical loading, respectively) vary with respect to the normalized stress due to internal pressure for preselected values of ȕ=3.5 and fM=0.5. Figure 6-14 shows how the mean partial 154 loads vary with respect to the normalized stress due to the mechanical loading for preselected values of ȕ=3.5 and fPB=5. Table 6-12: Parameters for the Computations for g6 Parameter Statistical Properties Range Recommended Value Distribution ȕ na na 1.5, 2, 3, 3.5, 4.5 na Mean NA 1 COV 0.05 to 0.10 0.10 fA Bias 1 to 1.05 1 Normal Mean 0.5 to 50* 0.5, 1, 3, 5, 10, 50 COV 0.10 to 0.14 0.13 fPB Bias NA 0.95 Lognormal Mean 0.5 to 2 0.5, 1, 2 COV NA 0.15 fM Bias 0.80 to 1.05 1 Lognormal Mean NA NA COV Table 3-5 Table 3-5 fy Bias Table 3-5 Table 3-5 Lognormal NA=Not Available, na=not applicable, *The value of 50 is shown, whereas the same results are obtained also for values greater than 50. 155 g 6 : Carbon Steel for T <=200oF (ȕ =3.5) 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f PB P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'PB Ȗ'Ȃ g 6 : Carbon Steel for T >200ȠF (ȕ =3.5) 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f PB P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'PB Ȗ'Ȃ 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f PB P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'PB Ȗ'Ȃ g 6 : Stainless Steel for any T (ȕ =3.5) Figure 6-13: Mean Partial Safety Factors with Respect to the Normalized Stress due to Internal Pressure for Preselected Value of ȕ=3.5 and fM=0.5 for g6 156 g 6: Carbon Steel for T <200oF (ȕ =3.5) 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 1 1.5 2 2.5 Normalized Mean Stress, f M P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'PB Ȗ'Ȃ g 6: Carbon Steel for T >200ȠF (ȕ =3.5) 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.5 1 1.5 2 2.5 Normalized Mean Stress, f M P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'PB Ȗ'Ȃ 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.5 1 1.5 2 2.5 Normalized Mean Stress, f M P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'PB Ȗ'Ȃ g 6 : Stainless Steel for any T (ȕ =3.5) Figure 6-14: Mean Partial Safety Factors with Respect to the Normalized Mechanical Load for Preselected Value of ȕ=3.5 and fPB=5 for g6 157 Table 6-13: Ranges and Recommended (1) Values for Adjusted Nominal Resistance Factors for g6 ijy for Ȗǹ=1.1, ȖM=1.2, and ȖPB=1.2 ȕ CARBON STEEL STAINLESS STEEL Room Temperature Room Temperature 2 0.91 (2) (0.89 to 0.94) 0.91 (2) (0.89 to 0.93) 3 For fPB”5, 0.97 (0.95 to 0.99) For fPB>5, 0.93 (0.91 to 0.96) 0.90 (0.88 to 0.93) 3.5 For fPB”5, 0.91 (0.89 to 0.94) For fPB>5, 0.87 (0.85 to 0.89) 0.82 (0.80 to 0.85) 4.5 For fPB ”5, 0.81 (0.78 to 0.84) For fPB >5, 0.75 (0.73 to 0.78) 0.69 (0.66 to 0.72) 200oF 200oF 2 0.89 (0.87 to 0.92) 0.94 (0.93 to 0.96) 3 For fPB”10, 0.79 (0.76 to 0.82) 0.79 (0.77 to 0.81) 3.5 For fPB”5, 0.74 (0.73 to 0.76) For fPBx>5, 0.72. (0.70 to 0.74) 0.72 (0.70 to 0.74) 4.5 For fPB”5, 0.66 (0.64 to 0.68) For fPB>5, 0.62 (0.60 to 0.64) For fPB”5, 0.61 (0.60 to 0.63) For fPB>5, 0.59 (0.57 to 0.61) 400oF 400oF 2 0.77 (0.76 to 0.79) 0.77 (0.76 to 0.79) 3 0.66 (0.64 to 0.68) 0.64 (0.63 to 0.66) 3.5 0.61 (0.58 to 0.63) 0.59 (0.57 to 0.61) 4.5 0.51 (0.49 to 0.54) 0.49 (0.47 to 0.51) 600oF 600oF 2 0.67 (0.66 to 0.68) 0.68 (0.68 to 0.70) 3 0.56 (0.55 to 0.58) 0.57 (0.56-0.59) 3.5 0.52 (0.50 to 0.54) 0.52 (0.51 to 0.54) 4.5 0.44 (0.42 to 0.46) 0.44 (0.42 to 0.46) 800oF 800oF 2 0.62 (0.61 to 0.64) 0.64 (0.63 to 0.66) 3 0.52 (0.51 to 0.54) 0.51 (0.52 to 0.55) 3.5 0.48 (0.47 to 0.50) 0.46 (0.48 to 0.51) 4.5 0.41 (0.39 to 0.43) 0.40 (0.39 to 0.42) (1) Recommended values are in bold face, (2) Ȗǹ=ȖM=ȖPB=1 158 6.3.3.2. Performance Function g7 This performance function considers the loading of pipes that are not pressurized, (e.g., auxiliary piping systems) and are subjected to seismic forces. The normalized performance function is given by Eq. (6-14), where fy is the normalized yield strength, fA the normalized sustained weight, and fO the normalized stress due to the Operating Basis Earthquake (OBE). OfAfyfg  7 (6-14) Table 6-14 provides the probabilistic characteristics for the variables under consideration and the recommended values used for the computation of partial safety factors. Table C-14 provides the calculated mean partial factors for all the examined cases. Table C-15 gives the adjusted nominal resistance factors for predefined load factors. The recommended nominal resistance factors are given in Table 6-15 by summarizing results in Table C-15. For this performance function the seismic loading is examined having two Coefficient Of Variation (COV), namely 0.50 and 0.80. Therefore, the influence of the Coefficient Of Variation (COV) of the earthquake loading on the results and the value of the target reliability index, ȕ, is examined. Figure 6-15 presents how the calculated mean factors (ijƍȊ, Ȗƍǹ, Ȗƍȅ for the yield strength, sustained weight, and stress due to earthquake, respectively) vary with respect to the normalized stress due to the OBE for selected values of ȕ=1.5 and 3.0. 159 Table 6-14: Parameters for the Computations for g7 Parameter Statistical Properties Range Recommended Value Distribution ȕ na na 1.5, 2, 3 na Mean NA 1 COV 0.05 to 0.10 0.10 fA Bias 1 to 1.05 1 Normal Mean 0.5 to 2.5 0.5, 1, 2, 2.5 COV 0.40 to 0.90 0.50, 0.80 fO Bias 0.65 to 1.0 0.75 Extreme II (Largest) Mean NA NA COV Table 3-5 Table 3-5 fy Bias Table 3-5 Table 3-5 Lognormal NA=Not Available, na=not applicable Table 6-15: Ranges and Recommended (1) Nominal Adjusted Resistance Factors for g7 ijy for Ȗǹ=1.1 & Ȗȅ=1.5 CARBON STEEL STAINLESS STEEL ȕ COV(fO)=0.50 COV(fO)=0.80 COV(fO)=0.50 COV(fO)=0.80 Room Temperature Room Temperature 1.5 0.90 (2) (0.86 to 0.96) 0.84 (2) (0.79 to 0.91) 0.96 (2) (0.93 to 1.01) 0.90 (2) (0.86 to 0.98) 2 0.86 (3) (0.80 to 0.95) 0.74 (3) (0.67 to 0.85) 0.92 (3) (0.86 to 1.00) 0.80 (3) (0.73 to 0.91) 3 0.57 (0.51 to 0.67) 0.39 (0.34 to 0.48) 0.60 (0.54 to 0.71) 0.40 (0.36 to 0.45) 200oF 200oF 1.5 0.74 (2) (0.71 to 0.79) 0.70 (2) (0.65 to 0.75) 0.86 (2) (0.81 to 0.94) 0.79 (2) (0.75 to 0.85) 2 0.86 (0.84 to 0.89) 0.75 (0.71 to 0.80) 0.96 (0.94 to 1.00) 0.83 (0.80 to 0.87) 3 0.47 (0.42 to 0.55) 0.32 (0.28 to 0.40) 0.52 (0.47 to 0.62) 0.35 (0.31 to 0.43) 400oF 400oF 1.5 0.97 (0.93 to 0.99) 0.90 (0.90 to 0.91) 0.96 (0.90 to 1.00) 0.90 (0.87 to 0.92) 2 0.78 (0.77 to 0.80) 0.68 (0.65 to 0.73) 0.77 (0.77 to 0.78) 0.67 (0.65 to 0.71) 3 0.42 (0.38 to 0.50) 0.29 (0.25 to 0.36) 0.42 (0.38 to 0.48) 0.29 (0.26 to 0.35) 160 Table 6-15: (Continued) CARBON STEEL STAINLESS STEEL ȕ COV(fO)=0.50 COV(fO)=0.80 COV(fO)=0.50 COV(fO)=0.80 600oF 600oF 1.5 0.83 (0.80 to 0.86) 0.78 0.85 (0.80 to 0.89) 0.80 (0.78 to 0.81) 2 0.67 (0.66 to 0.69) 0.59 (0.56 to 0.63) 0.69 0.60 (0.58 to 0.63) 3 0.37 (0.33 to 0.43) 0.25 (0.22 to 0.31) 0.37 (0.34 to 0.43) 0.26 (0.23 to 0.31) 800oF 800oF 1.5 0.78 (0.75 to 0.80) 0.72 (0.72 to 0.73) 0.80 (0.75 to 0.83) 0.75 (0.73 to 0.76) 2 0.63 (0.62 to 0.65) 0.55 (0.52 to 0.59) 0.64 (0.64 to 0.65) 0.56 (0.54 to 0.59) 3 0.34 (0.31 to 0.40) 0.23 (0.20 to 0.29) 0.35 (0.32 to 0.40) 0.24 (0.21 to 0.29) (1) Recommended values are in bold face, (2) Ȗǹ=1 and Ȗȅ=0.9, (3) Ȗǹ=1.1 and Ȗȅ=1.1 161 g 7: Carbon Steel for T <=200oF (ȕ =1.5) 0.85 1.05 1.25 1.45 1.65 1.85 2.05 0.5 1 1.5 2 2.5 Normalized Mean Stress, f O P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'O 0.65 1.65 2.65 3.65 4.65 5.65 6.65 7.65 8.65 0.5 1 1.5 2 2.5 Normalized Mean Stress, f O P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'O g 7: Carbon Steel for T<= 200oF (ȕ =3) g 7 : Carbon Steel for T >200ȠF (ȕ =1.5) 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 0.5 1 1.5 2 2.5 Normalized Mean Stress, f O P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'O g 7 : Carbon Steel for T> 200oF (ȕ= 3) 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 0.5 1 1.5 2 2.5 Normalized Mean Stress, f O P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'O g 7: Stainless Steel for any T (ȕ =1.5) 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.95 0.5 1 1.5 2 2.5 Normalized Mean Stress, f O P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'O 0.4 1.4 2.4 3.4 4.4 5.4 6.4 7.4 8.4 0.5 1 1.5 2 2.5 Normalized Mean Stress, f O P a rt ia l S a fe ty F a c to r ijy' Ȗǹ' Ȗ'O g 7: Stainless Steel for any T (ȕ =3) Figure 6-15: Mean Partial Factors with Respect to the Normalized Stress due to OBE for Selected Values of ȕ=1.5 and 3.0 for g7 162 Sensitivity Analysis Figures 6-16 and 6-17 show the variation of the adjusted nominal resistance factor with respect to temperature for values of ȕ=1.5 and 3 for carbon and stainless steel, respectively, and the two values of the COV for the earthquake loading. It can be seen that for the higher values of ȕ the difference becomes larger and as expected the higher coefficient of variation yields smaller values for ij (conservative). For ȕ=1.5 the diagrams are not smooth since the nominal resistance factor is evaluated for different sets of load factors. The COV of the earthquake loading has significant impact on the resultant value of ȕ, which can be as high as 0.5, when the target reliability index is equal to 3 and for operation at room temperature. Nevertheless, for the rest of performance functions the COV for earthquake loading is considered 0.80, as an estimated value considering the big range that the COV for earthquake loading attains. Carbon Steel (ȕ =1.5) 0 0.2 0.4 0.6 0.8 1 1.2 70 170 270 370 470 570 670 770 ȉemperature (oF) A d ju s te d R e s is a tn c e F a c to r, ij y COV=0.50 COV=0.80 Carbon Steel (ȕ =3) 0 0.1 0.2 0.3 0.4 0.5 0.6 70 170 270 370 470 570 670 770 ȉemperature (oF) A d ju s te d R e s is a tn c e F a c to r, ij y COV=0.50 COV=0.80 Figure 6-16: Variation with Temperature of the Recommended Nominal Resistance Factor for Carbon Steel and COV for Earthquake Loading 163 Stainless Steel (ȕ =1.5) 0 0.2 0.4 0.6 0.8 1 1.2 70 170 270 370 470 570 670 770 ȉemperature (oF) A d ju s te d R e s is a tn c e F a c to r, ij y COV=0.50 COV=0.80 Stainless Steel (ȕ =3) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 70 170 270 370 470 570 670 770 ȉemperature (oF) A d ju s te d R e s is a tn c e F a c to r, ij y COV=0.50 COV=0.80 Figure 6-17: Variation with Temperature of the Recommended Nominal Resistance Factor for Stainless Steel and COV for Earthquake Loading 6.3.3.3. Performance Function g8 This performance function checks pressurized pipes subjected to the OBE. Table 6-16 presents the probabilistic characteristics for the parameters in the performance function, given by Eq. (6-15), where fPO is the normalized stress due the pressure coincident with the OBE, fy the normalized yield strength of steel, and fO the normalized stress due to OBE. OfPOfAfyfg  8 (6-15) Table C-16 gives the mean partial load and resistance factors for different operating temperatures, values of ȕ, and types of steel. The calculated adjusted nominal resistance factor is presented for all cases in Table C-17. A summary with the recommended values for the adjusted nominal resistance factors is shown in Table 6-17. 164 Table 6-16: Parameters for the Calculations for g8 Parameter Statistical Properties Examined Range Recommended Value Distribution ȕ na na 1.5, 2, 3 na Mean NA 1 COV 0.05 to 0.10 0.10 fA Bias 1 to 1.05 1 Normal Mean 0.5 to 50* 0.5, 1, 3, 5, 10, 50 COV 0.10 to 0.14 0.13 fPO Bias NA 0.95 Lognormal Mean 0.5 to 2.5 0.5, 1, 2, 2.5 COV 0.40 to 0.90 0.50, 0.80 fO Bias 0.65 to 1 0.75 Extreme II (Largest) Mean NA NA COV Table 3-5 Table 3-5 fy Bias Table 3-5 Table 3-5 Lognormal NA=Not Available, na=not applicable, *The value of 50 is shown, whereas the same results are obtained also for values greater than 50. 165 Table 6-17: Recommended (1) Values and Ranges for the Nominal Resistance Factor for g8 ijy for Ȗǹ=1.1, ȖPO=1.2 & ȖO=1.5 ȕ CARBON STEEL STAINLESS STEEL Room Temperature Room Temperature 1.5 x For fPO”1 and fOt2 0.85(2) (0.82 to 0.87) x Otherwise 0.95(2) (0.90 to 1.01) x For fPO”1 and fOt2 0.91(2) (0.89 to 0.94) x Otherwise 1.00(2) (0.97 to 1.04) 2 x For fPO”1 and fOt2 0.63(2) (0.60 to 0.66) x For fPO”1 and 0.51 or for 11or 1200oF (ȕ =1.5) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 10.5 20.5 30.5 40.5 50.5 Normalized Mean Stress, f PO P a rt ia l S a fe ty F a c to r ij'Ȋ Ȗ'ǹ Ȗ'PO Ȗ'M Ȗ'O P.F. g 9 for Stainless Steel (ȕ =1.5) 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 10.5 20.5 30.5 40.5 50.5 Normalized Mean Stress, f PO P a rt ia l S a fe ty F a c to r ij'Ȋ Ȗ'ǹ Ȗ'PO Ȗ'M Ȗ'O Figure 6-18: Mean Partial Factors with Respect to the Normalized Stress Pressure Coincident with Earthquake for Selected Values of ȕ=1.5, fM=0.5, and fO=2 for g9 169 Table 6-19: Recommended (1) Values and Ranges for the Adjusted Nominal Resistance Factor for g9 ijy for Ȗǹ=1.1, ȖPO=ȖȂ=1.2 & ȖPO=1.5ȕ CARBON STEEL STAINLESS STEEL Room Temperature Room Temperature 1.5 x For fPO”1 and fOt2 0.88(2) (0.84 to 0.93) x Otherwise 0.98(2) (0.95 to 1.01) 0.98 (2) (0.93 to 1.04) 2.5 x For fPO”1 and fOt2 0.69(2) (0.62 to 0.76) x For 12 0.87 (0.83 to 0.89) x Otherwise 0.95 (0.92 to 1.02) x For fPO”1 and fOt2 0.54 (0.52 to 0.57) x For 12 or for fPO=5 and fO=1 0.75 (0.71 to 0.79) x Otherwise 0.91 (0.87 to 0.95) 3 x For fPO”1 and fOt2: 0.35 (0.34 to 0.38) x For 110 0.85 (0.82 to 0.88) x Otherwise 0.88 (0.85 to 0.92) 175 Table 6-23: (Continued) ȕ CARBON STEEL STAINLESS STEEL 400oF t400oF 2.5 x For fPO”1 and fOt2 0.44(2) (0.41 to 0.47) x For 110 0.85 (0.82 to 0.88) x For fPO=3 and fO=0.5 or for fPO=10 and fO”2 0.76 (0.71 to 0.81) x Otherwise 0.88 (0.85 to 0.92) 3 x For fPO”1 and fOt1 0.44 (0.37 to 0.51) x For 12 0.70(2) (0.65 to 0.74) x For fPO=3 and fO=0.5 0.95(2) x Otherwise 0.80(2) (0.78 to 0.82) 3 x For fPO”1 and fOt2: 0.37 (0.35 to 0.39) x For 11 0.93(2) (0.90 to 0.96) x For fPCt10 0.88(2) (0.86 to 0.90) x For fPC”1 and fM”1 0.98(2) (0.96 to 0.99) x For fPC”5 and fM=2 0.93(2) (0.92 to 0.95) x For fPCt10 0.86(2) (0.84 to 0.89) x Otherwise 0.91(2) (0.89 to 0.93) 4.5 x For fPC”1 and fM”1 0.87 (0.85 to 0.90) x For fPC”3 and fM=2 0.84 (0.82 to 0.87) x For fPC=3 and fM=1 0.78 x For fPCt10 0.70 (0.66 to 0.72) x Otherwise 0.74 (0.71 to 0.77) x For fPC”1 and fM”1 0.86 (0.84 to 0.89) x For fPC”5 and fM=2 0.82 (0.80 to 0.85) x For fPCt10 0.68 (0.65 to 0.71) x Otherwise 0.73 (0.70 to 0.76) 5.5 x For fPC”0.5 and fM”1 0.80 x For fPC”1 0.75 (0.73 to 0.77) x For fPC=3 and fM=1 0.69 x For fPCt10 0.55 (0.52 to 0.58) x Otherwise 0.61 (0.58 to 0.64) x For fPC”1 and fM”0.5 0.79 x For fPC”1 0.74 (0.72 to 0.76) x For fPCt10 0.54 (0.51 to 0.57) x For fPC=3 and fM=1 0.68 x Otherwise 0.60 (0.57 to 0.63) 200oF 200oF 3 x For fPC”1 and fM<2 0.95 (0.94 to 0.96) x For fPC”5 and fM=2 0.93 (0.91 to 0.95) x For fPCt10 0.87 (0.85 to 0.89) x Otherwise 0.90 (0.89 to 0.93) x For fPC”1 and fM”1 0.89 (0.88 to 0.90) x For fPC”5 and fM=2 0.87 (0.85 to 0.89) x For fPCt10 0.85 (0.83 to 0.87) x Otherwise 0.91 (0.89 to 0.93) 4.5 x For fPC”1 and fM<2 0.80 (0.78 to 0.82) x For fPC”3 and fMt2 0.77 (0.75 to 0.79) x For fPCt10 0.63 (0.60 to 0.66) x Otherwise 0.68 (0.65 to 0.71) x For fPC”1 and fM”1 0.74 (0.73 to 0.77) x For fPC”5 and fM=2 0.72 (0.70 to 0.74) x For fPCt10 0.58 (0.56 to 0.61) x Otherwise 0.63 (0.61 to 0.66) 5.5 x For fPC”1 and fM<2 0.70 (0.67 to 0.73) x For 110 and fO>1 0.88(2) (0.83 to 0.93) x For fPO”3 and fO>2 or for fPO”1 and fO=2 or for fPO”0.5 fM”1 and fO=1 0.55(2) (0.47 to 0.63) x Otherwise 0.73(2) (0.62 to 0.83) x For fPOt3 and fO”0.5 or fPOt5 and 0.5”fO”1 or fPO>10 and fO>1 or fPO=10 and fO=2 or fM>1, fPO”1 and fO”0.5 0.86(2) (0.80 to 0.92) x For fPO”1 and fOt2 or 0.51(2) (0.46 to 0.56) x Otherwise 0.65(2) (0.59 to 0.71) 3.5 x For fPO<10 and fO>1 or fPO”1 and fO>0.5 0.32 (0.22 to 0.42) x For fPO>10 or fPO=10 and fO”0.5 0.93 (0.90 to 0.97) x For fPO=10 and fO=1 or 11 or fPO”1 and fO>0.5 0.30 (0.21 to 0.42) x For fPO>10 or fPO=10 and fO”0.5 0.92 (0.89 to 0.96) x For fPO=10 and fO=1 or 110 and fO>2 0.97 (0.93 to 1.03) x For fPO”3 and fO>1 or 0.68 (0.58 to 0.79) Otherwise 0.78 (0.70 to 0.85) x For fPO”1 and fOt2 0.59 (0.54 to0.64) x For 11 or fPO”1 and fO>0.5 0.30 (0.20 to 0.39) x For fPO>10 or fPO=10 and fO”0.5 0.85 (0.82 to 0.89) x For fPO=10 and fO=1 or 11 or fPO”1 and fO>0.5 0.30 (0.19 to 0.36) x For fPO>10 or fPO=10 and fO”0.5 0.80 (0.77 to 0.83) x For fPO=10 and fO=1 or 110 and fO>1 or fM>1, fO”0.5, and fPO”0.5 0.85(1) (0.82 to 0.90) x For fPO”3 and fO>2 or for fPO”1 and fO=2 0.55(1) (0.47 to 0.63) Otherwise 0.73(1) (0.60 to 0.85) x For fPO”1 and fOt2 0.53 (0.49 to0.57) x For 11 or fPO”1 and fO>0.5 0.32 (0.22 to 0.42) x For fPO>10 or fPO=10 and fO”0.5 0.93 (0.90 to 0.97) x For fPO=10 and fO=1 or 11 or fPO”1 and fO>0.5 0.20 (0.16 to 0.31) x For fPO>10 or fPO=10 and fO”0.5 0.70 (0.67 to 0.74) x For fPO=10 and fO=1 or 110 and fO>2 0.97 (0.94 to 1.03) x For fPO”3 and fO>1 0.70 (0.60 to 0.81) x Otherwise 0.83 (0.72 to 0.93) 3.5 x For fPO<10 and fO>1 or fPO”1 and fO>0.5 0.31 (0.21 to 0.43) x For fPO>10 or fPO=10 and fO”0.5 0.83 (0.80 to 0.86) x For fPO=10 and fO=1 or 11 or fPO”1 and fO>0.5 0.24 (0.16 to 0.32) x For fPO>10 or fPO=10 and fO”0.5 0.66 (0.64 to 0.68) x For fPO=10 and fO=1 or 1200oF (ȕ =1.5) 0.9 1.1 1.3 1.5 1.7 1.9 2 2.5 3 3.5 4 Normalized Mean Stress, f S P a rt ia l S a fe ty F a c to r iju' Ȗǹ' Ȗ'S 0.4 1.4 2.4 3.4 4.4 5.4 6.4 7.4 8.4 2 2.5 3 3.5 4 Normalized Mean Stress, f S P a rt ia l S a fe ty F a c to r iju' Ȗǹ' Ȗ'S g 1 6: Carbon Steel T>200oF (ȕ=3) Figure 6-19: Mean Partial Safety Factors for ȕ=1.5 and 3 for g16 187 Table 6-29: Recommended (1) Values and Ranges for the Adjusted Nominal Resistance Factor for g16 iju for Ȗǹ=1.1 & ȖS=1.5 ȕ CARBON STEEL STAINLESS STEEL T”600oF Room Temperature 1.5 0.90 (2) (0.86 to 0.98) 0.95 (2) (0.92 to 0.96) 2 0.85 (0.78 to 0.89) 0.85 (0.84 to 0.87) 3 0.30 (0.25 to 0.36) 0.35 (0.32 to 0.35) 800oF 200oF 1.5 0.90 (0.93) 0.80 (2) (0.80 to 0.83) 2 0.65 (0.65 to 0.67) 0.75 (0.73 to 0.76) 3 0.25 (0.25 to 0.27) 0.30 (0.28 to 0.30) t400oF 1.5 0.94 (0.91 to 0.94) 2 0.65 (0.64 to 0.68) 3 0.25 (0.24 to 0.26) (1) Recommended values are in bold face, (2) Ȗǹ=1.1 and ȖS=1.1 6.3.5.2. Performance Function g17 This performance function checks pressurized pipes subjected to the Safe Shut-Down Earthquake (SSE). Table 6-30 presents the probabilistic characteristics for the parameters in the performance function, given by Eq. (6-22). SfPSfAfufg  17 (6-22) Table C-30 gives the mean partial load and resistance factors for different operating temperature, values of ȕ, and types of steel. The calculated adjusted nominal resistance factor is presented for all cases in Table C-31. A summary with the recommended values for the adjusted nominal resistance factor is shown herein in Table 6-31. 188 Table 6-30: Parameters for the Calculations for g17 Parameter Statistical Properties Examined Range Recommended Value Distribution ȕ na na 1.5, 2, 3 na Mean NA 1 COV 0.05 to 0.10 0.10 fA Bias 1 to 1.05 1 Normal Mean 0.5 to 50* 0.5, 1, 3, 5, 10, 50 COV 0.10 to 0.14 0.13 fPS Bias NA 0.95 Lognormal Mean 0.5 to 2.5 2, 3, 4, 4.5 COV 0.40 to 0.90 0.80 fS Bias 0.60 to 1.00 0.75 Extreme II (Largest) Mean NA NA COV Table 3-10 Table 3-10 fu Bias Table 3-10 Table 3-10 Lognormal NA=Not Available, na=not applicable, *The value of 50 is shown, whereas the same results are obtained also for values greater than 50. 189 Table 6-31: Recommended Values (1) and Ranges for the Adjusted Nominal Resistance Factor for g17 iju for Ȗǹ=1.1, ȖPS=1.2 & ȖS=1.5 ȕ CARBON STEEL STAINLESS STEEL Room Temperature Room Temperature 2 x For fPS”3 or for 3”fPS<10 and fSt3 0.68(2) (0.57to 0.79) x Otherwise 0.92(2) (0.85 to 0.95) x For fPS”3 or for 310 0.85 (0.85 to 0.87) x Otherwise 0.57 (0.50 to 0.65) 400oF t400oF 2 x For fPS”3 or for 3”fPS<10 and fSt3 0.68(2) (0.58 to 0.78) x Otherwise 0.89(2) (0.84 to 0.92) x For fPS”1 0.70 (0.65 to 0.74) x For 1200oF (ȕ =3) g 18: Carbon Steel for T >200 ȠF (ȕ =5.5) 0.55 0.9 1.25 1.6 1.95 2.3 2.65 3 3.35 3.7 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 Normalized Mean Stress, f PD P a rt ia l S a fe ty F a c to r iju' Ȗǹ' Ȗ'PD Ȗ'L Figure 6-20: Variation of Mean Partial Safety Factors with Respect to the Normalized Pressure for fL=1 and ȕ=3 and 5.5 194 0.65 0.85 1.05 1.25 1.45 1.65 1.85 2.05 0.5 1 1.5 2 Normalized Mean Stress, f L P a rt ia l S a fe ty F a c to r iju' Ȗǹ' Ȗ'PD Ȗ'L g 18: Carbon Steel for T<200oF & Stainless Steel (ȕ =3) g 18 : Carbon Steel for T <200oF & Stainless Steel (ȕ =5.5) 0.7 1.2 1.7 2.2 2.7 3.2 3.7 0.5 1 1.5 2 2.5 Normalized Mean Stress, f L P a rt ia l S a fe ty F a c to r iju' Ȗǹ' Ȗ'PD Ȗ'L 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 0.5 1 1.5 2 2.5 Normalized Mean Stress, f L P a rt ia l S a fe ty F a c to r iju' Ȗǹ' Ȗ'PD Ȗ'L g 18 : Carbon Steel for T >200oF (ȕ =3) g 18 : Carbon Steel for T >200ȠF (ȕ =5.5) 0.6 1 1.4 1.8 2.2 2.6 3 3.4 3.8 0.5 1 1.5 2 2.5 Normalized Mean Stress, f L P a rt ia l S a fe ty F a c to r iju' Ȗǹ' Ȗ'PD Ȗ'L Figure 6-21: Variation of Mean Partial Safety Factors with Respect to the Normalized Stress due to LOCA for fPB=5 and ȕ=3 and 5.5 195 Table 6-33: Ranges and Recommended (1) Adjusted Nominal Resistance Factors for g18 iju for Ȗǹ=1.1, ȖL=1.3 and ȖPD=1.2 ȕ CARBON STEEL STAINLESS STEEL Room Temperature Room Temperature 3 x For 0.51 or 0.97(2) (0.94 to 1.01) x For fPDt10 and fL”1 0.81(2) (0.79 to 0.83) x Otherwise 0.88(2) (0.85 to 0.91) 4 x For fPD”1 and fL”1 0.94 (0.91 to 0.97) x For fPDt10 and fL<2 or for fPD=50 and fL=2 0.70 (0.67 to 0.72) x For fPD”5 and fLt2 or for fPD=3 and fL=2 0.85 (0.82 to 0.87) x Otherwise 0.72 (0.74 to 0.78) x For fPD”1 and fL”1 or 0.92 (0.89 to 0.96) x For fPD=10 and fL”0.5 or for fPD>10 0.64 (0.61 to 0.67) x For fPD”5 and fL>1 or for fPD=3 and fL=1 0.84 (0.81 to 0.88) x Otherwise 0.74 (0.71 to 0.77) 5.5 x For fPD”0.5 and fL”0.5 0.79 x For fPD”1 and fL”1 0.72 (0.70 to 0.75) x For fPD”5 and fLt2 0.63 (0.60 to 0.66) x For fPDt10 or for fPD=5 and fL=0.5 0.50 (0.46 to 0.53) x Otherwise 0.57 (0.54 to 0.60) x For fPDt10 0.48 (0.45 to 0.52) x For fPD”5 and fL>1 0.62 (0.59 to 0.65) x For fPD”0.5 and fL”0.5 0.78 x For 0.51 or 0.84(2) (0.81 to 0.87) x For fPDt10 0.71(2) (0.69 to 0.74) x Otherwise 0.76(2) (0.74 to 0.79) 4 x For fPDt5 and fL”0.5 or fPDt10 0.65 (0.61 to 0.69) x For fPD”1 an1 fL”1 or fPD=3 an1 fL=2 0.85 (0.82 to 0.89) x Otherwise 0.73 (0.71 to 0.76) x For fPDt10 0.60 (0.57 to 0.64) x For fPD”5 and fL>1 or for fPD=3 and fL=1 0.73 (0.70 to 0.76) x For fPD”1 and fL”1 0.80 (0.77 to 0.83) x Otherwise 0.63 (0.60 to 0.66) 5.5 x For fPD>10 0.42 (0.42 to 0.43) x For fPD=10 or for fPDt5 and fL”0.5 0.46 (0.44 to 0.48) x For fPD”0.5 and and fL”0.5 0.72 x For fPD”1 and and fL”1 0.66 (0.64 to 0.68) x For fPD”5 and and fL>1 0.57 (0.54 to 0.60) x Otherwise 0.52 (0.50 to 0.55) x For fPDt10 0.42 (0.39 to 0.45) x For fPD”5 and fL>1 0.54 (0.51 to 0.57) x For fPD”0.5 and fL”0.5 0.68 x For 0.51 0.93(2) (0.90 to 0.96) Otherwise 0.87(2) (0.84 to 0.90) x For fPD”1 and fL”1 0.74(2) (0.71 to 0.78) x For fPDt10 0.62(2) (0.60 to 0.65) x For 1”fPD”5 and fL>1 0.71(2) (0.69 to 0.74) x Otherwise 0.67(2) (0.64 to 0.71) 4 x For fPDt10 0.65 (0.62 to 0.69) x For For fPD”0.5 and fL”0.5 0.90 x For fPD”1 and and fL”1 or for 1”fPD”3 and and fL>1 0.86 (0.84 to 0.89) x For 31 or for fPD”0.5 and and fL>1 0.79 x Otherwise 0.72 (0.70to 0.75) x For fPD”0.5 and fL”0.5 0.72 (0.72 to 0.74) x For fPD”1 and 0.51 0.63 (0.61 to 0.65) x For 31 or for fPD”0.5 and and fL>1 0.56 (0.56 to 0.57) x Otherwise 0.52 (0.49 to 0.56) x For fPD”10 or for fPD=5 and fL=0.5 0.37 (0.34 to 0.40) x For fPD”0.5 and fL”0.5 0.60 (0.59 to 0.61) x For fPD”1 and 0.51 0.48 (0.45 to 0.51) x Otherwise 0.43 (0.40 to 0.46) 600oF 3 x For fPD”1 and and fL”1 or for 1”fPD”5 and and fL>1 0.88(2) (0.85 to 0.91) x For fPDt10 0.77 (0.74 to 0.80) x Otherwise 0.82 (0.79 to 0.85) 4 x For fPD”1 and and fL”1 or for 1”fPD”3 and and fL>1 0.82(2) (0.79 to 0.85) x For fPDt10 0.6 (0.61 to 0.68) x For 3”fPD”5 and and fL”1 0.68(2) (0.67 to 0.70) x Otherwise 0.72(2) (0.70 to 0.75) 5.5 x For fPD”1 and and fL”1 0.64 (0.61 to 0.67) x For fPD”5 and and fL>2 or for 0.55 (0.53 to 0.58) x For fPDt10 0.44 (0.41 to 0.47) x Otherwise 0.52 (0.49 to 0.56) 197 Table 6-33: (Continued) ȕ CARBON STEEL 800oF 3 x For fPD”1 and and fL”1 or for fPD=3 and fL=2 0.72(2) (0.71 to 0.73) x For fPDt10 0.61(2) (0.59 to 0.64) x Otherwise 0.66(2) (0.63 to 0.69) 4 x For fPD”1 and and fL”1 0.66 (0.65 to 0.68) x For fPD”5 and and fL>2 or for 0.62 (0.60 to 0.64) x For fPDt10 0.52 (0.49 to 0.55) x Otherwise 0.56 (0.54 to 0.59) 5.5 x For fPD”1 and and fL”1 0.51 (0.48 to 0.54) x For fPD”5 and and fL>2 or for 0.44 (0.42 to 0.47) x For fPDt10 0.35 (0.33 to 0.38) x Otherwise 0.39 (0.37 to 0.42) (1) Recommended values are in bold face (2) Ȗǹ=1.1, ȖL=1.1 and ȖPD=1.1 6.3.5.4. Performance Function g19 This performance function checks pressurized pipes subjected to the SSE and loading resulted from LOCA. Table 6-34 presents the probabilistic characteristics for the parameters used in the performance function, given by Eq. (6-24) where fPS is the normalized stress due the pressure coincident with the SSE, fu the normalized ultimate strength of steel, fS the normalized stress due to SSE, and fL the normalized bending stress due to LOCA. Moreover, E is a factor applied to the combination of the dynamic loads that in this study is not examined and considered equal to 1 as explained in Chapter 5. )( 19 LfSfEPSfAfufg  (6-24) Table 6-34 gives the parameters used in the computations. Table C-34 gives the mean partial load and resistance factors for different operating temperature, values of ȕ, and types of steel. The calculated adjusted nominal resistance factors are presented for 198 all cases in Table C-35. A summary and the recommended values for the adjusted nominal resistance factor are shown herein in Table 6-35. Table 6-34: Parameters for the Calculations for g19 Parameter Statistical Properties Range Recommended Value Distribution ȕ na na 1.5, 2.5 na Mean NA 1 COV 0.05 to 0.10 0.10 fA Bias 1 to 1.05 1 Normal Mean 0.5 to 50* 0.5, 1, 3, 5, 10, 50 COV 0.10 to 0.14 0.13 fPS Bias NA 0.95 Lognormal Mean 2 to 4 2, 3, 4 COV 0.40 to 0.90 0.80 fS Bias 0.65 to 1 0.75 Extreme II (Largest) Mean 0.5 to 2 0.5, 1, 2 COV NA 0.20 fL Bias 0.80 to 1.05 0.85 Lognormal E NA NA 1.0 NA Mean NA NA COV Table 3-10 Table 3-10 fu Bias Table 3-10 Table 3-10 Lognormal NA=Not Available, na=not applicable, *The value of 50 is shown, whereas the same results are obtained also for values greater than 50 199 Table 6-35: Recommended (1) Values and Ranges for the Adjusted Nominal Resistance Factor for g19 iju for Ȗǹ=1.1, ȖPS=1.2, ȖL=1.3 & ȖS=1.5ȕ CARBON STEEL STAINLESS STEEL Room Temperature Room Temperature 2.5(2) x For fPS”1, fS>2, and fL”1 or for fPS=3, fS>2, and fL=0.5 or for fPS”1, fS=2, and fL=0.5 0.67 (0.63 to 0.71) x For fPS”10, fL>1and fS”1 or for 5”fPS”10, fL”1, and fS”1 0.98 (0.94 to 1.04) x For fPS>10 0.90 (0.88 to 0.94) x Otherwise 0.83 (0.72 to 0.94) x For fPS”1, fS>2, and fL”1 or for fPS=3, fS>2, and fL=0.5 or for fPS”1, fS=2, and fL=0.5 0.65 (0.61 to 0.69) x For fPS”10, fL>1and fS”1 or for 5”fPS”10, fL”1, and fS”1 0.98 (0.94 to 1.02) x For fPS>10 0.90 (0.87 to 0.92) x Otherwise 0.83 (0.71 to 0.95) 3.0 x For fPS>10 0.99 (0.98 to 1.02) x For fPS”1, and fSt2 or for fPS”0.5, fS”1, and fL”1 or for 12 or for fL”1 and fS=2 0.50 (0.39 to 0.60) x For 510 0.98 (0.96 to 1.01) x For fPS”1, and fSt2 or for fPS”0.5, fS”1, and fL”1 or for 12 or for fL”1 and fS=2 0.48 (0.38 to 0.59) x For 52, and fL”1 or for fPS=3, fS>2, and fL=0.5 or for fPS”1, fS=2, and fL=0.5 0.59 (0.51 to 0.67) x For fPS”10, fL>1and fS”1 or for 5”fPS”10, fL”1, and fS”1 0.91 (0.86 to 0.95) x Otherwise 0.77 (0.70 to 0.88) x For fPS”1, fS>2, and fL”1 or for fPS=3, fS>2, and fL=0.5 or for fPS”1, fS=2, and fL=0.5 0.65 (0.53 to 0.60) x For fPS”10, fL>1and fS”1 or for 5”fPS”10, fL”1, and fS”1 0.57 (0.70 to 0.89) x For fPS>10 0.77 (0.75 to 0.80) x Otherwise 0.72 (0.61 to 0.82) 3.0 x For fPS”1, fSt2, and fL”1 or for fPS=3, fS>2, and fL=0.5 or for fPS”0.5 and fS”1 0.41 (0.35 to 0.48) x For 3”fPS”10, fL>1and fS”1 or for 5”fPS”10, fL”1, and fS”1 0.85 (0.76 to 0.93) x For fPS>10 0.91 (0.89 to 0.94) x Otherwise 0.63 (0.50 to 0.77) x For fPS”1, fS>2, and fL”1 or for fPS=3, fS>2, and fL=0.5 or for fPS”1, fS=2, and fL=0.5 0.37 (0.33 to 0.41) x For 3”fPS”10, fL>1and fS”1 or for 5”fPS”10, fL”1, and fS”1 0.78 (0.69 to 0.87) x For fPS>10 0.85 (0.83 to 0.87) x Otherwise 0.58 (0.45 to 0.71) 200 Table 6-35: (Continued) ȕ CARBON STEEL STAINLESS STEEL 400oF t400oF 2.5(2) x For fPS”1, fS>2, and fL”1 or for fPS=3, fS>2, and fL=0.5 or for fPS”1, fS=2, and fL=0.5 0.67 (0.63 to 0.71) x For fPS”10, fL>1and fS”1 or for 5”fPS”10, fL”1, and fS”1 0.98 (0.94 to 1.02) x For fPS>10 0.85 (0.73 to 0.96) x Otherwise 0.83 (0.72 to 0.94) x For fPS”3 and fSt2 or for fPS”0.5, fS<2 and fL”1 or for 32 and fL>1 0.56 (0.46 to 0.66) x Otherwise 0.70 (0.62 to 0.80) 3.0 x For fPS>10 0.82 (0.80 to 0.85) x For fPS”1, and fSt2 or for fPS”0.5, fS”1, and fL”1 or for 12 or for fL”1 and fS=2 0.48 (0.39 to 0.56) x For 52 or for fPS”1, fS<2, and fL”1 0.37 (0.29 to 0.46) x For fPSt10 and fSt1 or or for fPS>10 0.75 (0.72 to 0.78) x Otherwise 0.58 (0.47 to 0.69) 600oF 2.5(2) x For fPS”1, and fSt2 or for fPS”0.5, fS”1, and fL”1 or for 12 or for fL”1 and fS=2 0.66 (0.59 to 0.73) x For fPS”10, fL>1and fS”1 or for 5”fPS”10, fL”1, and fS”1 0.92 (0.89 to 0.96) x Otherwise 0.77 (0.69 to 0.90) 3.0 x For fPS”1, fSt2, and fL”1 or for 12, 0.46 (0.36 to 0.56) x For 3”fPS”10, fL>1and fS”1 or for 5”fPS”10, fL”1, and fS”1 0.87 (0.76 to 0.98) x For fPS>10 0.90 (0.88 to 0.92) x Otherwise 0.69 (0.60 to 0.78) 201 Table 6-35: (Continued) ȕ CARBON STEEL 800oF 2.5(2) x For fPS”5, and fSt2 or for fPS”0.5, fS”1, and fL”1 or 0.56 (0.47 to 0.65) x For fPS>10, 0.66 (0.64 to 0.68) x Otherwise 0.70 (0.63 to 0.77) 3.0 x For fPS”1, fSt2, and fL”1 or for 12, 0.38 (0.29 to 0.46) x For 3”fPS”10, fL>1and fS”1 or for 5”fPS”10, fL”1, and fS”1 0.70 (0.61 to 0.78) x For fPS>10 0.71 (0.70 to 0.73) x Otherwise 0.58 (0.47 to 0.68) (1) Recommended values are in bold face, (2) Ȗǹ=1.1, ȖPS=1.0, ȖL=1.3 & ȖS=1.5 6.4. Computational Example Figure 6-22 shows a Class 2 straight piping segment anchored at both ends (e.g., pipe between two tanks) that should be designed to withstand the loads for Service Levels A, B, and D given in Table 6-36. Loading is given readily in the form of moments. 202 (a) (b) Figure 6-22: (a) A Piping Segment with Anchored Ends and (b) Cross-Section Used for Sample Computations Table 6-36: Data for Example Computations Material SA312, Type 304 Yield Strength, Sy*= 30 ksi Ultimate Strength, Su*= 75 ksi Do= 12.75 in t= 0.375 in L= 236 in Z= 43.82 in3 ZP= 57.46 in3 Design Temperature= 200 o F PDes= 820 psi Pmax= 845 psi PO= 1,080 psi PB= 1,080 psi PS= 1,350 psi MxA= 46,420 lb-in **MxM= 36,000 lb-in MxO= 75,000 lb-in MyO= 80,000 lb-in MxS= 140,000 lb-in MyS= 150,000 lb-in *Values obtained from ASME B&PV Code, Part II **Loading under Service Level B conditions Loading Nomenclature: MxA=Moment due to sustained weight MxO, MyO =Moments due to OBE MxS, MyS=Moments due to SSE MxM=Moment due to an instantaneous valve closure PDes=Design Pressure Pmax=Maximum Operating Pressure PB=Pressure under Service Limit B loading PO=Pressure coincident with OBE PS=Pressure coincident with SSE 203 Initially, in Table 6-37 the normalized nominal stresses are evaluated and using the appropriate bias factor, they are converted into mean normalized stresses. This procedure is necessary in order to choose the appropriate factors for the LRFD. Computation results according to the derived LRFD equations are given in Table 6-38. Values of the reliability index ȕ are arbitrary chosen. The symbols used are similar to the ones in the previous sections. Table 6-37: Nominal and Mean Normalized Stresses Symbol Nominal Normalized Stress [1] Bias [2] Mean Normalized Stress [1]x[2] fA 1 1 1 fPDes 6.58 1 6.6 fPmax 6.76 0.95 6.4 fPO=fPB 11.36 0.95 10.8 fPS 14.20 0.95 13.5 fM 0.77 1 0.8 fO 2.36 0.75 1.8 fS 4.42 0.75 3.3 204 Table 6-38: Computations with the Developed LRFD Equations Service Level P.F.* ȕ LRFD Equation g2 6 Z AM yS 2.145.0 t or psi20.271,1 82.43 420,46)2.1( psi500,13 t Design g3 3.5 t oDPDesP Z AM yS 4 2.11.170.0 t or )375.0(4 )75.12)(820(2.1 82.43 )420,46(1.1 000,21 t or 21,000psi 3.529,9t psi A g4 3.5 fPmax=6.4>5 t oDP Z AM yS 4 max 2.11.166.0 t or 3.784,9 )375.0(4 )75.12)(845(2.1 82.43 )420,46(1.1 9,800psi1 t psi g6 3.5 P Z MM t oDPBP PZ AM yS 2.1 4 2.11.172.0 t or 259,10 46.57 )000,36(2.1 )375.0(4 )75.12)(845(2.1 46.57 )420,46(1.1 600,21 tpsi psi g7 3 inlb660,109 2 )( 2 )(  o yM o xMOM PZ OM PZ AM yS 5.11.135.0 t or 751,3 46.57 )660,109(5.1 46.57 )420,46(1.1 500,10 t psi g8 3 fPO=10.8 and fO=1.8 PZ OM t oDOP PZ AM yS 5.1 4 2.11.174.0 t 767,14 46.57 )660,109( 5.1 )375.0(4 )75.12)(080,1( 2.1 46.57 )420,46( 1.1200,22 tpsi psi B g9 2.5 fPO=10.8, fO=1.8, and fM=0.8 PZ OM PZ MM t oDOP PZ AM yS 9.00.1 4 0.10.187.0 t 333,12 46.57 660,109 9.0 46.57 000,36 )375.0(4 )75.12)(080,1( 46.57 420,46 100,26 t psi *P.F.=Performance Function 205 Table 6-38: (Continued) Service Level P.F.* ȕ LRFD Equation g16 3 inlb180,205 2 )( 2 )(  o yM o xMS M PZ M PZ AM u S S5.11.130.0 t or 245,6 46.57 )180,205(5.1 46.57 )420,46(1.1 500,22 t psi D g17 3.0 fPS=13.5>10 PZ SM t oDsP PZ AMSu 5.1 4 2.11.185.0 t 015,20 46.57 )180,205(5.1 )375.0(4 )75.12)(350,1(2.1 46.57 420,46)1.1( 750,63 t psi *P.F.=Performance Function 6.5. Conclusions From the computation of partial safety factors in the previous sections the following can be deduced: 1. For the design only for internal pressure, the partial safety factors were computed for different types of steel and design temperatures. Linear interpolation can be used to obtain only approximate values of factors for other design temperatures or values of the target reliability index, ȕ. The adjusted nominal resistance factors were modified only in order to obtain rounded numbers, since the ranges of the mean values of the used variables yielded the same values for the mean partial safety factors. 2. For the performance functions with combined loading, the magnitude of loads significantly influences the adjusted nominal resistance factor. In order to achieve consistent reliability levels (ȕ in the space [ȕ-0.25, ȕ+0.25]) and for all the examined performance functions, several categories were derived for different magnitude of loading. Usually, the pressure alone or in combination with other 206 loads (e.g., earthquake) controls the separation in different categories. More categories usually result for higher values of the target reliability index. 3. For ratios of stresses of pressure to sustained weight greater or equal than 10, the resultant adjusted nominal resistance factor, in most of the cases examined, attains the highest values, meaning that for the same target reliability index, these loading cases are less critical. 4. Tables were provided with the ranges for the resultant nominal resistance factor for a predefined set of load factors, as well as recommended values. In order to obtain recommended mean partial factors, the nominal load and resistance partial factors listed in the tables should be devided by the considered recommended bias for each performance function. The relation of factors applied to the mean value of variables (mean factors) and those applied to nominal values (nominal factors) was explained in Chapter 4. Such calculations are not shown herein due to their simplicity. The need of having the recommended mean values may arise, should different bias factors than the ones used in this study, need to be applied. In the tables, the normalized load ratios also correspond to ratios of mean values. Dividing by the used bias, the nominal ratio may be obtained. In a computational example nominal normalized ratios were converted into equivalent mean ratios. 5. Figures showing the variation of mean factors are available for selected cases and performance functions, since there is a repeatability of results and any further the cases examined would be excessive. Nonetheless, computation results for the mean partial safety factors are listed comprehensively in tables of Appendix C. 207 From the figures showing the variation of mean factors, it can be seen that the difference between the partial mean factors is significant, e.g., in case of earthquake, or different levels of piping pressurization, whereas the transition in a predefined set of nominal load factors diminish differences and provides uniformer results. 6. The obtained nominal partial resistance and load factors correspond to the recommended probabilistic characteristics of this study. The impact of them (e.g., different coefficient of variation for the earthquake load, etc.) should be considered with respect to the selected value of the target reliability index. Differences become significantly less important for lower values of the target reliability index. 7. It can be noticed in the calculation results that for some cases the adjusted resistance factors are very close for carbon and stainless steel or for operation in different temperatures (e.g., design temperature at 400oF or at room temperature for Service Levels C and D, etc.). Therefore, for the target reliability index, derived according to the procedures described in Chapter 5, simplifications can be achieved that may diminish the volume of the adjusted nominal resistance factors. Such simplifications must take into consideration the impact on the target reliability index, β, and consequently the probability of piping failure. 8. For the performance functions for combined loading, it is important to perform the calculations for a specific predefined reliability index, since the derived categories for the adjusted resistance factor can vary with different reliability indeces or different design temperatures. Therefore, the use of approximate 208 methods such as the linear interpolation for immediate values of the target reliability index, ȕ, seems not efficient. 209 CHAPTER 7: FATIGUE DESIGN OF PIPING The fatigue design of piping belongs to Service Levels A and B. Although the design equations for all piping classes are similar with only different the allowable stress, design for fatigue is different for Class 1 piping compared to those of Class 2 and 3 piping. The chapter starts with a general discussion describing the fatigue design of all pipes according to the ASME B&PV Code, followed by a proposed probability-based framework for the design of Class 2 and 3 piping for bending cycling loading (e.g., thermal expansion). The proposed methodology is built on the design equations of the ASME B&PV Code (2001), Sections NC-3653.2 and ND-3653.2. 7.1. General Discussion Nuclear pipes are subjected to cycling loading. Significant cycles of stress in piping are produced by a) moment loading, b) pressure and c) thermal forces. Table 7-1 shows for example the design cycles of Class 1 piping. The produced stress cycles from these loadings are less than 10 5 and more often just few thousand during the service life of piping. This type of fatigue is called low-cycle fatigue in opposition to high-cycle fatigue that also occurs to piping components and is due for example to flow-induced vibrations, vibrating machines, etc. Low-cycle fatigue, the primary type of piping fatigue, significantly influences the high pressure injection makeup lines, the residual heat removal piping, surge lines, 210 feedwater lines, and the reactor coolant piping. It involves plastic strain and high stress levels in opposition to high-cycle fatigue, where the structure remains in the elastic region and the number of load cycles to cause failure is high. For low-cycle fatigue strain is the controlled variable, and not the stress, in fatigue tests that are conducted in order to determine the inherent decreasing strength of steel due to cycling loading. 211 Table 7-1: Design Transients and Cycles for the Reactor Coolant System (IAEA, 2003) Transient Number Description More Significant Contributor to CUF* Heat-up 250 From Tave = <200oF to >550 o F at <100 o F/h Cooldown 250 From Tave = >550oF to <200 o F at <100 o F/h Pressurizer cooldown 250 From Tpressurizer>650oF to <200 o F at 200 o F/h Loss of Load, without reactor trip 100 >15 to 0% of rated thermal power Loss of off-site ac 50 Loss of ac off-site electrical Reactor trip 500 100 to 0% of rated thermal power Reactor trip from full power 400 Primary hydrotest at 3125psi, 400 o F 10 Less Significant Contributor to CUF* Large step load decrease 200 100 to 0% of rated thermal power Inadvertent auxiliary spray actuation 10 Spray water temperature differential >320 o F Loss of flow in one loop 100 Loss of only one reactor coolant pump Pipe break 1 Break in reactor coolant system pipe>6in Operating Basis Earthquake 400 20 earthquakes with 20 cycles/earthquake Unloading between 0 and 15% power 500 Loading between 0 and 15% power 500 Plant unloading 5% full power/minute 13,200 Reduced temperature return to power 2,000 Step load increase at 10% full power 2,000 Step load decrease at 10% full power 2,000 Feedwater cycling 2,000 Primary side leak test 200 *CUF: Cumulative Usage Factor The fatigue design of Class 1 piping and those of Class 2 and 3 is different in the Code. More specifically, the fatigue design of Class 1 piping is based on fatigue curves (S-N) for the initiation of a crack on small size, polished specimens operating at room 212 temperature, while those of Class 2 and 3 is based on equations derived from fatigue curves calculated for the through the wall crack of full-scale pipes at room temperature. For Class 1 piping fatigue curves are plotted in the ASME Code’s Figures I-9.1 for carbon and low alloy steel and I-9.2 for austenitic stainless steel. For fatigue tests for Class 1 piping the strain for the initiation of the crack was measured and an equivalent stress was evaluated by multiplying the strain with the modulus of elasticity and dividing by two in order to obtain the stress amplitude. Moreover, the fatigue curves were lowered by a factor of 2 either on strain or of by 20 on the cycles, whichever resulted in a more conservative estimate for each curve point (for low cycle fatigue the factor of 20 usually dominates, whereas the factor of 2 on strain is dominant for high cycle fatigue points). These factors are not safety factors but rather aim to adjust the experimental results to real conditions of pipe functioning (temperature, not polished real size pipes). Moreover, using the modified Goodman diagram the mean stress effects were also considered for the construction of the Code fatigue curves. The cumulative fatigue damage based on Miner’s rule is also evaluated for these pipes. Figure 7-1 provides the nomenclature for a sinusoidal cycling load. Figure7-1: Nomenclature for Cycling Loading with Mean Stress (ASME Criteria, 1969) 213 For Class 1 piping research has been done in order to modify the curves for pipes operating in real LWR coolant environment conditions (e.g., effect of oxidized water, etc.) by Chopra, et al. (1999). Moreover, some studies exist in literature for the probabilistic design of piping. For example, Zhao, et al. (2000), propose an analysis based on the local-strain-approach life prediction and a performance function with respect to strains. Sudret, et al. (2005) quantifies the uncertainties of fatigue design by considering the cumulative damage as a criterion of failure due to crack initiation. 7.2. ASME Practice for Class 2 and 3 Piping Class 2 and 3 piping subjected to cyclic loading such as thermal expansion - contraction or anchor / support movements, are designed according to the ASME Boiler and Pressure Vessel Code, Section III, (2001) subsections NC-3653.2 and ND-3653.2, respectively. For the thermal loading, moment ranges from a flexibility analysis are considered and stresses are calculated in locations, where the pipe is restraint to move freely due to the displacements caused by thermal loading. Class 2 and 3 piping, unlike Class 1 piping, is not designed for peak stresses (e.g., radial gradient temperature, etc.) or cycles of alternating pressure (e.g., start-up, shut down, etc.). More specifically, the Code provides two equations, here Eqs. (7-1) and (7-2), for thermal expansion in subsections NC and ND-3653.2, which are identical for Classes 2 and 3 piping. Equation (7-1) allows considering only secondary stresses (stresses due to thermal expansion, and other support movements inducing moment loading), whereas Eq. (7-2) combines primary (sustained weight and internal pressure) with secondary stresses (thermal expansion stresses). 214 A C E SZ iM S d (7-1) where MC=range of resultant moments due to thermal expansion that also includes anchor movements of not reversing dynamic loading Z=elastic section modulus i=stress intensification factor SE=stress range Ah CA n o TE SSZ Mi Z Mi t DPS d¸ ¹ ·¨ © §¸ ¹ ·¨ © § 75.0 4 (7-2) where P=internal design pressure Do=outside diameter of pipe tn=nominal wall thickness Z=elastic section modulus MA=resultant moment due to weight and other sustained load MC=range of resultant moments due to thermal expansion i=stress intensification factor, while the factor (0.75i) shall not be less than 1. STE=stress range The stress intensification factor, i, is the ratio of the moment producing fatigue failure in a given number of cycles in a girth butt-welded pipe, shown in Figure 7-2, of nominal dimensions to that producing failure in the same number of cycles for the component under consideration. In Eqs. (7-1) and (7-2), the allowable stress range for cyclic stresses, SA, is given as a combination of the allowable stress at maximum temperature, Sh, and the minimum (cold) temperature allowable stress, Sc, of the cycle, as Eq. (7-3) shows, and also described in Table 2-1. In addition, a reduction factor for the stress, f, is used in order to account for the cycling nature of the load, which is dependent on thermal full cycles as Table 7-2 shows. 215 )25.025.1( hcA SSfS  (7-3) Figure7-2: Girth Butt-Welded Straight Pipe Table 7-2: Values for the Reduction Factor, f Number of equivalent full temperature cycles, N f d7,000 1.0 7,000 to 14,000 0.9 14,000 to 22,000 0.8 22,000 to 45,000 0.7 45,000 to 100,000 0.6 100,000 0.5 The thermal full cycles, N, are obtained using Eq. (7-4). nnE NrNrNrNN 5 2 5 21 5 1 ... (7-4) where NE=number of cycles at full temperature ǻTE for which expansion stress was calculated, N1, N2, …, Nn=number of cycles at lesser temperatures, ǻȉ1, ǻȉ2,..., ǻȉn, and r1, r2,…,rn = ratios of any lesser temperature cycles for which the expansion stress has been calculated, e.g., r1=ǻT1/ǻȉǼ, etc. The philosophy of the present design is to permit designers to avoid accounting reduction of fatigue cycles, when the latter are less than 7,000, which is a usual case. As shown later, significant conservatism is introduced in the calculations. 216 Fatigue design of Class 2 and 3 piping in the Code is based on the work done by Markl, et al. (1952), while Rodabaugh, et al. (1983) gives a commentary and presents a comparative study for the fatigue design of all piping Classes. Markl, et al. (1952) tested various cantilevered piping products (butt-welded pipes, straight pipes, elbows, tees, etc.) made of carbon steel A-106 Grade B, in actual dimensions, at room temperature, filled with water. Load was related to deflection by a calibration curve and nominal stresses, Sn, were estimated, using Eq. (7-5). Z LPSn (7-5) Sn=nominal stress P=load L=distance between failure point and load application point Z=elastic section modulus The approach of considering the nominal stress, which is the maximum stress due to applied loading at the crack location, for fatigue design is the simplest among others like the hot spot approach, or approaches where the material’s constitutive relations are taken into consideration. Markl, et al. (1952), moreover, derived a general best fit equation representing the results of the tested piping components, Eq. (7-6), which is applicable to all components by using different values of i, and thus the equation is in accordance with the whole philosophy of the Code design, where indices are used in order to obtain design equations for different piping components other than straight pipes. The equation here is expressed in terms of the stress range, whereas Markl, et al. (1952) presented them with the stress amplitude, instead. 217 bNaSi  (7-6a) where, S=applied stress range N=cycles to failure i=stress intensification factor as described above, for butt welded pipes equal to 1 Į=490,000 for butt girth welds b=0.2 By substituting in Eq. (7-6a) the above values for Į and b, Eq. (7-6b) is obtained: 2.0000,490  NSi (7-6b) Assuming a safety factor of 2 to be applied to the derived stress (Rodabaugh, 1983), reducing this way Į to 245,000, Eq. (7-6c) is obtained. 2.0000,245  NSi (7-6c) Figure 7-3 shows fatigue curves (log-log scale) developed by Markl, et al. (1952) -the test data includes experiments performed also by other researchers- while in Figure 7-4 Eqs. (7-3), (7-6b), and (7-6c) are plotted for butt-welded pipe (i=1). Equation (7-3) is plotted for carbon steel A106B and austenitic steel Type 304 for a temperature range between 70 o F and 200 o F. From this figure, it can be inferred that the safety factor varies considerably in the allowable stress design method, since in all cases it is greater than 2 and especially for values of N less than 7,000. The difference between carbon and stainless steel is not significant as shown in Figure 7-4. From additional fatigue tests on stainless steel reported by Rodabaugh (1983), it is derived that carbon steel specimens present greater resistance to fatigue as piping made of stainless steel. 218 Figure7-3: Fatigue Curves for Girth Butt-Welded Joints in Straight Pipes (Markl, et al., 1952), where the Nominal Stress Refers to the Stress Amplitude The stress intensification factor proposed by Markl, et al. (1952) for curved pipes is also used in the Code and more specifically is given by the following equation: 3/2 90.0 h i where h is a flexibility characteristic given by the formula: 2r Rth where t=pipe wall thickness, R=bend radius, and r=mean pipe radius. 219 1 10 100 1000 10 100 1,000 10,000 100,000 1,000,000 Number of Cycles to Failure, N S tr e s s R a n g e , S (k s i) Stainless Eq. (7-3) Eq. (7-6c) Eq. (7-6b) Carbon Eq.(7-3) Figure7-4: Allowable Fatigue Stress, SA, for Class 2 and 3 Piping, and Fatigue Test Stress versus Cycles to Failure for Girth Butt-Welded Pipe 7.3. Reliability-Based Fatigue Design Several structures are designed today for fatigue, based on reliability methods, e.g., bridges, ships, aircrafts, etc. In what follows, a probabilistic design method is suggested for Class 2 and 3 piping based on the fatigue curves proposed by Markl, where only cycling moment loading is considered. Therefore, for the consideration of loading other than cycling moments (e.g., thermal peak stresses, etc.) the proposed methodology is not recommended and pipes that need to be designed for such loads should be designed according to the provisions for Class 1 piping. For the reliability–based fatigue analysis of structures there are generally two approaches, a) the direct reliability method, and b) the Load and Resistance Factor Design method. In both approaches a limit state equation is first formulated. Then, for 220 the first method the design is checked by comparing the calculated reliability index for the specific design with a target reliability index. The second method is similar to the one used in the previous chapter, where load and resistance factors are calculated according to a target reliability index. The first approach is used herein, since it is the one usually adopted for the reliability fatigue design of structures and, moreover, it is not always possible to achieve uniform partial safety factors for all fatigue designs. Analysis is similar to that developed by Ayyub, et al. (2002) for fatigue of ship details. 7.3.1. Performance Function and Equivalent Stress Range Usually, performance functions for fatigue are either based on cycles, Eq. (7-7), or cumulative damage ratios, Eq. (7-8). NNg f  1 (7-7) DDg f  2 (7-8) where Nf=cycles that lead to failure N=expected cycles to act during service life of component Df=cumulative damage ratio that leads to failure D=expected accumulated damage ratio for the service life of component In this study the performance function g2 is used. The expected accumulated damage, D, the pipe will be subjected to, is calculated using the Miners’ rule, which is based on a linear damage hypothesis and more specifically in the following two assumptions (Meiner, 1945; Kesioglu, 1991): 1. Each component (in this case pipe) can absorb a maximum amount of energy before failure, W, which is the sum of the amount of energy absorbed Wi at each 221 stress range (level) Si, i=1 to nb that the component is subjected to during its service life. kWWWW  ...21 (7-9) 2. For a specific stress level the component absorbs energy that is proportional to the total energy, W, by an amount called usage factor, U, defined as the ratio of the expected cycles, ni, at a specific service to the total life (or cycles), Nfi, to failure for that stress level, as if it was the only one that was acting on the component. if i N nU (7-10) and therefore Eq. (7-9) yields: kf k ff N n W N nW N nWW  .... 2 2 1 1 or kf k ff N n N n N n  ....1 2 2 1 1 or 1 11 ¦¦ b i b n i f i n i i N n U (7-11) Based on the above discussion the performance function of Eq. (7-8) can be rewritten as: ¦ bn i if i f N n Dgg 1 2 (7-12) 222 where, ni=expected number of cycles at the ith stress range level Nfi=number of cycles to failure at the ith stress-range level nb=number of stress-range levels in a stress range histogram Generally, the Miner model is a good assumption if cycles of small and large stresses are evenly distributed throughout the service life of piping. If smaller ranges are applied first the cumulative damage ratio, Df, can reach values as high as 4 or 5 or be less than 1 in the opposite case, as then failure can be much accelerated, (ASME Criteria, 1969). Therefore, Df is actually a random variable with a high coefficient of variation, and a mean value considered to be 1, as explained later. The stress range, S, is also a random variable such as it can be described by a probability density functions fS(si), as Figure 7-5 shows. The probability density function can be composed by considering the anticipated structural behavior of piping to expected, in this case, moment loading. The random stress probability density distribution can be divided into a large number, nb, of narrow stress blocks of width 'S. In each block, the number of cycles, ni, is given by Eq. (7-13), where N denotes the total anticipated number of cycles during the service life of a pipe. SsNfn iSi ' )( (7-13) 223 fS (s) 'S fS (Si) S Si Figure7-5: Probability Density Function (PDF) for the Bending Stresses (Ayyub, et. al., 2002) By rewriting Markl’s general Eq. (7-6a) as in Eq (7-14) and solving for N, Eq. (7- 15) is obtained. bb NAN i aS  (7-14) m S AN ¸ ¹ ·¨ © § (7-15) where m=1/b and A=Į/i Equation (7-15) based on Markl’s fatigue tests correspond to constant range applied stress. Nevertheless, in real loading conditions cycles come with variant stress ranges. Considering Eqs. (7-13) and (7-15) and the failure cycles for the stress range Si, D in Eq. (7-12) becomes: ¦ ¸¸¹ · ¨¨© § ' ¦ bb n i m i iS n i if i S A SsfN N nD 11 )( (7-16) 224 As 'S goes to zero and by considering N, A, and m as constants, the sum in Eq. (7-16) becomes: ³ f 0 )( dssfS A ND S m m (7-17a) The integral expression of Eq. (7-17a) is the mean or the expected value, E(S m), of the random variable S m. Accordingly, the damage ratio D equals: )( mm SEA ND (7-17b) By equating Eq. (7-15), having the constant amplitude stress applied during the tests, with Eq. (7-17b) an expression of an equivalent for constant stress range loading, Se, is obtained as: m m e D SES )( (7-18a) Considering Eq. (7-18a) at failure and thus Df=D=1, Eq. (7-18a) is simplified: m n Ț i m i m m e b fSSES ¦ 1 )( (7-18b) where fi=ni/N and N is the expected number of applied cycles. Equation (7-18b) can be used to evaluate the equivalent mean stress, eS . 7.3.2. Strength and Loading Uncertainties For the probabilistic design of structures the probabilistic characteristics should be estimated. The applied stress range as well as the cumulative damage, Df, calculated 225 according to Miner’s rule are considered as variables. Considering that ks is a variable representing the stress range uncertainty factor, Eq. (7-18b) can be rewritten as: m n i m ii m s m mm se b SfkSEkS ∑== =1 )( (7-19) where fi is the fraction of cycles in the ith stress block, Si the stress in the ith block and nb number of stress blocks. As a result from the previous discussion the performance to be used in a reliability design is: ∑−= = bn i m m i m si f A SknDg 1 (7-20a) or m m e m s f A SkNDg −= (7-20b) In Eq. (7-20b), N is the total expected loading cycles (for variant stress range) and is considered constant. The random variables implicated in the performance function of Eq. (7-20) as well as their probabilistic characteristics are shown in Table 7-3. The probabilistic characteristics are only some recommendations based on usual values obtained from fatigue tests and not necessarily from piping. 226 Table 7-3: Variables for the Piping Fatigue Design Variable Description Mean COV Distribution ks Variable that considers uncertainty in fatigue stress range calculation 1 0.10 Normal A Variable that reflects the uncertainty in S-N relationship Depending on Component A=(490,000/i) 0.20 to 0.40 Lognormal Df Uncertainty introduced by the use of Miner’s rule 1 0.20 to 0.50 Lognormal Se Uncertainty arising from variables used in Se calculation Eq. (7-18b) 0.10 Lognormal m Constant 5 na na na=not applicable 7.3.3. Values for the Target Reliability Index, ȕ Usually, for fatigue the value of target reliability index, ȕ, is based on the method of inspection and even the ability or not to inspect the component under consideration. Table 7-4 presents values of ȕ for different structures and for the entire service life of structure. Table 7-4: Values of Target Reliability Index for Fatigue Design and Different Structures Target Reliability Index, ȕ Structure Reference 1.5 to 3.8 Buildings/Bridges ENV 1991-1 (1994) 2 to 3.5 Ships Mansour (1996) 7.3.4. Discussion and Evaluation The following limitations and disadvantages exist for the use of the developed performance function for Class 2 and 3 pipng: x It considers only cycling stresses produced by bending such as thermal expansion, 227 anchors' movements, etc. It can therefore substitute Eq. (7-1) of the Code that covers the same type of loading. x The effect of mean stress is neglected. Mean stress, when tensile, can reduce the strength capacity, since it helps the development of cracks. x Calculation of stresses refers to a linear elastic behavior of steel, therefore the elastic section modulus should be considered for the calculation of the stresses, whereas the plastic behavior of the material is ignored leading to conservative results. x The nominal stress approach for fatigue analysis is dependent on the model used. Markl (1952) selected the cantilever model as a more conservative case for the fatigue loading of straight pipes. Svacuzzo (2006) by testing pipes in a four-point fatigue bent test concluded that Markl’s equation and results in failure cycles are conservative. x As in the Code it is assumed that performance in fatigue tests do not change for temperature up to 600 o F. x Thermal stress analysis is needed for all temperature ranges in order to access the value of Si, while in the Code the thermal analysis is performed for only one thermal range and as explained equivalent thermal loads are evaluated for different temperature difference. The use of this probabilistic method, however, offers the following advantages: x The proposed method can substitute the one used in the Code, since here bending cyclic loading from various loads can be considered (thermal expansion, 228 earthquake, induced support movements, etc.) too. The reliability index helps understand the safety margins of the design. x Calibration with the ASME equations may provide the means for ameliorating the procedure and reducing conservatisms in a rational way. 7.3.5. Computation Procedure The steps for the reliability-based method presented herein are: 1. The cycles Ni and the stress range Si=Mi/Z are evaluated for different stress blocks of number nb. It should be clarified that the cycles are not full cycles as defined in the Code but the estimated cycles for the calculated stress range. 2. The performance function to use is either this of Eq. (7-20a) or (7-20b). 3. Determine A=Į/i, where Į=490,000 and i as defined in Table NC-3673.2(b)-1 of the Code. The probabilistic characteristics of variables A, ks, Df, and Si or Se are selected and using AFOSM a converged value of ȕc is produced. 4. Compare the computed ȕc with the target reliability index ȕ. If ȕc>ȕ the pipe’s geometrical properties are acceptable. In the opposite case, the elastic section modulus of pipe should be increased, and the above procedure should be repeated. An alternative procedure can be used to estimate an allowable Sƍe given the probabilistic characteristics of variables and a target reliability index. Then, if Seȕ ? NO Y E S Converged Reliability Index, ȕc Select Piping Dimensions & Evaluate Z END Target Reliability Index, ȕ Converged for ȕ Probabilistic Characteristics (Mean, COV, Distribution) for SP, SA, Su, ks, SeT Use the Performance Function: m m e m s f A SkN Dg  Probabilistic Characteristics (Mean, COV, Distribution) for A, Df and ks, and (COV, Distribution) for Se Evaluate Am=(490,000/i)m, i=1 for girth butt-welded pipe, m=5 eS ' Figure7-7: Flowchart Using Direct Reliability-Based Design for the Combination of Thermal Expansion Stresses with Primary Stresses 7.3.6.1. Example 1 This example shows calculations for the fatigue design of a pipe imposed on load cycles and corresponding stress ranges given in Table 7-5. The probabilistic characteristics of variables presented in Table 7-3 are used, and moreover a girth butt- 233 welded straight pipe is considered. Therefore, i=1 and the COV for both A and Df is considered to be 0.30. From Table 7-5, it is nb=6 and N=7,806. The target reliability index is considered 3.0. Table 7-5: Cycles and Stress Ranges for Example 1 Computations Stress Block Stress Range, Si(psi) Number of Cycles, Ni for 40 Years Service fi Sim fi 1 45,000 200 0.01937 4.72785E+21 2 90,000 10 0.001291 7.56457E+21 3 8,000 500 0.064566 2.0989E+18 4 4,000 7,000 0.903926 9.18268E+17 5 12,000 90 0.010331 2.86893+18 6 30,000 6 0.000517 1.86779+19 Using Eq. (7-18b), the equivalent mean stress for the different stress ranges is calculated as: 5 6 1 5¦ Ț iie fSS =26,188psi Convergence was achieved for ȕc=7.94>ȕ=3 and therefore the design is acceptable. By using the alternative procedure a converged mean value of equivalent stress for ȕ=3 is calculated as eS ' =52,538psi>26,188psi, showing again the adequacy of the design. 7.3.6.2. Example 2 For this example the performance function of Eq. (7-21) is used. Table 7-6 presents the total of cycles the pipe is subjected to, while from them, the thermal ones have the symbol (T) . Table 7-7 shows only the thermal cycles. The assumed probabilistic characteristics of variables are given in Table 7-8. The properties of ultimate strength of 234 steel are considered for the design temperature of the pipe. The target reliability index is ȕ=3. The pipe is straight and girth butt-welded. Table 7-6: Cycles and Stress Ranges for Example 2 Computations Stress Block Stress Range, Si(psi) Number of Cycles, Ni for 40 Years Service fi Sim fi 1 32,000 (T) 1,000 0.2257 7.57436E+21 2 80,000 (T) 80 0.0181 5.91747E+22 3 7,000 (T) 300 0.0677 1.13817E+18 4 14,000 (T) 3,000 0.6772 3.64215E+20 5 12,000 50 0.0113 2.80849E+18 Table 7-7: Thermal Cycles and Stress Ranges Stress Block Stress Range, Si(psi) Number of Cycles, Ni for 40 Years Service fi Sim fi 1 32,000 (T) 1,000 0.2283 7.66083+21 2 80,000 (T) 80 0.0183 5.98502+22 3 7,000 (T) 300 0.0685 1.15116E+18 4 14,000 (T) 3,000 0.6849 3.68373E+20 From the data in Table 7-7, showing only thermal cycles, the equivalent mean stress due to thermal loads, eTS , is evaluated as: 843,365 4 1 5 ¦ Ț iieT fSS psi (7-23) Following the procedure described in the previous example and for the data of Table 7-6 and ȕ=3, it produces eS ' =58,840psi. The performance function of Eq. (7-20b) is also satisfied since eS =36,759psi<58,840psi. 235 Table 7-8: Considered Probabilistic Characteristics of Variables in Eq. (7-21) na=not applicable The direct reliability method and AFOSM was used, given the data in Table 7-8, in order to evaluate the converged reliability index for the performance function of Eq. (7-21). It was found that ȕc=2.68<3, thus the elastic section modulus of the pipe should be increased. 7.3.7. Conclusions and Recommendations In this chapter reliability-based design equations were presented for the fatigue design of Class 2 and 3 piping. The derived equations are built up in the equations of the ASME B&PV Code, Part III, NC and ND-3653.2 Eqs. (10) and (11) or Eqs. (A-17) and (A-18) of Appendix A. It should be, moreover, clarified that the performance functions for the fatigue design necessitate that the designer provides the mean values of the variables used (not the nominal). In this study some probabilistic characteristics were proposed, while with future research new data may be used, too. Fatigue of Class 1 piping uses different fatigue curves that as briefly explained are based on measurements of plastic strains on small specimens and not on real pipes (Langer’s model). Class 2 and 3 piping may be also judged to be analyzed as Class 1 piping, permitting this way additional loads to be considered in the design (e.g., thermal peak stresses, etc.). A smaller target reliability index may be then considered for these pipes. Variable Mean COV Distribution ks 1 0.10 Normal SeT 36,843psi 0.10 Lognormal Sǹ 800psi 0.10 Normal SP 6,800psi 0.10 Lognormal Su 74,000psi 0.06 Lognormal Sƍe 58,840psi na na 236 CHAPTER 8: CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH This chapter presents a summary and conclusions of the dissertation. Moreover, it provides suggestions for further research on the reliability-based design of piping. 8.1. Summary The dissertation provides background information on the reliability–based design of nuclear piping. Allowable stress design equations used in the ASME B&PV Code are also explained. A comparison of the ASD and LRFD methods is shown and the advantages of the LRFD are demonstrated. The total bias of the resistance and the model uncertainty of strength models related to busrt or yielding of pipes due to internal pressure or the ultimate bending capacity is evaluated based on cited experimental results. Probabilistic characteristics of the design variables are summarized. Load combinations are provided for weight, internal pressure, dynamic loads (OBE, sudden valve closure, etc.), and accidental loads (SSE, pressure and mechanical loading due to Loss of Coolant Accident), and performance functions or else state limit equations are formulated. The mean partial load and resistance safety factors are evaluated and an adjusted nominal resistance factor is derived for a predefined set of partial load factors for each performance function. Computations are performed for at least two values of the target 237 reliability index for all the performance functions and cases examined. Neither calibration nor the other procedures described in Chapter 5 are used in order to access a unique value of the target reliability index for each performance function. Instead, partial safety factors are computed and recommended for ranges of target reliability indices. 8.2. Conclusions From the performed research, the following can be deduced: 1. The LRFD is possible for mechanical components such as piping in nuclear plants. This method of design can achieve, among others, more consistent (and known) reliability levels than the Allowable Stress Design that is used in the ASME Codes. The current design practice is expected to yield inconsistent levels of reliability for different design temperatures or loading conditions for piping. 2. A database with the probabilistic characteristics of design variables is created based on literature review and engineering judgement. There is inadequate information for some cases and especially concerning the uncertainties, i.e., bias, of the strength prediction models. 3. The degree of piping pressurization, as compared to sustained weight is critical for the selection of the adjusted nominal resistance factor. The adjusted nominal resistance factor can vary significantly under different loading conditions given a predefined set of load factors. 4. The partial safety factors for LRFD can become fewer compared to the database of allowable stresses in the Code. This reduction is possible since the reliability- based design effectively unifies materials with similar probabilistic characteristics. For the LRFD, the nominal values of steel resistance are 238 considered for all design temperatures as the minimum specified strength of steel at room temperature. These nominal values are therefore necessary for the design. 5. The proposed fatigue design for Class 2 and 3 piping can yield an estimate of the reliability of piping, which is not possible with the current design practices. 8.3. Suggestions for Future Research Some suggestions for future research on the reliability-based design of piping are provided below: 1. It is recommended to evaluate the target reliability index for each performance function. As explained in Chapter 5, a combination of methods should be utilized (e.g., code calibration, collection of historical data for failures of piping, etc.). This is a necessary task in order to provide uniform criteria for design, and to overcome conservatisms of the current design. 2. In order to access the uncertainties, i.e., bias, for the loads and strenght, analytical work is needed. It is, moreover, important to estimate the bias for other piping components (curved pipes, reducers, etc.), which can possibly differentiate the partial safety factors developed in this study. Moreover, with analytical work, the impact of combination of dynamic loads, which in this study was expressed by a factor E considered to be one, can also be examined. 3. The database of probabilistic characteristics of the steel strength and loads can be further expanded with the acquisition of new data (e.g., probabilistic properties of mechanical loads). Expert-opinion elicitation can provide useful and credible information. Also experimental studies can provide information and answers in 239 cases that information is inadequate (e.g., piping loading at elevated temperatures). 4. A reliability-based fatigue framework for Class 1 piping can be developed. Additionally, there are other equations mostly for Class 1 piping design that were not considered herein, e.g., thermal axial forces or thermal ratcheting, for which a probabilistic framework can be developed. Other loadings like the external pressure on piping due for example to vacuum, and other failure modes like buckling may be examined. 5. The LRFD equations developed in this study were built on the ASME B&PV Code. As explained, reliability theory provides the necessary means of systematically evaluating uncertainties of models; therefore amelioration of models is also possible. 6. The set of data space, the first step of the reliability based design, is important. In this study, a wide range of pipes categories and loads was examined. In order to lower the divergence from the target reliability index, the elimination of some cases or categorization of piping systems should be examined. 240 APPENDIX A: ASME B&PV CODE EQUATIONS Table A-1: Design Equations for Pipes According to ASME B&PV Code (2001) Design Equations Equation Class ASME BPV Code (2001) Eq. Number A yPmS PD mt   )(2 o (A-1) or )(2 22 PyPmS yPAAmSPd mt   1 NB-3641.1(1) NB-3641.1(2) A yPS PD mt   )(2 o (A-2) or )(2 22 PyPS APyASdP mt   2 NC-3641.1(3) NC-3641.1(4) D es ig n fo r i nt er na l p re ss ur e -w al l t hi ck ne ss c al cu la tio n- A yPES PD mt   )(2 o (A-3) or )(2 22 PyPES AyPAESdP mt   3 ND-3641.1(3) ND-3641.1(4) mS I iMoDB nt PD B 5.1 2 2 2 o 1 d (A-4) 1 NB-3652(9) D es ig n o f P ri m ar y St re ss In te ns ity L im it hS Z AMB nt PD BSLS 5.12 2 o 1 d (A-5) 2 & 3 NC-3652(8) ND-3652(8) Service Limits A&B Pr im ar y pl us se co nd ar y st re ss in te ns ity ra ng e. El as tic c yc lin g Eq .(6 ), si m pl ifi ed e la st ic -p la st ic an al ys is , E qs .( 7) & (8 ) mSbTbaaTaaabEC I iMoDC t oDoP CnS 33 2 2 2 1 d  (A-6) or mSiM I oDCeS 3* 2 2 d (A-7) and mSbTbaaTaabĮEC I iMoDC t oDoPC 33' 2 2 2 1 d (A-8) 1 NB-3653(10) NB-3653.6(12) NB-3653.6(13) 241 Table A-1: (Continued) Service Limits A&B (Continued) Equation Class ASME BPV Code (2001) Eq. Number Fa tig ue 2 1 1 33 13 )1(2 1 2 22 2 11 TaE Ȟ bTbaaTaaabECK TaEK Ȟ iM I oDCK t oDoP CKpS '   '   (A-9) and Salt = Sp/2 if (A-6) is satisfied (A-10) or Salt = Ke Sp/2 if (A-7)&(A-8) are satisfied (A-11) altSSĮ , from Figs. I-9.0 calculate N Ui = ni/Ni (A-12) 0.1 1 d¦ n i iUU (A-13) NB-3653.2(a)(11) NB-3653.3 NB-3653.6(14) NB-3653.4 NB-3222.4(e)(5) Th er m al St re ss Ra tc he tin g 4 7.0 ' 1 C aE ySy rangeT d' (A-14) NB-3653.7 Se rv ic e Li m it B )5.1,8.1min( 2 2 2 o 1 ySmS I iMoDB nt PD B d , P”1.1PĮ (A-15) 1 NB-3654.2(a) (NB-3654.1) O cc as io na l Lo ad s )5.1,8.1min( )( 2 2 omax 1 yShSZ BMAMB nt DP BOLS d   (A-16) NC-3653.1(9) ND-3653.1(9) Th er m al E xp an si on (r el at ed to e qu iv al en t t he rm al c yc le s, fa tig ue ) Eq s. (1 9) , ( 20 ) r ef er to se is m ic d yn am ic a nc ho r m ov em en t )25.025.1( hScSfAS Z CMi ES  d (A-17) or )(75.0 4 AShS Z cMi Z AMi nt oPD TES d (A-18) 0.75i • 1 and AS Z RiM RS 2d for Class 2 (A-19) and AS Z RiM RS 3d for Class 3 (A-20) NC-3653.2(a)(10) ND-3653.2(a)(10) NC-3653.2(c)(11) ND-3653.2(c)(11) NC-3652.2(d)(11a) ND-3652.2(d)(11a) Si ng le An ch or M ov em en t cS Z DiM 3d (A-21) 2&3 NC-3653.2(b)(10a) ND-3653.2(b)(11) 242 Table A-1: (Continued) Service Limit C Equation Class ASME BPV Code (2001) Eq. Number Eq . ( 22 ) c om bi ne s r ev er si ng a nd n ot re ve rs in g dy na m ic lo ad s, w hi le E qs (2 3) to (2 6) a re u se d on ly w he n re ve rs in g dy na m ic lo ad in gs n ee d to b e ta ke n in to ac co un t )8.1,25.2min( 2 2 2 o 1 ySmSI iMoDB nt PD B d , P”1.5PĮ (A-22) or mSWM I oDB 5.0 2 2 d (A-23) and mSmSEM I oDB nt oDDPB 10.2)70.0(3 2 '2 2 1 d (A-24) and mSmSS I oDAMMC 2.4)6(70.0170.0 2 2  (A-25) and mSS MA AMF 7.0270.0  (A-26) 1 NB-3655.2(a) NB-3655.2(b) Eq . ( 27 ) c on si de rs o nl y n ot re ve rs in g dy na m ic lo ad s, w hi le E qs (2 8) to (3 1) a re u se d on ly w he n re ve rs in g dy na m ic lo ad in gs n ee d to b e ta ke n in to a cc ou nt )8.1,25.2min( )( 2 2 omax 1 yShSZ BMAMB nt DP BOLS d   , P”1.5PĮ (A-27) or mSWM I oDB 5.0 2 2 d (A-28) and mSmSEM I oDB nt oDDPB 10.2)70.0(3 2 '2 2 1 d (A-29) and mSmSS I oDAMMC 2.4)6(70.0170.0 2 2  (A-30) and mSS MA AMF 7.0270.0  (A-31) 2&3 NC-3654.2(a) ND-3654.2(a) NC-3654.2(b) ND-3654.2(b) 243 Table A-1: (Continued) Service Limit D Equation Class ASME BPV Code (2001) Eq. Number Eq s. (3 3) to (3 7) c an b e us ed a lte rn at iv el y fo r p ip in g fa br ic at ed fr om m at er ia ls P. N o1 th ro ug h P. N o. 9 , D o/ tn ”5 0, T ab le 2 A, S ec tio n II , P ar t D )2,0.3min( )( 2 2 omax 1 ySmSZ BMAMB nt DP BOLS d   P”2PĮ (A-32) or PD”PDESIGN (A-33) and mSWM I oDB 5.0 2 2 d (A-34) and mSEM I oDB nt oDDPB 3 2 '2 2 1 d (A-35) and mS I oDAMMC 6 2 2  (A-36) and mSS MA AMF  2 (A-37) 1 NB-3656(a) NB-3656(b) Eq s. (3 9) to (4 3) c an b e us ed a lte rn at iv el y fo r p ip in g fa br ic at ed fr om m at er ia ls P .N o1 th ro ug h P. N o. 9 , D o/ tn ”5 0, T ab le 2 A, S ec tio n II , P ar t D )2,0.3min( )( 2 2 omax 1 yShSZ BMAMB nt DP BOLS d   P”2PĮ (A-38) or PD”PDESIGN (A-39) and mSWM I oDB 5.0 2 2 d (A-40) and mSEM I oDB nt oDDPB 3 2 '2 2 1 d (A-41) and mS I oDAMMC 6 2 2  (A-42) and mSS MA AMF  2 (A-43) 2&3 NC-3656(a) ND-3656(a) NC-3656(b) ND-3656(b) 244 Appendix Nomenclature A Additional thickness to provide for material removed in threading, corrosion or erosion allowance, and material required for structural strength of pipe during erection, as appropriate. PD Pressure occurring coincident with the reversing dynamic load, psi Pmax Peak operating pressure, psi B1, B2 Primary stress indices Do Outside diameter of pipe d Inside diameter of pipe I Moment of inertia Mi Resultant moment due to mechanical loads including earthquake loads and non-reversing dynamic loads P Internal design pressure C1, C2, C3 Secondary stress indices that according to Table NB-3681(a) S Maximum allowable stress for the material at the design temperature tn Nominal wall thickness of product Sh Material allowable stress at temperature consistent with loading ĮĮ, (Įb) Coefficient of thermal expansion on side Į(b) of a gross structural discontinuity or material discontinuity, at room temperature Sm Maximum allowable stress intensity for the material at the design SOL Stress due to operating conditions SSL Stresses due to pressure, weight, and other sustained loads Su Ultimate stress of material SUL Stress due to upset loading conditions Sy Material yield strength at temperature consistent with loading t Nominal wall thickness of product tm Minimum thickness of pipe tn Nominal wall thickness Mi* Same as Mi with only difference that includes only moments due to thermal expansion and thermal anchor movements C’3 Values in Table NB-3681(a)-1 E Joint efficiency for the type of longitudinal joint used or casting quality factor y Coefficient having a value of 0.4, except that for pipe with Do / tm ratio less than 6, o/ Dddy  Z Elastic section modulus Zp Plastic section modulus Į Coefficient of thermal expansion at room temperature 245 K1, K2, K3 Local stress indices for the specific component under investigation |ǻȉ1| Absolute value of the range of the temperature difference between To and Ti, assuming moment generating equivalent linear temperature |ǻȉ2| Absolute value of the range for the portion of the nonlinear thermal gradient, not included in ǻȉ1 To Outside surface temperature Ti Inside surface temperature EĮb Average modulus of elasticity of the two sides of a gross structural discontinuity C4 1.1 for ferritic material 1.3 for austentic material y' Maximum allowable range of thermal stress computed on an elastic basis divided by the yield stress of the material. MAM The range of the resultant moment resulting from the anchors motion due to earthquake and other reversing dynamic loading FAM The amplitude of the longitudinal force resulting from the anchors motion due to earthquake and other reversing dynamic loading Ke Correction factor that eliminates local plasticization effects when the fatigue analysis remains linear-elastic m, n Material parameters given in Table NB-3228.5(b)-1 Salt Alternating stress intensity Sn Primary plus secondary stress intensity value Sp Peak stress intensity value U Cumulative usage factor Mc Range of resultant moments due to thermal expansion i Stress intensification factor equal to C2 K2/2 f Stress range reduction factor for cyclic conditions for total number N of full temperature cycles during the service life of the pipe, (Table NC- 3611.2(e)-1). MD Resultant moment due to any single no repeated anchor movement MA Resultant moment due to weight and other sustained loads ME Resultant moment due to sustained weight and earthquake and other dynamic loads Mw Moment due to sustained weight 246 APPENDIX B: STEEL USED IN B&PV CODE, PART III The following table presents the Specified Minimum Yield Strength (SMYS) and the Specified Minimum Tensile Strength (SMTS) of steels, used in the ASME Code, Part III, for the design of piping. These values are considered as the nominal resistance of steel for the LRFD method. The table, moreover, provides information like the nominal composition, the class, type or grade of these steels, etc. Table B-1: Steel Used for Nuclear Piping Spec. Gr., Cl., Type Nominal Composition UNS # SMYS SMTS Notes* Common Name Ty S-Gr A C Stl K02504 48 30 Ty S-Gr B C-Mn Stl K03005 60 35 Ty E-Gr A C Stl K02504 48 30 SA-53 Ty E-Gr B C-Mn Stl K03005 60 35 welded or seamless black & hot- dipped zinc coated Gr A C-Si Stl K02501 48 30 Gr B C-Si Stl K03006 60 35 SA-106 Gr C C-Si Stl K03501 70 40 carbon seamless steel pipe for high- temperature service Gr TP304 18 Cr-8 Ni S30400 75 30 Gr TP304H 18 Cr-8 Ni S30409 75 30 Gr TP304L 18 Cr-8 Ni S30403 70 25 Gr TP304N 18 Cr-8 Ni-N S30451 80 35 Gr TP304LN 18 Cr-8 Ni-N S30453 75 30 Gr TP309S 23 Cr-12 Ni S30908 75 30 Gr TP309Cb 23 Cr-12 Ni-Cb S30940 75 30 Gr TP310S 25 Cr-20 Ni S31008 75 30 Gr TP310Cb 25 Cr-20 Ni-Cb S31040 75 30 Gr TP316 16Cr-12Ni-2Mo S31600 75 30 Gr TP316H 16Cr-12Ni-2Mo S31609 75 30 Gr TP316L 16Cr-12Ni-2Mo S31603 70 25 SA-312 Gr TP316N 16Cr-12Ni-2Mo- N S31651 80 35 austentic stainless steel seamless or welded 247 Table B-1: (Continued) Spec. Gr., Cl., Type Nominal Composition UNS # SMYS SMTS Notes Common Name Gr TP316LN 16Cr-12Ni-2Mo- N S31653 75 30 austenitic stainless steel Gr TP317 18Cr-13Ni- 3Mo S31700 75 30 seamless or welded Gr TP321 18 Cr-10 Ni-Ti S32100 75 30 Sm<3/8in Gr TP321H 18 Cr-10 Ni-Ti S32109 75 30 Sm<3/8in Gr TP347 18 Cr-10 Ni-Cb S34700 75 30 Gr TP347 H 18 Cr-10 Ni-Cb S34709 75 30 Gr TP348 18 Cr-10 Ni-Cb S34800 75 30 Gr TP348 H 18 Cr-10 Ni-Cb S34809 75 30 SA-312 Gr TP XM19 22Cr-13Ni-Mn S20910 100 55 nitronic 50 or 22-13-5 Gr 1 C- Mn Stl K03008 55 30 Gr 6 C- Mn-Ci Stl K03006 60 35 Gr 8 9Ni K81340 100 75 SA-333 Gr9 2Ni-1Cu K22035 63 46 seamless or welded pipe for low-temperature service Gr P1 C- 1/2Mo K11522 55 30 Gr P2 1/2Cr- 1/2Mo K11547 55 30 Gr P5 5Cr- 1/2Mo K41545 60 30 Gr P9 9Cr- 1Mo K81590 60 30 Gr P11 11/4Cr-1/2Mo-Si K11597 60 30 Gr P12 1Cr- 1/2Mo K11562 60 30 Gr P21 3Cr- 1/2Mo K31545 60 30 SA-335 Gr P22 21/4Cr- 1Mo K21590 60 30 ferritic alloy, seamless steel pipe for high- temperature service Gr 304 18Cr- 8Ni S30400 75 30 Gr 304L 18Cr- 8Ni S30403 70 25 Gr 304N 18Cr- 8Ni-N S30451 80 35 Gr 304LN 18Cr- 8Ni-N S30453 75 30 Gr 304H 18Cr- 8Ni S30409 75 30 Gr 309 23Cr- 12Ni S30900 75 30 Gr 310 25Cr- 20Ni S31000 75 30 Gr 316 16Cr- 12Ni-2Mo S31600 75 30 Gr 316L 16Cr- 12Ni-2Mo S31603 70 25 Gr 316H 16Cr- 12Ni-2Mo S31609 75 30 Gr 316N 16Cr- 12Ni-2Mo- N S31651 80 35 Gr 316N 16Cr- 12Ni-2Mo- N S31653 75 30 Gr 321 18Cr-10Ni-Ti S32100 75 30 Gr 347 18Cr-10Ni-Cb S34700 75 30 Gr 348 18Cr-10Ni-Cb S34800 75 30 SA-358 Gr XM-19 22Cr-13Ni-5Mn S22100 100 55 electric-fusion welded austentic chromium-nickel alloy steel pipe low-temperature service Gr FP1 C-1/2Mo K11522 55 30 Gr FP2 1/2Cr-1/2Mo K11547 55 30 Gr FP5 5Cr-1/2Mo K41545 60 30 Gr FP9 9Cr-1Mo K90941 60 30 Gr FP11 11/4Cr-1/2Mo-Si K11597 60 30 Gr FP12 1Cr-1/2Mo K11562 60 30 Gr FP21 3Cr-1Mo K31545 60 30 SA-369 Gr FP22 21/4Cr-1Mo K21590 60 30 carbon and ferretic alloy steel for high- temperature service 248 Table B-1: (Continued) Spec. Gr., Cl., Type Nominal Composition UNS # SMYS SMTS Notes Common Name Gr TP304 18Cr- 8Ni S30400 75 30 Gr TP304H 18Cr-8Ni S30409 75 30 Gr TP304N 18Cr-8Ni-N S30451 80 35 Gr TP304LN 18Cr-8Ni-N S30453 75 30 Gr TP316 16Cr- 12Ni-2Mo S31600 75 30 Gr TP316H 16Cr- 12Ni-2Mo S31609 75 30 Gr TP316N 16Cr- 12Ni-2Mo-N S31651 80 35 Gr TP316LN 16Cr- 12Ni-2Mo-N S31653 75 30 austentic seamless steel pipe for high temperature (central station service) Gr TP321 18Cr-10Ni-Ti S32100 75 30 <3/8in Gr TP321 18Cr-10Ni-Ti S32100 70 25 >3/8in Gr TP321H 18Cr-10Ni-Ti S32109 75 30 <3/8in Gr TP321H 18Cr-10Ni-Ti S32109 70 25 >3/8in Gr TP347 18Cr-10Ni-Cb S34700 75 30 Gr TP347H 18Cr-10Ni-Cb S34709 75 30 SA-376 Gr TP 348 18Cr-10Ni-Cb S34800 75 30 Gr TP304 18Cr- 8Ni S30400 75 30 Gr TP304L 18Cr-8Ni S30403 70 25 Gr TP316 16Cr-12Ni-2Mo S31600 75 30 Gr TP316L 16Cr-12Ni-2Mo S31603 70 25 Gr TP321 18Cr-10Ni-Ti S32100 75 30 Gr TP347 18Cr-10Ni-Cb S34700 75 30 SA-409 Gr TP348 18Cr-10Ni-Ti S34800 75 30 welded pipe of large diameter austentic steel for corrosive or high-temperature service Gr CP1 C-1/2Mo J12521 65 35 Gr CP2 1/2Cr-1/2Mo J11547 60 30 Gr CP5 5Cr-1/2Mo J42045 90 60 Gr CP9 9Cr-1Mo J82090 90 60 Gr CP11 11/4Cr-1/2Mo J12072 70 40 Gr CP12 1Cr-1/2Mo J11562 60 30 Gr CP21 3Cr-1Mo J31545 60 30 Gr CP22 21/4Cr-1Mo J21890 70 40 SA-426 Gr CPCA15 13Cr J91150 90 65 centrifugally cast ferritic alloy steel for high- temperature service Gr FP304 18Cr-8Ni S30400 70 30 Gr FP304H 18Cr-8Ni S30409 70 30 Gr FP304N 18Cr-8Ni-N S03451 75 35 Gr FP316 16Cr-12Ni-2Mo S31600 70 30 Gr FP316H 16Cr-12Ni-2Mo S31609 70 30 Gr FP316N 16Cr-12Ni-2Mo-N S31651 75 35 Gr FP321 18Cr-10Ni-Ti S32100 70 30 Gr FP321H 18Cr-10Ni-Ti S32109 70 30 Gr FP347 18Cr-10Ni-Cb S34700 70 30 SA-430 Gr FP347H 18Cr-10Ni-Cb S34709 70 30 austentic steel for high- temperature service Gr TP304H 18Cr-8Ni S30409 75 30 Gr TP347H 18Cr-10Ni-Cb S34709 75 30 SA-452 Gr TP316H 16Cr- 12Ni-2Mo S31609 75 30 centrifugally cast austentic steel for high temperature Gr WCA C-Si Stl J02504 60 30 Gr WCB C-Si Stl J03003 70 36 SA-660 Gr WCC C-Mn-Si Stl J02505 70 40 centrifugally cast carbon steel for high temperature 249 Table B-1: (Continued) Spec. Gr., Cl., Type Nominal Composition UNS # SMYS SMTS Notes Common Name Gr CPF3 18Cr-8Ni J92500 70 30 Gr CPF3A 18Cr-8Ni J92500 77 25 Gr CPF3M 16Cr-12Ni-2Mo J92800 70 30 Gr CPF8 18Cr-8Ni J92600 70 30 Gr CPF8A 18Cr-8Ni J92600 77 35 Gr CPF8M 16Cr-12Ni-2Mo J92900 70 30 Gr CPF8C 18Cr-10Ni-Cb J92700 70 30 Gr CPH8 25Cr-12Ni J93400 65 28 Gr CPK20 25Cr-20Ni J94202 65 28 SA-451 Gr CPH20 25Cr-12Ni J93402 70 30 centrifugally cast austentic steel for high- temperature service Gr CA55 C Stl K02801 55 30 SA-515 Gr60 Gr CB60 C-Si Stl K02401 60 32 SA-515 Gr65 Gr CB65 C-Si Stl K02800 65 35 SA-515 Gr70 Gr CB70 C-Si Stl KO3101 70 38 SA-516 Gr60 Gr CC60 C-Mn-Si Stl K02100 60 32 Gr CC65 C-Mn-Si Stl K02403 65 35 SA-516 Gr65 Gr CC70 C-Mn-Si Stl K02700 70 38 SA-516 Gr70 Gr CD70 C-Mn-Si Stl K02400 70 50 SA-537 Cl 1 Gr CD80 C-Mn-Si Stl K02400 80 60 SA-537 Cl 2 Gr CE55 C-Mn-Si Stl KO2202 55 30 SA-442 Cr 55 Gr CE60 C-Mn-Si Stl K02402 60 32 SA-442 Cr 60 SA-671 Gr CK75 C-Mn-Si Stl K02803 75 40 electric-fusion welded pipe SA-299 Gr A45 C Stl K01700 45 24 SA-285 Gr A Gr A50 C Stl K02200 50 27 SA-285 Gr B Gr A55 C Stl K02801 55 30 SA-285 Gr C Gr B55 C-Si Stl K02001 55 30 SA-515 Gr55 Gr B60 C-Si Stl K02401 60 32 SA-515 Gr60 Gr B65 C-Si Stl K02800 65 35 SA-515 Gr65 Gr B70 C-Si Stl K03101 70 38 SA-515 Gr70 Gr C55 C-Si Stl K01800 55 30 SA-515 Gr55 Gr C60 C-Mn-Si Stl K02100 60 32 SA-516 Gr60 Gr C65 C-Mn-Si Stl K02403 65 35 SA-516 Gr65 Gr C70 C-Mn-Si Stl K02700 70 38 SA-516 Gr70 Gr D70 C-Mn-Si Stl K02400 70 50 SA-537 Cl 1 SA-672 Gr D80 C-Mn-Si Stl K02400 80 60 electric-fusion welded pipe for high- pressure service at moderate temperature SA-537 Cl 2 250 Table B-1: (Continued) Spec. Gr., Cl., Type Nominal Composition UNS # SMYS SMTS Notes Common Name Gr E55 C-Mn-Si Stl K02202 55 30 SA-442 Gr55 Gr E60 C-Mn-Si Stl K02402 60 32 SA-442 Gr 60 Gr H75 Mn-1/2Mo K12021 75 45 SA-302 Gr A Gr J80 Mn-1/2Mo-1/2Ni K12539 80 50 SA-533 Gr B, Cl1 Gr J90 Mn-1/2Mo-1/2Ni K12539 90 70 SA-533 Gr B, Cl2 Gr J100 Mn-1/2Mo-1/2Ni K12539 100 83 SA-533 Gr B, Cl3 Gr L65 C-1/2Mo K11820 65 37 SA-204 Gr A Gr L70 C-1/2Mo K12020 70 40 SA-204 Gr B Gr L75 C-1/2Mo K12320 75 43 SA-204 Gr C SA-672 Gr N75 C-Mn-Si Stl K02803 75 40 electric-fusion welded pipe for high-pressure service at moderate temperature SA-299 Gr CM65 C-1/2Mo K11820 65 37 A204 Gr A Gr CM70 C-1/2Mo K12020 70 40 A204 Gr B Gr CM75 C-1/2Mo K12320 75 43 A204 Gr C Gr CMSH- 70 C-Mn-Si Stl K02400 70 50 A537 Cl1 SA-691 Gr CMS-75 C-Mn-Si Stl K02803 75 40 Gr TPXM- 33 27Cr-1Mo-Ti S44626 65 40 SA-731 Gr TPXM- 33 27Cr-1Mo S44627 65 40 martensitic stainless steel seamless and welded pipe Gr TP304 18Cr-8Ni S30400 75 30 Gr TP304H 18Cr-8Ni S30409 75 30 Gr TP304L 18Cr-8Ni S30403 70 25 Gr TP304N 18Cr-8Ni-N S30451 80 32 Gr TP304LN 18Cr-8Ni-N S30453 75 30 Gr TP309S 23Cr-12Ni S30908 75 30 Gr TP316 16Cr-12Ni-2Mo S31600 75 30 Gr TP316H 16Cr-12Ni-2Mo S31609 75 30 Gr TP316L 16Cr-12Ni-2Mo S31603 70 25 Gr TP316N 16Cr-12Ni-2Mo-N S31651 80 32 Gr TP321 18Cr-10Ni-Ti S32100 75 30 Gr TP321H 18Cr-10Ni-Ti S32109 75 30 Gr TP347 18Cr-10Ni-Cb S34700 75 30 Gr TP347H 18Cr-10Ni-Cb S34709 75 30 Gr TP348 18Cr-10Ni-Cb S34800 75 30 SA-813 Gr TP348H 18Cr-10Ni-Cb S34809 75 30 single or double welded austentic stainless steel 251 Table B-1: (Continued) Spec. Gr., Cl., Type Nominal Composition UNS # SMYS SMTS Notes Common Name Gr TP304 18Cr-8Ni S30400 75 30 Gr TP304H 18Cr-8Ni S30409 75 30 Gr TP304L 18Cr-8Ni S30403 70 25 Gr TP304N 18Cr-8Ni-N S30451 80 35 Gr TP304LN 18Cr-8Ni-N S30453 75 30 Gr TP316 16Cr-12Ni-2Mo S31600 75 30 Gr TP316H 16Cr-12Ni-2Mo S31609 75 30 Gr TP316L 16Cr-12Ni-2Mo S31603 70 25 Gr TP316N 16Cr-12Ni-2Mo-N S31651 80 35 Gr TP321 18Cr-10Ni-Ti S32100 75 30 Gr TP321H 18Cr-10Ni-Ti S32109 75 30 Gr TP347H 18Cr-10Ni-Cb S34709 75 30 Gr TP348 18Cr-10Ni-Cb S34800 75 30 SA-814 Gr TP348H 18Cr-10Ni-Cb S34809 75 30 cold-worked welded austentic stainless steel Spec.=Specification, Gr.=Grade, Cl.=Class, Sm=Seamless, SMYS=Specified Minimum Yield Strength, SMTS=Specified Minimum Tensile Strength 252 APPENDIX C: PARTIAL MEAN RESISTANCE FACTORS AND ADJUSTED NOMINAL RESISTANCE FACTORS This appendix provides in tabulated format the calculated mean partial safety factors and the nominal adjusted resistance factors for the performance functions of Table 5-2 under separate headings. Some plots of the mean partial safety factors as well as the summary of the recommended adjusted nominal resistance factors were presented for each performance function in Chapter 6. C.1. Performance Functions g1, g5, g10, and g15 Tables C-1 to C-4 provide the calculated mean values of partial safety factors and Table C-5 the adjusted resistance factor for a given total nominal load factor 1.2. 253 Table C-1: Mean Partial Load and Resistance Factors for g1 Design Temperature ( o F)ȕ Partial Factor R.T. * 200 400 600 800 Stainless steel ijƍy 0.824 0.824 0.824 0.824 0.824 Ȗƍȇ 1.076 1.076 1.076 1.076 1.076 1.5 ȖƍȂ 1.020 1.020 1.020 1.020 1.020 ijƍy 0.774 0.774 0.774 0.774 0.774 Ȗƍȇ 1.100 1.100 1.100 1.100 1.100 2.0 ȖƍȂ 1.027 1.027 1.027 1.027 1.027 ijƍy 0.726 0.726 0.726 0.726 0.726 Ȗƍȇ 1.123 1.123 1.123 1.123 1.123 2.5 ȖƍȂ 1.034 1.034 1.034 1.034 1.034 ijƍy 0.682 0.682 0.682 0.682 0.682 Ȗƍȇ 1.146 1.146 1.146 1.146 1.146 3.0 ȖƍȂ 1.040 1.040 1.040 1.040 1.040 ijƍy 0.639 0.639 0.639 0.639 0.639 Ȗƍȇ 1.168 1.168 1.168 1.168 1.168 3.5 ȖƍȂ 1.047 1.047 1.047 1.047 1.047 ijƍy 0.562 0.562 0.562 0.562 0.562 Ȗƍȇ 1.210 1.210 1.210 1.210 1.210 4.5 ȖƍȂ 1.060 1.060 1.060 1.060 1.060 Carbon Steel ijƍy 0.926 0.926 0.853 0.853 0.853 Ȗƍȇ 1.104 1.104 1.083 1.083 1.083 1.5 ȖƍȂ 1.028 1.028 1.022 1.022 1.022 ijƍy 0.902 0.902 0.809 0.809 0.809 Ȗƍȇ 1.137 1.137 1.109 1.109 1.109 2.0 ȖƍȂ 1.038 1.038 1.029 1.029 1.029 ijƍy 0.879 0.879 0.768 0.768 0.768 Ȗƍȇ 1.169 1.169 1.134 1.134 1.134 2.5 ȖƍȂ 1.047 1.047 1.037 1.037 1.037 ijƍy 0.855 0.855 0.728 0.728 0.728 Ȗƍȇ 1.200 1.200 1.159 1.159 1.159 3.0 ȖƍȂ 1.057 1.057 1.044 1.044 1.044 ijƍy 0.832 0.832 0.690 0.690 0.690 Ȗƍȇ 1.231 1.231 1.183 1.183 1.183 3.5 ȖƍȂ 1.067 1.067 1.051 1.051 1.051 ijƍy 0.785 0.785 0.619 0.619 0.619 Ȗƍȇ 1.290 1.290 1.229 1.229 1.229 4.5 ȖƍȂ 1.086 1.086 1.066 1.066 1.066 * R.T.=Room Temperature 254 Table C-2: Mean Partial Load and Resistance Factors for g5 Design Temperature ( o F)ȕ Partial Factor R.T. * 200 400 600 800 Stainless steel ijƍy 0.834 0.834 0.834 0.834 0.834 Ȗƍȇ 1.116 1.116 1.116 1.116 1.116 1.5 ȖƍȂ 1.019 1.019 1.019 1.019 1.019 ijƍy 0.786 0.786 0.786 0.786 0.786 Ȗƍȇ 1.152 1.152 1.152 1.152 1.152 2.0 ȖƍȂ 1.025 1.025 1.025 1.025 1.025 ijƍy 0.739 0.739 0.739 0.739 0.739 Ȗƍȇ 1.186 1.186 1.186 1.186 1.186 2.5 ȖƍȂ 1.032 1.032 1.032 1.032 1.032 ijƍy 0.695 0.695 0.695 0.695 0.695 Ȗƍȇ 1.219 1.219 1.219 1.219 1.219 3.0 ȖƍȂ 1.038 1.038 1.038 1.038 1.038 ijƍy 0.653 0.653 0.653 0.653 0.653 Ȗƍȇ 1.251 1.251 1.251 1.251 1.251 3.5 ȖƍȂ 1.045 1.045 1.045 1.045 1.045 ijƍy 0.576 0.576 0.576 0.576 0.576 Ȗƍȇ 1.313 1.313 1.313 1.313 1.313 4.5 ȖƍȂ 1.057 1.057 1.057 1.057 1.057 Carbon Steel ijƍy 0.934 0.934 0.863 0.863 0.863 Ȗƍȇ 1.150 1.150 1.125 1.125 1.125 1.5 ȖƍȂ 1.025 1.025 1.020 1.020 1.020 ijƍy 0.912 0.912 0.821 0.821 0.821 Ȗƍȇ 1.197 1.197 1.163 1.163 1.163 2.0 ȖƍȂ 1.034 1.034 1.027 1.027 1.027 ijƍy 0.890 0.890 0.781 0.781 0.781 Ȗƍȇ 1.243 1.243 1.200 1.200 1.200 2.5 ȖƍȂ 1.043 1.043 1.034 1.034 1.034 ijƍy 0.867 0.867 0.742 0.742 0.742 Ȗƍȇ 1.287 1.287 1.236 1.236 1.236 3.0 ȖƍȂ 1.052 1.052 1.041 1.041 1.041 ijƍy 0.845 0.845 0.705 0.705 0.705 Ȗƍȇ 1.330 1.330 1.271 1.271 1.271 3.5 ȖƍȂ 1.061 1.061 1.049 1.049 1.049 ijƍy 0.800 0.800 0.634 0.634 0.634 Ȗƍȇ 1.413 1.413 1.337 1.337 1.337 4.5 ȖƍȂ 1.080 1.080 1.063 1.063 1.063 * R.T.=Room Temperature 255 Table C-3: Mean Partial Load and Resistance Factors for g10 Design Temperature ( o F)ȕ Partial Factor R.T. * 200 400 600 800 Stainless steel ijƍy 0.972 0.972 0.972 0.972 0.972 Ȗƍȇ 1.218 1.218 1.218 1.218 1.218 1.5 ȖƍȂ 1.018 1.018 1.018 1.018 1.018 ijƍy 0.965 0.965 0.965 0.965 0.965 Ȗƍȇ 1.337 1.337 1.337 1.337 1.337 2.0 ȖƍȂ 1.023 1.023 1.023 1.023 1.023 ijƍy 0.959 0.959 0.959 0.959 0.959 Ȗƍȇ 1.478 1.478 1.478 1.478 1.478 2.5 ȖƍȂ 1.027 1.027 1.027 1.027 1.027 ijƍy 0.954 0.954 0.954 0.954 0.954 Ȗƍȇ 1.643 1.643 1.643 1.643 1.643 3.0 ȖƍȂ 1.031 1.031 1.031 1.031 1.031 ijƍy 0.947 0.947 0.947 0.947 0.947 Ȗƍȇ 1.832 1.832 1.832 1.832 1.832 3.5 ȖƍȂ 1.035 1.035 1.035 1.035 1.035 ijƍy 0.934 0.934 0.934 0.934 0.934 Ȗƍȇ 2.281 2.281 2.281 2.281 2.281 4.5 ȖƍȂ 1.044 1.044 1.044 1.044 1.044 Carbon Steel ijƍy 0.972 0.972 0.927 0.927 0.927 Ȗƍȇ 1.218 1.218 1.195 1.195 1.195 1.5 ȖƍȂ 1.018 1.018 1.017 1.017 1.017 ijƍy 0.965 0.965 0.911 0.911 0.911 Ȗƍȇ 1.337 1.337 1.304 1.304 1.304 2.0 ȖƍȂ 1.023 1.023 1.022 1.022 1.022 ijƍy 0.959 0.959 0.896 0.896 0.896 Ȗƍȇ 1.478 1.478 1.435 1.435 1.435 2.5 ȖƍȂ 1.027 1.027 1.026 1.026 1.026 ijƍy 0.954 0.954 0.882 0.882 0.882 Ȗƍȇ 1.643 1.643 1.588 1.588 1.588 3.0 ȖƍȂ 1.031 1.031 1.030 1.030 1.030 ijƍy 0.947 0.947 0.867 0.867 0.867 Ȗƍȇ 1.832 1.832 1.762 1.762 1.762 3.5 ȖƍȂ 1.035 1.035 1.034 1.034 1.034 ijƍy 0.934 0.934 0.835 0.835 0.835 Ȗƍȇ 2.281 2.281 2.172 2.172 2.172 4.5 ȖƍȂ 1.044 1.044 1.042 1.042 1.042 * R.T.=Room Temperature 256 Table C-4: Mean Partial Load and Resistance Factors for g15 Design Temperature ( o F)ȕ Partial Factor R.T. * 200 400 600 800 Stainless steel ijƍy 0.977 0.977 0.977 0.977 0.977 Ȗƍȇ 1.303 1.303 1.303 1.303 1.303 1.5 ȖƍȂ 1.015 1.015 1.015 1.015 1.015 ijƍy 0.971 0.971 0.971 0.971 0.971 Ȗƍȇ 1.465 1.465 1.465 1.465 1.465 2.0 ȖƍȂ 1.019 1.019 1.019 1.019 1.019 ijƍy 0.965 0.965 0.965 0.965 0.965 Ȗƍȇ 1.656 1.656 1.656 1.656 1.656 2.5 ȖƍȂ 1.023 1.023 1.023 1.023 1.023 ijƍy 0.960 0.960 0.960 0.960 0.960 Ȗƍȇ 1.879 1.879 1.879 1.879 1.879 3.0 ȖƍȂ 1.027 1.027 1.027 1.027 1.027 ijƍy 0.954 0.954 0.954 0.954 0.954 Ȗƍȇ 2.134 2.134 2.134 2.134 2.134 3.5 ȖƍȂ 1.031 1.031 1.031 1.031 1.031 ijƍy 0.940 0.940 0.940 0.940 0.940 Ȗƍȇ 2.739 2.739 2.739 2.739 2.739 4.5 ȖƍȂ 1.040 1.040 1.040 1.040 1.040 Carbon Steel ijƍy 0.977 0.977 0.939 0.939 0.939 Ȗƍȇ 1.303 1.303 1.281 1.281 1.281 1.5 ȖƍȂ 1.015 1.015 1.014 1.014 1.014 ijƍy 0.971 0.971 0.924 0.924 0.924 Ȗƍȇ 1.465 1.465 1.433 1.433 1.433 2.0 ȖƍȂ 1.019 1.019 1.018 1.018 1.018 ijƍy 0.965 0.965 0.910 0.910 0.910 Ȗƍȇ 1.656 1.656 1.614 1.614 1.614 2.5 ȖƍȂ 1.023 1.023 1.022 1.022 1.022 ijƍy 0.960 0.960 0.896 0.896 0.896 Ȗƍȇ 1.879 1.879 1.823 1.823 1.823 3.0 ȖƍȂ 1.027 1.027 1.026 1.026 1.026 ijƍy 0.954 0.954 0.881 0.881 0.881 Ȗƍȇ 2.134 2.134 2.061 2.061 2.061 3.5 ȖƍȂ 1.031 1.031 1.030 1.030 1.030 ijƍy 0.940 0.940 0.849 0.849 0.849 Ȗƍȇ 2.739 2.739 2.618 2.618 2.618 4.5 ȖƍȂ 1.040 1.040 1.038 1.038 1.038 * R.T.=Room Temperature 257 Table C-5a: Adjusted Nominal Resistance Factors, ij, for Total Load Factor Ȗ=1.2 and Carbon Steel for g1, g5, g10, and g15. Adjusted Value of ij for ȕ: Service Level Temper. ( o F) 1.5 2 2.5 3 3.5 4.5 70 0.99 0.93 0.87 0.82 0.77 0.68 200 0.81 0.76 0.71 0.67 0.63 0.56 400 0.72 0.66 0.61 0.56 0.52 0.51 600 0.62 0.57 0.52 0.48 0.45 0.44 Design and A 800 0.58 0.53 0.49 0.45 0.42 0.41 1.01 0.94 0.87 0.82 0.76 0.67 0.83 0.77 0.72 0.67 0.63 0.55 0.74 0.67 0.62 0.57 0.52 0.44 0.64 0.58 0.53 0.49 0.45 0.38 B 70 200 400 600 800 0.59 0.54 0.50 0.45 0.42 0.36 1.21 1.09 0.98 0.87 0.77 0.61 1.11 1.00 0.89 0.79 0.70 0.55 1.20 1.07 0.96 0.85 0.75 0.58 1.13 1.01 0.90 0.80 0.70 0.55 C 70 200 400 600 800 0.90 0.81 0.72 0.64 0.56 0.44 1.21 1.07 0.94 0.82 0.71 0.54 1.11 0.98 0.85 0.75 0.65 0.50 1.21 1.06 0.92 0.80 0.69 0.52 1.13 1.00 0.87 0.75 0.65 0.49 D 70 200 400 600 800 0.91 0.80 0.69 0.60 0.52 0.39 258 Table C-5b: Adjusted Nominal Resistance Factors, ij, for Total Load Factor Ȗ=1.2 and Stainless Steel for g1, g5, g10, and g15. Adjusted Value of ij for ȕ: Service Level Temper. ( o F) 1.5 2 2.5 3 3.5 4.5 70 1.01 0.92 0.84 0.77 0.70 0.59 200 0.88 0.81 0.74 0.67 0.62 0.52 400 0.72 0.66 0.60 0.55 0.50 0.42 600 0.64 0.59 0.54 0.49 0.45 0.38 Design and A 800 0.60 0.55 0.50 0.46 0.42 0.35 1.04 0.95 0.86 0.78 0.71 0.59 0.91 0.83 0.75 0.68 0.62 0.51 0.74 0.67 0.61 0.56 0.51 0.42 0.66 0.60 0.54 0.50 0.45 0.37 B 70 200 400 600 800 0.62 0.58 0.51 0.46 0.42 0.35 1.19 1.07 0.96 0.86 0.76 0.60 1.03 0.93 0.83 0.74 0.66 0.52 0.93 0.83 0.75 0.67 0.59 0.46 0.93 0.83 0.75 0.67 0.59 0.46 C 70 200 400 600 800 0.90 0.81 0.72 0.64 0.57 0.45 1.19 1.050 0.92 0.80 0.70 0.53 1.03 0.911 0.80 0.70 0.61 0.46 0.93 0.82 0.72 0.63 0.54 0.41 0.93 0.82 0.72 0.63 0.54 0.41 D 70 200 400 600 800 0.90 0.79 0.69 0.60 0.53 0.40 C.2. Performance Function g2 Table C-6 presents the calculated mean load and resistance factors for the performance function g2. 259 Table C-6: Load and Resistance Factors Applied to Mean Values of Variables for P.F. g2 Carbon Steel Stainless Steel Temperature, T (oF) ȕ ȝy I'y Ȗ'ǹ ȝy I'y Ȗ'ǹ ”200 2.007 0.697 1.400 2.856 0.448 1.279 >200 6 2.569 0.508 1.306 2.856 0.448 1.279 ”200 2.232 0.652 1.456 3.379 0.390 1.317 >200 7 2.984 0.452 1.348 3.379 0.390 1.317 ”200 2.478 0.609 1.511 3.992 0.339 1.355 >200 8 3.462 0.401 1.389 3.992 0.339 1.355 Table C-7: Adjusted Nominal Values of Load and Resistance Factors for g2 Carbon Steel Stainless Steel Temperature T (oF) ȕ Iy Ȗǹ Iy Ȗǹ Room Temperature 0.68 1.2 0.53 1.2 200 0.56 1.2 0.46 1.2 400 0.41 1.2 0.38 1.2 t600 6 0.34 1.2 0.33 1.2 Room Temperature 0.61 1.2 0.45 1.2 200 0.50 1.2 0.39 1.2 400 0.35 1.2 0.32 1.2 t600 7 0.29 1.2 0.27 1.2 Room Temperature 0.55 1.2 0.38 1.2 200 0.45 1.2 0.33 1.2 400 0.30 1.2 0.27 1.2 t600 8 0.25 1.2 0.23 1.2 C.3. Performance Function g3 Table C-8 provides the calculated mean load and resistance factors for performance function g3. In this table, ȝfy is the converged mean value of the steel resistance. Table C-9 presents the adjusted nominal resistance factor for Ȗǹ=1.1 and ȖPDes=1.2. T ab le C -8 : M ea n L o ad a n d R es is ta n ce F ac to rs f o r D if fe re n t O p er at in g T em p er at u re , T, f o r g 3 C ar b o n S te el ( T” 2 0 0 o F ) C ar b o n S te el ( T> 2 0 0 o F ) S ta in le ss S te el ( fo r an y T ) ȕ ȝ f y I’ y Ȗ' ǹ Ȗ' P D ȝ f y I’ y Ȗ' ǹ Ȗ' P D ȝ f y I’ y Ȗ' ǹ Ȗ' P D 2 .0 1 .8 6 0 .8 8 3 1 .1 1 5 1 .0 5 7 2 .0 3 0 .7 9 0 1 .0 8 5 1 .0 4 0 2 .1 1 0 .7 5 5 1 .0 7 6 1 .0 3 5 3 .0 2 .0 7 0 .8 2 9 1 .1 6 9 1 .0 9 0 2 .3 5 0 .7 0 3 1 .1 2 4 1 .0 6 2 2 .4 9 0 .6 5 9 1 .1 1 1 1 .0 5 5 3 .5 2 .1 8 0 .8 0 3 1 .1 9 4 1 .1 0 7 2 .5 3 0 .6 6 4 1 .1 4 3 1 .0 7 4 2 .7 0 0 .6 1 5 1 .1 2 8 1 .0 6 5 4 .5 2 .4 1 0 .7 5 2 1 .2 4 3 1 .1 4 3 2 .9 3 0 .5 9 0 1 .1 7 9 1 .0 9 7 3 .1 8 0 .5 3 6 1 .1 6 2 1 .0 8 6 f PDes =0.5 5 .5 2 .6 7 0 .7 0 4 1 .2 9 0 1 .1 8 0 3 .3 8 0 .5 2 4 1 .2 1 4 1 .1 2 1 3 .7 4 0 .4 6 6 1 .1 9 3 1 .1 0 7 2 .0 2 .4 7 0 .8 8 3 1 .0 8 8 1 .0 9 4 2 .7 0 0 .7 8 9 1 .0 6 4 1 .0 6 5 2 .8 0 0 .7 5 4 1 .0 5 8 1 .0 5 7 3 .0 2 .7 4 0 .8 2 9 1 .1 2 6 1 .1 5 3 .1 2 0 .7 0 3 1 .0 9 4 1 .1 0 3 3 .3 0 0 .6 5 8 1 .0 8 4 1 .0 9 0 3 .5 2 .8 9 0 .8 0 4 1 .1 4 5 1 .1 8 3 .3 6 0 .6 6 3 1 .1 0 8 1 .1 2 2 3 .5 9 0 .6 1 5 1 .0 9 7 1 .1 0 7 4 .5 3 .2 1 0 .7 5 5 1 .1 8 0 1 .2 4 3 .8 9 0 .5 9 1 1 .1 3 5 1 .1 6 3 4 .2 2 0 .5 3 6 1 .1 2 2 1 .1 4 2 f PDes =1 5 .5 3 .5 6 0 .7 0 9 1 .2 1 1 1 .3 1 1 4 .5 0 0 .5 2 6 1 .1 6 0 1 .2 0 5 4 .9 7 0 .4 6 7 1 .1 4 5 1 .1 8 0 2 .0 7 .5 7 0 .8 9 4 1 .0 2 5 1 .1 4 9 8 .2 1 0 .7 9 9 1 .0 2 0 1 .1 0 9 8 .5 2 0 .7 6 3 1 .0 1 8 1 .0 9 7 3 .0 8 .5 2 0 .8 4 7 1 .0 3 5 1 .2 3 6 9 .6 0 0 .7 1 8 1 .0 2 8 1 .1 7 2 1 0 .1 2 0 .6 7 1 1 .0 2 6 1 .1 5 3 3 .5 9 .0 3 0 .8 2 5 1 .0 4 0 1 .2 8 3 1 0 .3 7 0 .6 8 0 1 .0 3 2 1 .2 0 5 1 1 .0 3 0 .6 2 9 1 .0 2 9 1 .1 8 2 4 .5 1 0 .1 7 0 .7 8 2 1 .0 4 8 1 .3 8 1 1 2 .1 2 0 .6 1 1 1 .0 3 9 1 .2 7 4 1 3 .1 0 0 .5 5 3 1 .0 3 6 1 .2 4 3 f PDes =5 5 .5 1 1 .4 5 0 .7 4 2 1 .0 5 4 1 .4 8 9 1 4 .1 6 0 .5 5 0 1 .0 4 5 1 .3 4 8 1 5 .5 7 0 .4 8 7 1 .0 4 2 1 .3 0 8 2 .0 1 4 .0 1 0 .8 9 8 1 .0 1 3 1 .1 5 6 1 5 .1 6 0 .8 0 3 1 .0 1 0 1 .1 1 6 1 5 .7 1 0 .7 6 7 1 .0 0 9 1 .1 0 4 3 .0 1 5 .8 3 0 .8 5 2 1 .0 1 8 1 .2 4 7 1 7 .7 7 0 .7 2 3 1 .0 1 5 1 .1 8 3 1 8 .7 2 0 .6 7 6 1 .0 1 4 1 .1 6 4 3 .5 1 6 .8 2 0 .8 3 1 1 .0 2 0 1 .2 9 5 1 9 .2 4 0 .6 8 6 1 .0 1 7 1 .2 1 8 2 0 .4 4 0 .6 3 4 1 .0 1 5 1 .1 9 5 4 .5 1 9 .0 2 0 .7 8 9 1 .0 2 4 1 .3 9 8 2 2 .5 6 0 .6 1 8 1 .0 2 0 1 .2 9 1 2 4 .3 6 0 .5 5 9 1 .0 1 9 1 .2 6 0 f PDes =10 5 .5 2 1 .5 0 0 .7 5 1 .0 2 8 1 .5 0 9 2 6 .4 6 0 .5 5 6 1 .0 2 3 1 .3 6 9 2 9 .0 3 0 .4 9 3 1 .0 2 2 1 .3 2 9 2 .0 6 5 .5 6 0 .9 0 1 1 .0 0 3 1 .1 6 1 7 0 .8 1 0 .8 0 7 1 .0 0 2 1 .1 2 2 7 3 .3 3 0 .7 7 1 1 .0 0 2 1 .1 1 0 3 .0 7 4 .4 0 0 .8 5 7 1 .0 0 4 1 .2 5 5 8 3 .2 9 0 .7 2 8 1 .0 0 3 1 .1 9 2 8 7 .6 6 0 .6 8 0 1 .0 0 3 1 .1 7 3 3 .5 7 9 .2 6 0 .8 3 6 1 .0 0 4 1 .3 0 5 9 0 .3 4 0 .6 9 1 1 .0 0 3 1 .2 2 9 9 5 .8 4 0 .6 3 9 1 .0 0 3 1 .2 0 5 4 .5 8 9 .9 6 0 .7 9 5 1 .0 0 5 1 .4 1 0 1 0 6 .2 8 0 .6 2 3 1 .0 0 4 1 .3 0 5 1 1 4 .5 8 0 .5 6 5 1 .0 0 4 1 .2 7 4 f PDes =50 5 .5 1 0 2 .1 2 0 .7 5 6 1 .0 0 6 1 .5 2 4 1 2 5 .0 4 0 .5 6 3 1 .0 0 5 1 .3 8 7 1 3 6 .9 9 0 .4 9 8 1 .0 0 5 1 .3 4 6 260 261 Table C-9: Evaluated Nominal Resistance Factor, ijy, for Ȗǹ=1.1 and ȖPDes=1.2 for g3 Carbon Steel Stainless Steel ȕ R. T. 200oF 400oF 600oF 800oF R. T. 200oF 400oF 600oF 800oF 2.0 0.94* 0.85 0.73 0.63 0.59 0.93* 0.89 0.73 0.64 0.60 3.0 0.93 0.76 0.63 0.54 0.51 0.86 0.75 0.61 0.55 0.51 3.5 0.88 0.73 0.58 0.50 0.47 0.79 0.69 0.57 0.50 0.47 4.5 0.80 0.66 0.50 0.44 0.41 0.67 0.59 0.48 0.43 0.40 f P D es = 0 .5 5.5 0.72 0.59 0.44 0.38 0.35 0.57 0.50 0.41 0.36 0.34 2.0 0.96* 0.87 0.74 0.64 0.60 0.95* 0.90 0.74 0.66 0.62 3.0 0.95 0.78 0.64 0.55 0.52 0.88 0.77 0.63 0.56 0.52 3.5 0.90 0.74 0.60 0.51 0.48 0.81 0.70 0.58 0.51 0.48 4.5 0.81 0.67 0.51 0.44 0.41 0.69 0.60 0.49 0.44 0.41 f P D es = 1 5.5 0.73 0.60 0.44 0.38 0.36 0.58 0.51 0.42 0.37 0.35 2.0 0.97* 0.87 0.75 0.65 0.61 0.96* 0.92 0.75 0.67 0.63 3.0 0.94 0.78 0.64 0.55 0.52 0.88 0.77 0.63 0.56 0.53 3.5 0.89 0.73 0.60 0.51 0.48 0.81 0.71 0.58 0.51 0.48 4.5 0.79 0.65 0.51 0.44 0.41 0.68 0.60 0.49 0.43 0.41 f P D es = 5 5.5 0.70 0.58 0.44 0.38 0.35 0.57 0.50 0.41 0.36 0.34 2.0 0.97* 0.87 0.75 0.65 0.60 0.96 0.92 0.75 0.67 0.63 3.0 0.94 0.77 0.64 0.55 0.52 0.88 0.77 0.63 0.56 0.52 3.5 0.88 0.72 0.59 0.51 0.48 0.81 0.70 0.58 0.51 0.48 4.5 0.78 0.64 0.51 0.44 0.41 0.68 0.59 0.48 0.43 0.40 f P D es = 1 0 5.5 0.69 0.57 0.43 0.37 0.35 0.57 0.50 0.41 0.36 0.34 2.0 0.97* 0.87 0.75 0.65 0.60 0.96 0.92 0.75 0.67 0.62 3.0 0.93 0.76 0.64 0.55 0.51 0.88 0.77 0.63 0.56 0.52 3.5 0.87 0.72 0.59 0.51 0.47 0.80 0.70 0.57 0.51 0.48 4.5 0.77 0.63 0.50 0.43 0.40 0.67 0.59 0.48 0.43 0.40 f P D es = 5 0 5.5 0.68 0.56 0.43 0.37 0.34 0.56 0.49 0.40 0.36 0.33 * For these factors Ȗǹ=1.0 and ȖPD=1.1, R. T.=Room Temperature C.4. Performance Function g4 Table C-10 gives the mean load and resistance factors for the performance function g4. In this table, ȝfy is the converged mean value of the steel resistance. Table C-11 shows the evaluated adjusted nominal resistance factors for Ȗǹ=1.1 and ȖPmax=1.2. T ab le C -1 0 : M ea n L o ad a n d R es is ta n ce F ac to rs f o r D if fe re n t O p er at in g T em p er at u re , T, f o r g 4 C ar b o n S te el ( T” 2 0 0 o F ) C ar b o n S te el ( T> 2 0 0 o F ) S ta in le ss S te el ( fo r an y T ) ȕ ȝ f y I’ y Ȗ' ǹ Ȗ' P m ax ȝ f y I’ y Ȗ' ǹ Ȗ' P m ax ȝ f y I’ y Ȗ' ǹ Ȗ' P m ax 2 .0 1 .8 7 0 .8 8 9 1 .1 0 8 1 .0 9 8 2 .0 3 0 .7 9 4 1 .0 8 3 1 .0 5 7 2 .1 1 0 .7 5 8 1 .0 7 5 1 .0 4 6 3 .0 2 .0 9 0 .8 4 3 1 .1 4 9 1 .2 2 1 2 .3 6 0 .7 1 1 1 .1 1 8 1 .1 1 9 2 .4 9 0 .6 6 5 1 .1 0 8 1 .0 9 7 3 .5 2 .2 1 0 .8 2 4 1 .1 6 4 1 .3 1 7 2 .5 5 0 .6 7 4 1 .1 3 4 1 .1 6 2 2 .7 1 0 .6 2 3 1 .1 2 3 1 .1 3 0 f Pmax =0.5 4 .5 2 .5 0 0 .7 9 4 1 .1 8 0 1 .6 0 5 2 .9 7 0 .6 0 9 1 .1 6 1 1 .2 9 3 3 .2 1 0 .5 4 9 1 .1 5 0 1 .2 2 5 2 .0 2 .5 1 0 .9 0 0 1 .0 7 1 1 .1 8 7 2 .7 1 0 .8 0 2 1 .0 5 8 1 .1 1 6 2 .8 1 0 .7 6 5 1 .0 5 4 1 .0 9 7 3 .0 2 .8 6 0 .8 7 0 1 .0 8 6 1 .4 0 5 3 .1 9 0 .7 3 2 1 .0 7 8 1 .2 5 5 3 .3 5 0 .6 8 1 1 .0 7 3 1 .2 1 0 3 .5 3 .0 8 0 .8 5 8 1 .0 8 9 1 .5 5 1 3 .4 7 0 .7 0 4 1 .0 8 4 1 .3 5 7 3 .6 7 0 .6 4 7 1 .0 8 0 1 .2 9 3 f Pmax =1 4 .5 3 .6 0 0 .8 3 4 1 .0 9 3 1 .9 0 7 4 .1 4 0 .6 5 7 1 .0 9 0 1 .6 2 9 4 .4 3 0 .5 9 0 1 .0 8 9 1 .5 2 7 2 .0 5 .2 1 0 .9 2 1 1 .0 2 6 1 .2 5 8 5 .5 7 0 .8 2 6 1 .0 2 4 1 .1 9 1 5 .7 5 0 .7 8 7 1 .0 2 3 1 .1 6 7 3 .0 6 .1 9 0 .8 9 8 1 .0 2 9 1 .5 1 0 6 .7 5 0 .7 7 3 1 .0 2 8 1 .3 9 7 7 .0 4 0 .7 2 2 1 .0 2 8 1 .3 5 2 3 .5 6 .8 0 0 .8 8 7 1 .0 3 0 1 .6 6 8 7 .4 8 0 .7 5 1 1 .0 3 0 1 .5 2 9 7 .8 5 0 .6 9 4 1 .0 2 9 1 .4 7 3 f Pmax =3 4 .5 8 .2 8 0 .8 6 4 1 .0 3 1 2 .0 4 3 9 .3 0 0 .7 0 6 1 .0 3 1 1 .8 4 6 9 .8 4 0 .6 4 3 1 .0 3 1 1 .7 6 6 2 .0 7 .9 5 0 .9 2 6 1 .0 1 6 1 .2 6 9 8 .4 6 0 .8 3 4 1 .0 1 5 1 .2 0 8 8 .7 2 0 .7 9 6 1 .0 1 4 1 .1 8 5 3 .0 9 .5 6 0 .9 0 4 1 .0 1 8 1 .5 2 6 1 0 .3 7 0 .7 8 4 1 .0 1 7 1 .4 2 3 1 0 .8 0 0 .7 3 4 1 .0 1 7 1 .3 8 1 3 .5 1 0 .5 7 0 .8 9 3 1 .0 1 8 1 .6 8 5 1 1 .5 7 0 .7 6 2 1 .0 1 8 1 .5 5 9 1 2 .1 0 0 .7 0 7 1 .0 1 8 1 .5 0 7 f PDes =5 4 .5 1 3 .0 2 0 .8 7 0 1 .0 1 9 2 .0 6 3 1 4 .5 4 0 .7 1 7 1 .0 1 9 1 .8 8 1 1 5 .3 5 0 .6 5 5 1 .0 1 8 1 .8 0 7 2 .0 1 4 .8 1 0 .9 3 0 1 .0 0 8 1 .2 7 6 1 5 .7 1 0 .8 4 1 1 .0 0 7 1 .2 2 0 1 6 .1 8 0 .8 0 3 1 .0 0 7 1 .1 9 8 3 .0 1 8 .0 1 0 .9 0 9 1 .0 0 9 1 .5 3 5 1 9 .4 6 0 .7 9 3 1 .0 0 9 1 .4 4 2 2 0 .2 3 0 .7 4 3 1 .0 0 9 1 .4 0 2 3 .5 2 0 .0 1 0 .8 9 8 1 .0 0 9 1 .6 9 6 2 1 .8 2 0 .7 7 0 1 .0 0 9 1 .5 8 0 2 2 .7 8 0 .7 1 6 1 .0 0 9 1 .5 1 f PDes =10 4 .5 2 4 .8 8 0 .8 7 5 1 .0 0 9 2 .0 7 6 2 7 .6 7 0 .7 2 6 1 .0 0 9 1 .9 0 6 2 9 .1 7 0 .6 6 4 1 .0 0 9 1 .8 3 6 2 .0 6 9 .7 5 0 .9 3 3 1 .0 0 2 1 .2 8 1 7 3 .8 0 0 .8 4 7 1 .0 0 2 1 .2 3 0 7 5 .9 0 0 .8 0 9 1 .0 0 1 1 .2 0 9 3 .0 8 5 .6 4 0 .9 1 2 1 .0 0 2 1 .5 4 2 9 2 .2 6 0 .8 0 0 1 .0 0 2 1 .4 5 5 9 5 .7 7 0 .7 5 1 1 .0 0 2 1 .4 1 8 3 .5 9 5 .5 7 0 .9 0 2 1 .0 0 2 1 .7 0 3 1 0 3 .8 9 0 .7 7 7 1 .0 0 2 1 .5 6 1 0 8 .3 4 0 .7 2 4 1 .0 0 2 1 .5 5 0 f PDes =50 4 .5 1 1 9 .8 0 0 .8 7 9 1 .0 0 2 2 .0 8 6 1 3 2 .8 3 0 .7 3 2 1 .0 0 2 1 .9 2 5 1 3 9 .8 4 0 .6 7 2 1 .0 0 2 1 .8 5 8 262 263 Table C-11: Evaluated Nominal Resistance Factor, ijy, for Ȗǹ=1.1 and ȖPmax=1.2 for g4 Carbon Steel Stainless Steel ȕ R. T. 200oF 400oF 600oF 800oF R. T. 200oF 400oF 600oF 800oF 2.0 0.95* 0.86 0.74 0.64 0.60 0.94* 0.90 0.74 0.66 0.62 3.0 0.94 0.77 0.64 0.55 0.51 0.88 0.76 0.63 0.56 0.52 3.5 0.89 0.73 0.59 0.51 0.48 0.81 0.70 0.58 0.51 0.48 f P m ax = 0 .5 4.5 0.78 0.64 0.51 0.44 0.41 0.68 0.59 0.49 0.43 0.40 2.0 0.97* 0.88 0.76 0.65 0.61 0.97* 0.93 0.76 0.67 0.63 3.0 0.93 0.77 0.64 0.56 0.52 0.89 0.78 0.63 0.56 0.53 3.5 0.87 0.71 0.59 0.51 0.48 0.81 0.71 0.58 0.52 0.48 f P m ax = 1 4.5 0.74 0.61 0.50 0.43 0.40 0.67 0.59 0.48 0.43 0.40 2.0 0.97* 0.87 0.76 0.66 0.61 0.98* 0.94 0.77 0.68 0.61 3.0 0.89 0.73 0.63 0.54 0.51 0.84 0.76 0.63 0.56 0.50 3.5 0.78 0.67 0.57 0.49 0.46 0.75 0.69 0.56 0.50 0.45 f P m ax = 3 4.5 0.64 0.55 0.46 0.39 0.37 0.60 0.55 0.45 0.40 0.36 2.0 0.97* 0.87 0.76 0.66 0.61 0.98* 0.94 0.77 0.68 0.64 3.0 0.88 0.72 0.62 0.54 0.50 0.87 0.76 0.62 0.55 0.51 3.5 0.79 0.65 0.56 0.48 0.45 0.77 0.67 0.55 0.49 0.46 f P m ax = 5 4.5 0.64 0.53 0.44 0.38 0.36 0.61 0.53 0.43 0.39 0.36 2.0 0.96* 0.86 0.76 0.66 0.61 0.98* 0.93 0.76 0.68 0.64 3.0 0.86 0.71 0.61 0.53 0.49 0.86 0.75 0.61 0.54 0.51 3.5 0.78 0.64 0.55 0.47 0.44 0.76 0.66 0.54 0.48 0.45 f P m ax = 1 0 4.5 0.62 0.51 0.43 0.37 0.35 0.59 0.52 0.42 0.38 0.35 2.0 0.95* 0.86 0.76 0.65 0.61 0.98* 0.93 0.76 0.68 0.63 3.0 0.85 0.70 0.61 0.52 0.49 0.85 0.74 0.60 0.54 0.50 3.5 0.76 0.63 0.54 0.46 0.43 0.75 0.65 0.53 0.47 0.44 f P m ax = 5 0 4.5 0.61 0.50 0.42 0.36 0.34 0.58 0.51 0.41 0.37 0.34 * Ȗǹ=1.0 and ȖPmax=1.1 C.5. Performance Function g6 Table C-12 gives the calculated mean load and resistance factors for performance function g6. In this table, ȝfy is the converged mean value of the steel resistance. Table C-13 presents the adjusted nominal resistance factors for nominal load factors Ȗǹ=1.1 and ȖȂ=ȖPB=1.2. T ab le C -1 2 a: C al cu la te d M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el , T” 2 0 0 o F f o r g 6 C ar b o n S te el T ”2 0 0 o F f M = 0 .5 f M = 1 .0 f M = 2 .0 ȕ ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P B Ȗƍ M ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P B Ȗƍ M ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P B Ȗƍ M 2 2 .4 7 0 .8 8 1 .0 8 8 1 .0 7 3 1 .1 0 1 3 .1 1 0 .8 8 8 1 .0 6 5 1 .0 5 0 1 .1 7 3 4 .4 6 0 .9 0 0 1 .0 4 0 1 .0 2 6 1 .2 3 0 3 2 .7 4 0 .8 3 1 .1 2 7 1 .1 1 7 1 .1 6 6 3 .4 8 0 .8 4 0 1 .0 9 2 1 .0 7 8 1 .2 8 8 5 .0 6 0 .8 5 8 1 .0 5 4 1 .0 4 0 1 .3 8 1 3 .5 2 .8 8 0 .8 0 1 .1 4 6 1 .1 4 0 1 .2 0 2 3 .6 8 0 .8 1 7 1 .1 0 4 1 .0 9 1 1 .3 5 3 5 .3 9 0 .8 3 8 1 .0 6 0 1 .0 4 6 1 .4 6 6 f PB =0.5 4 .5 3 .2 0 0 .7 5 1 .1 8 1 1 .1 8 7 1 .2 7 8 4 .1 1 0 .7 7 4 1 .1 2 5 1 .1 1 4 1 .5 0 0 6 .1 4 0 .8 0 0 1 .0 7 0 1 .0 5 5 1 .6 5 7 2 3 .0 9 0 .8 5 1 .0 6 8 1 .1 2 9 1 .0 7 3 3 .7 2 0 .8 8 6 1 .0 5 6 1 .0 9 9 1 .1 3 9 5 .0 4 0 .8 9 6 1 .0 3 7 1 .0 5 9 1 .2 0 8 3 3 .4 4 0 .8 3 1 .0 9 8 1 .2 0 9 1 .1 1 7 4 .1 4 0 .8 3 6 1 .0 7 9 1 .1 5 7 1 .2 2 9 5 .6 8 0 .8 5 1 1 .0 5 1 1 .0 8 9 1 .3 4 6 3 .5 3 .6 3 0 .8 1 1 .1 1 2 1 .2 5 3 1 .1 3 9 4 .3 8 0 .8 1 2 1 .0 9 0 1 .1 8 6 1 .2 7 8 6 .0 4 0 .8 3 0 1 .0 5 7 1 .1 0 3 1 .4 2 3 f PB =1 4 .5 4 .0 4 0 .7 6 1 .1 3 6 1 .3 5 0 1 .1 8 4 4 .8 8 0 .7 6 7 1 .1 1 0 1 .2 4 7 1 .3 8 7 6 .8 2 0 .7 9 0 1 .0 6 8 1 .1 2 6 1 .5 9 9 2 5 .7 2 0 .9 0 1 .0 3 2 1 .2 0 0 1 .0 2 5 6 .3 0 0 .8 9 5 1 .0 3 0 1 .1 8 4 1 .0 6 1 7 .5 3 0 .8 9 3 1 .0 2 6 1 .1 5 0 1 .1 2 4 3 6 .4 8 0 .8 5 1 .0 4 3 1 .3 2 5 1 .0 4 0 7 .1 0 0 .8 5 0 1 .0 4 1 1 .3 0 0 1 .0 9 4 8 .4 6 0 .8 4 6 1 .0 3 6 1 .2 4 1 1 .1 9 8 3 .5 6 .8 9 0 .8 3 1 .0 4 9 1 .3 9 3 1 .0 4 6 7 .5 4 0 .8 2 8 1 .0 4 6 1 .3 6 4 1 .1 0 9 8 .9 7 0 .8 2 3 1 .0 4 1 1 .2 9 0 1 .2 3 6 f PB =3 4 .5 7 .8 2 0 .8 0 1 .0 5 7 1 .5 4 4 1 .0 5 8 8 .5 1 0 .7 8 7 1 .0 5 5 1 .5 0 4 1 .1 3 7 1 0 .0 8 0 .7 8 1 1 .0 4 9 1 .3 9 6 1 .3 1 5 2 8 .4 0 0 .9 0 1 .0 2 0 1 .2 2 6 1 .0 0 0 8 .9 7 0 .9 0 2 1 .0 1 9 1 .2 0 7 1 .0 3 4 1 0 .1 4 0 .8 9 8 1 .0 1 8 1 .1 8 6 1 .0 7 8 3 9 .5 9 0 .8 6 1 .0 2 7 1 .3 6 5 1 .0 0 5 1 0 .1 9 0 .8 5 9 1 .0 2 7 1 .3 3 6 1 .0 5 3 1 1 .4 6 0 .8 5 3 1 .0 2 5 1 .3 0 2 1 .1 2 0 3 .5 1 0 .2 5 0 .8 4 1 .0 3 0 1 .4 4 1 1 .0 0 6 1 0 .8 7 0 .8 3 9 1 .0 3 0 1 .4 0 6 1 .0 6 1 1 2 .1 9 0 .8 3 2 1 .0 2 8 1 .3 6 6 1 .1 4 0 f PB =5 4 .5 1 1 .7 3 0 .8 1 1 .0 3 5 1 .6 0 6 1 .0 0 9 1 2 .3 8 0 .8 0 0 1 .0 3 5 1 .5 6 1 1 .0 7 4 1 3 .8 0 0 .7 9 1 1 .0 3 3 1 .5 0 7 1 .1 7 6 2 1 5 .1 3 0 .9 1 1 .0 1 0 1 .2 3 4 0 .9 9 1 1 5 .6 8 0 .9 0 8 1 .0 1 0 1 .2 2 3 1 .0 1 2 1 6 .8 1 0 .9 0 5 1 .0 1 0 1 .2 1 4 1 .0 3 5 3 1 7 .4 2 0 .8 7 1 .0 1 4 1 .3 7 7 0 .9 9 2 1 8 .0 0 0 .8 6 8 1 .0 1 4 1 .3 6 0 1 .0 2 1 1 9 .2 0 0 .8 6 4 1 .0 1 4 1 .3 4 6 1 .0 5 4 3 .5 1 8 .7 0 0 .8 5 1 .0 1 5 1 .4 5 5 0 .9 9 2 1 9 .3 0 0 .8 4 9 1 .0 1 5 1 .4 3 4 1 .0 2 4 2 0 .5 3 0 .8 4 4 1 .0 1 5 1 .4 1 9 1 .0 6 2 f PB =10 4 .5 2 1 .5 6 0 .8 1 1 .0 1 8 1 .6 2 4 0 .9 9 3 2 2 .1 9 0 .8 1 2 1 .0 1 8 1 .5 9 8 1 .0 3 0 2 3 .4 9 0 .8 0 6 1 .0 1 8 1 .5 7 7 1 .0 7 6 2 6 9 .0 6 0 .9 1 1 .0 0 2 1 .2 1 5 1 .0 1 1 6 9 .6 0 0 .9 1 5 1 .0 0 2 1 .2 3 4 0 .9 9 4 7 0 .6 9 0 .9 1 4 1 .0 0 2 1 .2 3 2 0 .9 9 8 3 8 0 .2 1 0 .8 8 1 .0 0 3 1 .3 4 9 1 .0 2 0 8 0 .7 8 0 .8 7 7 1 .0 0 3 1 .3 7 6 0 .9 9 5 8 1 .9 2 0 .8 7 6 1 .0 0 3 1 .3 7 4 1 .0 0 2 3 .5 8 6 .4 5 0 .8 6 1 .0 0 3 1 .4 2 2 1 .0 2 4 8 7 .0 3 0 .8 5 8 1 .0 0 3 1 .4 5 4 0 .9 9 6 8 8 .1 9 0 .8 5 7 1 .0 0 3 1 .4 5 2 1 .0 0 3 f PB =50 4 .5 1 0 0 .4 4 0 .8 2 1 .0 0 4 1 .5 8 1 1 .0 3 0 1 0 1 .0 5 0 .8 2 3 1 .0 0 4 1 .6 2 3 0 .9 9 7 1 0 2 .2 7 0 .8 2 1 1 .0 0 4 1 .6 2 0 1 .0 0 5 264 T ab le C -1 2 b : M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el , T> 2 0 0 o F f o r g 6 C ar b o n S te el , T> 2 0 0 o F f M = 0 .5 f M = 1 .0 f M = 2 .0 ȕ ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P B Ȗƍ M ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P B Ȗƍ M ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P B Ȗƍ M 2 2 .6 9 0 .7 8 9 1 .0 6 4 1 .0 4 9 1 .0 6 8 3 .3 8 0 .7 9 4 1 .0 5 0 1 .0 3 5 1 .1 1 9 4 .8 2 0 .8 0 5 1 .0 3 2 1 .0 1 9 1 .1 6 9 3 3 .1 2 0 .7 0 3 1 .0 9 4 1 .0 8 0 1 .1 1 1 3 .9 3 0 .7 1 0 1 .0 7 1 1 .0 5 6 1 .1 9 5 5 .6 6 0 .7 2 7 1 .0 4 5 1 .0 3 1 1 .2 7 8 3 .5 3 .3 5 0 .6 6 3 1 .1 0 8 1 .0 9 5 1 .1 3 3 4 .2 4 0 .6 7 2 1 .0 8 1 1 .0 6 6 1 .2 3 7 6 .1 4 0 .6 9 1 1 .0 5 1 1 .0 3 6 1 .3 3 8 f PB =0.5 4 .5 3 .8 8 0 .5 9 1 1 .1 3 5 1 .1 2 7 1 .1 8 1 4 .9 3 0 .6 0 2 1 .1 0 0 1 .0 8 6 1 .3 2 9 7 .2 3 0 .6 2 6 1 .0 6 1 1 .0 4 6 1 .4 7 1 2 3 .3 7 0 .7 9 0 1 .0 5 1 1 .0 8 8 1 .0 4 9 4 .0 5 .7 9 2 1 .0 4 2 1 .0 6 9 1 .0 9 5 5 .4 6 0 .8 0 0 1 .0 2 9 1 .0 4 4 1 .1 4 9 3 3 .9 1 0 .7 0 6 1 .0 7 4 1 .1 4 2 1 .0 8 1 4 .7 0 0 .7 0 8 1 .0 6 1 1 .1 1 0 1 .1 5 5 6 .3 9 0 .7 2 0 1 .0 4 2 1 .0 6 8 1 .2 4 5 3 .5 4 .2 1 0 .6 6 7 1 .0 8 5 1 .1 7 1 1 .0 9 6 5 .0 6 0 .6 6 9 1 .0 6 9 1 .1 3 1 1 .1 8 8 6 .9 1 0 .6 8 3 1 .0 4 7 1 .0 8 0 1 .2 9 8 f PB =1 4 .5 4 .8 7 0 .5 9 5 1 .1 0 5 1 .2 3 2 1 .1 2 8 5 .8 8 0 .5 9 8 1 .0 8 6 1 .1 7 4 1 .2 5 7 8 .0 9 0 .6 1 6 1 .0 5 7 1 .1 0 1 1 .4 1 5 2 6 .1 9 0 .8 0 4 1 .0 2 5 1 .1 4 7 1 .0 1 8 6 .8 3 0 .8 0 0 1 .0 2 3 1 .1 3 3 1 .0 4 4 8 .1 7 0 .7 9 8 1 .0 2 0 1 .1 0 8 1 .0 8 9 3 7 .2 6 0 .7 2 5 1 .0 3 6 1 .2 3 7 1 .0 3 0 7 .9 9 0 .7 2 0 1 .0 3 3 1 .2 1 5 1 .0 7 1 9 .5 4 0 .7 1 6 1 .0 2 8 1 .1 7 2 1 .1 4 3 3 .5 7 .8 7 0 .6 8 8 1 .0 4 0 1 .2 8 6 1 .0 3 6 8 .6 4 0 .6 8 3 1 .0 3 8 1 .2 6 0 1 .0 8 3 1 0 .3 0 0 .6 7 9 1 .0 3 2 1 .2 0 7 1 .1 7 0 f PB =3 4 .5 9 .2 5 0 .6 2 2 1 .0 4 8 1 .3 9 2 1 .0 4 6 1 0 .1 2 0 .6 1 5 1 .0 4 6 1 .3 5 6 1 .1 0 7 1 2 .0 3 0 .6 1 0 1 .0 4 0 1 .2 7 9 1 .2 2 8 2 9 .0 6 0 .8 1 0 1 .0 1 6 1 .0 1 6 1 .0 0 7 9 .6 8 0 .8 0 7 1 .0 1 6 1 .1 5 5 1 .0 2 5 1 0 .9 8 0 .8 0 3 1 .0 1 4 1 .1 3 6 1 .0 5 8 3 1 0 .7 0 0 .7 3 4 1 .0 2 3 1 .0 2 3 1 .0 1 5 1 1 .4 0 0 .7 2 9 1 .0 2 2 1 .2 5 0 1 .0 4 1 1 2 .8 7 0 .7 2 3 1 .0 2 0 1 .2 2 0 1 .0 9 1 3 .5 1 1 .6 3 0 .6 9 9 1 .0 2 6 1 .0 2 6 1 .0 1 8 1 2 .3 7 0 .6 9 3 1 .0 2 5 1 .3 0 1 1 .0 4 8 1 3 .9 3 0 .6 8 8 1 .0 2 3 1 .2 6 5 1 .1 0 7 f PB =5 4 .5 1 3 .7 5 0 .6 3 3 1 .0 3 1 1 .0 3 1 1 .0 2 4 1 4 .5 8 0 .6 2 7 1 .0 3 0 1 .4 1 2 1 .0 6 1 1 6 .3 5 0 .6 1 9 1 .0 2 8 1 .3 6 2 1 .1 3 9 2 1 6 .2 5 0 .8 1 7 1 .0 0 9 1 .0 0 9 0 .9 9 9 1 6 .8 6 0 .8 1 5 1 .0 0 9 1 .1 7 3 1 .0 0 8 1 8 .1 2 0 .8 1 1 1 .0 0 8 1 .1 6 3 1 .0 2 7 3 1 9 .3 3 0 .7 4 3 1 .0 1 2 1 .0 1 2 1 .0 0 2 2 0 .0 1 0 .7 4 0 1 .0 1 2 1 .2 7 7 1 .0 1 6 2 1 .4 1 0 .7 3 4 1 .0 1 1 1 .2 6 2 1 .0 4 3 3 .5 2 1 .0 9 0 .7 0 8 1 .0 1 3 1 .0 1 3 1 .0 0 4 2 1 .8 0 0 .7 0 5 1 .0 1 3 1 .3 3 4 1 .0 1 9 2 3 .2 8 0 .6 9 9 1 .0 1 3 1 .3 1 6 1 .0 5 0 f PB =10 4 .5 2 5 .1 1 0 .6 4 4 1 .0 1 6 1 .0 1 6 1 .0 0 7 2 5 .9 0 0 .6 4 1 1 .0 1 6 1 .4 5 5 1 .0 2 5 2 7 .5 4 0 .6 3 4 1 .0 1 5 1 .4 3 1 1 .0 6 3 2 7 3 .9 2 0 .8 2 4 1 .0 0 2 1 .0 0 2 0 .9 9 1 7 4 .5 2 0 .8 2 3 1 .0 0 2 1 .1 8 7 0 .9 9 3 7 5 .7 3 0 .8 2 2 1 .0 0 2 1 .1 8 5 0 .9 9 7 3 8 8 .5 9 0 .7 5 1 1 .0 0 2 1 .0 0 2 0 .9 9 2 8 9 .2 5 0 .7 5 0 1 .0 0 2 1 .3 0 0 0 .9 9 4 9 0 .5 8 0 .7 4 9 1 .0 0 2 1 .2 9 7 1 .0 0 0 3 .5 9 6 .9 8 0 .7 1 7 1 .0 0 3 1 .0 0 3 0 .9 9 2 9 7 .6 7 0 .7 1 7 1 .0 0 3 1 .3 6 0 0 .9 9 5 9 9 .0 7 0 .7 1 5 1 .0 0 3 1 .3 5 6 1 .0 0 1 f PB =50 4 .5 1 1 6 .2 5 0 .6 5 4 1 .0 0 3 1 .0 0 3 0 .9 9 3 1 1 7 .0 1 0 .6 5 3 1 .0 0 3 1 .4 8 9 0 .9 9 6 1 1 8 .5 4 0 .6 5 2 1 .0 0 3 1 .4 8 5 1 .0 0 3 265 T ab le C -1 2 c: M ea n L o ad a n d R es is ta n ce F ac to rs f o r S ta in le ss S te el a n d a n y T f o r g 6 S ta in le ss S te el f M = 0 .5 f M = 1 .0 f M = 2 .0 ȕ ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P B Ȗƍ M ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P B Ȗƍ M ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P B Ȗƍ M 2 2 .8 0 0 .7 5 4 1 .0 5 8 1 .0 4 3 1 .0 5 9 3 .5 1 0 .7 5 8 1 .0 4 5 1 .0 3 1 1 .1 0 4 4 .9 9 0 .7 6 9 1 .0 2 9 1 .0 1 7 1 .1 5 0 3 3 .2 9 0 .6 5 8 1 .0 8 5 1 .0 7 0 1 .0 9 6 4 .1 5 0 .6 6 4 1 .0 6 5 1 .0 5 0 1 .1 7 1 5 .9 6 0 .6 7 9 1 .0 4 2 1 .0 2 8 1 .2 4 6 3 .5 3 .5 8 0 .6 1 4 1 .0 9 7 1 .0 8 3 1 .1 1 6 4 .5 2 0 .6 2 2 1 .0 7 4 1 .0 5 9 1 .2 0 7 6 .5 2 0 .6 3 9 1 .0 4 7 1 .0 3 3 1 .2 9 9 f PB =0.5 4 .5 4 .2 7 0 .5 3 6 1 .1 2 2 1 .1 1 1 1 .1 5 7 5 .3 5 0 .5 4 5 1 .0 9 2 1 .0 7 7 1 .2 8 6 7 .7 9 0 .5 6 6 1 .0 5 7 1 .0 4 2 1 .4 1 5 2 3 .5 0 0 .7 5 6 1 .0 4 6 1 .0 7 7 1 .0 4 3 4 .2 0 0 .7 5 7 1 .0 3 8 1 .0 6 1 1 .0 8 3 5 .6 6 0 .7 6 4 1 .0 2 7 1 .0 3 9 1 .1 3 2 3 4 .1 3 0 .6 6 0 1 .0 6 7 1 .1 2 4 1 .0 7 1 4 .9 6 0 .6 6 2 1 .0 5 5 1 .0 9 7 1 .1 3 6 6 .7 3 0 .6 7 3 1 .0 3 8 1 .0 6 1 1 .2 1 6 3 .5 4 .4 8 0 .6 1 7 1 .0 7 7 1 .1 4 9 1 .0 8 5 5 .4 0 0 .6 1 9 1 .0 6 3 1 .1 1 5 1 .1 6 4 7 .3 4 0 .6 3 2 1 .0 4 3 1 .0 7 2 1 .2 6 2 f PB =1 4 .5 5 .2 9 0 .5 3 9 1 .0 9 5 1 .2 0 2 1 .1 1 3 6 .3 7 0 .5 4 2 1 .0 7 8 1 .1 5 3 1 .2 2 4 8 .7 4 0 .5 5 7 1 .0 5 3 1 .0 9 3 1 .3 6 3 2 6 .4 1 0 .7 6 8 1 .0 2 3 1 .1 3 1 1 .0 1 5 7 .0 8 0 .7 6 4 1 .0 2 1 1 .1 1 8 1 .0 3 9 8 .4 7 0 .7 6 3 1 .0 1 8 1 .0 9 5 1 .0 7 9 3 7 .6 5 0 .6 7 7 1 .0 3 3 1 .2 1 1 1 .0 2 9 8 .4 2 0 .6 7 3 1 .0 3 1 1 .1 9 1 1 .0 6 3 1 0 .0 6 0 .6 7 0 1 .0 2 6 1 .1 5 3 1 .1 2 6 3 .5 8 .3 5 0 .6 3 6 1 .0 3 7 1 .2 5 4 1 .0 3 2 9 .1 9 0 .6 3 1 1 .0 3 5 1 .2 3 0 1 .0 7 5 1 0 .9 6 0 .6 2 8 1 .0 3 0 1 .1 8 3 1 .1 5 1 f PB =3 4 .5 9 .9 7 0 .5 6 2 1 .0 4 5 1 .3 4 7 1 .0 4 2 1 0 .9 3 0 .5 5 6 1 .0 4 3 1 .3 1 3 1 .0 9 7 1 3 .0 1 0 .5 5 2 1 .0 3 6 1 .2 4 7 1 .2 0 2 2 9 .3 7 0 .7 7 4 1 .0 1 5 1 .1 4 7 1 .0 0 6 1 0 .0 3 0 .7 7 1 1 .0 1 4 1 .1 3 9 1 .0 2 2 1 1 .3 8 0 .7 6 7 1 .0 1 3 1 .1 2 2 1 .0 5 1 3 1 1 .2 4 0 .6 8 6 1 .0 2 1 1 .2 3 7 1 .0 1 3 1 1 .9 9 0 .6 8 2 1 .0 2 0 1 .2 2 3 1 .0 3 7 1 3 .5 6 0 .6 7 6 1 .0 1 9 1 .1 9 6 1 .0 8 2 3 .5 1 2 .3 2 0 .6 4 6 1 .0 2 4 1 .2 8 5 1 .0 1 6 1 3 .1 2 0 .6 4 1 1 .0 2 3 1 .2 6 9 1 .0 4 4 1 4 .8 0 0 .6 3 5 1 .0 2 1 1 .2 3 5 1 .0 9 7 f PB =5 4 .5 1 4 .7 9 0 .5 7 3 1 .0 2 9 1 .3 8 8 1 .0 2 2 1 5 .7 1 0 .5 6 8 1 .0 2 8 1 .3 6 6 1 .0 5 6 1 7 .6 7 0 .5 6 0 1 .0 2 6 1 .3 2 0 1 .1 2 6 2 1 6 .8 0 0 .7 8 1 1 .0 0 8 1 .1 6 1 0 .9 9 8 1 7 .4 4 0 .7 7 8 1 .0 0 8 1 .1 5 6 1 .0 0 7 1 8 .7 5 0 .7 7 5 1 .0 0 0 8 1 .1 4 7 1 .0 2 4 3 2 0 .2 8 0 .6 9 5 1 .0 1 1 1 .2 5 8 1 .0 0 1 2 1 .0 1 0 .6 9 2 1 .0 1 1 1 .2 5 0 1 .0 1 4 2 2 .5 0 0 .6 8 6 1 .0 1 1 1 .2 3 6 1 .0 3 9 3 .5 2 2 .2 9 0 .6 5 5 1 .0 1 3 1 .3 0 9 1 .0 0 3 2 3 .0 7 0 .6 5 2 1 .0 1 2 1 .3 0 1 1 .0 1 7 2 4 .6 6 0 .6 4 6 1 .0 1 2 1 .2 8 3 1 .0 4 6 f PB =10 4 .5 2 6 .9 4 0 .5 8 3 1 .0 1 5 1 .4 2 0 1 .0 0 6 2 7 .8 2 0 .5 8 0 1 .0 1 5 1 .4 0 9 1 .0 2 3 2 9 .6 2 0 .5 7 4 1 .0 1 4 1 .3 8 6 1 .0 5 8 2 7 6 .3 2 0 .7 8 8 1 .0 0 2 1 .1 7 2 0 .9 9 1 7 6 .9 5 0 .7 8 7 1 .0 0 2 1 .1 7 1 0 .9 9 3 7 8 .2 2 0 .7 8 6 1 .0 0 2 1 .1 6 9 0 .9 9 6 3 9 2 .8 0 0 .7 0 3 1 .0 0 2 1 .2 7 5 0 .9 9 2 9 3 .5 1 0 .7 0 2 1 .0 0 2 1 .2 7 3 0 .9 9 4 9 4 .9 3 0 .7 0 1 1 .0 0 2 1 .2 7 0 0 .9 9 9 3 .5 1 0 2 .3 4 0 .6 6 4 1 .0 0 3 1 .3 2 9 0 .9 9 2 1 0 3 .0 8 0 .6 6 3 1 .0 0 3 1 3 2 8 0 .9 9 5 1 0 4 .5 9 0 .6 6 2 1 .0 0 3 1 .3 2 4 1 .0 0 0 f PB =50 4 .5 1 2 4 .4 6 0 .5 9 3 1 .0 0 3 1 .4 4 6 0 .9 9 2 1 2 5 .3 0 0 .5 9 2 1 .0 0 3 1 .4 4 4 0 .9 9 6 1 2 6 .9 9 0 .5 9 1 1 .0 0 3 1 .4 4 0 1 .0 0 3 266 T ab le C -1 3 a: A d ju st ed N o m in al R es is ta n ce F ac to rs f o r C ar b o n S te el a n d Ȗ ǹ = 1 .1 , Ȗ Ȃ = Ȗ P B= 1 .2 f o r g 6 C ar b o n S te el f M = 0 .5 f M = 1 .0 f M = 2 .0 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 2 0 .9 4 (1 ) 0 .8 9 0 .7 6 0 .6 6 0 .6 1 0 .9 4 0 .9 0 0 .7 7 0 .6 6 0 .6 2 0 .9 2 0 .8 9 0 .7 7 0 .6 6 0 .6 2 3 0 .9 7 0 .8 0 0 .6 6 0 .5 7 0 .5 3 0 .9 7 0 .8 0 0 .6 6 0 .5 7 0 .5 3 0 .9 5 0 .7 8 0 .6 5 0 .5 6 0 .5 3 3 .5 0 .9 3 0 .7 6 0 .6 1 0 .5 3 0 .4 9 0 .9 2 0 .7 6 0 .6 1 0 .5 3 0 .4 9 0 .8 9 0 .7 3 0 .6 0 0 .5 2 0 .4 9 f PB =0.5 4 .5 0 .8 3 0 .6 9 0 .5 3 0 .4 6 0 .4 3 0 .8 2 0 .6 8 0 .5 3 0 .4 6 0 .4 3 0 .7 8 0 .6 4 0 .5 1 0 .4 4 0 .4 1 2 0 .9 4 (1 ) 0 .9 0 0 .7 7 0 .6 7 0 .6 2 0 .9 4 0 .9 1 0 .7 8 0 .6 7 0 .6 3 0 .9 3 0 .9 0 0 .7 8 0 .6 7 0 .6 3 3 0 .9 8 0 .8 1 0 .6 7 0 .5 7 0 .5 4 0 .9 9 0 .8 1 0 .6 7 0 .5 8 0 .5 4 0 .9 7 0 .8 0 0 .6 7 0 .5 7 0 .5 4 3 .5 0 .9 3 0 .7 7 0 .6 2 0 .5 3 0 .5 0 0 .9 4 0 .7 7 0 .6 2 0 .5 4 0 .5 0 0 .9 1 0 .7 5 0 .6 2 0 .5 3 0 .5 0 f PB =1 4 .5 0 .8 4 0 .6 9 0 .5 3 0 .4 6 0 .4 3 0 .8 4 0 .6 9 0 .5 4 0 .4 6 0 .4 3 0 .8 1 0 .6 7 0 .5 3 0 .4 5 0 .4 2 2 0 .9 3 (1 ) 0 .9 0 0 .7 8 0 .6 7 0 .6 2 0 .9 3 0 .9 1 0 .7 8 0 .6 8 0 .6 3 0 .9 4 0 .9 2 0 .7 9 0 .6 8 0 .6 4 3 0 .9 6 0 .7 9 0 .6 6 0 .5 7 0 .5 3 0 .9 8 0 .8 1 0 .6 7 0 .5 8 0 .5 4 0 .9 9 0 .8 2 0 .6 8 0 .5 8 0 .5 4 3 .5 0 .9 1 0 .7 5 0 .6 1 0 .5 3 0 .4 9 0 .9 2 0 .7 6 0 .6 2 0 .5 3 0 .5 0 0 .9 3 0 .7 7 0 .6 3 0 .5 4 0 .5 0 f PB =3 4 .5 0 .8 0 0 .6 6 0 .5 2 0 .4 5 0 .4 2 0 .8 2 0 .6 7 0 .5 3 0 .4 6 0 .4 3 0 .8 3 0 .6 8 0 .5 4 0 .4 6 0 .4 3 2 0 .9 1 (1 ) 0 .8 9 0 .7 7 0 .6 7 0 .6 2 0 .9 2 0 .9 0 0 .7 8 0 0 .6 7 0 .6 3 0 .9 3 0 .9 1 0 .7 9 0 .6 8 0 .6 3 3 0 .9 5 0 .7 8 0 .6 5 0 .5 6 0 .5 3 0 .9 6 0 .7 9 0 .6 6 0 .5 6 0 .5 3 0 .9 8 0 .8 1 0 .6 7 0 .5 8 0 .5 4 3 .5 0 .8 9 0 .7 3 0 .6 0 0 .5 2 0 .4 8 0 .9 0 0 .7 4 0 .6 1 0 .5 2 0 .4 9 0 .9 2 0 .7 6 0 .6 2 0 .5 4 0 .5 0 f PB =5 4 .5 0 .7 8 0 .6 4 0 .5 1 0 .4 4 0 .4 1 0 .7 9 0 .6 5 0 .5 2 0 .4 3 0 .4 2 0 .8 1 0 .6 7 0 .5 3 0 .4 6 0 .4 3 2 0 .9 0 (1 ) 0 .8 8 0 .7 7 0 .6 6 0 .6 2 0 .9 1 0 .8 9 0 .7 7 0 .6 6 0 .6 2 0 .9 2 0 .9 0 0 .7 8 0 .6 7 0 .6 3 3 0 .8 3 0 .7 7 0 .6 5 0 .5 6 0 .5 2 0 .9 4 0 .7 7 0 .6 5 0 .5 5 0 .5 2 0 .9 6 0 .7 9 0 .6 6 0 .5 7 0 .5 3 3 .5 0 .8 7 0 .7 1 0 .5 9 0 .5 1 0 .4 8 0 .8 8 0 .7 2 0 .6 0 0 .5 0 0 .4 8 0 .8 9 0 .7 4 0 .6 1 0 .5 2 0 .4 9 f PB =10 4 .5 0 .7 5 0 .6 2 0 .5 0 0 .4 3 0 .4 0 0 .7 6 0 .6 3 0 .5 0 0 .4 2 0 .4 1 0 .7 8 0 .6 4 0 .5 1 0 .4 4 0 .4 1 2 0 .8 9 (1 ) 0 .8 7 0 .7 6 0 .6 6 0 .6 1 0 .8 9 0 .8 8 0 .7 6 0 .6 7 0 .6 2 0 .8 9 0 .8 8 0 .7 7 0 .6 6 0 .6 2 3 0 .9 1 0 .7 5 0 .6 4 0 .5 5 0 .5 1 0 .9 2 0 .7 5 0 .6 4 0 .5 7 0 .5 1 0 .9 2 0 .7 6 0 .6 4 0 .5 5 0 .5 2 3 .5 0 .8 5 0 .7 0 0 .5 8 0 .5 0 0 .4 7 0 .8 5 0 .7 0 0 .5 8 0 .5 3 0 .4 7 0 .8 6 0 .7 0 0 .5 9 0 .5 1 0 .4 7 f PB =50 4 .5 0 .7 3 0 .6 0 0 .4 9 0 .4 2 0 .3 9 0 .7 3 0 .6 0 0 .4 9 0 .4 5 0 .3 9 0 .7 4 0 .6 1 0 .4 9 0 .4 2 0 .3 9 (1 ) F o r th es e fa ct o rs Ȗ ǹ = Ȗ Ȃ = Ȗ P B = 1 267 T ab le C -1 3 b : A d ju st ed N o m in al R es is ta n ce F ac to rs f o r Ȗ ǹ = 1 .1 , Ȗ Ȃ = Ȗ P B= 1 .2 a n d S ta in le ss S te el f o r g 6 S ta in le ss S te el f M = 0 .5 f M = 1 .0 f M = 2 .0 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 2 0 .9 2 (1 ) 0 .9 3 0 .7 6 0 .6 8 0 .6 3 0 .9 3 0 .9 4 0 .7 7 0 .6 8 0 .6 4 0 .9 2 0 .9 4 0 .7 7 0 .6 8 0 .6 4 3 0 .9 1 0 .7 9 0 .6 5 0 .5 7 0 .5 4 0 .9 1 0 .7 9 0 .6 5 0 .5 8 0 .5 4 0 .9 0 0 .7 9 0 .6 4 0 .5 7 0 .5 4 3 .5 0 .8 3 0 .7 3 0 .5 9 0 .5 3 0 .5 0 0 .8 3 0 .7 3 0 .6 0 0 .5 3 0 .5 0 0 .8 2 0 .7 2 0 .6 4 0 .5 2 0 .4 9 f PB =0.5 4 .5 0 .7 1 0 .6 2 0 .5 1 0 .4 5 0 .4 2 0 .7 1 0 .6 2 0 .5 0 0 .4 5 0 .4 2 0 .6 9 0 .6 0 0 .4 9 0 .4 4 0 .4 1 2 0 .9 3 (1 ) 0 .9 4 0 .7 7 0 .6 8 0 .6 4 0 .9 3 0 .9 5 0 .7 8 0 .6 9 0 .6 5 0 .9 3 0 .9 5 0 .7 8 0 .6 9 0 .6 5 3 0 .9 1 0 .8 0 0 .6 5 0 .5 8 0 .5 4 0 .9 2 0 .8 0 0 .6 6 0 .5 8 0 .5 5 0 .9 2 0 .8 0 0 .6 5 0 .5 8 0 .5 4 3 .5 0 .8 4 0 .7 4 0 .6 0 0 .5 3 0 .5 0 0 .8 5 0 .7 4 0 .6 0 0 .5 4 0 .5 0 0 .8 4 0 .7 3 0 .6 0 0 .5 3 0 .5 0 f PB =1 4 .5 0 .7 1 0 .6 2 0 .5 1 0 .4 5 0 .4 2 0 .7 2 0 .6 3 0 .5 1 0 .4 6 0 .4 3 0 .7 0 0 .6 2 0 .5 0 0 .4 5 0 .4 2 2 0 .9 2 (1 ) 0 .9 5 0 .7 8 0 .6 9 0 .6 5 0 .9 3 0 .9 6 0 .7 8 0 .7 0 0 .6 5 0 .9 3 0 .9 6 0 .7 9 0 .7 0 0 .6 6 3 0 .9 1 0 .7 9 0 .6 5 0 .5 8 0 .5 4 0 .9 2 0 .8 0 0 .6 6 0 .5 8 0 .5 5 0 .9 3 0 .8 1 0 .6 6 0 .5 9 0 .5 5 3 .5 0 .8 3 0 .7 3 0 .6 0 0 .5 3 0 .5 0 0 .8 4 0 .7 4 0 .6 0 0 .5 4 0 .5 0 0 .8 5 0 .7 4 0 .6 1 0 .5 4 0 .5 1 f PB =3 4 .5 0 .7 0 0 .6 1 0 .5 0 0 .4 4 0 .4 2 0 .7 1 0 .6 2 0 .5 1 0 .4 5 0 .4 2 0 .7 2 0 .6 3 0 .5 1 0 .4 6 0 .4 3 2 0 .9 0 (1 ) 0 .9 4 0 .7 7 0 .6 9 0 .6 4 0 .9 2 0 .9 5 0 .7 8 0 .6 9 0 .6 5 0 .9 3 0 .9 6 0 .7 9 0 .7 0 0 .6 6 3 0 .9 0 0 .7 9 0 .6 4 0 .5 7 0 .5 4 0 .9 1 0 .8 0 0 .6 5 0 .5 8 0 .5 4 0 .9 2 0 .8 1 0 .6 6 0 .5 9 0 .5 5 3 .5 0 .8 2 0 .7 2 0 .5 8 0 .5 2 0 .4 9 0 .8 3 0 .7 3 0 .6 0 0 .5 3 0 .5 0 0 .8 5 0 .7 4 0 .6 0 0 .5 4 0 .5 0 f PB =5 4 .5 0 .6 9 0 .6 0 0 .4 8 0 .4 4 0 .4 1 0 .7 0 0 .6 1 0 .5 0 0 .4 4 0 .4 1 0 .7 1 0 .6 2 0 .5 1 0 .4 5 0 .4 2 2 0 .9 0 (1 ) 0 .9 4 0 .7 7 0 .6 8 0 .6 4 0 .9 1 0 .9 5 0 .7 7 0 .6 9 0 .6 4 0 .9 2 0 .9 5 0 .7 8 0 .6 9 0 .6 5 3 0 .8 9 0 .7 8 0 .6 4 0 .5 7 0 .5 3 0 .9 0 0 .7 9 0 .6 4 0 .5 7 0 .5 4 0 .9 1 0 .7 9 0 .6 5 0 .5 8 0 .5 4 3 .5 0 .8 1 0 .7 1 0 .5 9 0 .5 2 0 .4 8 0 .8 2 0 .7 1 0 .5 8 0 .5 2 0 .4 9 0 .8 3 0 .7 3 0 .5 9 0 .5 3 0 .4 9 f PB =10 4 .5 0 .6 7 0 .5 9 0 .4 9 0 .4 3 0 .4 0 0 .6 8 0 .5 9 0 .4 9 0 .4 3 0 .4 0 0 .6 9 0 .6 0 0 .4 9 0 .4 4 0 .4 1 2 0 .8 9 (1 ) 0 .9 4 0 .7 7 0 .6 8 0 .6 4 0 .9 0 0 .9 4 0 .7 7 0 .6 8 0 .6 4 0 .9 0 0 .9 4 0 .7 7 0 .6 8 0 .6 4 3 0 .8 8 0 .7 7 0 .6 3 0 .5 6 0 .5 2 0 .8 8 0 .7 7 0 .6 3 0 .5 6 0 .5 3 0 .8 9 0 .7 7 0 .6 3 0 .5 6 0 .5 3 3 .5 0 .8 0 0 .7 0 0 .5 7 0 .5 1 0 .4 8 0 .8 0 0 .7 0 0 .5 7 0 .5 1 0 .4 8 0 .8 0 0 .7 0 0 .5 7 0 .5 1 0 .4 8 f PB =50 4 .5 0 .6 6 0 .5 7 0 .4 7 0 .4 2 0 .3 9 0 .6 6 0 .5 8 0 .4 7 0 .4 2 0 .3 9 0 .6 6 0 .5 8 0 .4 7 0 .4 2 0 .3 9 (1 ) F o r th es e fa ct o rs Ȗ ǹ = Ȗ Ȃ = Ȗ P B= 1 268 C.6. Performance Function g7 Table C-14 provides the calculated mean load and resistance factors for performance function g7. In this table, ȝfy is the converged mean value of the steel resistance. Table C-15 shows the evaluated adjusted nominal resistance factors for nominal load factors Ȗǹ=1.1 and ȖO=1.5. 269 T ab le C -1 3 b : A d ju st ed N o m in al R es is ta n ce F ac to rs f o r Ȗ ǹ = 1 .1 , Ȗ Ȃ = Ȗ P B= 1 .2 a n d S ta in le ss S te el f o r g 6 S ta in le ss S te el f M = 0 .5 f M = 1 .0 f M = 2 .0 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 2 0 .9 2 (1 ) 0 .9 3 0 .7 6 0 .6 8 0 .6 3 0 .9 3 0 .9 4 0 .7 7 0 .6 8 0 .6 4 0 .9 2 0 .9 4 0 .7 7 0 .6 8 0 .6 4 3 0 .9 1 0 .7 9 0 .6 5 0 .5 7 0 .5 4 0 .9 1 0 .7 9 0 .6 5 0 .5 8 0 .5 4 0 .9 0 0 .7 9 0 .6 4 0 .5 7 0 .5 4 3 .5 0 .8 3 0 .7 3 0 .5 9 0 .5 3 0 .5 0 0 .8 3 0 .7 3 0 .6 0 0 .5 3 0 .5 0 0 .8 2 0 .7 2 0 .6 4 0 .5 2 0 .4 9 f PB =0.5 4 .5 0 .7 1 0 .6 2 0 .5 1 0 .4 5 0 .4 2 0 .7 1 0 .6 2 0 .5 0 0 .4 5 0 .4 2 0 .6 9 0 .6 0 0 .4 9 0 .4 4 0 .4 1 2 0 .9 3 (1 ) 0 .9 4 0 .7 7 0 .6 8 0 .6 4 0 .9 3 0 .9 5 0 .7 8 0 .6 9 0 .6 5 0 .9 3 0 .9 5 0 .7 8 0 .6 9 0 .6 5 3 0 .9 1 0 .8 0 0 .6 5 0 .5 8 0 .5 4 0 .9 2 0 .8 0 0 .6 6 0 .5 8 0 .5 5 0 .9 2 0 .8 0 0 .6 5 0 .5 8 0 .5 4 3 .5 0 .8 4 0 .7 4 0 .6 0 0 .5 3 0 .5 0 0 .8 5 0 .7 4 0 .6 0 0 .5 4 0 .5 0 0 .8 4 0 .7 3 0 .6 0 0 .5 3 0 .5 0 f PB =1 4 .5 0 .7 1 0 .6 2 0 .5 1 0 .4 5 0 .4 2 0 .7 2 0 .6 3 0 .5 1 0 .4 6 0 .4 3 0 .7 0 0 .6 2 0 .5 0 0 .4 5 0 .4 2 2 0 .9 2 (1 ) 0 .9 5 0 .7 8 0 .6 9 0 .6 5 0 .9 3 0 .9 6 0 .7 8 0 .7 0 0 .6 5 0 .9 3 0 .9 6 0 .7 9 0 .7 0 0 .6 6 3 0 .9 1 0 .7 9 0 .6 5 0 .5 8 0 .5 4 0 .9 2 0 .8 0 0 .6 6 0 .5 8 0 .5 5 0 .9 3 0 .8 1 0 .6 6 0 .5 9 0 .5 5 3 .5 0 .8 3 0 .7 3 0 .6 0 0 .5 3 0 .5 0 0 .8 4 0 .7 4 0 .6 0 0 .5 4 0 .5 0 0 .8 5 0 .7 4 0 .6 1 0 .5 4 0 .5 1 f PB =3 4 .5 0 .7 0 0 .6 1 0 .5 0 0 .4 4 0 .4 2 0 .7 1 0 .6 2 0 .5 1 0 .4 5 0 .4 2 0 .7 2 0 .6 3 0 .5 1 0 .4 6 0 .4 3 2 0 .9 0 (1 ) 0 .9 4 0 .7 7 0 .6 9 0 .6 4 0 .9 2 0 .9 5 0 .7 8 0 .6 9 0 .6 5 0 .9 3 0 .9 6 0 .7 9 0 .7 0 0 .6 6 3 0 .9 0 0 .7 9 0 .6 4 0 .5 7 0 .5 4 0 .9 1 0 .8 0 0 .6 5 0 .5 8 0 .5 4 0 .9 2 0 .8 1 0 .6 6 0 .5 9 0 .5 5 3 .5 0 .8 2 0 .7 2 0 .5 8 0 .5 2 0 .4 9 0 .8 3 0 .7 3 0 .6 0 0 .5 3 0 .5 0 0 .8 5 0 .7 4 0 .6 0 0 .5 4 0 .5 0 f PB =5 4 .5 0 .6 9 0 .6 0 0 .4 8 0 .4 4 0 .4 1 0 .7 0 0 .6 1 0 .5 0 0 .4 4 0 .4 1 0 .7 1 0 .6 2 0 .5 1 0 .4 5 0 .4 2 2 0 .9 0 (1 ) 0 .9 4 0 .7 7 0 .6 8 0 .6 4 0 .9 1 0 .9 5 0 .7 7 0 .6 9 0 .6 4 0 .9 2 0 .9 5 0 .7 8 0 .6 9 0 .6 5 3 0 .8 9 0 .7 8 0 .6 4 0 .5 7 0 .5 3 0 .9 0 0 .7 9 0 .6 4 0 .5 7 0 .5 4 0 .9 1 0 .7 9 0 .6 5 0 .5 8 0 .5 4 3 .5 0 .8 1 0 .7 1 0 .5 9 0 .5 2 0 .4 8 0 .8 2 0 .7 1 0 .5 8 0 .5 2 0 .4 9 0 .8 3 0 .7 3 0 .5 9 0 .5 3 0 .4 9 f PB =10 4 .5 0 .6 7 0 .5 9 0 .4 9 0 .4 3 0 .4 0 0 .6 8 0 .5 9 0 .4 9 0 .4 3 0 .4 0 0 .6 9 0 .6 0 0 .4 9 0 .4 4 0 .4 1 2 0 .8 9 (1 ) 0 .9 4 0 .7 7 0 .6 8 0 .6 4 0 .9 0 0 .9 4 0 .7 7 0 .6 8 0 .6 4 0 .9 0 0 .9 4 0 .7 7 0 .6 8 0 .6 4 3 0 .8 8 0 .7 7 0 .6 3 0 .5 6 0 .5 2 0 .8 8 0 .7 7 0 .6 3 0 .5 6 0 .5 3 0 .8 9 0 .7 7 0 .6 3 0 .5 6 0 .5 3 3 .5 0 .8 0 0 .7 0 0 .5 7 0 .5 1 0 .4 8 0 .8 0 0 .7 0 0 .5 7 0 .5 1 0 .4 8 0 .8 0 0 .7 0 0 .5 7 0 .5 1 0 .4 8 f PB =50 4 .5 0 .6 6 0 .5 7 0 .4 7 0 .4 2 0 .3 9 0 .6 6 0 .5 8 0 .4 7 0 .4 2 0 .3 9 0 .6 6 0 .5 8 0 .4 7 0 .4 2 0 .3 9 (1 ) F o r th es e fa ct o rs Ȗ ǹ = Ȗ Ȃ = Ȗ P B= 1 270 271 Table C-15a: Adjusted Nominal Resistance Factor for Ȗǹ=1.1 and ȖO=1.5 and Carbon Steel for g7 Carbon Steel fO ȕ R.T. 200oF 400oF 600oF 800oF 1.5 0.96 (1) 0.79 (1) 0.93 0.80 0.75 2 0.95 (2) 0.89 0.80 0.69 0.65 0.5 3 0.67 0.55 0.50 0.43 0.40 1.5 0.91 (1) 0.75 (1) 0.97 0.83 0.78 2 0.87 (2) 0.87 0.79 0.68 0.63 1 3 0.58 0.47 0.43 0.37 0.35 1.5 0.88 (1) 0.72 (1) 0.99 0.85 0.79 2 0.81 (2) 0.85 0.77 0.67 0.62 2 3 0.52 0.43 0.39 0.34 0.31 1.5 0.86 (1) 0.71 (1) 0.99 0.86 0.80 2 0.80 (2) 0.84 0.77 0.66 0.62 CO V( f O )= 0 .5 0 2.5 3 0.51 0.42 0.38 0.33 0.31 1.5 0.91 (1) 0.75 (1) 0.90 0.78 0.72 2 0.85 (2) 0.80 0.73 0.63 0.59 0.5 3 0.48 0.40 0.36 0.31 0.29 1.5 0.87 (1) 0.70 (1) 0.91 0.78 0.73 2 0.76 0.75 0.69 0.59 0.55 1.0 3 0.40 0.33 0.30 0.26 0.24 1.5 0.80 (1) 0.72 (1) 0.91 0.78 0.73 2 0.69 (2) 0.72 0.66 0.57 0.53 2 3 0.35 0.29 0.26 0.23 0.21 1.5 0.79 (1) 0.65 (1) 0.91 0.78 0.73 2 0.67 (2) 0.71 0.65 0.56 0.52 CO V( f O )= 0 .8 0 2.5 3 0.34 0.28 0.25 0.22 0.20 (1) Ȗǹ=1 and Ȗȅ=0.9, (2) Ȗǹ=1.1 and Ȗȅ=1.1, R.T.=Room Temperature 272 Table C-15b: Adjusted Nominal Resistance Factor for Ȗǹ=1.1 and ȖO=1.5 and Stainless Steel for g7 Stainless Steel fO ȕ R.T. 200oF 400oF 600oF 800oF 1.5 1.01 (1) 0.94 (1) 0.90 0.80 0.75 2 1.00 (2) 1.00 0.78 0.69 0.65 0.5 3 0.71 0.62 0.48 0.43 0.40 1.5 0.98 (1) 0.86 (1) 0.95 0.85 0.80 2 0.93 (2) 0.95 0.78 0.69 0.65 1 3 0.62 0.52 0.43 0.38 0.36 1.5 0.94 (1) 0.85 (1) 0.99 0.88 0.82 2 0.87 (2) 0.95 0.77 0.69 0.64 2 3 0.56 0.48 0.39 0.35 0.33 1.5 0.93 (1) 0.81 (1) 1.00 0.89 0.83 2 0.86 (2) 0.94 0.77 0.69 0.64 C O V (f O )= 0 .5 0 2.5 3 0.54 0.47 0.38 0.34 0.32 1.5 0.98 (1) 0.85 (1) 0.87 0.78 0.73 2 0.91 (2) 0.87 0.71 0.63 0.59 0.5 3 0.45 0.43 0.35 0.31 0.29 1.5 0.92 (1) 0.80 (1) 0.90 0.80 0.75 2 0.82 (2) 0.83 0.68 0.61 0.57 1.0 3 0.43 0.36 0.30 0.26 0.25 1.5 0.87 (1) 0.76 (1) 0.91 0.81 0.76 2 0.74 (2) 0.81 0.66 0.59 0.55 2 3 0.37 0.32 0.26 0.23 0.22 1.5 0.86 (1) 0.75 (1) 0.92 0.81 0.76 2 0.73 (2) 0.80 0.65 0.58 0.54 C O V (f O )= 0 .8 0 2.5 3 0.36 0.31 0.26 0.23 0.21 (1) Ȗǹ=1 and Ȗȅ=0.9, (2) Ȗǹ=1.1 and Ȗȅ=1.1, R.T.=Room Temperature C.7. Performance Function g8 Table C-16 provides the calculated mean load and resistance factors for performance function g8. In this table, ȝfy is the converged mean value of the steel resistance. Table C-17 presents the evaluated adjusted nominal resistance factors for nominal load factors Ȗǹ=1.1, ȖPO=1.2, and ȖO=1.5. T ab le C -1 6 a: M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T ”2 0 0 o F f o r g 8 C ar b o n S te el f o r T” 2 0 0 o F f O = 0 .5 f O = 1 f O = 2 ȕ ȝ f y I' fy Ȗƍ ǹ Ȗƍ O Ȗƍ P O ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ O Ȗƍ P O ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ O Ȗƍ P O 1 .5 2 .5 0 0 .9 6 2 1 .0 2 3 1 .7 4 6 1 .0 1 1 3 .4 2 0 .9 7 3 1 .0 1 1 1 .8 1 8 1 .0 0 1 5 .3 0 0 .9 7 9 1 .0 0 6 1 .8 4 0 0 .9 9 6 2 2 .9 5 0 .9 6 6 1 .0 1 7 2 .6 6 0 1 .0 0 6 4 .3 4 0 .9 7 4 1 .0 0 8 2 .7 2 1 0 .9 9 9 7 .1 4 0 .9 7 8 1 .0 0 4 2 .7 4 1 0 .9 9 5 f Po = 0.5 3 .0 5 .4 4 0 .9 7 5 1 .0 0 7 7 .5 9 5 0 .9 9 7 9 .3 4 0 .9 7 8 1 .0 0 3 7 .6 3 3 0 .9 9 4 1 7 .1 5 0 .9 7 9 1 .0 0 2 7 .6 4 9 0 .9 9 3 1 .5 3 .0 3 0 .9 5 4 1 .0 2 4 1 .6 6 9 1 .0 3 3 3 .9 4 0 .9 7 0 1 .0 1 1 1 .7 9 7 1 .0 1 1 5 .8 1 0 .9 7 7 1 .0 0 6 1 .8 3 3 1 .0 0 1 2 3 .4 8 0 .9 6 0 1 .0 1 8 2 .5 9 8 1 .0 2 2 4 .8 5 0 .9 7 2 1 .0 0 8 2 .7 0 3 1 .0 0 6 7 .6 5 0 .9 7 7 1 .0 0 4 2 .7 3 5 0 .9 9 9 f PO =1 3 .0 5 .9 5 0 .9 7 3 1 .0 0 7 7 .5 6 2 1 .0 0 3 9 .8 5 0 .9 7 7 1 .0 0 3 7 .6 2 1 0 .9 9 7 1 7 .6 6 0 .9 7 9 1 .0 0 2 7 .6 4 4 0 .9 9 4 1 .5 5 .3 4 0 .9 2 8 1 .0 2 3 1 .1 0 5 1 .1 2 7 6 .0 9 0 .9 5 4 1 .0 1 2 1 .6 2 3 1 .0 5 6 7 .8 8 0 .9 7 0 1 .0 0 6 1 .7 8 8 1 .0 2 1 2 5 .7 3 0 .9 2 2 1 .0 2 3 1 .7 4 8 1 .1 3 0 6 .9 7 0 .9 6 0 1 .0 0 9 2 .5 6 6 1 .0 3 9 9 .7 1 0 .9 7 2 1 .0 0 4 2 .6 9 6 1 .0 1 3 f PO =3 3 .0 8 .0 5 0 .9 6 3 1 .0 0 7 7 .3 3 4 1 .0 2 8 1 1 .9 1 0 .9 7 2 1 .0 0 3 7 .5 5 2 1 .0 0 9 1 9 .7 0 0 .9 7 7 1 .0 0 2 7 .6 1 9 1 .0 0 0 1 .5 7 .8 0 0 .9 2 8 1 .0 1 5 0 .9 4 4 1 .1 5 0 8 .3 8 0 .9 3 7 1 .0 1 2 1 .2 7 8 1 .1 1 3 1 0 .0 2 0 .9 6 2 1 .0 0 6 1 .7 0 8 1 .0 4 3 2 8 .3 4 0 .9 0 7 1 .0 2 0 1 .0 1 5 1 .2 0 8 9 .2 0 0 .9 4 4 1 .0 1 0 2 .2 5 4 1 .0 8 4 1 1 .8 2 0 .9 6 6 1 .0 0 4 2 .6 3 3 1 .0 2 9 f PO =5 3 .0 1 0 .2 3 0 .9 5 3 1 .0 0 7 6 .8 8 7 1 .0 5 8 1 4 0 .9 6 8 1 .0 0 3 7 .4 4 5 1 .0 2 1 2 1 .7 5 0 .9 7 5 1 .0 0 2 7 .5 8 6 1 .0 0 6 1 .5 1 4 .0 2 0 .9 3 1 1 .0 0 8 0 .8 6 7 1 .1 6 2 1 4 .5 1 0 .9 3 1 1 .0 0 8 0 .9 5 0 1 .1 5 5 1 5 .6 9 0 .9 4 0 1 .0 0 6 1 .3 0 7 1 .1 1 3 2 1 5 .0 5 0 .9 1 1 1 .0 1 0 0 .8 8 6 1 .2 2 5 1 5 .5 7 0 .9 1 1 1 .0 1 0 1 .0 2 5 1 .2 1 5 1 7 .3 3 0 .9 4 8 1 .0 0 5 2 .2 9 5 1 .0 8 2 f PO = 10 3 .0 1 7 .3 4 0 .8 7 1 1 .0 1 4 0 .9 2 5 1 .3 6 4 1 9 .4 0 0 .9 5 5 1 .0 0 4 6 .9 4 8 1 .0 5 7 2 6 .9 7 0 .9 6 9 1 .0 0 2 7 .4 6 1 1 .0 2 1 1 .5 6 4 0 .9 3 5 1 .0 0 2 0 .8 2 2 1 .1 6 9 6 4 .4 4 0 .9 3 5 1 .0 0 2 0 .8 3 2 1 .1 6 8 6 5 .3 4 0 .9 3 4 1 .0 0 2 0 .8 5 6 1 .1 6 7 2 6 8 .9 7 0 .9 1 5 1 .0 0 2 0 .8 2 4 1 .2 3 4 6 9 .4 2 0 .9 1 5 1 .0 0 2 0 .8 3 8 1 .2 3 4 7 0 .3 5 0 .9 1 4 1 .0 0 2 0 .8 7 0 1 .2 3 2 f PO = 50 3 .0 8 0 .1 1 0 .8 7 7 1 .0 0 3 0 .8 2 9 1 .3 7 7 8 0 .5 9 0 .8 7 7 1 .0 0 3 0 .8 4 8 1 .3 7 6 8 1 .5 8 0 .8 7 6 1 .0 0 3 0 .8 9 6 1 .3 7 4 273 T ab le C -1 6 a: ( C o n ti n u ed ) C ar b o n S te el f o r T” 2 0 0 o F f O = 2 .5 ȕ ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ O Ȗƍ P O 1 .5 6 .2 4 0 .9 8 1 .0 0 4 1 .8 4 3 0 .9 9 5 2 8 .5 4 0 .9 7 9 1 .0 0 3 2 .7 4 4 0 .9 9 4 f Po = 0.5 3 .0 2 1 .0 6 0 .9 8 1 .0 0 1 7 .6 5 2 0 .9 9 3 1 .5 6 .7 5 0 .9 7 8 1 .0 0 4 1 .8 3 8 0 .9 9 9 2 9 .0 5 0 .9 7 8 1 .0 0 3 2 .7 4 0 0 .9 9 7 f PO =1 3 .0 2 1 .5 6 0 .9 7 9 1 .0 0 1 7 .6 4 8 0 .9 9 4 1 .5 8 .8 1 0 .9 7 2 1 .0 0 5 1 .8 0 8 1 .0 1 5 2 1 1 .1 0 0 .9 7 4 1 .0 0 3 2 .7 1 3 1 .0 0 9 f PO =3 3 .0 2 3 .6 0 0 .9 7 8 1 .0 0 1 7 .6 3 0 0 .9 9 8 1 .5 1 0 .9 2 0 .9 6 6 1 .0 0 5 1 .7 5 7 1 .0 3 2 2 1 3 .1 9 0 .9 6 9 1 .0 0 3 2 .6 7 2 1 .0 2 1 f PO =5 3 .0 2 5 .6 5 0 .9 7 6 1 .0 0 1 7 .6 0 6 1 .0 0 3 1 .5 1 6 .4 3 0 .9 4 9 1 .0 0 5 1 .5 0 5 1 .0 8 4 2 1 8 .5 9 0 .9 5 6 1 .0 0 4 2 .4 7 8 1 .0 5 8 f PO = 10 3 .0 3 0 .8 3 0 .9 7 2 1 .0 0 1 7 .5 2 1 1 .0 1 5 1 .5 6 5 .8 0 0 .9 3 4 1 .0 0 2 0 .8 6 9 1 .1 6 6 2 7 0 .8 3 0 .9 1 4 1 .0 0 2 0 .8 8 8 1 .2 3 1 f PO = 50 3 .0 8 2 .1 1 0 .8 7 6 1 .0 0 3 0 .9 2 7 1 .3 7 2 274 T ab le C -1 6 b : M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T > 2 0 0 o F f o r g 8 C ar b o n S te el f o r T> 2 0 0 o F f O = 0 .5 f O = 1 f O = 2 ȕ ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ O Ȗƍ P O ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ O Ȗƍ P O ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ O Ȗƍ P O 1 .5 2 .5 8 0 .9 0 3 1 .0 2 4 1 .6 0 4 1 .0 1 2 3 .5 1 0 .9 3 1 1 .0 1 1 1 .7 5 3 1 .0 0 1 5 .4 0 0 .9 4 5 1 .0 0 6 1 .8 0 1 0 .9 9 6 2 3 .0 4 0 .9 1 2 1 .0 1 8 2 .5 0 6 1 .0 0 7 4 .4 4 0 .9 3 4 1 .0 0 9 2 .6 4 2 0 .9 9 9 7 .2 9 0 .9 4 4 1 .0 0 4 2 .6 8 8 0 .9 9 5 f Po = 0.5 3 .0 5 .5 7 0 .9 3 4 1 .0 0 7 7 .3 9 8 0 .9 9 7 9 .5 4 0 .9 4 3 1 .0 0 3 7 .4 9 0 0 .9 9 4 1 7 .4 9 0 .9 4 7 1 .0 0 2 7 .5 2 9 0 .9 9 3 1 .5 3 .1 5 0 .8 8 4 1 .0 2 5 1 .4 5 7 1 .0 3 5 4 .0 5 0 .9 2 2 1 .0 1 2 1 .7 1 1 1 .0 1 1 5 .9 3 0 .9 4 0 1 .0 0 6 1 .7 8 7 1 .0 0 1 2 3 .6 0 0 .8 9 6 1 .0 1 9 2 .3 7 0 1 .0 2 4 4 .9 8 0 .9 2 7 1 .0 0 9 2 .6 0 3 1 .0 0 6 7 .8 1 0 .9 4 1 1 .0 0 4 2 .6 7 4 0 .9 9 9 f PO =1 3 .0 6 .1 1 0 .9 2 9 1 .0 0 7 7 .3 2 4 1 .0 0 3 1 0 .0 7 0 .9 4 0 1 .0 0 3 7 .4 6 2 0 .9 9 7 1 8 .0 2 0 .9 4 5 1 .0 0 2 7 .5 1 6 0 .9 9 4 1 .5 5 .6 5 0 .8 5 2 1 .0 1 9 0 .9 9 9 1 .1 0 0 6 .3 4 0 .8 8 3 1 .0 1 2 1 .4 0 9 1 .0 5 9 8 .1 1 0 .9 2 2 1 .0 0 6 1 .7 0 1 1 .0 2 2 2 6 .1 5 0 .8 1 4 1 .0 2 4 1 .1 5 6 1 .1 3 4 7 .2 3 0 .8 9 4 1 .0 1 0 2 .3 2 8 1 .0 4 3 9 .9 7 0 .9 2 7 1 .0 0 4 2 .5 9 6 1 .0 1 4 f PO =3 3 .0 8 .3 2 0 .9 0 4 1 .0 0 7 6 .8 5 5 1 .0 3 0 1 2 .2 2 0 .9 2 9 1 .0 0 3 7 .3 1 3 1 .0 0 9 2 0 .1 3 0 .9 4 0 1 .0 0 2 7 .4 5 9 1 .0 0 0 1 .5 8 .2 5 0 .8 5 4 1 .0 1 3 0 .9 1 0 1 .1 1 6 8 .8 3 0 .8 6 1 1 .0 1 1 1 .0 9 5 1 .0 9 9 1 0 .3 7 0 .9 0 2 1 .0 0 6 1 .5 6 1 1 .0 4 6 2 8 .9 8 0 .8 1 3 1 .0 1 6 0 .9 5 6 1 .1 6 1 9 .6 9 0 .8 4 2 1 .0 1 2 1 .6 0 9 1 .1 0 8 1 2 .2 0 0 .9 1 1 1 .0 0 5 2 .4 7 4 1 .0 3 2 f PO =5 3 .0 1 0 .6 4 0 .7 3 8 1 .0 2 2 1 .1 0 2 1 .2 5 6 1 4 .4 2 0 .9 1 7 1 .0 0 4 7 .1 0 1 1 .0 2 2 2 2 .2 7 0 .9 3 4 1 .0 0 2 7 .3 8 9 1 .0 0 6 1 .5 1 4 .8 1 0 .8 5 8 1 .0 0 7 0 .8 5 7 1 .1 2 7 1 5 .3 3 0 .8 5 8 1 .0 0 7 0 .9 1 6 1 .1 2 2 1 6 .4 9 0 .8 6 5 1 .0 0 6 1 .1 2 0 1 .1 0 2 2 1 6 .1 6 0 .8 1 8 1 .0 0 9 0 .8 7 2 1 .1 7 7 1 6 .7 1 0 .8 1 8 1 .0 0 8 0 .9 6 5 1 .1 7 0 1 8 .1 9 0 .8 5 4 1 .0 0 6 1 .7 4 9 1 .1 0 3 f PO = 10 3 .0 1 9 .2 3 0 .7 4 4 1 .0 1 2 0 .9 0 3 1 .2 8 4 1 9 .9 0 0 .7 4 4 1 .0 1 2 1 .1 3 2 1 .2 6 7 2 7 .7 4 0 .9 2 0 1 .0 0 2 7 .1 4 4 1 .0 2 2 1 .5 6 7 .4 3 0 .8 6 3 1 .0 0 1 0 .8 2 0 1 .1 3 6 6 7 .9 1 0 .8 6 3 1 .0 0 1 0 .8 2 9 1 .1 3 5 6 8 .8 8 0 .8 6 2 1 .0 0 1 0 .8 4 8 1 .1 3 4 2 7 3 .8 2 0 .8 2 4 1 .0 0 2 0 .8 2 3 1 .1 8 8 7 4 .3 2 0 .8 2 4 1 .0 0 2 0 .8 3 4 1 .1 8 7 7 5 .3 5 0 .8 2 3 1 .0 0 2 0 .8 6 0 1 .1 8 5 f PO = 50 3 .0 8 8 .4 7 0 .7 5 1 1 .0 0 2 0 .8 2 7 1 .3 0 1 8 9 .0 3 0 .7 5 1 1 .0 0 2 0 .8 4 3 1 .3 0 0 9 0 .1 8 0 .7 5 0 1 .0 0 2 0 .8 8 2 1 .2 9 7 275 T ab le C -1 6 b : (C o n ti n u ed ) C ar b o n S te el f o r T> 2 0 0 o F f O = 2 .5 ȕ ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ O Ȗƍ P O 1 .5 6 .3 6 0 .9 4 7 1 .0 0 5 1 .8 0 9 0 .9 9 5 2 8 .7 1 0 .9 4 6 1 .0 0 3 2 .6 9 6 0 .9 9 4 f Po = 0.5 3 .0 2 1 .4 7 0 .9 4 7 1 .0 0 1 7 .5 3 6 0 .9 9 3 1 .5 6 .8 9 0 .9 4 4 1 .0 0 5 1 .7 9 8 0 .9 9 9 2 9 .2 4 0 .9 4 3 1 .0 0 3 2 .6 8 5 0 .9 9 7 f PO =1 3 .0 2 1 .9 9 0 .9 4 6 1 .0 0 1 7 .5 2 6 0 .9 9 4 1 .5 9 .0 4 0 .9 2 9 1 .0 0 5 1 .7 3 9 1 .0 1 5 2 1 1 .3 8 0 .9 3 2 1 .0 0 3 2 .6 3 0 1 .0 0 9 f PO =3 3 .0 2 4 .1 0 0 .9 4 2 1 .0 0 1 7 .4 8 3 0 .9 9 8 1 .5 1 1 .2 6 0 .9 1 4 1 .0 0 5 1 .6 4 8 1 .0 3 3 2 1 3 .5 7 0 .9 2 0 1 .0 0 4 2 .5 5 0 1 .0 2 2 f PO =5 3 .0 2 6 .2 3 0 .9 3 8 1 .0 0 1 7 .4 3 2 1 .0 0 3 1 .5 1 7 .1 8 0 .8 7 6 1 .0 0 5 1 .2 7 7 1 .0 8 5 2 1 9 .3 4 0 .8 8 4 1 .0 0 4 2 .1 7 5 1 .0 6 5 f PO = 10 3 .0 3 1 .6 5 0 .9 2 6 1 .0 0 1 7 .2 6 3 1 .0 1 5 1 .5 6 9 .3 8 0 .8 6 2 1 .0 0 1 0 .8 5 8 1 .1 3 3 2 7 5 .8 8 0 .8 2 2 1 .0 0 2 0 .8 7 4 1 .1 8 4 f PO = 50 3 .0 9 0 .7 7 0 .7 4 9 1 .0 0 2 0 .9 0 6 1 .2 9 5 276 T ab le C -1 6 c: M ea n L o ad a n d R es is ta n ce F ac to rs f o r S ta in le ss S te el a n d a n y T f o r g 8 S ta in le ss S te el f o r an y T f O = 0 .5 f O = 1 f O = 2 ȕ ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ O Ȗƍ P O ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ O Ȗƍ P O ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ O Ȗƍ P O 1 .5 2 .6 3 0 .8 7 3 1 .0 2 4 1 .5 3 2 1 .0 1 2 3 .5 5 0 .9 1 0 1 .0 1 2 1 .7 2 0 1 .0 0 1 5 .4 6 0 .9 2 7 1 .0 0 6 1 .7 8 1 0 .9 9 6 2 3 .0 9 0 .8 8 4 1 .0 1 9 2 .4 2 2 1 .0 0 7 4 .5 0 0 .9 1 3 1 .0 0 9 2 .6 0 0 0 .9 9 9 7 .3 6 0 .9 2 6 1 .0 0 4 2 .6 6 0 0 .9 9 5 f Po = 0.5 3 .0 5 .6 4 0 .9 1 4 1 .0 0 7 7 .2 9 4 0 .9 9 7 9 .6 5 0 .9 2 4 1 .0 0 3 7 .4 1 5 0 .9 9 4 1 7 .6 7 0 .9 3 0 1 .0 0 2 7 .4 6 6 0 .9 9 3 1 .5 3 .2 2 0 .8 5 0 1 .0 2 5 1 .3 6 0 1 .0 3 5 4 .1 1 0 .8 9 8 1 .0 1 2 1 .6 6 6 1 .0 1 2 6 .0 0 0 .9 2 2 1 .0 0 6 1 .7 6 3 1 .0 0 1 2 3 .6 8 0 .8 6 1 1 .0 2 0 2 .2 4 0 1 .0 2 6 5 .0 5 0 .9 0 3 1 .0 0 9 2 .5 4 9 1 .0 0 7 7 .9 0 0 .9 2 2 1 .0 0 4 2 .6 4 2 0 .9 9 9 f PO =1 3 .0 6 .1 9 0 .9 0 6 1 .0 0 7 7 .1 9 8 1 .0 0 3 1 0 .1 8 0 .9 2 1 1 .0 0 3 7 .3 7 7 0 .9 9 7 1 8 .2 1 0 .9 2 8 1 .0 0 2 7 .4 4 9 0 .9 9 4 1 .5 5 .8 0 0 .8 2 3 1 .0 1 7 0 .9 7 3 1 .0 9 0 6 .4 8 0 .8 5 0 1 .0 1 2 1 .3 1 6 1 .0 5 9 8 .2 3 0 .8 9 8 1 .0 0 6 1 .6 5 6 1 .0 2 2 2 6 .3 6 0 .7 7 6 1 .0 2 2 1 .0 8 7 1 .1 2 3 7 .3 8 0 .8 5 9 1 .0 1 0 2 .1 9 1 1 .0 4 5 1 0 .1 1 0 .9 0 3 1 .0 4 2 .5 4 2 1 .0 1 4 f PO =3 3 .0 8 .4 8 0 .8 7 2 1 .0 0 8 6 .5 8 2 1 .0 3 2 1 2 .3 8 0 .9 0 6 1 .0 0 4 7 .1 8 6 1 .0 0 9 2 0 .3 7 0 .9 2 1 1 .0 0 2 7 .3 7 5 1 .0 0 0 1 .5 8 .4 7 0 .8 2 5 1 .0 1 2 0 .9 0 0 1 .1 0 5 9 .0 6 0 .8 3 0 1 .0 1 0 1 .0 5 1 1 .0 9 1 1 0 .5 6 0 .8 7 2 1 .0 0 6 1 .4 8 8 1 .0 4 6 2 9 .2 9 0 .7 7 7 1 .0 1 5 0 .9 3 9 1 .1 4 6 9 .9 8 0 .7 9 4 1 .0 1 2 1 .3 5 1 1 .1 1 3 1 2 .4 0 0 .8 8 2 1 .0 0 5 2 .3 8 7 1 .0 3 3 f PO =5 3 .0 1 1 .1 7 0 .6 9 0 1 .0 2 1 1 .0 5 3 1 .2 3 2 1 4 .6 4 0 .8 9 0 1 .0 0 4 6 .9 1 3 1 .0 2 3 2 2 .5 5 0 .9 1 3 1 .0 0 2 7 .2 8 5 1 .0 0 6 1 .5 1 5 .1 9 0 .8 2 9 1 .0 0 6 0 .8 5 3 1 .1 1 6 1 5 .7 2 0 .8 2 8 1 .0 0 6 0 .9 0 6 1 .1 1 1 1 6 .9 0 0 .8 3 4 1 .0 0 5 1 .0 7 3 1 .0 9 5 2 1 6 .7 0 .7 8 2 1 .0 0 8 0 .8 6 7 1 .1 6 1 1 7 .2 8 0 .7 8 1 1 .0 0 8 0 .9 4 8 1 .1 5 4 1 8 .7 1 0 .8 0 5 1 .0 0 6 1 .4 6 2 1 .1 1 2 f PO = 10 3 .0 2 0 .1 8 0 .6 9 6 1 .0 1 1 0 .8 9 6 1 .2 5 8 2 0 .8 8 0 .6 9 6 1 .0 1 1 1 .0 7 6 1 .2 4 4 2 8 .1 6 0 .8 9 4 1 .0 0 2 6 .9 7 2 1 .0 2 3 1 .5 6 9 .1 1 0 .8 3 4 1 .0 0 1 0 .8 2 0 1 .1 2 4 6 9 .6 1 0 .8 3 3 1 .0 0 1 0 .8 2 8 1 .1 2 4 7 0 .6 1 0 .8 3 3 1 .0 0 1 0 .8 4 5 1 .1 2 2 2 7 6 .2 1 0 .7 8 8 1 .0 0 2 0 .8 2 2 1 .1 7 2 7 6 .7 4 0 .7 8 7 1 .0 0 2 0 .8 3 2 1 .1 7 1 7 7 .8 1 0 .7 8 6 1 .0 0 2 0 .8 5 6 1 .1 6 9 f PO = 50 3 .0 9 2 .6 8 0 .7 0 3 1 .0 0 2 0 .8 2 6 1 .2 7 5 9 3 .2 7 0 .7 0 3 1 .0 0 2 0 .8 4 1 1 .2 7 4 9 4 .5 0 0 .7 0 2 1 .0 0 2 0 .8 7 7 1 .2 7 1 277 T ab le C -1 6 c: ( C o n ti n u ed ) S ta in le ss S te el f o r an y T f O = 2 .5 ȕ ȝ f y I’ fy Ȗƍ ǹ Ȗƍ O Ȗƍ P O 1 .5 6 .4 2 0 .9 3 1 1 .0 0 5 1 .7 9 1 0 .9 9 5 2 8 .8 0 0 .9 2 9 1 .0 0 3 2 .6 7 1 0 .9 9 4 f Po = 0.5 3 .0 2 1 .6 9 0 .9 3 1 1 .0 0 1 7 .4 7 5 0 .9 9 3 1 .5 6 .9 6 0 .9 2 6 1 .0 0 5 1 .7 7 8 0 .9 9 9 2 9 .3 4 0 .9 2 6 1 .0 0 3 2 .6 5 7 0 .9 9 7 f PO =1 3 .0 2 2 .2 2 0 .9 2 9 1 .0 0 1 7 .4 6 2 0 .9 9 4 1 .5 9 .1 6 0 .9 0 7 1 .0 0 5 1 .7 0 3 1 .0 1 6 2 1 1 .5 2 0 .9 1 1 1 .0 0 4 2 .5 8 6 1 .0 1 0 f PO =3 3 .0 2 4 .3 7 0 .9 2 4 1 .0 0 1 7 .4 0 6 0 .9 9 8 1 .5 1 1 .4 4 0 .8 8 7 1 .0 0 5 1 .5 9 2 1 .0 3 4 2 1 3 .7 7 0 .8 9 5 1 .0 0 4 2 .4 8 4 1 .0 2 3 f PO =5 3 .0 2 6 .5 5 0 .9 1 8 1 .0 0 1 7 .3 4 0 1 .0 0 3 1 .5 1 7 .5 8 0 .8 4 3 1 .0 0 5 1 .2 0 0 1 .0 8 2 2 1 9 .7 7 0 .8 4 4 1 .0 0 4 1 .9 9 1 1 .0 7 0 f PO = 10 3 .0 3 2 .0 9 0 .9 0 3 1 .0 0 1 7 .1 2 4 1 .0 1 6 1 .5 7 1 .1 2 0 .8 3 2 1 .0 0 1 0 .8 5 5 1 .1 2 1 2 7 8 .3 6 0 .7 8 6 1 .0 0 2 0 .8 6 9 1 .1 6 8 f PO = 50 3 .0 9 5 .1 3 0 .7 0 1 1 .0 0 2 0 .8 9 9 1 .2 6 9 278 T ab le C -1 7 a: A d ju st ed N o m in al R es is ta n ce F ac to r fo r g 8 a n d C ar b o n S te el f o r Ȗ ǹ = 1 .1 , Ȗ P O = 1 .2 a n d Ȗ O = 1 .5 C ar b o n S te el f O = 0 .5 f O = 1 .0 f O = 2 .0 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 1 .5 0 .9 6 (1 ) 0 .7 9 (1 ) 0 .9 2 0 .7 9 0 .7 4 0 .9 0 (1 ) 0 .7 4 (1 ) 0 .9 2 0 .8 0 0 .7 4 0 .8 4 (1 ) 0 .6 9 (1 ) 0 .9 2 0 .8 0 0 .7 4 2 0 .8 1 (1 ) 0 .8 6 0 .7 8 0 .6 7 0 .6 3 0 .7 1 (1 ) 0 .8 0 0 .7 3 0 .6 3 0 .5 9 0 .6 2 (1 ) 0 .7 5 0 .6 8 0 .5 9 0 .5 5 f PO = 0.5 3 .0 0 .5 7 0 .4 7 0 .4 3 0 .3 7 0 .3 4 0 .4 5 0 .3 7 0 .3 4 0 .2 9 0 .2 7 0 .3 8 0 .3 1 0 .2 9 0 .2 5 0 .2 3 1 .5 0 .9 9 (1 ) 0 .8 1 (1 ) 0 .9 3 0 .8 0 0 .7 5 0 .9 3 (1 ) 0 .7 7 (1 ) 0 .9 4 0 .8 1 0 .7 5 0 .8 7 (1 ) 0 .7 1 (1 ) 0 .9 3 0 .8 0 0 .7 5 2 0 .8 6 (1 ) 0 .9 0 0 .8 1 0 .7 0 0 .6 5 0 .7 6 (1 ) 0 .8 4 0 .7 6 0 .6 6 0 .6 1 0 .6 6 (1 ) 0 .7 7 0 .7 1 0 .6 1 0 .5 7 f PO =1 3 .0 0 .6 4 0 .5 3 0 .4 8 0 .4 1 0 .3 9 0 .5 0 0 .4 1 0 .3 8 0 .3 2 0 .3 0 0 .4 1 0 .3 4 0 .3 1 0 .2 6 0 .2 5 1 .5 1 .0 1 (1 ) 0 .8 3 (1 ) 0 .9 1 0 .7 8 0 .7 3 0 .9 9 (1 ) 0 .8 2 (1 ) 0 .9 5 0 .8 2 0 .7 6 0 .9 4 (1 ) 0 .7 7 (1 ) 0 .9 5 0 .8 2 0 .7 7 2 0 .9 4 (1 ) 0 .9 6 0 .8 3 0 .7 2 0 .6 7 0 .8 7 (1 ) 0 .9 2 0 .8 3 0 .7 1 0 .6 7 0 .7 6 (1 ) 0 .8 5 0 .7 8 0 .6 7 0 .6 2 f PO =3 3 .0 0 .8 3 0 .6 8 0 .6 2 0 .5 3 0 .5 0 0 .6 5 0 .5 4 0 .4 9 0 .4 2 0 .3 9 0 .5 1 0 .4 2 0 .3 8 0 .3 3 0 .3 1 1 .5 0 .9 9 (1 ) 0 .8 2 (1 ) 0 .8 9 0 .7 7 0 .7 1 1 .0 1 (1 ) 0 .8 3 (1 ) 0 .9 3 0 .8 0 0 .7 5 0 .9 8 (1 ) 0 .8 0 (1 ) 0 .9 6 0 .8 3 0 .7 7 2 0 .9 3 (1 ) 0 .9 4 0 .8 2 0 .7 0 0 .6 6 0 .9 2 (1 ) 0 .9 5 0 .8 5 0 .7 3 0 .6 8 0 .8 3 (1 ) 0 .9 0 0 .8 1 0 .7 0 0 .6 6 f PO =5 3 .0 0 .9 3 0 .7 7 0 .6 9 0 .5 9 0 .5 5 0 .7 6 0 .6 3 0 .5 7 0 .4 9 0 .4 6 0 .5 9 0 .4 9 0 .4 5 0 .3 8 0 .3 6 1 .5 0 .9 8 (1 ) 0 .8 0 (1 ) 0 .8 7 0 .7 5 0 .7 0 0 .9 9 (1 ) 0 .8 2 (1 ) 0 .8 9 0 .7 7 0 .7 2 1 .0 0 (1 ) 0 .8 3 (1 ) 0 .9 4 0 .8 1 0 .7 5 2 0 .9 1 (1 ) 0 .9 1 0 .7 9 0 .6 8 0 .6 4 0 .9 2 (1 ) 0 .9 4 0 .8 2 0 .7 1 0 .6 6 0 .9 1 (1 ) 0 .9 5 0 .8 5 0 .7 3 0 .6 8 f PO =10 3 .0 0 .9 6 0 .7 9 0 .6 7 0 .5 7 0 .5 4 0 .9 2 0 .7 5 0 .6 9 0 .5 9 0 .5 5 0 .3 1 0 .6 1 0 .5 6 0 .4 8 0 .4 5 1 .5 0 .9 6 (1 ) 0 .7 9 (1 ) 0 .8 4 0 .7 3 0 .6 8 0 .9 6 (1 ) 0 .7 9 (1 ) 0 .8 5 0 .7 3 0 .6 8 0 .9 7 (1 ) 0 .8 0 (1 ) 0 .8 6 0 .7 4 0 .6 9 2 0 .8 9 (1 ) 0 .8 8 0 .7 7 0 .6 6 0 .6 2 0 .8 9 (1 ) 0 .8 9 0 .7 8 0 .6 7 0 .6 2 0 .9 0 (1 ) 0 .9 0 0 .7 9 0 .6 8 0 .6 3 f PO =50 3 .0 0 .9 2 0 .7 6 0 .6 4 0 .5 5 0 .5 2 0 .9 3 0 .7 6 0 .6 5 0 .5 6 0 .5 2 0 .9 5 0 .7 8 0 .6 6 0 .5 7 0 .5 3 (1 ) F o r th es e fa ct o rs Ȗ ǹ = Ȗ P B= 1 a n d Ȗ O = 0 .9 279 T ab le C -1 7 a: ( C o n ti n u ed ) C ar b o n S te el f O = 2 .5 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 1 .5 0 .8 2 (1 ) 0 .6 7 (1 ) 0 .9 2 0 .7 9 0 .7 4 2 0 .6 0 (1 ) 0 .7 3 0 .6 7 0 .5 8 0 .5 4 f PO = 0.5 3 .0 0 .3 6 0 .3 0 0 .2 7 0 .2 4 0 .2 2 1 .5 0 .8 5 (1 ) 0 .7 0 (1 ) 0 .9 3 0 .8 0 0 .7 5 2 0 .6 3 (1 ) 0 .7 6 0 .6 9 0 .6 0 0 .5 6 f PO =1 3 .0 0 .3 9 0 .3 2 0 .2 9 0 .2 5 0 .2 3 1 .5 0 .9 2 (1 ) 0 .7 6 (1 ) 0 .9 5 0 .8 2 0 .7 7 2 0 .7 3 (1 ) 0 .8 3 0 .7 6 0 .6 5 0 .6 1 f PO =3 3 .0 0 .4 7 0 .3 9 0 .3 6 0 .3 1 0 .2 9 1 .5 0 .9 6 (1 ) 0 .7 9 (1 ) 0 .9 6 0 .8 3 0 .7 7 2 0 .7 9 (1 ) 0 .8 8 0 .8 0 0 .6 9 0 .6 4 f PO =5 3 .0 0 .5 5 0 .4 5 0 .4 1 0 .3 6 0 .3 3 1 .5 1 .0 0 (1 ) 0 .8 2 (1 ) 0 .9 5 0 .8 2 0 .7 6 2 0 .8 8 (1 ) 0 .9 4 0 .8 4 0 .7 3 0 .6 8 f PO =10 3 .0 0 .6 9 0 .5 7 0 .5 1 0 .4 4 0 .4 1 1 .5 0 .9 7 (1 ) 0 .8 0 (1 ) 0 .8 7 0 .7 5 0 .7 0 2 0 .9 0 (1 ) 0 .9 1 0 .7 9 0 .6 8 0 .6 4 f PO =50 3 .0 0 .9 5 0 .7 8 0 .6 6 0 .5 7 0 .5 3 (1 ) F o r th es e fa ct o rs Ȗ ǹ = Ȗ P B = 1 a n d Ȗ O = 0 .9 280 T ab le C -1 7 b : A d ju st ed N o m in al R es is ta n ce F ac to r fo r g 8 a n d S ta in le ss S te el f o r Ȗ ǹ = 1 .1 , Ȗ P O = 1 .2 a n d Ȗ O = 1 .5 S ta in le ss S te el f O = 0 .5 f O = 1 .0 f O = 2 .0 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 1 .5 1 .0 2 (1 ) 0 .8 9 (1 ) 0 .9 3 0 .8 3 0 .7 8 0 .9 7 0 .8 4 0 .9 5 0 .8 4 0 .7 9 0 .9 1 0 .7 9 0 .9 4 0 .8 4 0 .7 9 2 0 .8 7 (1 ) 0 .9 7 0 .8 0 0 .7 1 0 .6 6 0 .7 6 0 .9 1 0 .7 5 0 .6 6 0 .6 2 0 .6 7 0 .8 6 0 .7 0 0 .6 2 0 .5 8 f PO = 0.5 3 .0 0 .6 1 0 .5 3 0 .4 4 0 .3 9 0 .3 6 0 .4 9 0 .4 3 0 .3 5 0 .3 1 0 .2 9 0 .4 1 0 .3 6 0 .2 9 0 .2 6 0 .2 4 1 .5 1 .0 4 (1 ) 0 .9 1 (1 ) 0 .9 4 0 .8 4 0 .7 8 1 .0 0 0 .8 7 0 .9 6 0 .8 5 0 .8 0 0 .9 4 0 .8 2 0 .9 5 0 .8 5 0 .8 0 2 0 .9 1 (1 ) 1 .0 1 0 .8 2 0 .7 3 0 .6 9 0 .8 1 0 .9 5 0 .7 8 0 .6 9 0 .6 5 0 .7 1 0 .8 9 0 .7 2 0 .6 4 0 .6 0 f PO =1 3 .0 0 .6 8 0 .6 0 0 .4 9 0 .4 3 0 .4 1 0 .5 4 0 .4 7 0 .3 9 0 .3 4 0 .3 2 0 .4 4 0 .3 8 0 .3 1 0 .2 8 0 .2 6 1 .5 1 .0 3 (1 ) 0 .9 0 (1 ) 0 .9 1 0 .8 1 0 .7 6 1 .0 4 0 .9 1 0 .9 6 0 .8 5 0 .8 0 1 .0 0 0 .8 8 0 .9 7 0 .8 6 0 .8 1 2 0 .9 4 (1 ) 1 .0 2 0 .8 3 0 .7 4 0 .6 9 0 .9 1 1 .0 3 0 .8 4 0 .7 5 0 .7 0 0 .8 2 0 .9 7 0 .7 9 0 .7 0 0 .6 6 f PO =3 3 .0 0 .8 8 0 .7 6 0 .6 3 0 .5 6 0 .5 2 0 .7 0 0 .6 1 0 .5 0 0 .4 5 0 .4 2 0 .5 5 0 .4 8 0 .3 9 0 .3 5 0 .3 3 1 .5 1 .0 2 (1 ) 0 .8 9 (1 ) 0 .8 9 0 .7 9 0 .7 5 1 .0 4 0 .9 1 0 .9 4 0 .8 3 0 .7 8 1 .0 3 0 .9 0 0 .9 7 0 .8 6 0 .8 1 2 0 .9 3 (1 ) 1 .0 0 .8 2 0 .7 2 0 .6 8 0 .9 4 1 .0 4 0 .8 5 0 .7 5 0 .7 1 0 .8 8 1 .0 1 0 .8 3 0 .7 4 0 .6 9 f PO =5 3 .0 0 .9 5 0 .8 3 0 .6 8 0 .6 0 0 .5 7 0 .8 1 0 .7 1 0 .5 8 0 .5 1 0 .4 8 0 .6 4 0 .5 6 0 .4 6 0 .4 0 0 .3 8 1 .5 1 .0 1 (1 ) 0 .8 8 (1 ) 0 .8 7 0 .7 8 0 .7 3 1 .0 2 0 .8 9 0 .9 0 0 .8 0 0 .7 5 1 .0 4 0 .9 1 0 .9 4 0 .8 4 0 .7 9 2 0 .9 1 (1 ) 0 .9 7 0 .7 9 0 .7 1 0 .6 6 0 .9 3 1 .0 0 0 .8 2 0 .7 3 0 .6 8 0 .9 4 1 .0 4 0 .8 5 0 .7 6 0 .7 1 f PO =10 3 .0 0 .9 2 0 .8 0 0 .6 6 0 .5 8 0 .5 5 0 .9 5 0 .8 3 0 .6 8 0 .6 0 0 .5 7 0 .7 9 0 .6 9 0 .5 7 0 .5 0 0 .4 7 1 .5 0 .9 9 (1 ) 0 .8 6 (1 ) 0 .8 5 0 .7 6 0 .7 1 0 .9 9 0 .8 7 0 .8 6 0 .7 6 0 .7 1 1 .0 0 0 .8 7 0 .8 7 0 .7 7 0 .7 3 2 0 .9 0 (1 ) 0 .9 4 0 .7 7 0 .6 9 0 .6 4 0 .9 0 0 .9 5 0 .7 8 0 .6 9 0 .6 5 0 .9 1 0 .9 6 0 .7 9 0 .7 0 0 .6 6 f PO =50 3 .0 0 .8 9 0 .7 7 0 .6 3 0 .5 6 0 .5 3 0 .9 0 0 .7 8 0 .6 4 0 .5 7 0 .5 3 0 .9 1 0 .7 9 0 .6 5 0 .5 8 0 .5 4 (1 ) F o r th es e fa ct o rs Ȗ ǹ = Ȗ P B= 1 a n d Ȗ O = 0 .9 281 T ab le C -1 7 b : (C o n ti n u ed ) S ta in le ss S te el f O = 2 .5 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 1 .5 0 .8 9 (1 ) 0 .7 8 (1 ) 0 .9 4 0 .8 4 0 .7 9 2 0 .6 5 (1 ) 0 .8 4 0 .6 9 0 .6 1 0 .5 7 f PO = 0.5 3 .0 0 .3 9 0 .3 4 0 .2 8 0 .2 5 0 .2 3 1 .5 0 .9 1 (1 ) 0 .8 0 (1 ) 0 .9 5 0 .8 5 0 .7 9 2 0 .6 8 (1 ) 0 .8 7 0 .7 1 0 .6 3 0 .5 9 f PO =1 3 .0 0 .4 2 0 .3 6 0 .3 0 0 .2 7 0 .2 5 1 .5 0 .9 8 (1 ) 0 .8 6 (1 ) 0 .9 7 0 .8 6 0 .8 1 2 0 .7 8 (1 ) 0 .9 4 0 .7 7 0 .6 9 0 .6 4 f PO =3 3 .0 0 .5 1 0 .4 5 0 .3 7 0 .3 2 0 .3 0 1 .5 1 .0 2 (1 ) 0 .8 9 (1 ) 0 .9 8 0 .8 7 0 .8 1 2 0 .8 5 (1 ) 0 .9 9 0 .8 1 0 .7 2 0 .6 8 f PO =5 3 .0 0 .5 9 0 .5 1 0 .4 2 0 .3 7 0 .3 5 1 .5 1 .0 4 (1 ) 0 .9 1 (1 ) 0 .9 6 0 .8 5 0 .8 0 2 0 .9 3 (1 ) 1 .0 4 0 .8 5 0 .7 6 0 .7 1 f PO =10 3 .0 0 .7 4 0 .6 4 0 .5 3 0 .4 7 0 .4 4 1 .5 1 .0 0 (1 ) 0 .8 8 (1 ) 0 .8 8 0 .7 8 0 .7 3 2 0 .9 1 (1 ) 0 .9 7 0 .8 0 0 .7 1 0 .6 6 f PO =50 3 .0 0 .9 2 0 .8 0 0 .6 6 0 .5 8 0 .5 5 (1 ) Ȗ ǹ = Ȗ P B= 1 a n d Ȗ O = 0 .9 282 283 C.8. Performance Function g9 Table C-18 gives the calculated mean load and resistance factors for performance function g9. In this table, ȝfy is the converged mean value of the yield strength of steel. Table C-19 provides the evaluated adjusted nominal resistance factors for nominal load factors Ȗǹ=1.1, ȖPO=1.2, ȖM=1.2, and ȖO=1.5. T ab le C -1 8 a: M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T ”2 0 0 o F f o r g 9 C ar b o n S te el , T” 2 0 0 o F ȕ= 1 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o 0 .5 3 .0 2 0 .9 5 5 1 .0 2 3 1 .0 1 2 1 .0 1 5 1 .6 9 0 3 .5 6 0 .9 5 1 .0 2 0 .0 1 1 .0 5 1 .5 9 4 .7 1 0 .9 3 1 .0 2 1 .0 1 1 .1 2 1 .2 4 1 3 .9 3 0 .9 7 0 1 .0 1 1 1 .0 0 1 1 .0 0 1 1 .8 0 1 4 .4 5 0 .9 7 1 .0 1 1 .0 0 1 .0 1 1 .7 8 5 .5 2 0 .9 6 1 .0 1 1 .0 0 1 .0 4 1 .6 9 2 5 .8 0 0 .9 7 7 1 .0 0 6 0 .9 9 6 0 .9 9 5 1 .8 3 4 6 .3 2 0 .9 8 1 .0 1 1 .0 0 1 .0 0 1 .8 3 7 .3 5 0 .9 7 1 .0 1 1 .0 0 1 .0 1 1 .8 0 0.5 2 .5 6 .7 4 0 .9 7 8 1 .0 0 4 0 .9 9 5 0 .9 9 4 1 .8 3 9 7 .2 5 0 .9 8 1 .0 0 1 .0 0 1 .0 0 1 .8 3 8 .2 8 0 .9 7 1 .0 0 1 .0 0 1 .0 1 1 .8 2 0 .5 3 .5 6 0 .9 4 7 1 .0 2 4 1 .0 3 4 1 .0 2 1 .6 0 4 .1 0 0 .9 4 1 .0 2 1 .0 3 1 .0 5 1 .4 9 5 .2 6 0 .9 3 1 .0 2 1 .0 3 1 .1 1 1 .1 8 1 4 .4 5 0 .9 7 1 .0 1 1 .0 1 1 .0 0 1 .7 8 4 .9 8 0 .9 6 1 .0 1 1 .0 1 1 .0 1 1 .7 5 6 .0 5 0 .9 5 1 .0 1 1 .0 1 1 .0 4 1 .6 6 2 6 .3 2 0 .9 8 1 .0 1 1 .0 0 1 .0 0 1 .8 3 6 .8 3 0 .9 7 1 .0 1 1 .0 0 1 .0 0 1 .8 2 7 .8 7 0 .9 7 1 .0 1 1 .0 0 1 .0 1 1 .8 0 1 2 .5 7 .2 5 0 .9 8 1 .0 0 1 .0 0 0 .9 9 1 .8 3 7 .7 6 0 .9 8 1 .0 0 1 .0 0 1 .0 0 1 .8 3 8 .7 9 0 .9 7 1 .0 0 1 .0 0 1 .0 1 1 .8 1 0 .5 5 .8 8 0 .9 2 1 .0 2 1 .1 2 1 .0 1 0 .6 8 6 .4 3 0 .9 2 1 .0 2 1 .1 1 1 .0 4 1 .0 5 7 .5 9 0 .9 2 1 .0 2 1 .1 0 1 .0 8 0 .9 9 1 6 .6 1 0 .9 5 1 .0 1 1 .0 6 1 .0 0 1 .5 9 7 .1 4 0 .9 5 1 .0 1 1 .0 6 1 .0 2 1 .5 6 8 .2 4 0 .9 4 1 .0 1 1 .0 6 1 .0 5 1 .4 4 2 8 .4 0 0 .9 7 1 .0 1 1 .0 2 1 .0 0 1 .7 8 8 .9 1 0 .9 7 1 .0 1 1 .0 2 1 .0 0 1 .7 7 9 .9 6 0 .9 6 1 .0 1 1 .0 2 1 .0 2 1 .7 4 3 2 .5 9 .3 2 0 .9 7 1 .0 0 1 .0 1 0 .9 9 1 .8 0 9 .8 3 0 .9 7 1 .0 0 1 .0 1 1 .0 0 1 .8 0 1 0 .8 7 0 .9 7 1 .0 0 1 .0 2 1 .0 1 1 .7 8 0 .5 8 .3 4 0 .9 3 1 .0 2 1 .1 5 1 .0 1 0 .9 4 8 .8 8 0 .9 2 1 .0 1 1 .1 4 1 .0 2 0 .9 3 1 0 .0 1 0 .9 2 1 .0 1 1 .1 3 1 .0 5 0 .9 2 1 8 .9 1 0 .9 3 1 .0 1 1 .1 1 1 .0 0 1 .2 5 9 .4 5 0 .9 3 1 .0 1 1 .1 1 1 .0 2 1 .2 2 1 0 .5 6 0 .9 3 1 .0 1 1 .1 1 1 .0 4 1 .1 6 2 1 0 .5 4 0 .9 6 1 .0 1 1 .0 4 1 .0 0 1 .7 0 1 1 .0 6 0 .9 6 1 .0 1 1 .0 4 1 .0 0 1 .6 8 1 2 .1 1 0 .9 5 1 .0 1 1 .0 4 1 .0 2 1 .6 5 5 2 .5 1 1 .4 3 0 .9 7 1 .0 0 1 .0 3 0 .9 9 1 .7 5 1 1 .9 5 0 .9 6 1 .0 0 1 .0 3 1 .0 0 1 .7 4 1 3 .0 0 0 .9 6 1 .0 0 1 .0 3 1 .0 1 1 .7 2 0 .5 1 4 .5 6 0 .9 3 1 .0 1 1 .1 6 1 .0 0 0 .8 7 1 5 .1 0 0 .9 3 1 .0 1 1 .1 6 1 .0 1 0 .8 7 1 6 .1 9 0 .9 3 1 .0 1 1 .1 5 1 .0 2 0 .8 6 1 1 5 .0 4 0 .9 3 1 .0 1 1 .1 5 1 .0 0 0 .9 5 1 5 .5 8 0 .9 3 1 .0 1 1 .1 5 1 .0 1 0 .9 4 1 6 .6 8 0 .9 3 1 .0 1 1 .1 4 1 .0 2 0 .9 4 2 1 6 .2 1 0 .9 4 1 .0 1 1 .1 1 1 .0 0 1 .2 9 1 6 .7 5 0 .9 4 1 .0 1 1 .1 1 1 .0 0 1 .2 8 1 7 .8 3 0 .9 3 1 .0 1 1 .1 1 1 .0 2 1 .2 5 10 2 .5 1 6 .9 6 0 .9 5 1 .0 0 1 .0 8 0 .9 9 1 .4 9 1 7 .4 8 0 .9 5 1 .0 0 1 .0 8 1 .0 0 1 .4 8 1 8 .5 3 0 .9 4 1 .0 0 1 .0 8 1 .0 1 1 .4 5 0 .5 6 4 .5 3 0 .9 3 1 .0 0 1 .1 7 0 .9 9 0 .8 2 6 5 .0 6 0 .9 3 1 .0 0 1 .1 7 0 .9 9 0 .8 2 6 6 .1 2 0 .9 3 1 .0 0 1 .1 7 1 .0 0 0 .8 2 1 6 4 .9 7 0 .9 3 1 .0 0 1 .1 7 0 .9 9 0 .8 3 6 5 .5 0 0 .9 3 1 .0 0 1 .1 7 0 .9 9 0 .8 3 6 6 .5 6 0 .9 3 1 .0 0 1 .1 7 1 .0 0 0 .8 3 2 6 5 .8 7 0 .9 3 1 .0 0 1 .1 7 0 .9 9 0 .8 6 6 6 .4 0 0 .9 3 1 .0 0 1 .1 7 0 .9 9 0 .8 6 6 7 .4 7 0 .9 3 1 .0 0 1 .1 6 1 .0 0 0 .8 6 f Po 50 2 .5 6 6 .3 3 0 .9 3 1 .0 0 1 .1 7 0 .9 9 0 .8 7 6 6 .8 6 0 .9 3 1 .0 0 1 .1 6 0 .9 9 0 .8 7 6 7 .9 3 0 .9 3 1 .0 0 1 .1 6 1 .0 0 0 .8 7 284 T ab le C -1 8 a: ( C o n ti n u ed ) C ar b o n S te el , T” 2 0 0 o F ȕ= 2 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o 0 .5 4 .3 0 0 .9 7 1 .0 1 1 .0 0 1 .0 0 4 .3 0 4 .8 2 0 .9 6 1 .0 1 1 .0 0 1 .0 1 4 .2 4 5 .9 0 0 .9 5 1 .0 1 1 .0 0 1 .0 5 4 .0 6 1 6 .5 4 0 .9 7 1 .0 1 1 .0 0 1 .0 0 4 .3 7 7 .0 5 0 .9 7 1 .0 1 1 .0 0 1 .0 0 4 .3 5 8 .0 9 0 .9 7 1 .0 1 1 .0 0 1 .0 1 4 .3 1 2 1 1 .0 3 0 .9 8 1 .0 0 0 .9 9 0 .9 9 4 .4 0 1 1 .5 4 0 .9 8 1 .0 0 0 .9 9 1 .0 0 4 .3 9 1 2 .5 6 0 .9 8 1 .0 0 0 .9 9 1 .0 0 4 .3 7 0.5 2 .5 1 3 .2 8 0 .9 8 1 .0 0 0 .9 9 0 .9 9 4 .4 0 1 3 .7 8 0 .9 8 1 .0 0 0 .9 9 0 .9 9 4 .4 0 1 4 .8 0 0 .9 8 1 .0 0 0 .9 9 1 .0 0 4 .3 8 0 .5 4 .8 2 0 .9 6 1 .0 1 1 .0 1 1 .0 0 4 .2 5 5 .3 5 0 .9 6 1 .0 1 1 .0 1 1 .0 2 4 .1 9 6 4 3 0 .9 5 1 .0 1 1 .0 1 1 .0 5 3 .9 8 1 7 .0 5 0 .9 7 1 .0 1 1 .0 0 1 .0 0 4 .3 5 7 .5 6 0 .9 7 1 .0 1 1 .0 0 1 .0 0 4 .3 4 8 .6 1 0 .9 7 1 .0 1 1 .0 0 1 .0 1 4 .2 9 2 1 1 .5 4 0 .9 8 1 .0 0 1 .0 0 0 .9 9 4 .3 9 1 2 .0 5 0 .9 8 1 .0 0 1 .0 0 1 .0 0 4 .3 8 1 3 .0 7 0 .9 7 1 .0 0 1 .0 0 1 .0 0 4 .3 7 1 2 .5 1 3 .7 9 0 .9 8 1 .0 0 1 .0 0 0 .9 9 4 .4 0 1 4 .2 9 0 .9 8 1 .0 0 1 .0 0 0 .9 9 4 .3 9 1 5 .3 1 0 .9 8 1 .0 0 1 .0 0 1 .0 0 4 .3 8 0 .5 6 .9 9 0 .9 5 1 .0 1 1 .0 6 1 .0 0 3 .8 2 7 .5 3 0 .9 4 1 .0 1 1 .0 6 1 .0 2 3 .7 1 8 .6 4 0 .9 3 1 .0 1 1 .0 7 1 .0 6 3 .3 2 1 9 .1 4 0 .9 7 1 .0 1 1 .0 2 1 .0 0 4 .2 5 9 .6 6 0 .9 6 1 .0 1 1 .0 2 1 .0 0 4 .2 3 1 0 .7 0 0 .9 6 1 .0 1 1 .0 2 1 .0 2 4 .1 7 2 1 3 .5 9 0 .9 7 1 .0 0 1 .0 1 0 .9 9 4 .3 6 1 4 .1 0 0 .9 7 1 .0 0 1 .0 1 1 .0 0 4 .3 5 1 5 .1 3 0 .9 7 1 .0 0 1 .0 1 1 .0 0 4 .3 3 3 2 .5 1 5 .8 3 0 .9 8 1 .0 0 1 .0 0 0 .9 9 4 .3 7 1 6 .3 4 0 .9 7 1 .0 0 1 .0 0 0 .9 9 4 .3 7 1 7 .3 6 0 .9 7 1 .0 0 1 .0 0 1 .0 0 4 .3 5 0 .5 9 .5 0 0 .8 8 1 .0 2 1 .2 6 1 .0 1 1 .1 1 1 0 .0 8 0 .8 8 1 .0 2 1 .2 5 1 .0 4 1 .0 9 1 1 .3 0 0 .8 7 1 .0 2 1 .2 3 1 .0 9 1 .0 5 1 1 1 .2 9 0 .9 6 1 .0 1 1 .0 4 1 .0 0 4 .0 7 1 1 .8 1 0 .9 5 1 .0 1 1 .0 5 1 .0 0 4 .0 4 1 2 .8 7 0 .9 5 1 .0 1 1 .0 5 1 .0 2 3 .9 6 2 1 5 .6 7 0 .9 7 1 .0 0 1 .0 2 0 .9 9 4 .3 1 1 6 .1 8 0 .9 7 1 .0 0 1 .0 2 1 .0 0 4 .3 0 1 7 .2 2 0 .9 7 1 .0 0 1 .0 2 1 .0 0 4 .2 8 5 2 .5 1 7 .9 0 0 .9 7 1 .0 0 1 .0 1 0 .9 9 4 .3 4 1 8 .4 1 0 .9 7 1 .0 0 1 .0 1 0 .9 9 4 .3 0 1 9 .4 4 0 .9 7 1 .0 0 1 .0 1 1 .0 0 4 .3 2 0 .5 1 6 .7 1 0 .8 9 1 .0 1 1 .2 9 1 .0 0 0 .9 0 1 7 .2 8 0 .8 9 1 .0 1 1 .2 8 1 .0 2 0 .9 0 1 8 .4 5 0 .8 8 1 .0 1 1 .2 7 1 .0 4 0 .9 0 1 1 7 .2 7 0 .8 9 1 .0 1 1 .2 7 1 .0 0 1 .1 4 1 7 .8 4 0 .8 9 1 .0 1 1 .2 7 1 .0 2 1 .1 3 1 9 .0 0 0 .8 8 1 .0 1 1 .2 6 1 .0 4 1 .1 1 2 2 1 .0 1 0 .9 6 1 .0 0 1 .0 4 0 .9 9 4 .1 1 2 1 .5 3 0 .9 6 1 .0 0 1 .0 4 1 .0 0 4 .1 0 2 2 .5 7 0 .9 6 1 .0 0 1 .0 4 1 .0 0 4 .0 7 10 2 .5 2 3 .1 7 0 .9 6 1 .0 0 1 .0 3 0 .9 9 4 .2 1 2 3 .6 9 0 .9 6 1 .0 0 1 .0 3 0 .9 9 4 .2 0 2 4 .7 2 0 .9 6 1 .0 0 1 0 3 1 .0 0 4 .1 9 0 .5 7 4 .8 8 0 .9 0 1 .0 0 1 .3 0 0 .9 9 0 .8 3 7 5 .4 4 0 .9 0 1 .0 0 1 .3 0 0 .9 9 0 .8 3 7 6 .5 5 0 .8 9 1 .0 0 1 .3 0 1 .0 0 0 .8 3 1 7 5 .3 5 0 .9 0 1 .0 0 1 .3 0 0 .9 9 0 .8 4 7 5 .9 0 0 .8 9 1 .0 0 1 .3 0 0 .9 9 0 .8 4 7 7 .0 2 0 .8 9 1 .0 0 1 .3 0 1 .0 0 0 .8 4 2 7 6 .3 1 0 .8 9 1 .0 0 1 .3 0 0 .9 9 0 .8 8 7 6 .8 7 0 .8 9 1 .0 0 1 .3 0 0 .9 9 0 .8 8 7 7 .9 8 0 .8 9 1 .0 0 1 .3 0 1 .0 0 0 .8 8 f Po 50 2 .5 7 6 .8 1 0 .8 9 1 .0 0 1 .3 0 0 .9 9 0 .9 1 7 7 .3 7 0 .8 9 1 .0 0 1 .3 0 0 .9 9 0 .9 1 7 8 .4 8 0 .8 9 1 .0 0 1 .3 0 1 .0 0 0 .9 1 285 T ab le C -1 8 b : M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T > 2 0 0 o F f o r g 9 C ar b o n S te el , T> 2 0 0 o F ȕ= 1 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o 0 .5 3 .1 4 0 .8 8 1 .0 2 1 .0 1 1 .0 2 1 .4 8 3 .7 3 0 .8 7 1 .0 2 1 .0 1 1 .0 5 1 .3 1 4 .9 9 0 .8 5 1 .0 2 1 .0 1 1 .1 0 1 .0 6 1 4 .0 4 0 .9 2 1 .0 1 1 .0 0 1 .0 0 1 .7 2 4 .5 9 0 .9 1 1 .0 1 1 .0 0 1 .0 2 1 .6 6 5 .7 3 0 .8 9 1 .0 1 1 .0 0 1 .0 5 1 .5 1 2 5 .9 3 0 .9 4 1 .0 1 1 .0 0 1 .0 0 1 7 9 6 .4 6 0 .9 4 1 .0 1 1 .0 0 1 .0 0 1 .7 7 7 .5 5 0 .9 3 1 .0 1 1 .0 0 1 .0 2 1 .7 3 0.5 2 .5 6 .8 8 0 .9 4 1 .0 0 1 .0 4 0 .9 9 1 .8 0 7 .4 1 0 .9 4 1 .0 0 1 .0 0 1 .0 0 1 .7 9 8 .4 8 0 .9 3 1 .0 0 1 .0 0 1 .0 1 1 .7 6 0 .5 3 .7 3 0 .8 7 1 .0 2 1 .0 1 1 .0 2 1 .3 3 4 .3 3 0 .8 5 1 .0 2 1 .0 3 1 .0 5 1 .1 9 5 .5 9 0 .8 5 1 .0 2 1 .0 3 1 .0 9 1 .0 2 1 4 .5 9 0 .9 1 1 .0 1 1 .0 0 1 .0 0 1 .6 7 5 .1 5 0 .9 0 1 .0 1 1 .0 1 1 .0 2 1 .6 1 6 .3 0 0 .8 8 1 .0 1 1 .0 1 1 .0 5 1 .4 5 2 6 .4 6 0 .9 4 1 .0 1 1 .0 0 1 .0 0 1 .7 7 7 .0 0 0 .9 3 1 .0 1 1 .0 0 1 .0 0 1 .7 5 8 .0 9 0 .9 2 1 .0 1 1 .0 0 1 .0 2 1 .7 1 1 2 .5 7 .4 1 0 .9 4 1 .0 0 1 .0 9 0 .9 9 1 .7 9 7 .9 4 0 .9 4 1 .0 0 1 .0 0 1 .0 0 1 .7 8 9 .0 2 0 .9 3 1 .0 0 1 .0 0 1 .0 1 1 .7 4 0 .5 6 .2 4 0 .8 5 1 .0 2 1 .0 6 1 .0 1 0 .9 8 6 .8 4 0 .8 4 1 .0 2 1 .0 9 1 .0 3 0 .9 6 8 .0 8 0 .8 4 1 .0 1 1 .0 7 1 .0 6 0 .9 3 1 6 .9 1 0 .8 7 1 .0 1 1 .0 2 1 .0 0 1 .3 5 7 .4 9 0 .8 7 1 .0 1 1 .0 6 1 .0 2 1 .2 6 8 .6 9 0 .8 6 1 .0 1 1 .0 6 1 .0 4 1 .1 6 2 8 .6 5 0 .9 2 1 .0 1 1 .0 2 1 .0 0 1 .6 8 9 .1 9 0 .9 1 1 .0 1 1 .0 2 1 .0 0 1 .6 6 1 0 .3 0 0 .9 0 1 .0 1 1 .0 2 1 .0 2 1 .6 0 3 2 .5 9 .5 7 0 .9 3 1 .0 0 1 .1 1 0 .9 9 1 .7 3 1 0 .1 1 0 .9 2 1 .0 0 1 .0 2 1 .0 0 1 .7 1 1 1 .2 0 0 .9 1 1 .0 0 1 .0 2 1 .0 1 1 .6 7 0 .5 8 .8 4 0 .8 5 1 .0 1 1 .1 0 1 .0 0 0 .9 1 9 .4 3 0 .8 5 1 .0 1 1 .1 0 1 .0 2 0 .9 0 1 0 .6 4 0 .8 5 1 .0 1 1 .0 9 1 .0 4 0 .8 9 1 9 .4 1 0 .8 6 1 .0 1 1 .0 5 1 .0 0 1 .0 7 1 0 .0 0 0 .8 5 1 .0 1 1 .0 9 1 .0 1 1 .0 5 1 1 .2 0 0 .8 5 1 .0 1 1 .0 8 1 .0 3 1 .0 1 2 1 0 .9 2 0 .9 0 1 .0 1 1 .0 3 1 .0 0 1 .5 4 1 1 .4 8 0 .8 9 1 .0 1 1 .0 5 1 .0 0 1 .5 1 1 2 .6 2 0 .8 8 1 .0 1 1 .0 5 1 .0 2 1 .4 4 5 2 .5 1 1 .8 0 0 .9 1 1 .0 0 1 .1 2 0 .9 9 1 .6 3 1 2 .3 5 0 .9 1 1 .0 0 1 .0 3 1 .0 0 1 .6 1 1 3 .4 6 0 .9 0 1 .0 0 1 .0 3 1 .0 1 1 .5 6 0 .5 1 5 .3 9 0 .8 6 1 .0 1 1 .1 2 1 .0 0 0 .8 6 1 5 .9 7 0 .8 5 1 .0 1 1 .1 2 1 .0 0 0 .8 5 1 7 .1 6 0 .8 5 1 .0 1 1 .1 1 1 .0 2 0 .8 5 1 1 5 .9 0 0 .8 6 1 .0 1 1 .1 0 1 .0 0 0 .9 1 1 6 .4 9 0 .8 5 1 .0 1 1 .1 2 1 .0 0 0 .9 1 1 7 .6 7 0 .8 5 1 .0 1 1 .1 1 1 .0 2 0 .9 0 2 1 7 .0 7 0 .8 6 1 .0 1 1 .0 8 1 .0 0 1 .1 1 1 7 .6 5 0 .8 6 1 .0 1 1 .1 0 1 .0 0 1 .1 0 1 8 .8 2 0 .8 6 1 .0 1 1 .1 0 1 .0 1 1 .0 7 10 2 .5 1 7 .7 5 0 .8 7 1 .0 0 1 .1 3 0 .9 9 1 .2 6 1 8 .3 2 0 .8 7 1 .0 0 1 .0 8 1 .0 0 1 .2 4 1 9 .4 8 0 .8 7 1 .0 0 1 .0 8 1 .0 1 1 .2 0 0 .5 6 8 .0 1 0 .8 6 1 .0 0 1 .1 3 0 .9 9 0 .8 2 6 8 .5 8 0 .8 6 1 .0 0 1 .1 3 0 .9 9 0 .8 2 6 9 .7 3 0 .8 6 1 .0 0 1 .1 3 1 .0 0 0 .8 2 1 6 8 .4 8 0 .8 6 1 .0 0 1 .1 3 0 .9 9 0 .8 3 6 9 .0 6 0 .8 6 1 .0 0 1 .1 3 0 .9 9 0 .8 3 7 0 .2 1 0 .8 6 1 .0 0 1 .1 3 1 .0 0 0 .8 3 2 6 9 .4 6 0 .8 6 1 .0 0 1 .1 3 0 .9 9 0 .8 5 7 0 .0 3 0 .8 6 1 .0 0 1 .1 3 0 .9 9 0 .8 5 7 1 .1 9 0 .8 6 1 .0 0 1 .1 3 1 .0 0 0 .8 5 f Po 50 2 .5 6 9 .9 5 0 .8 6 1 .0 0 1 .1 3 0 .9 9 0 .8 6 7 0 .5 3 0 .8 6 1 .0 0 1 .1 3 0 .9 9 0 .8 6 7 1 .6 8 0 .8 6 1 .0 0 1 .1 3 0 .9 9 0 .8 6 286 T b le C -1 8 b : (C o n ti n u ed ) C ar b o n S te el , T> 2 0 0 o F ȕ= 2 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o 0 .5 4 .4 3 0 .9 1 1 .0 1 1 .0 0 1 .0 0 4 .0 8 4 .9 9 0 .9 0 1 .0 1 1 .0 0 1 .0 2 3 .9 6 6 .1 5 0 .8 8 1 .0 1 1 .0 0 1 .0 5 3 .5 5 1 6 .7 0 0 .9 3 1 .0 1 1 .0 0 1 .0 0 4 .2 5 7 .2 3 0 .9 3 1 .0 1 1 .0 0 1 .0 0 4 .2 1 8 .3 2 0 .9 2 1 .0 1 1 .0 0 1 .0 2 4 .1 2 2 1 1 .2 6 0 .9 4 1 .0 0 0 .9 9 0 .9 9 4 .3 1 1 1 .7 9 0 .9 4 1 .0 0 0 .9 9 1 .0 0 4 .3 0 1 2 .8 5 0 .9 4 1 .0 0 0 .9 9 1 .0 0 4 .2 6 0.5 2 .5 1 3 .5 5 0 .9 4 1 .0 0 0 .9 9 0 .9 9 4 .3 2 1 4 .0 7 0 .9 4 1 .0 0 0 .9 9 0 .9 9 4 .3 1 1 5 .1 3 0 .9 4 1 .0 0 0 .9 9 1 .0 0 4 .2 9 0 .5 4 .9 9 0 .9 0 1 .0 1 1 .0 1 1 .0 0 3 .9 7 5 .5 5 0 .8 9 1 .0 1 1 .0 1 1 .0 2 3 .8 2 6 .7 3 0 .8 6 1 .0 1 1 .0 2 1 .0 6 3 .3 2 1 7 .2 3 0 .9 3 1 .0 1 1 .0 0 1 .0 0 4 .2 1 7 .7 7 0 .9 2 1 .0 1 1 .0 0 1 .0 0 4 .1 7 8 .8 7 0 .9 1 1 .0 1 1 .0 0 1 .0 2 4 .0 7 2 1 1 .7 9 0 .9 4 1 .0 0 1 .0 0 0 .9 9 4 .3 0 1 2 .3 2 0 .9 4 1 .0 0 1 .0 0 1 .0 0 4 .2 8 1 3 .3 8 0 .9 3 1 .0 0 1 .0 0 1 .0 0 4 .2 5 1 2 .5 1 4 .0 7 0 .9 4 1 .0 0 1 .0 0 0 .9 9 4 .3 1 1 4 .6 0 0 .9 4 1 .0 0 1 .0 0 0 .9 9 4 .3 0 1 5 .6 6 0 .9 4 1 .0 0 1 .0 0 1 .0 0 4 .2 8 0 .5 7 .3 7 0 .8 3 1 .0 2 1 .0 9 1 .0 1 2 .6 8 8 .0 2 0 .7 7 1 .0 2 1 .1 4 1 .0 5 1 .3 1 9 .4 3 0 .7 6 1 .0 2 1 .1 3 1 .1 0 1 .1 0 1 9 .4 3 0 .9 1 1 .0 1 1 .0 2 1 .0 1 4 .0 0 9 .9 8 0 .9 0 1 .0 1 1 .0 2 1 .0 0 3 .9 4 1 1 .1 1 0 .8 9 1 .0 1 1 .0 2 1 .0 2 3 .8 0 2 1 3 .9 3 0 .9 3 1 .0 0 1 .0 1 0 .9 9 4 .2 3 1 4 .4 6 0 .9 3 1 .0 0 1 .0 1 1 .0 0 4 .2 1 1 5 .5 4 0 .9 2 1 .0 0 1 .0 1 1 .0 0 4 .1 7 3 2 .5 1 6 .2 0 0 .9 4 1 .0 0 1 .0 0 0 .9 9 4 .2 6 1 6 .7 3 0 .9 3 1 .0 0 1 .0 0 0 .9 9 4 .2 5 1 7 .8 0 0 .9 3 1 .0 0 1 .0 0 1 .0 0 4 .2 2 0 .5 1 0 .4 2 0 .7 7 1 .0 2 1 .2 0 1 .0 1 1 .0 0 1 1 .0 8 0 .7 7 1 .0 2 1 .1 9 1 .0 3 0 .9 9 1 2 .4 6 0 .7 6 1 .0 2 1 .1 7 1 .0 7 0 .9 6 1 1 1 .7 4 0 .8 8 1 .0 1 1 .0 5 1 .0 0 3 .6 2 1 2 .3 1 0 .8 8 1 .0 1 1 .0 5 1 .0 0 3 .5 2 1 3 .4 8 0 .8 6 1 .0 1 1 .0 6 1 .0 2 0 .2 7 2 1 6 .1 1 0 .9 2 1 .0 0 1 .0 2 0 .9 9 4 .1 3 1 6 .6 5 0 .9 2 1 .0 0 1 .0 2 1 .0 0 4 .1 1 1 7 .7 4 0 .9 1 1 .0 0 1 .0 2 1 .0 0 4 .0 6 5 2 .5 1 8 .3 6 0 .9 3 1 .0 0 1 .0 1 0 .9 9 4 .2 0 1 8 .9 0 0 .9 3 1 .0 0 1 .0 1 0 .9 9 4 .1 8 1 9 .9 8 0 .9 2 1 .0 0 1 .0 1 1 .0 0 4 .1 5 0 .5 1 8 .2 7 0 .7 8 1 .0 1 1 .2 2 1 .0 0 0 .8 9 1 8 .9 1 0 .7 7 1 .0 1 1 .2 2 1 .0 1 0 .8 8 2 0 .2 4 0 .7 7 1 .0 1 1 .2 1 1 .0 3 0 .8 8 1 1 8 .8 7 0 .7 8 1 .0 1 1 .2 1 1 .0 0 1 .0 2 1 9 .5 2 0 .7 7 1 .0 1 1 .2 1 1 .0 1 1 .0 2 2 0 .8 5 0 .7 7 1 .0 1 1 .2 0 1 .0 3 1 .0 0 2 2 1 .8 0 0 .8 9 1 .0 0 1 .0 5 0 .9 9 3 .7 3 2 2 .3 5 0 .8 9 1 .0 0 1 .0 5 1 .0 0 3 .7 0 2 3 .4 8 0 .8 8 1 .0 0 1 .0 5 1 .0 0 3 .6 1 10 2 .5 2 3 .9 3 0 .9 1 1 .0 0 1 .0 3 0 .9 9 3 .9 5 2 4 .4 8 0 .9 0 1 .0 0 1 .0 3 0 .9 9 0 .9 2 2 5 .5 9 0 .9 0 1 .0 0 1 .0 4 1 .0 0 3 .8 8 0 .5 8 1 .4 4 0 .7 9 1 .0 0 1 .2 4 0 .9 9 0 .8 2 8 2 .0 7 0 .7 9 1 .0 0 1 .2 4 0 .9 9 0 .8 2 8 3 .3 4 0 .7 8 1 .0 0 1 .2 4 1 .0 0 0 .8 2 1 8 1 .9 7 0 .7 9 1 .0 0 1 .2 4 0 .9 9 0 .8 4 8 2 .6 0 0 .7 8 1 .0 0 1 .2 4 0 .9 9 0 .8 4 8 3 .8 7 0 .7 8 1 .0 0 1 .2 4 1 .0 0 0 .8 4 2 8 3 .0 6 0 .7 8 1 .0 0 1 .2 4 0 .9 9 0 .8 7 8 3 .6 9 0 .7 8 1 .0 0 1 .2 4 0 .9 9 0 .8 7 8 4 .9 6 0 .7 8 1 .0 0 1 .2 4 1 .0 0 0 .8 7 f Po 50 2 .5 8 3 .6 2 0 .7 8 1 .0 0 1 .2 4 0 .9 9 0 .8 9 8 4 .2 5 0 .7 8 1 .0 0 1 .2 4 0 .9 9 0 .8 9 8 5 .5 2 0 .7 8 1 .0 0 1 .2 3 1 .0 0 0 .8 9 287 T ab le C -1 8 c: M ea n L o ad a n d R es is ta n ce F ac to rs f o r S ta in le ss S te el a n d a n y T f o r g 9 S ta in le ss S te el f o r an y T ȕ= 1 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o 0 .5 3 .2 1 0 .8 5 1 .0 2 1 .0 1 1 .0 2 1 .3 8 3 .8 2 0 .8 3 1 .0 2 1 .0 1 1 .0 5 1 .2 2 5 .1 2 0 .8 2 1 .0 2 1 .0 1 1 .0 9 1 .0 2 1 4 .1 0 0 .9 0 1 .0 1 1 .0 0 1 .0 0 1 .6 7 4 .6 7 0 .8 9 1 .0 1 1 .0 0 1 .0 2 1 .6 1 5 .8 5 0 .8 6 1 .0 1 1 .0 0 1 .0 5 1 .4 3 2 6 .0 0 0 .9 2 1 .0 1 1 .0 0 1 .0 0 1 .7 6 6 .5 4 0 .9 2 1 .0 1 1 .0 0 1 .0 0 1 .7 4 7 .6 5 0 .9 0 1 .0 1 1 .0 0 1 .0 2 1 .6 9 0.5 2 .5 6 .9 6 0 .9 3 1 .0 0 1 .0 0 0 .9 9 1 .7 8 7 .5 0 0 .9 2 1 .0 0 1 .0 0 1 .0 0 1 .7 6 8 .5 9 0 .9 1 1 .0 0 1 .0 0 1 .0 1 1 .7 3 0 .5 3 .8 2 0 .8 3 1 .0 2 1 .0 3 1 .0 2 1 .2 3 4 .4 4 0 .8 2 1 .0 2 1 .0 3 1 .0 4 1 .1 1 5 .7 4 0 .8 2 1 .0 2 1 .0 2 1 .0 8 0 .9 9 1 4 .6 7 0 .8 9 1 .0 1 1 .0 1 1 .0 0 1 .6 1 5 .2 4 0 .8 7 1 .0 1 1 .0 1 1 .0 2 1 .5 4 6 .4 4 0 .8 5 1 .0 1 1 .0 1 1 .0 5 1 .3 5 2 6 .5 4 0 .9 2 1 .0 1 1 .0 0 1 .0 0 1 .7 4 7 .0 9 0 .9 1 1 .0 1 1 .0 0 1 .0 0 1 .7 2 8 .2 1 0 .9 0 1 .0 1 1 .0 0 1 .0 2 1 .6 6 1 2 .5 7 .5 0 0 .9 2 1 .0 0 1 .0 0 0 .9 9 1 .7 6 8 .0 4 0 .9 2 1 .0 0 1 .0 0 1 .0 0 1 .7 5 9 .1 4 0 .9 1 1 .0 0 1 .0 0 1 .0 1 1 .7 1 0 .5 6 .4 1 0 .8 2 1 .0 2 1 .0 8 1 .0 1 0 .9 6 7 .0 3 0 .8 2 1 .0 2 1 .0 8 1 .0 2 0 .9 4 8 .3 1 0 .8 1 1 .0 1 1 .0 6 1 .0 5 0 .9 1 1 7 .0 7 0 .8 4 1 .0 1 1 .0 6 1 .0 0 1 .2 5 7 .6 7 0 .8 3 1 .0 1 1 .0 6 1 .0 2 1 .1 9 8 .9 2 0 .8 2 1 .0 1 1 .0 5 1 .0 4 1 .1 0 2 8 .7 8 0 .8 9 1 .0 1 1 .0 2 1 .0 0 1 .6 3 9 .3 4 0 .8 9 1 .0 1 1 .0 2 1 .0 0 1 .6 0 1 0 .4 9 0 .8 7 1 .0 1 1 .0 2 1 .0 2 1 .5 3 3 2 .5 9 .7 1 0 .9 0 1 .0 0 1 .0 2 0 .9 9 1 .6 8 1 0 .2 6 0 .9 0 1 .0 0 1 .0 2 1 .0 0 1 .6 6 1 1 .3 8 0 .8 9 1 .0 0 1 .0 2 1 .0 1 1 .6 2 0 .5 9 .0 8 0 .8 2 1 .0 1 1 .1 0 1 .0 0 0 .8 9 9 .6 9 0 .8 2 1 .. 0 1 1 .0 9 1 .0 1 0 .8 9 1 0 .9 4 0 .8 2 1 .0 1 1 .0 8 1 .0 3 0 .8 8 1 9 .6 6 0 .8 3 1 .0 1 1 .0 9 1 .0 0 1 .0 3 1 0 .2 7 0 .8 2 1 .0 1 1 .0 8 1 .0 1 1 .0 1 1 1 .5 1 0 .8 2 1 .0 1 1 .0 8 1 .0 3 0 .9 8 2 1 1 .1 4 0 .8 7 1 .0 1 1 .0 5 1 .0 0 1 .4 6 1 1 .7 1 0 .8 6 1 .0 1 1 .0 5 1 .0 0 1 .4 2 1 2 .8 9 0 .8 5 1 .0 1 1 .0 5 1 .0 2 1 .3 4 5 2 .5 1 2 .0 0 0 .8 8 1 .0 0 1 .0 3 0 .9 9 1 .5 7 1 2 .5 7 0 .8 8 1 .0 0 1 .0 3 1 .0 0 1 .5 4 1 3 .7 2 0 .8 7 1 .0 0 1 .0 3 1 .0 1 1 .4 9 0 .5 1 5 .7 9 0 .8 3 1 .0 1 1 .1 1 1 .0 0 0 .8 5 1 6 .4 0 0 .8 2 1 .0 1 1 .1 1 1 .0 0 0 .8 5 1 7 .6 2 0 .8 2 1 .0 1 1 .1 0 1 .0 1 0 .8 5 1 1 6 .3 2 0 .8 3 1 .0 1 1 .1 1 1 .0 0 0 .9 0 1 6 .9 3 0 .8 2 1 .0 1 1 .1 1 1 .0 0 0 .9 0 1 8 .1 5 0 .8 2 1 .0 1 1 .1 0 1 .0 1 0 .8 9 2 1 7 .5 0 0 .8 3 1 .0 1 1 .0 9 0 .9 9 1 .0 6 1 8 .1 0 0 .8 3 1 .0 1 1 .0 9 1 .0 0 1 .0 5 1 9 .3 1 0 .8 3 1 .0 1 1 .0 9 1 .0 1 1 .0 3 10 2 .5 1 8 .. 1 7 0 .8 4 1 .0 0 1 .0 8 0 .9 9 1 .1 8 1 8 .7 6 0 .8 4 1 .0 0 1 .0 8 1 .0 0 1 .1 7 1 9 .9 6 0 .8 3 1 .0 0 1 .0 8 1 .0 1 1 .1 3 0 .5 6 9 .7 1 0 .8 3 1 .0 0 1 .1 2 0 .9 9 0 .8 2 7 0 .3 0 0 .8 3 1 .0 0 1 .1 2 0 .9 9 0 .8 2 7 1 .5 0 0 .8 3 1 .0 0 1 .1 2 0 .9 9 0 .8 2 1 7 0 .2 0 0 .8 3 1 .0 0 1 .1 2 0 .9 9 0 .8 3 7 0 .8 0 0 .8 3 1 .0 0 1 .1 2 0 .9 9 0 .8 3 7 1 .9 9 0 .8 3 1 .0 0 1 .1 2 0 .9 9 0 .8 3 2 7 1 .2 1 0 .8 3 1 .0 0 1 .1 2 0 .9 9 0 .8 5 7 1 .8 0 0 .8 3 1 .0 0 1 .1 2 0 .9 9 0 .8 4 7 3 .0 0 0 .8 3 1 .0 0 1 .1 2 0 .9 9 0 .8 4 f Po 50 2 .5 7 1 .7 2 0 .8 3 1 .0 0 1 .1 2 0 .9 9 0 .8 5 7 2 .3 1 0 .8 3 1 .0 0 1 .1 2 0 .9 9 0 .8 5 7 3 .5 1 0 .8 3 1 .0 0 1 .1 2 0 .9 9 0 .8 5 288 T ab le C -1 8 c: ( C o n ti n u ed ) S ta in le ss S te el f o r an y T ȕ= 2 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f y Iƍ fy Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o 0 .5 4 .5 1 0 .8 9 1 .0 1 1 .0 0 1 .0 0 3 .9 7 5 .0 8 0 .8 7 1 .0 1 1 .0 0 1 .0 2 3 .8 0 6 .2 9 0 .8 3 1 .0 1 1 .0 0 1 .0 6 3 .2 1 1 6 .7 8 0 .9 1 1 .0 1 1 .0 0 1 .0 0 4 .1 9 7 .3 3 0 .9 1 1 .0 1 1 .0 0 1 .0 0 4 .1 4 8 .4 5 0 .8 9 1 .0 1 1 .0 0 1 .0 2 4 .0 1 2 1 1 .3 9 0 .9 2 1 .0 0 0 .9 9 0 .9 9 4 .2 7 1 1 .9 2 0 .9 2 1 .0 0 0 .9 9 1 .0 0 4 .2 5 1 3 .0 1 0 .9 2 1 .0 0 0 .9 9 1 .0 0 4 .2 1 0.5 2 .5 1 3 .6 9 0 .9 3 1 .0 0 0 .9 9 0 .9 9 4 .2 8 1 4 .2 3 0 .9 2 1 .0 0 0 .9 9 0 .9 9 4 .2 7 1 5 .3 1 0 .9 2 1 .0 0 0 .9 9 1 .0 0 4 .2 4 0 .5 5 .0 8 0 .8 7 1 .0 1 1 .0 1 1 .0 0 3 .8 1 5 .6 6 0 .8 6 1 .0 1 1 .0 1 1 .0 2 3 .6 1 6 .9 1 0 .8 0 1 .0 2 1 .0 0 2 1 .0 7 2 .7 5 1 7 .3 3 0 .9 1 1 .0 1 1 .0 0 1 .0 0 4 .1 4 7 .8 8 0 .9 0 1 .0 1 1 .0 0 1 .0 0 4 .0 9 9 .0 1 0 .8 9 1 .0 1 1 .0 0 1 .0 2 3 .9 5 2 1 1 .9 2 0 .9 2 1 .0 0 1 .0 0 0 .9 9 4 .2 5 1 2 .4 6 0 .9 2 1 .0 0 1 .0 0 1 .0 0 4 .2 3 1 3 .5 5 0 .9 1 1 .0 0 1 .0 0 1 .0 0 4 .1 9 1 2 .5 1 4 .2 3 0 .9 2 1 .0 0 1 .0 0 0 .9 9 4 .2 7 1 4 .7 0 .9 2 1 .0 0 1 .0 0 0 .9 9 4 .2 5 1 5 .8 5 0 .9 2 1 .0 0 1 .0 0 1 .0 0 4 .2 2 0 .5 7 .6 6 0 .7 3 1 .0 2 1 .1 4 1 .0 2 1 .2 5 8 .3 7 0 .7 2 1 .0 2 1 .1 3 1 .0 4 1 .1 5 9 .8 6 0 .7 2 1 .0 2 1 .1 1 1 .0 9 1 .0 4 1 9 .6 0 0 .8 8 1 .0 1 1 .0 2 1 .0 0 3 .8 6 1 0 .1 7 0 .8 7 1 .0 1 1 .0 2 1 .0 0 3 .7 8 1 1 .3 4 0 .8 6 1 .0 1 1 .0 3 1 .0 2 3 .5 8 2 1 4 .1 1 0 .9 1 1 .0 0 1 .0 1 0 .9 9 4 .1 6 1 4 .6 6 0 .9 1 1 .0 0 1 .0 1 1 .0 0 4 .1 4 1 5 .7 6 0 .9 0 1 .0 0 1 .0 1 1 .0 0 4 .0 8 3 2 .5 1 6 .4 0 0 .9 2 1 .0 0 1 .0 0 0 .9 9 4 .2 0 1 6 .9 5 0 .9 1 1 .0 0 1 .0 0 0 .9 9 4 .1 9 1 8 .0 4 0 .9 1 1 .0 0 1 .0 0 1 .0 0 4 .1 5 0 .5 1 0 .8 7 0 .7 3 1 .0 2 1 .1 8 1 .0 1 0 .9 8 1 1 .5 7 0 .7 2 1 .0 2 1 .1 7 1 .0 3 0 .9 6 1 3 .0 2 0 .7 2 1 .0 2 1 .1 5 1 .0 6 0 .9 4 1 1 2 .0 1 0 .8 4 1 .0 1 1 .0 6 1 .0 0 3 .3 3 1 2 .6 0 0 .8 3 1 .0 1 1 .0 6 1 .0 1 3 .1 7 1 3 .8 5 0 .8 0 1 .0 1 1 .0 7 1 .0 3 2 .6 3 2 1 6 .3 5 0 .9 0 1 .0 0 1 .0 2 0 .9 9 4 .0 4 1 6 .9 1 0 .8 9 1 .0 0 1 .0 2 1 .0 0 4 .0 1 1 8 .0 3 0 .8 9 1 .0 0 1 .0 2 1 .0 0 3 .9 4 5 2 .5 1 8 .6 2 0 .9 1 1 .0 0 1 .0 1 0 .9 9 4 .1 2 1 9 .1 7 0 .9 0 1 .0 0 1 .0 1 0 .9 9 4 .1 0 2 0 .2 7 0 .9 0 1 .0 0 1 .0 1 1 .0 0 4 .0 5 0 .5 1 9 .0 3 0 .7 3 1 .0 1 1 .2 0 1 .0 0 0 .8 8 1 9 .7 2 0 .7 3 1 .0 1 1 .2 0 1 .0 1 0 .8 8 2 1 .1 1 0 .7 3 1 .0 1 1 .1 0 1 .0 3 0 .8 7 1 1 9 .6 6 0 .7 3 1 .0 1 1 .1 9 1 .0 0 0 .9 9 2 0 .3 5 0 .7 3 1 .0 1 1 .1 9 1 .0 1 0 .9 9 2 1 .7 5 0 .7 3 1 .0 1 1 .1 8 1 .0 3 0 .9 8 2 2 2 .2 4 0 .8 6 1 .0 0 1 .0 5 0 .9 9 3 .5 0 2 2 .8 2 0 .8 5 1 .0 0 1 .0 5 1 .0 0 3 .4 5 2 4 .0 1 0 .8 4 1 .0 0 1 .0 6 1 .0 0 3 .3 2 10 2 .5 2 4 .. 3 6 0 .8 8 1 .0 0 1 .0 4 0 .9 9 3 .8 0 2 4 .9 2 0 .8 7 1 .0 0 1 .0 4 0 .9 9 3 .7 6 2 6 .0 7 0 .8 7 1 .0 0 1 .0 4 1 .0 0 3 .7 0 0 .5 8 4 .7 1 0 .7 4 1 .0 0 1 .2 2 0 .9 9 0 .8 2 8 5 .3 7 0 .7 4 1 .0 0 1 .2 2 0 .9 9 0 .8 2 8 6 .7 2 0 .7 4 1 .0 0 1 .2 2 1 .0 0 0 .8 2 1 8 5 .2 7 0 .7 4 1 .0 0 1 .2 2 0 .9 9 0 .8 4 8 5 .9 3 0 .7 4 1 .0 0 1 .2 2 0 .9 9 0 .8 4 8 7 .2 8 0 .7 4 1 .0 0 1 .2 2 1 .0 0 0 .8 4 2 8 6 .4 1 0 .7 4 1 .0 0 1 .2 2 0 .9 9 0 .8 7 8 7 .0 8 0 .7 4 1 .0 0 1 .2 2 0 .9 9 0 .8 7 8 8 .4 2 0 .7 4 1 .0 0 1 .2 1 1 .0 0 0 .8 7 f Po 50 2 .5 8 7 .0 0 0 .7 4 1 .0 0 1 .2 2 0 .9 9 0 .8 8 8 7 .6 7 0 .7 4 1 .0 0 1 .2 2 0 .9 9 0 .8 8 8 9 .0 2 0 .7 4 1 .0 0 1 .2 1 1 .0 0 0 .8 8 289 T ab le C -1 9 a: A d ju st ed N o m in al R es is ta n ce F ac to rs f o r C ar b o n S te el a n d Ȗ ǹ = 1 .1 , Ȗ P O = Ȗ M = 1 .2 , Ȗ ȅ = 1 .5 f o r g 9 C ar b o n S te el ȕ= 1 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F 0 .5 0 .9 8 (1 ) 0 .8 1 (1 ) 0 .9 2 0 .8 0 0 .7 4 0 .9 9 (1 ) ) 0 .8 2 (1 ) 0 .9 2 0 .7 9 0 .7 4 0 .9 9 (1 ) 0 .8 1 (1 ) 0 .8 9 0 .7 7 0 .7 2 1 0 .9 3 (1 ) 0 .7 6 (1 ) 0 .9 3 0 .8 0 0 .7 5 0 .9 5 (1 ) 0 .7 8 (1 ) 0 .9 3 0 .8 1 0 .7 5 0 .9 7 (1 ) 0 .8 0 (1 ) 0 .9 3 0 .8 0 0 .7 5 2 0 .8 6 (1 ) 0 .7 1 (1 ) 0 .9 3 0 .8 0 0 .7 5 0 .8 8 (1 ) 0 .7 2 (1 ) 0 .9 3 0 .8 0 0 .7 5 0 .9 1 (1 ) 0 .7 5 (1 ) 0 .9 4 0 .8 1 0 .7 5 0.5 2 .5 0 .8 4 (1 ) 0 .8 1 (1 ) 0 .9 3 0 .8 0 0 .7 5 0 .8 6 (1 ) 0 .7 1 (1 ) 0 .9 3 0 .8 0 0 .7 5 0 .8 9 (1 ) 0 .7 3 (1 ) 0 .9 4 0 .8 1 0 .7 5 0 .5 1 .0 0 (1 ) 0 .8 2 (1 ) 0 .9 2 0 .8 0 0 .7 4 1 .0 1 (1 ) 0 .8 3 (1 ) 0 .9 2 0 .7 9 0 .7 4 1 .0 0 (1 ) 0 .8 2 (1 ) 0 .9 0 0 .7 7 0 .7 2 1 0 .9 5 (1 ) 0 .7 8 (1 ) 0 .9 4 0 .8 1 0 .7 6 0 .9 6 (1 ) 0 .7 9 (1 ) 0 .9 4 0 .8 1 0 .7 6 0 .9 8 (1 ) 0 .8 1 (1 ) ) 0 .9 3 0 .8 1 0 .7 5 2 0 .8 9 (1 ) 0 .7 3 (1 ) 0 .9 4 0 .8 1 0 .7 5 0 .9 0 (1 ) 0 .7 4 (1 ) 0 .9 4 0 .8 1 0 .7 6 0 .9 3 (1 ) 0 .7 6 (1 ) ) 0 .9 4 0 .8 1 0 .7 5 1 2 .5 0 .8 7 (1 ) 0 .7 1 (1 ) 0 .9 3 0 .8 1 0 .7 5 0 .8 8 (1 ) 0 .7 3 (1 ) 0 .9 4 0 .8 1 0 .7 5 0 .9 1 (1 ) 0 .7 5 (1 ) 0 .9 4 0 .8 1 0 .7 6 0 .5 1 .0 1 (1 ) 0 .8 3 (1 ) 0 .9 0 0 .7 8 0 .7 3 1 .0 1 (1 ) 0 .8 3 (1 ) 0 .9 0 0 .7 8 0 .7 3 1 .0 1 (1 ) 0 .8 3 (1 ) 0 .8 9 0 .7 7 0 .7 2 1 1 .0 0 (1 ) 0 .8 2 (1 ) 0 .9 4 0 .8 1 0 .7 6 1 .0 1 (1 ) 0 .8 3 (1 ) ) 0 .9 4 0 .8 1 0 .7 6 1 .0 1 (1 ) 0 .8 3 (1 ) 0 .9 3 0 .8 0 0 .7 5 2 0 .9 5 (1 ) 0 .7 8 (1 ) 0 .9 5 0 .8 2 0 .7 7 0 .9 6 (1 ) 0 .7 9 (1 ) 0 .9 6 0 .8 2 0 .7 7 0 .9 7 (1 ) 0 .8 0 (1 ) 0 .9 5 0 .8 2 0 .7 7 3 2 .5 0 .9 3 (1 ) 0 .7 6 (1 ) 0 .9 5 0 .8 2 0 .7 7 0 .9 4 (1 ) 0 .7 7 (1 ) ) 0 .9 5 0 .8 2 0 .7 7 0 .9 5 (1 ) 0 .7 8 (1 ) 0 .9 5 0 .8 2 0 .7 7 0 .5 1 .0 0 (1 ) 0 .8 2 (1 ) 0 .8 9 0 .7 5 0 .7 1 1 .0 0 (1 ) 0 .8 2 (1 ) 0 .8 9 0 .7 6 0 .7 1 1 .0 0 (1 ) 0 .8 2 (1 ) 0 .8 8 0 .7 6 0 .7 1 1 1 .0 1 (1 ) 0 .8 3 (1 ) 0 .9 3 0 .8 0 0 .7 5 1 .0 1 (1 ) 0 .8 3 (1 ) 0 .9 2 0 .8 0 0 .7 4 1 .0 1 (1 ) 0 .8 3 (1 ) 0 .9 2 0 .7 9 0 .7 4 2 0 .9 8 (1 ) 0 .8 1 (1 ) 0 .9 3 0 .8 3 0 .7 7 0 .9 9 (1 ) 0 .8 1 (1 ) 0 .9 6 0 .8 2 0 .7 7 0 .9 9 (1 ) 0 .8 2 (1 ) 0 .9 5 0 .8 2 0 .7 7 5 2 .5 0 .9 7 (1 ) 0 .7 9 (1 ) 0 .9 6 0 .8 3 0 .7 7 0 .9 7 (1 ) 0 .8 0 (1 ) 0 .9 6 0 .8 3 0 .7 7 0 .9 8 (1 ) 0 .8 1 (1 ) 0 .9 6 0 .8 3 0 .7 7 0 .5 0 .9 8 (1 ) 0 .8 1 (1 ) 0 .8 7 0 .7 5 0 .7 0 0 .9 8 (1 ) ) 0 .8 1 (1 ) 0 .8 7 0 .7 5 0 .7 0 0 .9 9 (1 ) 0 .8 1 (1 ) 0 .8 7 0 .7 5 0 .7 0 1 0 .9 9 (1 ) 0 .8 2 (1 ) 0 .8 9 0 .7 7 0 .7 2 1 .0 0 (1 ) 0 .8 2 (1 ) 0 .8 9 0 .7 7 0 .7 2 1 .0 0 (1 ) 0 .8 2 (1 ) 0 .8 9 0 .7 7 0 .7 2 2 1 .0 1 (1 ) 0 .8 3 (1 ) ) 0 .9 3 0 .8 1 0 .7 5 1 .0 0 (1 ) 0 .8 3 (1 ) 0 .9 3 0 .8 0 0 .7 5 1 .0 1 (1 ) 0 .8 3 (1 ) 0 .9 3 0 .8 0 0 .7 5 10 2 .5 1 .0 0 (1 ) 0 .8 2 (1 ) ) 0 .9 5 0 .8 2 0 .7 0 1 .0 0 (1 ) 0 .8 3 (1 ) 0 .9 5 0 .8 2 0 .7 6 1 .0 1 (1 ) 0 .8 3 (1 ) 0 .9 4 0 .8 1 0 .7 6 0 .5 0 .9 6 (1 ) 0 .7 9 (1 ) 0 .8 4 0 .7 3 0 .6 8 0 .9 6 (1 ) 0 .7 9 (1 ) 0 .8 4 0 .7 3 0 .6 8 0 .9 6 (1 ) 0 .7 9 (1 ) 0 .8 4 0 .7 3 0 .6 8 1 0 .9 6 (1 ) 0 .7 9 (1 ) 0 .8 5 0 .7 3 0 .6 8 0 .9 6 (1 ) 0 .7 9 (1 ) 0 .8 5 0 .7 3 0 .6 8 0 .9 6 (1 ) ) 0 .7 9 (1 ) 0 .8 5 0 .7 3 0 .6 8 2 0 .9 7 (1 ) 0 .8 0 (1 ) 0 .8 6 0 .7 4 0 .6 9 0 .9 7 (1 ) 0 .8 0 (1 ) 0 .8 6 0 .7 5 0 .6 9 0 .9 7 (1 ) 0 .8 0 (1 ) 0 .8 6 0 .7 4 0 .6 9 f Po 50 2 .5 0 .9 7 (1 ) 0 .8 0 (1 ) 0 .8 7 0 .7 5 0 .7 0 0 .9 7 (1 ) 0 .8 0 (1 ) ) 0 .8 7 0 .7 6 0 .7 0 0 .9 8 (1 ) 0 .8 0 (1 ) 0 .8 7 0 .7 5 0 .7 0 (1 ) Ȗ ǹ = Ȗ Ȃ = Ȗ P ȅ = 1 .0 , Ȗ ȅ = 0 .9 290 T ab le C -1 9 a: ( C o n ti n u ed ) C ar b o n S te el ȕ= 2 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F 0 .5 0 .8 8 0 .7 2 0 .6 5 0 .5 6 0 .5 3 0 .9 2 0 .7 6 0 .6 9 0 .5 9 0 .5 5 0 .9 8 0 .8 1 0 .7 3 0 .6 3 0 .5 8 1 0 .7 5 0 .6 2 0 .5 6 0 .4 8 0 .4 5 0 .7 8 0 .6 5 0 .5 9 0 .5 1 0 .4 8 0 .8 6 0 .7 0 0 .6 4 0 .5 5 0 .5 2 2 0 .6 5 0 .5 3 0 .4 9 0 .4 2 0 .3 9 0 .7 2 0 .5 6 0 .5 1 0 .4 4 0 .4 1 0 .7 5 0 .6 0 0 .5 5 0 .4 7 0 .4 4 0.5 2 .5 0 .6 2 0 .5 1 0 .4 7 0 .4 1 0 .3 8 0 .7 1 0 .5 4 0 .4 9 0 .4 2 0 .3 9 0 .7 3 0 .5 7 0 .5 3 0 .4 5 0 .4 2 0 .5 0 .9 3 0 .7 6 0 .6 9 0 .6 0 0 .5 6 0 .9 6 0 .7 9 0 .7 2 0 .6 2 0 .5 8 1 .0 1 0 .8 3 0 .7 5 0 .6 4 0 .6 0 1 0 .8 0 0 .6 5 0 .6 0 0 .5 1 0 .4 8 0 .8 3 0 .6 8 0 .6 2 0 .5 4 0 .5 0 0 .8 9 0 .7 3 0 .6 6 0 .5 7 0 .5 3 2 0 .6 8 0 .5 6 0 .5 1 0 .4 4 0 .4 1 0 .7 1 0 .5 8 0 .5 3 0 .4 6 0 .4 3 0 .7 6 0 .6 2 0 .5 7 0 .4 9 0 .4 6 1 2 .5 0 .6 5 0 .5 4 0 .4 9 0 .4 2 0 .4 7 0 .6 8 0 .5 6 0 .5 1 0 .4 4 0 .4 1 0 .7 2 0 .5 9 0 .5 4 0 .4 7 0 .4 4 0 .5 1 .0 5 0 .8 6 0 .7 7 0 .6 6 0 .6 2 1 .0 6 0 .8 8 0 .7 7 0 .6 6 0 .6 2 1 .0 8 0 .8 9 0 .7 6 0 .6 6 0 .6 2 1 0 .9 3 0 .6 6 0 .6 9 0 .6 0 0 .5 6 0 .9 5 0 .7 8 0 .7 1 0 .6 1 0 .5 7 0 .9 8 0 .8 1 0 .7 3 0 .6 3 0 .5 9 2 0 .7 9 0 .6 5 0 .5 9 0 .5 1 0 .4 8 0 .8 1 0 .6 7 0 .6 1 0 .5 2 0 .4 9 0 .8 4 0 .6 9 0 .6 3 0 .5 4 0 .5 1 3 2 .5 0 .7 5 0 .6 2 0 .5 6 0 .4 9 0 .4 5 0 .7 7 0 .6 3 0 .5 8 0 .5 0 0 .4 6 0 .8 0 0 .6 6 0 .6 0 0 .5 2 0 .4 8 0 .5 1 .0 7 0 .8 8 0 .7 5 0 .6 5 0 .6 1 1 .0 8 0 .8 9 0 .7 6 0 .6 5 0 .6 1 1 .0 8 0 .8 9 0 .7 6 0 .6 5 0 .6 1 1 1 .0 0 0 .8 3 0 .7 4 0 .6 4 0 .6 0 1 .0 2 0 .8 4 0 .7 5 0 .6 5 0 .6 0 1 .0 4 0 .8 5 0 .7 6 0 .6 6 0 .6 1 2 0 .8 7 0 .7 1 0 .6 5 0 .5 6 0 .5 2 0 .8 8 0 .7 3 0 .6 6 0 .5 7 0 .5 3 0 .9 1 0 .7 5 0 .6 8 0 .5 8 0 .5 5 5 2 .5 0 .8 2 0 .6 8 0 .6 2 0 .5 3 0 .5 0 0 .8 4 0 .6 9 0 .6 3 0 .5 4 0 .5 0 0 .8 6 0 .7 1 0 .6 5 0 .5 6 0 .5 2 0 .5 1 .0 4 0 .8 5 0 .7 3 0 .6 3 0 .5 9 1 .0 4 0 .8 6 0 .7 3 0 .6 3 0 .5 9 1 .0 5 0 .8 6 0 .7 4 0 .6 3 0 .5 9 1 1 .0 7 0 .8 8 0 .7 5 0 .6 5 0 .6 1 1 .0 7 0 .8 8 0 .7 5 0 .6 5 0 .6 1 1 .0 8 0 .8 9 0 .7 6 0 .6 5 0 .6 1 2 0 .9 9 0 .8 1 0 .7 3 0 .6 3 0 .5 9 0 .9 9 0 .8 2 0 .7 4 0 .6 4 0 .5 9 1 .0 1 0 .8 3 0 .7 5 0 .6 4 0 .6 0 10 2 .5 0 .9 4 0 .7 8 0 .7 0 0 .6 1 0 .5 7 0 .9 5 0 .7 8 0 .7 1 0 .6 6 0 .5 7 0 .9 7 0 .7 9 0 .7 2 0 .6 2 0 .5 8 0 .5 0 .9 9 0 .8 2 0 .7 0 0 .6 1 0 .5 7 1 .0 0 0 .8 2 0 .7 0 0 .6 1 0 .5 7 1 .0 0 0 .8 2 0 .7 1 0 .6 1 0 .5 7 1 1 .0 0 0 .8 3 0 .7 1 0 .6 1 0 .5 7 1 .0 0 0 .8 3 0 .7 1 0 .6 1 0 .5 7 1 .0 1 0 .8 3 0 .7 1 0 .6 1 0 .5 7 2 1 .0 2 0 .8 4 0 .7 2 0 .6 2 0 .5 8 1 .0 2 0 .8 4 0 .7 2 0 .6 2 0 .5 8 1 .0 2 0 .8 4 0 .7 2 0 .6 2 0 .5 8 f Po 50 2 .5 1 .0 3 0 .8 5 0 .7 3 0 .6 3 0 .5 8 1 .0 3 0 .8 5 0 .7 3 0 .6 3 0 .5 9 1 .0 3 0 .8 5 0 .7 3 0 .6 3 0 .5 9 291 T ab le C -1 9 b : A d ju st ed N o m in al R es is ta n ce F ac to rs f o r S ta in le ss S te el a n d Ȗ ǹ = 1 .1 , Ȗ P O = Ȗ M = 1 .2 , Ȗ ȅ = 1 .5 f o r g 9 S ta in le ss S te el ȕ= 1 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F 0 .5 1 .0 3 (1 ) 0 .9 0 (1 ) 0 .9 3 0 .8 3 0 .7 8 1 .0 3 (1 ) 0 .9 0 (1 ) 0 .9 3 0 .8 2 0 .7 7 1 .0 2 (1 ) 0 .8 9 (1 ) 0 .9 0 0 .8 0 0 .7 5 1 0 .9 9 (1 ) 0 .8 7 (1 ) 0 .9 5 0 .8 5 0 .7 9 1 .0 1 (1 ) 0 .8 8 (1 ) 0 .9 5 0 .8 4 0 .7 9 1 .0 2 (1 ) 0 .8 9 0 .9 4 0 .8 4 0 .7 9 2 0 .9 3 (1 ) 0 .8 1 (1 ) 0 .9 5 0 .8 4 0 .7 9 0 .9 5 (1 ) 0 .8 3 (1 ) 0 .9 5 0 .8 5 0 .7 9 0 .9 8 (1 ) 0 .8 5 0 .9 6 0 .8 5 0 .8 0 0.5 2 .5 0 .9 1 (1 ) 0 .7 9 (1 ) 0 .9 5 0 .8 4 0 .7 9 0 .9 3 (1 ) 0 .8 1 (1 ) 0 .9 5 0 .8 5 0 .7 9 0 .9 6 (1 ) 0 .8 4 0 .9 6 0 .8 5 0 .8 0 0 .5 1 .0 4 (1 ) 0 .9 1 (1 ) 0 .9 3 0 .8 3 0 .7 8 1 .0 4 (1 ) 0 .9 0 (1 ) 0 .9 2 0 .8 2 0 .7 7 1 .0 2 (1 ) 0 .8 9 0 .9 0 0 .8 0 0 .7 5 1 1 .0 1 (1 ) 0 .8 8 (1 ) 0 .9 6 0 .8 5 0 .8 0 1 .0 2 (1 ) 0 8 9 (1 ) 0 .9 6 0 .8 5 0 .8 0 1 .0 3 (1 ) 0 .9 0 0 .9 5 0 .8 4 0 .7 9 2 0 .9 5 (1 ) 0 .8 3 (1 ) 0 .9 6 0 .8 5 0 .8 0 0 .9 7 (1 ) 0 .8 5 (1 ) 0 .9 6 0 .8 5 0 .8 0 0 .9 9 (1 ) 0 .9 6 0 .9 6 0 .8 5 0 .8 0 1 2 .5 0 .9 3 (1 ) 0 .8 1 (1 ) 0 .9 6 0 .8 5 0 .8 0 0 .9 5 (1 ) 0 .8 3 (1 ) 0 .9 6 0 .8 5 0 .8 0 0 .9 7 (1 ) 0 .8 5 0 .9 6 0 .8 5 0 .8 0 0 .5 1 .0 3 (1 ) 0 .9 0 (1 ) 0 .9 1 0 .8 1 0 .7 6 1 .0 3 (1 ) 0 .9 0 (1 ) 0 .9 1 0 .8 1 0 .7 6 1 .0 2 (1 ) 0 .8 9 0 .9 0 0 .8 0 0 .7 5 1 1 .0 4 (1 ) 0 .9 1 (1 ) 0 .9 5 0 .8 5 0 .7 9 1 .0 4 (1 ) 0 .9 1 (1 ) 0 .9 5 0 .8 4 0 .7 9 1 .0 4 (1 ) 0 .9 1 0 .9 4 0 .8 3 0 .7 8 2 1 .0 1 (1 ) 0 .8 8 (1 ) 0 .9 7 0 .8 6 0 .8 1 1 .0 2 (1 ) 0 .8 9 (1 ) 0 .9 7 0 .8 6 0 .8 1 1 .0 3 (1 ) 0 .9 0 0 .9 7 0 .8 6 0 .8 1 3 2 .5 0 .9 9 (1 ) 0 .8 7 (1 ) 0 .9 7 0 .8 6 0 .8 1 1 .0 0 (1 ) 0 .8 7 (1 ) 0 .9 7 0 .8 6 0 .8 1 1 .0 1 (1 ) 0 .8 9 0 .9 7 0 .8 6 0 .8 1 0 .5 1 .0 2 (1 ) 0 .8 9 (1 ) 0 .8 9 0 .7 9 0 .7 4 1 .0 2 (1 ) 0 .8 9 (1 ) 0 .8 9 0 .7 9 0 .7 4 1 .0 2 0 .8 9 0 .8 9 0 .7 9 0 .7 4 1 1 .0 4 (1 ) 0 .9 1 (1 ) 0 .9 3 0 .8 3 0 .7 8 1 .0 4 (1 ) 0 .9 1 (1 ) 0 .9 3 0 .8 3 0 .7 8 1 .0 4 0 .9 0 0 .9 2 0 .8 2 0 .7 7 2 1 .0 4 (1 ) 0 .9 0 (1 ) 0 .9 7 0 .8 6 0 .8 1 1 .0 4 (1 ) 0 .9 1 (1 ) 0 .9 7 0 .8 6 0 .8 1 1 .0 4 0 .9 1 0 .9 6 0 .8 6 0 .8 0 5 2 .5 1 .0 3 (1 ) 0 .8 9 (1 ) 0 .9 8 0 .8 7 0 .8 1 1 .0 3 (1 ) 0 .9 0 (1 ) 0 .9 7 0 .8 7 0 .8 1 1 .0 3 0 .9 0 0 .9 7 0 .8 6 0 .8 1 0 .5 1 .0 1 (1 ) 0 .8 8 (1 ) 0 .8 7 0 .7 8 0 .7 3 1 .0 1 (1 ) 0 .8 8 (1 ) 0 .8 7 0 .7 8 0 .7 3 1 .0 1 0 .8 8 0 .8 8 0 .7 8 0 .7 3 1 1 .0 2 (1 ) 0 .8 9 (1 ) 0 .9 0 0 .8 0 0 .7 5 1 .0 2 (1 ) 0 .8 (1 ) 9 0 .9 0 0 .8 0 0 .7 7 1 .0 2 0 .8 9 0 .9 0 0 .8 0 0 .7 5 2 1 .0 4 (1 ) 0 .9 1 (1 ) 0 .9 4 0 .8 4 0 .7 9 1 .0 4 (1 ) 0 .9 1 (1 ) 0 .9 4 0 .8 4 0 .7 8 1 .0 4 0 .9 1 0 .9 4 0 .8 3 0 .7 8 10 2 .5 1 .0 4 (1 ) 0 .9 1 (1 ) 0 .9 6 0 .8 5 0 .8 0 1 .0 4 (1 ) 0 .9 1 (1 ) 0 .9 6 0 .8 5 0 .8 0 1 .0 4 0 .9 1 0 .9 5 0 .8 5 0 .7 9 0 .5 0 .9 9 (1 ) 0 .8 6 (1 ) 0 .8 5 0 .7 6 0 .7 1 0 .9 9 (1 ) 0 .8 6 (1 ) 0 .8 5 0 .7 6 0 .7 1 0 .9 9 0 .8 7 0 .8 5 0 .7 6 0 .7 1 1 0 .9 9 (1 ) 0 .8 7 (1 ) 0 .8 6 0 .7 6 0 .7 1 0 .9 9 (1 ) 0 .8 7 (1 ) 0 .8 6 0 .7 6 0 .7 1 0 .9 9 0 .8 7 0 .8 6 0 .7 6 0 .7 2 2 1 .0 0 (1 ) 0 .8 7 (1 ) 0 .8 7 0 .7 7 0 .7 3 1 .0 0 (1 ) 0 .8 7 (1 ) 0 .8 7 0 .7 7 0 .7 3 1 .0 0 0 .8 7 0 .8 7 0 .7 7 0 .7 3 f Po 50 2 .5 1 .0 0 (1 ) 0 .8 8 (1 ) 0 .8 8 0 .7 8 0 .7 3 1 .0 0 (1 ) 0 .8 8 (1 ) 0 .8 8 0 .7 9 0 .7 3 1 .0 0 0 .8 8 0 .8 8 0 .7 8 0 .7 3 (1 ) Ȗ ǹ = Ȗ Ȃ = Ȗ P ȅ = 1 .0 , Ȗ ȅ = 0 .9 292 T ab le C -1 9 b : (C o n ti n u ed ) S ta in le ss S te el ȕ= 2 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F 0 .5 0 .9 3 0 .8 1 0 .6 6 0 .5 9 0 .5 5 0 .9 8 0 .8 5 0 .7 0 0 .6 2 0 .5 8 1 .0 3 0 .9 0 0 .7 3 0 .6 5 0 .6 1 1 0 .8 0 0 .7 0 0 .5 7 0 .5 1 0 .4 8 0 .8 5 0 .7 4 0 .6 1 0 .5 4 0 .5 0 0 .9 1 0 .8 0 0 .6 5 0 .5 8 0 .5 4 2 0 .7 0 0 .6 1 0 .5 0 0 .4 4 0 .4 2 0 .7 3 0 .6 1 0 .5 2 0 .4 7 0 .4 4 0 .7 9 0 .6 9 0 .5 6 0 .5 0 0 .4 7 0.5 2 .5 0 .6 7 0 .5 9 0 .4 8 0 .4 3 0 .4 0 0 .7 0 0 .6 1 0 .5 0 0 .4 5 0 .4 2 0 .7 5 0 .6 6 0 .5 4 0 .4 8 0 .4 5 0 .5 0 .9 8 0 .8 6 0 .7 0 0 .6 2 0 .5 9 1 .0 2 0 .8 9 0 .7 3 0 .6 4 0 .6 0 1 .0 5 0 .9 2 0 .7 5 0 .6 7 0 .6 3 1 0 .8 5 0 .7 4 0 .6 1 0 .5 4 0 .5 1 0 .8 9 0 .7 8 0 .6 4 0 .5 6 0 .5 3 0 .9 5 0 .8 3 0 .6 8 0 .6 0 0 .5 6 2 0 .7 4 0 .6 4 0 .5 3 0 .4 7 0 .4 4 0 .7 6 0 .6 7 0 .5 5 0 .4 9 0 .4 6 0 .8 1 0 .7 1 0 .5 8 0 .5 2 0 .4 9 1 2 .5 0 .7 1 0 .6 2 0 .5 0 0 .4 5 0 .4 2 0 .7 3 0 .6 4 0 .5 2 0 .4 6 0 .4 3 0 .7 8 0 .6 8 0 .5 5 0 .4 9 0 .4 6 0 .5 1 .0 7 0 .9 3 0 .7 6 0 .6 8 0 .6 4 1 .0 7 0 .9 3 0 .7 6 0 .6 8 0 .6 4 1 .0 6 0 .9 2 0 .7 6 0 .6 7 0 .6 3 1 0 .9 8 0 .8 6 0 .7 0 0 .6 2 0 .5 9 1 .0 0 0 .8 7 0 .7 2 0 .6 4 0 .6 0 1 .0 3 0 .9 0 0 .7 4 0 .6 6 0 .6 1 2 0 .8 5 0 .7 4 0 .6 1 0 .5 4 0 .5 0 0 .8 7 0 .7 6 0 .6 2 0 .5 5 0 .5 2 0 .9 0 0 .7 9 0 .6 4 0 .5 7 0 .5 4 3 2 .5 0 .8 1 0 .7 0 0 .5 8 0 .5 1 0 .4 8 0 .8 2 0 .7 2 0 .5 9 0 .5 2 0 .4 9 0 .8 6 0 .7 5 0 .6 1 0 .5 4 0 .5 1 0 .5 1 .0 5 0 .9 1 0 .7 5 0 .6 6 0 .6 2 1 .0 5 0 .9 1 0 .7 5 0 .6 6 0 .6 2 1 .0 5 0 .9 1 0 .7 5 0 .6 6 0 .6 2 1 1 .0 5 0 .9 2 0 .7 5 0 .6 7 0 .6 3 1 .0 6 0 .9 3 0 .7 6 0 .6 7 0 .6 3 1 .0 7 0 .9 4 0 .7 7 0 .6 8 0 .6 4 2 0 .9 3 0 .8 1 0 .6 6 0 .5 9 0 .5 5 0 .9 4 0 .8 2 0 .6 7 0 .6 0 0 .5 6 0 .9 7 0 .8 4 0 .6 9 0 .6 1 0 .5 7 5 2 .5 0 .8 8 0 .7 7 0 .6 3 0 .5 6 0 .5 2 0 .8 9 0 .8 9 0 .6 4 0 .5 7 0 .5 3 0 .9 2 0 .8 0 0 .6 6 0 .5 8 0 .5 5 0 .5 1 .0 2 0 .8 9 0 .7 3 0 .6 4 0 .6 0 1 .0 2 0 .8 9 0 .7 3 0 .6 5 0 .6 1 1 .0 2 0 .8 9 0 .7 3 0 .6 5 0 .6 1 1 1 .0 5 0 .9 1 0 .7 5 0 .6 6 0 .6 2 1 .0 5 0 .9 2 0 .7 5 0 .6 7 0 .6 2 1 .0 5 0 .9 2 0 .7 5 0 .6 7 0 .6 3 2 1 .0 4 0 .9 1 0 .7 4 0 .6 6 0 .6 2 1 .0 5 0 .9 1 0 .7 5 0 .6 6 0 .6 2 1 .0 6 0 .9 2 0 .7 5 0 .6 7 0 .6 3 10 2 .5 1 .0 0 0 .8 7 0 .7 1 0 .6 3 0 .6 0 1 .0 1 0 .8 8 0 .7 2 0 .6 4 0 .6 0 1 .0 2 0 .8 9 0 .7 3 0 .6 5 0 .6 1 0 .5 0 .9 8 0 .8 6 0 .7 0 0 .6 2 0 .5 8 0 .9 8 0 .8 4 0 .7 0 0 .6 2 0 .5 8 0 .9 8 0 .8 6 0 .7 0 0 .6 2 0 .5 9 1 0 .9 9 0 .8 6 0 .7 1 0 .6 3 0 .5 9 0 .9 9 0 .8 5 0 .7 1 0 .6 3 0 .5 9 0 .9 9 0 .8 7 0 .7 1 0 .6 3 0 .5 9 2 1 .0 0 0 .8 8 0 .7 2 0 .6 4 0 .6 0 1 .0 1 0 .8 6 0 .7 2 0 .6 4 0 .6 0 1 .0 1 0 .8 8 0 .7 2 0 .6 4 0 .6 0 f Po 50 2 .5 1 .0 1 0 .8 8 0 .7 2 0 .6 4 0 .6 0 1 .0 1 0 .8 7 0 .7 2 0 .6 4 0 .6 0 1 .0 1 0 .8 9 0 .7 2 0 .6 4 0 .6 0 293 294 C.9. Performance Function g11 Table C-20 provides the calculated mean load and resistance factors for performance function g11. In this table, ȝfu is the converged mean value of the ultimate strength of steel. Table C-21 gives the evaluated adjusted nominal resistance factors for nominal load factors Ȗǹ=1.1 and ȖO=1.5. Table C-20: Mean Partial Safety Factors for g11 Carbon Steel (T”200oF) & Stainless Steel Carbon Steel (T>200oF) ȕ ȝfu Iƍu Ȗ'ǹ ȖƍO ȝfu Iƍu Ȗ'ǹ ȖƍO 1.5 1.96 0.983 1.023 1.813 2.00 0.953 1.023 1.758 2 2.42 0.984 1.017 2.724 2.46 0.956 1.017 2.662 f O = 0 .5 3 4.90 0.987 1.007 7.662 4.97 0.964 1.007 7.564 1.5 2.89 0.987 1.011 1.844 2.93 0.964 1.011 1.815 2 3.81 0.987 1.008 2.750 3.86 0.64 1.008 2.712 f O = 1 3 8.79 0.988 1.003 7.678 8.90 0.967 1.003 7.599 1.5 4.77 0.989 1.006 1.853 4.82 0.969 1.006 1.834 2 6.60 0.988 1.004 2.758 6.68 0.968 1.004 2.730 f O = 2 3 16.56 0.989 1.002 7.684 16.75 0.969 1.002 7.614 1.5 5.70 0.989 1.004 1.855 5.77 0.971 1.004 1.837 2 7.99 0.989 1.003 2.760 8.09 0.969 1.003 2.733 f O = 2 .5 3 20.45 0.989 1.001 7.685 20.68 0.969 1.001 7.617 295 Table C-21a: Adjusted Nominal Resistance Factor for Ȗǹ=1.1 and ȖS=1.5 and Carbon Steel for g11 Carbon Steel fO ȕ R.T. 200oF 400oF 600oF 800oF 1.5 0.98 (1) 0.89 (1) 0.98 (1) 0.92 (1) 0.92 2 1.00 0.91 1.00 0.94 0.75 0.5 3 0.49 0.45 0.49 0.46 0.37 1.5 0.93 (1) 0.85 (1) 0.93 (1) 0.88 (1) 0.93 2 0.92 0.85 0.94 0.88 0.71 1 3 0.41 0.37 0.41 0.38 0.31 1.5 0.88 (1) 0.81 (1) 0.89 (1) 0.84 (1) 0.93 2 0.89 0.81 0.89 0.84 0.67 2 3 0.35 0.32 0.36 0.33 0.27 1.5 0.87 (1) 0.80 (1) 0.88 (1) 0.70 (1) 0.93 2 0.88 0.80 0.88 0.83 0.66 2.5 3 0.34 0.31 0.35 0.32 0.26 (1) Ȗǹ=1.0 and ȖO=1.0, R.T.=Room Temperature Table C-21b: Adjusted Nominal Resistance Factor for Ȗǹ=1.1 and ȖO=1.5 and Stainless Steel for g11 Stainless Steel fO ȕ R.T. 200oF 400oF 600oF 800oF 1.5 0.96 (1) 0.83 (1) 0.94 0.94 0.91 2 0.98 0.85 0.76 0.76 0.74 0.5 3 0.48 0.42 0.38 0.38 0.36 1.5 0.91 (1) 0.79 (1) 0.94 0.94 0.91 2 0.92 0.80 0.72 0.72 0.69 1 3 0.40 0.35 0.31 0.31 0.30 1.5 0.87 (1) 0.75 (1) 0.94 0.94 0.91 2 0.87 0.76 0.68 0.68 0.66 2 3 0.35 0.30 0.27 0.27 0.26 1.5 0.86 (1) 0.75 (1) 0.94 0.94 0.91 2 0.86 0.75 0.67 0.67 0.65 2.5 3 0.34 0.29 0.26 0.26 0.25 (1) Ȗǹ=1.0 and ȖO=1.0, R.T.=Room Temperature 296 C.10. Performance Function g12 Table C-22 gives the calculated mean load and resistance factors for performance function g12. In this table, ȝfu is the converged mean value of the ultimate strength of steel. Table C-23 shows the evaluated adjusted nominal resistance factors for load factors Ȗǹ=1.1, ȖPO=1.2, and ȖM=1.2. T ab le C -2 2 a: M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T ”2 0 0 o F a n d S ta in le ss S te el f o r g 1 2 C ar b o n S te el f o r T” 2 0 0 o F a n d S ta in le ss S te el f O = 0 .5 f O = 1 f O = 2 ȕ ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ O Ȗƍ P O ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ O Ȗƍ P O ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ O Ȗƍ P O 2 2 .9 3 0 .9 8 1 1 .0 1 7 2 .7 0 0 1 .0 0 6 4 .3 1 0 .9 8 5 1 .0 0 8 2 .7 4 2 0 .9 9 9 7 .1 0 0 .9 8 8 1 .0 0 4 2 .7 5 5 0 .9 9 5 2 .5 3 .7 6 0 .9 8 1 .0 1 4 .3 8 1 .0 0 5 .9 9 0 .9 9 1 .0 1 4 .4 1 1 .0 0 1 0 .4 7 0 .9 9 1 .0 0 4 .4 2 0 .9 9 f Po =0 .5 3 5 .4 1 0 .9 8 6 1 .0 0 7 7 .6 4 7 0 .9 9 7 9 .2 9 0 .9 8 7 1 .0 0 3 7 .6 7 2 0 .9 9 4 1 7 .0 6 0 .9 8 8 1 .0 0 2 7 .6 8 2 0 .9 9 3 2 3 .4 4 0 .9 7 8 1 .0 1 7 2 .6 5 6 1 .0 2 1 4 .8 2 0 .9 8 4 1 .0 0 8 2 .7 3 0 1 .0 0 6 7 .6 0 0 .9 8 7 1 .0 0 4 2 .7 5 6 1 0 .9 9 9 2 .5 4 .2 7 0 .9 8 1 .0 1 4 .3 5 1 .0 1 6 .5 0 0 .9 9 1 .0 1 4 .4 0 1 .0 0 1 0 .9 7 0 .9 9 1 .0 0 4 .4 2 1 .0 0 f PO =1 3 5 .9 1 0 .9 8 4 1 .0 0 7 7 .6 2 5 1 .0 3 3 9 .7 9 0 .9 8 7 1 .0 0 3 7 .6 6 4 0 .9 9 7 1 7 .5 6 0 .9 8 8 1 .0 0 2 7 .6 7 9 0 .9 9 4 2 5 .6 3 0 .9 5 9 1 .0 2 1 2 .0 7 7 1 .1 1 4 6 .9 0 0 .9 7 7 1 .0 0 9 2 .6 2 6 1 .0 3 8 9 .6 5 0 .9 8 4 1 .0 0 4 2 .7 2 3 1 .0 1 3 2 .5 6 .3 9 0 .9 7 1 .0 1 4 .0 5 1 .0 6 8 .5 6 0 .9 8 1 .0 1 4 .3 3 1 .0 2 1 3 .0 0 0 .9 9 1 .0 0 4 .4 0 1 .0 1 f PO =3 3 7 .9 8 0 .9 7 9 1 .0 0 7 7 .4 5 7 1 .0 2 7 1 1 .8 3 0 .9 8 4 1 .0 0 3 7 .6 1 6 1 .0 0 8 1 9 .5 8 0 .9 8 7 1 .0 0 2 7 .6 6 2 1 .0 0 0 2 8 .1 5 0 .9 4 2 1 .0 2 1 1 .0 4 7 1 .2 2 8 9 .0 8 0 .9 6 9 1 .0 0 9 2 .3 9 0 1 .0 7 9 1 1 .7 2 0 .9 8 1 1 .0 0 4 2 .6 7 4 1 .0 2 9 2 .5 8 .6 9 0 .9 3 1 .0 2 1 .2 2 1 .2 9 1 0 .6 6 0 .9 8 1 .0 1 4 .1 9 1 .0 4 1 5 .0 5 0 .9 8 1 .0 0 4 .3 6 1 .0 2 f PO =5 3 1 0 .1 1 0 .9 7 4 1 .0 0 7 7 .1 1 6 1 .0 5 6 1 3 .8 9 0 .9 8 2 1 .0 0 3 7 .5 3 5 1 .0 2 1 2 1 .6 2 0 .9 8 6 1 .0 0 2 7 .6 3 8 1 .0 0 6 2 1 4 .7 2 0 .9 4 5 1 .0 1 1 0 .8 9 2 1 .2 4 6 1 5 .2 3 0 .9 4 5 1 .0 1 1 1 .0 5 6 1 .2 3 3 1 7 .1 1 0 .9 7 1 1 .0 0 5 2 .4 1 3 1 .0 7 9 3 1 5 .7 3 0 .9 3 1 .0 1 0 .9 1 1 .3 2 1 6 .2 8 0 .9 3 1 .0 1 1 .2 5 1 .2 9 2 0 .3 8 0 .9 8 1 .0 0 4 .2 1 1 .0 4 f PO =1 0 4 .5 1 6 .8 0 0 .9 2 0 1 .0 1 5 0 .9 3 4 1 .3 9 8 1 9 .1 9 0 .9 7 5 1 .0 0 4 7 .1 5 1 1 .0 5 5 2 6 .7 6 0 .9 8 3 1 .0 0 2 7 .5 4 5 1 .0 2 0 2 6 7 .5 6 0 .9 4 8 1 .0 0 2 0 .8 2 5 1 .2 5 3 6 8 .0 0 0 .9 4 8 1 .0 0 2 0 .8 4 0 1 .2 5 2 6 8 .9 0 0 .9 4 8 1 .0 0 2 0 .8 7 3 1 .2 5 1 2 .5 7 2 .4 6 0 .9 4 1 .0 0 0 .8 3 1 .3 3 7 2 .9 1 0 .9 4 1 .0 0 0 .8 5 1 .3 3 7 3 .8 4 0 .9 4 1 .0 0 0 .8 9 1 .3 3 f PO =5 0 3 7 7 .7 3 0 .9 2 4 1 .0 0 3 0 .8 3 0 1 .4 0 9 7 8 .1 9 0 .9 2 4 1 .0 0 3 0 .8 5 0 1 .4 0 8 7 9 .1 3 0 .9 2 4 1 .0 0 3 0 .9 0 2 1 .4 0 5 297 T ab le C -2 2 a: ( C o n ti n u ed ) C ar b o n S te el f o r T” 2 0 0 o F a n d S ta in le ss S te el f O = 2 .5 ȕ ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ O Ȗƍ P O 2 8 .4 9 0 .9 8 8 1 .0 0 3 2 .7 5 7 0 .9 9 4 2 .5 1 2 .7 0 0 .9 9 1 .0 0 4 .4 2 0 .9 9 f Po = 0.5 3 2 0 .9 5 0 .9 8 8 1 .0 0 1 7 .6 8 3 0 .9 9 3 2 9 .0 0 0 .9 8 8 1 .0 0 3 2 .7 5 5 0 .9 9 7 2 .5 1 3 .2 1 0 .9 9 1 .0 0 4 .4 2 1 .0 0 f PO =1 3 2 1 .4 5 0 .9 8 8 1 .0 0 1 7 .6 8 1 0 .9 9 4 2 1 1 .0 3 0 .9 8 5 1 .0 0 3 2 .7 3 6 1 .0 0 9 2 .5 1 5 .2 3 0 .9 9 1 .0 0 4 .4 1 1 .0 0 f PO =3 3 2 3 .4 7 0 .9 8 7 1 .0 0 1 7 .6 6 9 0 .9 9 8 2 1 3 .0 9 0 .9 8 3 1 .0 0 3 2 .7 0 4 1 .0 2 1 2 .5 1 7 .2 7 0 .9 8 1 .0 0 4 .3 8 1 .0 1 f PO =5 3 2 5 .4 9 0 .9 8 6 1 .0 0 1 7 .6 5 3 1 .0 0 3 2 1 8 .3 9 0 .9 7 6 1 .0 0 4 2 .5 5 2 1 .0 5 6 3 2 2 .4 7 0 .9 8 1 .0 0 4 .2 8 1 .0 3 f PO = 10 4 .5 3 0 .6 1 0 .9 8 4 1 .0 0 1 7 .5 9 0 1 .0 1 4 2 6 9 .3 7 0 .9 4 7 1 .0 0 2 0 .8 9 4 1 .2 5 0 2 .5 7 4 .3 2 0 .9 4 1 .0 0 0 .9 1 1 .3 2 f PO = 50 3 7 9 .6 3 0 .9 2 3 1 .0 0 3 0 .9 3 6 1 .4 0 4 298 T ab le C -2 2 b : M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T > 2 0 0 o F f o r g 1 2 C ar b o n S te el f o r T> 2 0 0 o F f O = 0 .5 f O = 1 f O = 2 ȕ ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ O Ȗƍ P O ȝ f y Iƍ fu Ȗƍ ǹ Ȗƍ O Ȗƍ P O ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ O Ȗƍ P O 2 2 .9 8 0 .9 4 7 1 .0 1 7 2 .6 0 7 1 .0 6 6 4 .3 8 0 .9 6 0 1 .0 0 9 2 .6 9 4 0 .9 9 9 7 .1 9 0 .9 6 6 1 .0 0 4 2 .7 2 3 0 .9 9 5 2 .5 3 .8 2 0 .9 5 1 .0 1 4 .2 8 1 .0 0 6 .0 8 0 .9 6 1 .0 1 4 .3 5 1 .0 0 1 0 .6 0 0 .9 7 1 .0 0 4 .3 7 0 .9 9 f Po = 0.5 3 5 .4 8 0 .9 6 1 1 .0 0 7 7 .5 2 7 0 .9 9 7 9 .4 1 0 .9 6 6 1 .0 0 3 7 .5 8 4 0 .9 9 4 1 7 .2 7 0 .9 6 8 1 .0 0 2 7 .6 0 7 0 .9 9 3 2 3 .5 2 0 .9 3 8 1 .0 1 8 2 .5 2 1 1 .0 2 3 4 .9 0 0 .9 5 6 1 .0 0 9 2 .6 6 8 1 .0 0 6 7 .7 1 0 .9 6 4 1 .0 0 4 2 .7 1 4 0 .9 9 9 2 .5 4 .3 5 0 .9 5 1 .0 1 4 .2 2 1 .0 1 6 .6 0 0 .9 6 1 .0 1 4 .3 3 1 .0 0 1 1 .1 1 0 .9 7 1 .0 0 4 .3 7 1 .0 0 f PO =1 3 6 .0 1 0 .9 5 7 1 .0 0 7 7 .4 8 0 1 .0 0 3 9 .9 3 0 .9 6 4 1 .0 0 3 7 .5 6 6 0 .9 9 7 1 7 .7 8 0 .9 6 7 1 .0 0 2 7 .6 0 0 0 .9 9 4 2 5 .8 7 0 .8 7 6 1 .0 2 5 1 .3 8 1 1 .1 4 4 7 .0 6 0 .9 3 7 1 .0 0 9 2 .4 8 7 1 .0 4 0 9 .8 0 0 .9 5 6 1 .0 0 4 2 .6 6 2 1 .0 1 4 2 .5 6 .5 7 0 .9 2 1 .0 1 3 .6 7 1 .0 6 8 .7 1 0 .9 5 1 .0 1 4 .2 0 1 .0 2 1 3 .1 9 0 .9 6 1 .0 0 4 .3 2 1 .0 1 f PO =3 3 8 .1 4 0 .9 4 3 1 .0 0 7 7 .1 7 4 1 .0 2 8 1 2 .0 1 0 .9 5 7 1 .0 0 3 7 .4 7 0 1 .0 0 9 1 9 .8 5 0 .9 6 4 1 .0 0 2 7 .5 6 4 1 .0 0 0 2 8 .5 7 0 .8 7 0 1 .0 1 8 0 .9 8 8 1 .1 8 8 9 .3 6 0 .9 1 1 1 .0 1 0 2 .0 5 6 1 .0 9 1 1 1 .9 5 0 .9 4 7 1 .0 0 4 2 .5 7 9 1 .0 3 0 2 .5 9 .2 3 0 .8 4 1 .0 2 1 .0 7 1 .2 4 1 0 .9 1 0 .9 4 1 .0 1 3 .9 6 1 .0 5 1 5 .3 0 0 .9 5 1 .0 0 4 .2 6 1 .0 2 f PO =5 3 9 .9 5 0 .8 1 6 1 .0 2 5 1 .2 5 0 1 .2 9 4 1 4 .1 4 0 .9 5 0 1 .0 0 3 7 .3 2 8 1 .0 2 1 2 1 .9 3 0 .9 6 1 1 .0 0 2 7 .5 1 8 1 .0 0 6 2 1 5 .4 5 0 .8 7 4 1 .0 1 0 0 .8 8 0 1 .2 0 5 1 5 .9 8 0 .8 7 4 1 .0 0 9 0 .9 9 8 1 .1 9 6 1 7 .6 1 0 .9 1 7 1 .0 0 5 2 .1 2 8 1 .0 8 8 3 1 6 .6 8 0 .8 5 1 .0 1 0 .9 0 1 .2 7 1 7 .2 6 0 .8 5 1 .0 1 1 .0 9 1 .2 5 2 0 .7 4 0 .9 4 1 .0 0 4 .0 0 1 .0 5 f PO = 10 4 .5 1 8 .0 1 0 .8 2 0 1 .0 1 3 0 .9 1 6 1 .3 3 0 1 8 .6 5 0 .8 2 3 1 .0 1 2 1 .3 4 5 1 .2 9 9 2 7 .2 3 0 .9 5 2 1 .0 0 2 7 .3 5 4 1 .0 2 1 2 7 0 .7 0 0 .8 7 9 1 .0 0 2 0 .8 2 4 1 .2 1 5 7 1 .1 7 0 .8 7 9 1 .0 0 2 0 .8 3 6 1 .2 1 4 7 2 .1 4 0 .8 7 8 1 .0 0 2 0 .8 6 6 1 .2 1 3 2 .5 7 6 .6 3 0 .8 5 1 .0 0 0 .8 3 1 .2 8 7 7 .1 2 0 .8 5 1 .0 0 0 .8 4 1 .2 8 7 8 .1 3 0 .8 5 1 .0 0 0 .8 8 1 .2 8 f PO = 50 3 8 3 .0 7 0 .8 2 7 1 .0 0 3 0 .8 2 8 1 .3 4 6 8 3 .5 7 0 .8 2 6 1 .0 0 3 0 .8 4 6 1 .3 4 4 8 4 .6 2 0 .8 2 5 1 .0 0 3 0 .8 9 1 1 .3 4 1 299 T ab le C -2 2 b : (C o n ti n u ed ) C ar b o n S te el f o r T> 2 0 0 o F f O = 2 .5 ȕ ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ O Ȗƍ P O 2 8 .6 0 0 .9 6 8 1 .0 0 3 2 .7 2 8 0 .9 9 4 2 .5 1 2 .8 6 0 .9 7 1 .0 0 4 .3 8 0 .9 9 f Po = 0.5 3 2 1 .2 0 0 .9 6 8 1 .0 0 1 7 .6 1 2 0 .9 9 3 2 9 .1 1 0 .9 6 6 1 .0 0 3 2 .7 2 1 0 .9 9 7 2 .5 1 3 .3 7 0 .9 7 1 .0 0 4 .3 7 1 .0 0 f PO =1 3 2 1 .7 1 0 .9 6 8 1 .0 0 1 7 .6 0 6 0 .9 9 4 2 1 1 .2 0 0 .9 5 9 1 .0 0 3 2 .6 8 5 1 .0 0 9 2 .5 1 5 .4 4 0 .9 6 1 .0 0 4 .3 4 1 .0 0 f PO =3 3 2 3 .7 7 0 .9 6 5 1 .0 0 1 7 .5 7 9 0 .9 9 8 2 1 3 .3 2 0 .9 5 2 1 .0 0 3 2 .6 3 0 1 .0 2 2 2 .5 1 7 .5 4 0 .9 5 8 1 .0 0 2 4 .2 9 9 1 .0 1 1 f PO =5 3 2 5 .8 5 0 .9 6 3 1 .0 0 1 7 .5 4 6 1 .0 3 3 2 1 8 .8 4 0 .9 3 2 1 .0 0 4 2 .3 7 9 1 .0 6 0 2 .5 2 2 .9 1 0 .9 5 1 .0 0 4 .1 3 1 .0 3 f PO = 10 3 3 1 .1 1 0 .9 5 6 1 .0 0 1 7 .4 3 3 1 .0 1 5 2 7 2 .6 3 0 .8 7 8 1 .0 0 2 0 .8 8 2 1 .2 1 1 2 .5 7 8 .6 5 0 .8 5 1 1 .0 0 2 0 .9 0 0 1 .2 7 4 f PO = 50 3 8 5 .1 7 0 .8 2 5 1 .0 0 3 0 .9 1 9 1 .3 3 9 300 T ab le C -2 3 a: A d ju st ed N o m in al R es is ta n ce F ac to r fo r an d C ar b o n S te el a n d Ȗ ǹ = 1 .1 , Ȗ P O = 1 .2 a n d Ȗ O = 1 .5 f o r g 1 2 C ar b o n S te el f O = 0 .5 f O = 1 .0 f O = 2 .0 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 2 0 .8 3 (1 ) 0 .7 6 (1 ) 0 .8 3 (1 ) 0 .7 8 (1 ) 0 .8 1 0 .7 3 (1 ) 0 .6 6 (1 ) 0 .7 3 (1 ) 0 .6 8 (1 ) 0 .7 5 0 .6 4 (1 ) 0 .5 8 (1 ) 0 .6 4 (1 ) 0 .6 0 (1 ) 0 .7 0 2 .5 0 .6 5 (1 ) 0 .8 9 0 .6 5 (1 ) 0 .6 1 (1 ) 0 .6 3 0 .5 2 (1 ) 0 .6 5 0 .5 2 (1 ) 0 .4 9 (1 ) 0 .5 4 0 .4 3 (1 ) 0 .5 7 0 .4 3 (1 ) 0 .4 1 (1 ) 0 .4 8 f PO = 0.5 3 0 .5 8 0 .5 3 0 .5 8 0 .5 5 0 .4 4 0 .4 6 0 .4 2 0 .4 6 0 .4 4 0 .3 5 0 .3 9 0 .3 5 0 .3 9 0 .3 7 0 .2 9 2 0 .8 9 (1 ) 0 .8 1 (1 ) 0 .8 8 (1 ) 0 8 3 (1 ) 0 .8 4 0 .7 8 (1 ) 0 .7 1 (1 ) 0 .7 8 (1 ) 0 .7 3 (1 ) 0 .7 8 0 .6 7 (1 ) 0 .6 2 (1 ) 0 .6 8 (1 ) 0 .6 4 (1 ) 0 .7 3 2 .5 0 .7 0 (1 ) 0 .8 3 0 .7 1 (1 ) 0 .6 7 (1 ) 0 .6 8 0 .5 8 (1 ) 0 .7 0 0 .5 8 (1 ) 0 .5 4 (1 ) 0 .5 8 0 .4 7 (1 ) 0 .6 1 0 .4 7 (1 ) 0 .4 4 (1 ) 0 .5 0 f PO =1 3 0 .6 5 0 .6 0 0 .6 5 0 .6 2 0 .4 9 0 .5 1 0 .4 7 0 .5 1 0 .4 8 0 .3 9 0 .4 2 0 .3 8 0 .4 2 0 .3 9 0 .3 1 2 0 .9 7 (1 ) 0 .8 9 (1 ) 0 .9 5 (1 ) 0 .8 9 (1 ) 0 .8 8 0 .8 9 (1 ) 0 .8 2 (1 ) 0 .8 9 (1 ) 0 .8 3 (1 ) 0 .8 6 0 .7 8 (1 ) 0 .7 1 (1 ) 0 .7 8 (1 ) 0 .7 4 (1 ) 0 .8 0 2 .5 0 .8 6 (1 ) 0 .9 7 0 .8 5 (1 ) 0 .8 0 (1 ) 0 .7 9 0 .7 2 (1 ) 0 .7 9 0 .7 2 (1 ) 0 .6 8 (1 ) 0 .7 0 0 .5 8 (1 ) 0 .7 2 0 .5 8 (1 ) 0 .5 5 (1 ) 0 .5 9 f PO =3 3 0 .8 5 0 .7 7 0 .8 5 0 .8 0 0 .6 4 0 .6 7 0 .6 1 0 .6 7 0 .6 3 0 .5 0 0 .5 2 0 .4 8 0 .5 2 0 .4 9 0 .3 9 2 0 .9 7 (1 ) 0 .8 8 (1 ) 0 .9 4 (1 ) 0 .8 8 (1 ) 0 .8 6 0 .9 5 (1 ) 0 .8 6 (1 ) 0 .9 3 (1 ) 0 .8 8 (1 ) 0 .8 9 0 .8 5 (1 ) 0 .7 8 (1 ) 0 .8 5 (1 ) 0 .8 0 (1 ) 0 .8 4 2 .5 0 .9 1 (1 ) 1 .0 2 0 .8 7 (1 ) 0 .8 2 (1 ) 0 .8 0 0 .8 0 (1 ) 0 .9 3 1 .0 1 (1 ) 0 .9 5 (1 ) 0 .6 2 0 .6 6 (1 ) 0 .8 0 0 .6 6 (1 ) 0 .6 2 (1 ) 0 .6 6 f PO =5 3 0 .9 6 0 .8 7 0 .9 9 0 .9 3 0 .7 4 0 .7 8 0 .7 1 0 .7 8 0 .7 3 0 .5 9 0 .6 1 0 .5 5 0 .6 1 0 .5 7 0 .4 6 2 0 .9 5 (1 ) 0 .8 6 (1 ) 0 .9 2 (1 ) 0 .8 6 (1 ) 0 .8 4 0 .9 6 (1 ) 0 .8 8 (1 ) 0 .9 3 (1 ) 0 .8 8 (1 ) 0 .8 7 0 .9 4 (1 ) 0 .8 5 (1 ) 0 .9 3 (1 ) 0 .8 7 (1 ) 0 .8 9 3 0 .8 9 (1 ) 0 .9 8 0 .8 5 (1 ) 0 .8 0 (1 ) 0 .7 8 0 .9 0 (1 ) 1 .0 1 0 .8 6 (1 ) 0 .8 1 (1 ) 0 .8 0 0 .7 9 (1 ) 0 .9 2 0 .7 9 (1 ) 0 .7 4 (1 ) 0 .7 5 f PO =10 4 .5 1 .0 1 0 .9 2 0 .9 6 0 .9 0 0 .7 2 0 .9 4 0 .8 6 0 .9 9 0 .9 3 0 .7 4 0 .7 6 0 .7 0 0 .7 6 0 .7 2 0 .5 7 2 0 .9 2 (1 ) 0 .8 4 (1 ) 0 .9 0 (1 ) 0 .8 4 (1 ) 0 .8 1 0 .9 3 (1 ) 0 .8 5 (1 ) 0 .9 0 (1 ) 0 .8 5 (1 ) 0 .8 2 0 .9 4 (1 ) 0 .8 5 (1 ) 0 .9 1 (1 ) 0 .8 5 (1 ) 0 .8 3 2 .5 0 .8 6 (1 ) 0 .9 5 0 .8 3 (1 ) 0 .7 8 (1 ) 0 .7 2 0 .8 6 (1 ) 0 .9 5 0 .8 3 (1 ) 0 .7 8 (1 ) 0 .7 6 0 .8 7 (1 ) 0 .9 7 0 .8 4 (1 ) 0 .7 9 (1 ) 0 .7 7 f PO =50 3 0 .9 7 0 .8 8 0 .9 2 0 .8 6 0 .6 9 0 .9 7 0 .8 9 0 .9 3 0 .8 7 0 .7 0 0 .9 9 0 .9 1 0 .9 4 0 .8 9 0 .7 1 (1 ) Ȗ ǹ = Ȗ P 0= 1 a n d Ȗ O = 0 .9 301 T ab le C -2 3 a: ( C o n ti n u ed ) C ar b o n S te el f O = 2 .5 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 2 0 .6 1 (1 ) 0 .5 6 (1 ) 0 .6 2 (1 ) 0 .5 8 (1 ) 0 .6 9 2 .5 0 .4 1 (1 ) 0 .5 6 0 .4 1 (1 ) 0 .3 9 (1 ) 0 .4 6 f PO = 0.5 3 0 .3 7 0 .3 4 0 .3 7 0 .3 5 0 .2 8 2 0 .6 5 (1 ) 0 .5 9 (1 ) 0 .6 5 (1 ) 0 .6 1 (1 ) 0 .7 1 2 .5 0 .4 4 (1 ) 0 .5 9 0 .4 4 (1 ) 0 .4 2 (1 ) 0 .4 8 f PO =1 3 0 .3 9 0 .3 6 0 .4 0 0 .3 7 0 .3 0 2 0 .7 5 (1 ) 0 .6 8 (1 ) 0 .7 5 (1 ) 0 .7 0 (1 ) 0 .7 8 2 .5 0 .5 4 (1 ) 0 .6 8 0 .5 4 (1 ) 0 .5 1 (1 ) 0 .5 6 f PO =3 3 0 .4 8 0 .4 4 0 .4 9 0 .4 6 0 .3 7 2 0 .8 1 (1 ) 0 .7 4 (1 ) 0 .8 1 (1 ) 0 .7 6 (1 ) 0 .8 2 2 .5 0 .6 2 (1 ) 0 .7 5 0 .6 2 (1 ) 0 .5 8 (1 ) 0 .6 2 f PO =5 3 0 .5 6 0 .5 1 0 .5 6 0 .5 3 0 .4 2 2 0 .9 1 (1 ) 0 .8 3 (1 ) 0 .9 0 (1 ) 0 .8 5 (1 ) 0 .8 7 3 0 .7 4 (1 ) 0 .8 8 0 .7 4 (1 ) 0 .7 0 (1 ) 0 .7 2 f PO =10 4 .5 0 .7 0 0 .6 4 0 .7 0 0 .6 6 0 .5 3 2 0 .9 4 (1 ) 0 .8 6 (1 ) 0 .9 1 (1 ) 0 .8 6 (1 ) 0 .8 4 2 .5 0 .8 8 (1 ) 0 .9 8 0 .8 4 (1 ) 0 .7 9 (1 ) 0 .7 7 f PO =50 3 1 .0 0 0 .9 1 0 .9 5 0 .8 9 0 .7 2 (1 ) F o r th es e fa ct o rs Ȗ ǹ = Ȗ P O = 1 a n d Ȗ O = 0 .9 302 T ab le C -2 3 b : A d ju st ed N o m in al R es is ta n ce F ac to r fo r g 1 2 a n d S ta in le ss S te el f o r Ȗ ǹ = 1 .1 , Ȗ P O = 1 .2 a n d Ȗ O = 1 .5 S ta in le ss S te el f O = 0 .5 f O = 1 .0 f O = 2 .0 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 2 0 .8 2 (1 ) 0 .7 1 (1 ) 0 .8 2 0 .8 2 0 .7 9 0 .7 1 (1 ) 0 .6 2 (1 ) 0 .7 6 0 .7 6 0 .7 4 0 .6 2 (1 ) 0 .5 4 (1 ) 0 .7 1 0 .7 1 0 .6 9 2 .5 0 .6 4 (1 ) 0 .7 1 0 .6 4 0 .6 4 0 .6 2 0 .5 1 (1 ) 0 .6 1 0 .5 5 0 .5 5 0 .5 3 0 .4 2 (1 ) 0 .5 4 0 .4 8 0 .4 8 0 .4 7 f PO = 0.5 3 0 .5 7 0 .4 9 0 .4 2 0 .4 2 0 .4 3 0 .4 5 0 .3 9 0 .3 5 0 .3 5 0 .3 4 0 .3 8 0 .3 3 0 .3 0 0 .3 0 0 .2 9 2 0 .8 7 (1 ) 0 .7 6 (1 ) 0 .8 6 0 .8 6 0 .8 3 0 .7 6 (1 ) 0 .6 6 (1 ) 0 .8 0 0 .8 0 0 .7 7 0 .6 6 (1 ) 0 .5 7 (1 ) 0 .7 4 0 .7 4 0 .7 1 2 .5 0 .7 1 (1 ) 0 .7 7 0 .6 9 0 .6 9 0 .7 7 0 .5 7 (1 ) 0 .6 6 0 .5 9 0 .5 9 0 .5 7 0 .4 6 (1 ) 0 .5 7 0 .5 1 0 .5 1 0 .4 9 f PO =1 3 0 .6 4 0 .5 6 0 .5 0 0 .5 0 0 .4 8 0 .5 0 0 .4 4 0 .3 9 0 .3 9 0 .3 8 0 .4 1 0 .6 0 .3 2 0 .3 2 0 .3 1 2 0 .9 5 (1 ) 0 .8 3 (1 ) 0 .9 2 0 .9 2 0 .8 9 0 .8 8 (1 ) 0 .7 6 (1 ) 0 .8 8 0 .8 8 0 .8 5 0 .7 7 (1 ) 0 .6 7 (1 ) 0 .8 1 0 .8 1 0 .7 8 2 .5 0 .8 4 (1 ) 0 .9 0 0 .8 1 0 .8 1 0 .7 8 0 .7 1 (1 ) 0 .7 9 0 .7 1 0 .7 1 0 .6 8 0 .5 7 (1 ) 0 .6 7 0 .6 0 0 .6 0 0 .6 0 f PO =3 3 0 .8 3 0 .7 2 0 .6 5 0 .6 5 0 .6 3 0 .6 6 0 .5 7 0 .5 1 0 .5 1 0 .5 0 0 .5 7 0 .5 0 0 .4 0 0 .4 0 0 .3 9 2 0 .9 5 (1 ) 0 .8 3 (1 ) 0 .9 1 0 .9 1 0 .8 8 0 .9 3 (1 ) 0 .8 1 (1 ) 0 .9 1 0 .9 1 0 .8 8 0 .8 4 (1 ) 0 .7 2 (1 ) 0 .8 6 0 .8 6 0 .8 3 2 .5 0 .8 9 (1 ) 0 .9 5 0 .8 5 0 .8 5 0 .8 2 0 .7 9 (1 ) 0 .8 7 0 .7 8 0 .7 8 0 .7 5 0 .6 5 (1 ) 0 .7 4 0 .6 7 0 .6 7 0 .6 4 f PO =5 3 0 .9 4 0 .8 2 0 .7 3 0 .7 3 0 .7 1 0 .7 7 0 .6 6 0 .6 0 0 .6 0 0 .5 8 0 .6 0 0 .5 2 0 .4 6 0 .4 6 0 .4 5 2 0 .9 3 (1 ) 0 .8 1 (1 ) 0 .8 8 0 .8 8 0 .8 5 0 .9 4 (1 ) 0 .8 2 (1 ) 0 .9 1 0 .9 1 0 .8 8 0 .9 2 (1 ) 0 .8 0 (1 ) 0 .9 1 0 .9 1 0 .8 8 3 0 .8 7 (1 ) 0 .9 2 0 .8 2 0 .8 2 0 .8 0 0 .8 8 (1 ) 0 .9 5 0 .8 5 0 .8 5 0 .8 2 0 .7 8 (1 ) 0 .8 6 0 .7 7 0 .7 7 0 .7 4 f PO =10 4 .5 0 .9 9 0 .8 6 0 .7 7 0 .7 7 0 .7 5 0 .9 3 0 .8 0 0 .7 2 0 .7 2 0 .7 0 0 .7 5 0 .6 5 0 .5 8 0 .5 8 0 .5 6 2 0 .9 1 (1 ) 0 .7 9 (1 ) 0 .8 5 0 .8 5 0 .8 2 0 .9 1 (1 ) 0 .7 9 (1 ) 0 .8 6 0 .8 6 0 .8 3 0 .9 2 (1 ) 0 .8 0 (1 ) 0 .8 7 0 .8 7 0 .8 4 2 .5 0 .8 5 (1 ) 0 .8 8 0 .7 9 0 .7 9 0 .7 7 0 .8 5 (1 ) 0 .8 9 0 .8 0 0 .8 0 0 .7 1 0 .8 6 (1 ) 0 .9 1 0 .8 1 0 .8 1 0 .7 9 f PO =50 3 0 .9 5 0 .8 2 0 .7 4 0 .7 4 0 .7 1 0 .9 6 0 .8 3 0 .7 5 0 .7 5 0 .7 2 0 .9 7 0 .8 5 0 .7 6 0 .7 6 0 .7 3 (1 ) F o r th es e fa ct o rs Ȗ ǹ = Ȗ P O = 1 a n d Ȗ O = 0 .9 303 T ab le C -2 3 b : (C o n ti n u ed ) S ta in le ss S te el f O = 2 .5 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 2 0 .6 0 (1 ) 0 .5 2 (1 ) 0 .7 0 0 .7 0 0 .6 7 2 .5 0 .4 0 (1 ) 0 .5 2 0 .4 7 0 .4 7 0 .4 5 f PO = 0.5 3 0 .3 6 0 .3 1 0 .2 8 0 .2 8 0 .2 7 2 0 .6 3 (1 ) 0 .5 5 (1 ) 0 .7 2 0 .7 2 0 .7 0 2 .5 0 .4 3 (1 ) 0 .5 5 0 .4 9 0 .4 9 0 .4 7 f PO =1 3 0 .3 9 0 .3 4 0 .3 0 0 .3 0 0 .2 9 2 0 .7 3 (1 ) 0 .6 4 (1 ) 0 .7 9 0 .7 9 0 .7 6 2 .5 0 .5 3 (1 ) 0 .6 4 0 .5 7 0 .5 7 0 .5 5 f PO =3 3 0 .4 8 0 .4 1 0 .3 7 0 .3 7 0 .3 6 2 0 .8 0 (1 ) 0 .6 9 (1 ) 0 .8 3 0 .8 3 0 .8 1 2 .5 0 .6 1 (1 ) 0 .7 0 0 .6 3 0 .6 3 0 .6 1 f PO =5 3 0 .5 5 0 .4 8 0 .4 3 0 .4 3 0 .4 1 2 0 .8 9 (1 ) 0 .7 7 (1 ) 0 .9 0 0 .9 0 0 .8 7 2 .5 0 .7 3 (1 ) 0 .8 2 0 .7 3 0 .7 3 0 .7 1 f PO =10 3 0 .6 9 0 .6 0 0 .5 4 0 .5 4 0 .5 2 2 0 .9 2 (1 ) 0 .8 0 (1 ) 0 .8 8 0 .8 8 0 .8 5 2 .5 0 .8 6 (1 ) 0 .9 1 0 .8 2 0 .8 2 0 .7 9 f PO =50 3 0 .9 8 0 .8 5 0 .7 7 0 .7 7 0 .7 4 (1 ) F o r th es e fa ct o rs Ȗ ǹ = Ȗ P O = 1 a n d Ȗ O = 0 .9 304 305 C.11. Performance Function g13 Table C-24 gives the calculated mean load and resistance factors for performance function g13. In this table, ȝfu is the converged mean value of the ultimate strength of steel. Table C-25 shows the evaluated adjusted nominal resistance factors for load factors Ȗǹ=1.1, ȖPC=1.2, and ȖM=1.2. T ab le C -2 4 a: M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T ”2 0 0 o F a n d S ta in le ss S te el f o r g 1 3 C ar b o n S te el T ”2 0 0 o F & S ta in le ss S te el f M = 0 .5 f M = 1 .0 f M = 2 .0 ȕ ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P C Ȗƍ M ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P C Ȗƍ M ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P C Ȗƍ M 3 2 .6 3 0 .8 9 1 .1 3 1 .2 7 1 .1 7 3 .3 4 0 .9 0 1 .1 0 1 .1 3 1 .3 3 4 .8 7 0 .9 1 1 .0 6 1 .0 4 1 .4 3 4 .5 3 .0 7 0 .8 6 1 .1 6 1 .7 4 1 .2 2 3 .8 9 0 .8 5 1 .1 3 1 .2 7 1 .5 6 5 .8 3 0 .8 7 1 .0 7 1 .0 7 1 .7 4 f PC = 0.5 5 .5 3 .4 4 0 .8 4 1 .1 6 2 .2 3 1 .2 4 4 .3 2 0 .8 3 1 .1 5 1 .4 1 1 .7 3 6 .5 9 0 .8 5 1 .0 8 1 .0 8 1 .9 9 3 3 .4 1 0 .9 2 1 .0 8 1 .5 1 1 .0 8 4 .0 4 0 .9 1 1 .0 7 1 .4 0 1 .2 0 5 .4 8 0 .9 1 1 .0 5 1 .1 6 1 .3 8 4 .5 4 .2 0 0 .8 9 1 .0 8 2 .1 3 1 .0 9 4 .8 5 0 .8 8 1 .0 8 1 .9 7 1 .2 3 6 .5 0 0 .8 7 1 .0 7 1 .3 2 1 .6 3 f PC =1 5 .5 4 .8 9 0 .8 8 1 .0 8 2 .6 7 1 .0 9 5 .5 6 0 .8 7 1 .0 8 2 .4 9 1 .2 4 7 .3 1 0 .8 5 1 .0 7 1 .5 0 1 .8 0 3 6 .8 5 0 .9 4 1 .0 3 1 .6 3 1 .0 2 7 .4 0 0 .9 4 1 .0 3 1 .6 2 1 .0 5 8 .5 6 0 .9 3 1 .0 3 1 .5 6 1 .1 2 4 .5 9 .1 2 0 .9 2 1 .0 3 2 .2 8 1 .0 2 9 .6 9 0 .9 2 1 .0 3 2 .2 6 1 .0 5 1 0 .8 8 0 .9 1 1 .0 3 2 .1 9 1 .1 3 f PC =3 5 .5 1 1 .1 2 0 .9 0 1 .0 3 2 .8 4 1 .0 2 1 1 .7 0 0 .9 0 1 .0 3 2 .8 1 1 .0 5 1 2 .9 2 0 .8 9 1 .0 3 2 .7 4 1 .1 4 3 1 0 .3 3 0 .9 5 1 .0 2 1 .6 5 1 .0 1 1 0 .8 7 0 .9 4 1 .0 2 1 .6 4 1 .0 2 1 1 .9 8 0 .9 4 1 .0 2 1 .6 2 1 .0 6 4 .5 1 4 .1 0 0 .9 2 1 .0 2 2 .3 0 1 .0 1 1 4 .6 5 0 .9 2 1 .0 2 2 .2 9 1 .0 3 1 5 .7 9 0 .9 2 1 .0 2 2 .2 6 1 .0 7 f PC =5 5 .5 1 7 .4 2 0 .9 1 1 .0 2 2 .8 6 1 .0 1 1 7 .9 8 0 .9 1 1 .0 2 2 .8 5 1 .0 3 1 9 .1 4 0 .9 0 1 .0 2 2 .8 2 1 .0 7 3 1 9 .0 7 0 .9 5 1 .0 1 1 .6 6 1 .0 0 1 9 .6 0 0 .9 5 1 .0 1 1 .6 6 1 .0 1 2 0 .6 7 0 .9 5 1 .0 1 1 .6 5 1 .0 2 4 .5 2 6 .5 6 0 .9 3 1 .0 1 2 .3 2 1 .0 0 2 7 .1 0 0 .9 3 1 .0 1 2 .3 1 1 .0 1 2 8 .2 0 0 .9 2 1 .0 1 2 .3 0 1 .0 3 f PC = 10 5 .5 3 3 .1 7 0 .9 1 1 .0 1 2 .8 8 1 .0 0 3 3 .7 2 0 .9 1 1 .0 1 2 .8 7 1 .0 1 3 4 .8 4 0 .9 1 1 .0 1 2 .8 6 1 .0 3 3 8 9 .0 2 0 .9 5 1 .0 0 1 .6 7 0 .9 9 8 9 .5 4 0 .9 5 1 .0 0 1 .6 6 0 .9 9 9 0 .5 9 0 .9 5 1 .0 0 1 .6 6 1 .0 0 4 .5 1 2 6 .3 0 .9 3 1 .0 0 2 .3 2 0 .9 9 1 2 6 .9 0 .9 3 1 .0 0 2 .3 2 0 .9 9 1 2 7 .9 0 .9 3 1 .0 0 2 .3 2 1 .0 0 f PC =5 0 5 .5 1 5 9 .3 0 .9 2 1 .0 0 2 .8 9 0 .9 9 1 5 9 .8 0 .9 2 1 .0 0 2 .8 8 0 .9 9 1 6 0 .9 0 .9 1 1 .0 0 2 .8 8 1 .0 0 306 T ab le C -2 4 b : M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T > 2 0 0 o F f o r g 1 3 C ar b o n S te el , T> 2 0 0 o F f M = 0 .5 f M = 1 .0 f M = 2 .0 ȕ ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P C Ȗƍ M ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P C Ȗƍ M ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P C Ȗƍ M 3 2 .8 8 0 .7 8 1 .1 1 1 .1 5 1 .1 3 3 .6 4 0 .7 9 1 .0 8 1 .0 9 1 .2 4 5 .2 7 0 .8 0 1 .0 5 1 .0 3 1 .3 3 4 .5 3 .4 8 0 .7 0 1 .1 4 1 .3 9 1 .2 0 4 .4 1 0 .7 0 1 .1 1 1 .1 7 1 .4 1 6 .5 1 0 .7 3 1 .0 7 1 .0 6 1 .5 7 f PC = 0.5 5 .5 3 .9 6 0 .6 6 1 .1 6 1 .7 1 1 .2 2 5 .0 2 0 .6 5 1 .1 3 1 .2 5 1 .5 3 7 .5 2 0 .6 8 1 .0 7 1 .0 7 1 .7 6 3 3 .6 8 0 .8 1 1 .0 7 1 .3 6 1 .0 8 4 .3 8 0 .8 0 1 .0 6 1 .2 5 1 .1 7 5 .9 3 0 .8 0 1 .0 5 1 .1 1 1 .2 9 4 .5 4 .6 2 0 .7 6 1 .0 8 1 .8 7 1 .0 9 5 .4 0 0 .7 3 1 .0 8 1 .6 6 1 .2 2 7 .2 9 0 .7 2 1 .0 6 1 .2 2 1 .4 9 f PC =1 5 .5 5 .4 6 0 .7 2 1 .0 8 2 .3 2 1 .0 9 6 .2 8 0 .7 0 1 .0 8 2 .0 8 1 .2 3 8 .3 9 0 .6 7 1 .0 7 1 .3 2 1 .6 3 3 7 .2 3 0 .8 5 1 .0 3 1 .5 4 1 .0 2 7 .8 4 0 .8 4 1 .0 3 1 .5 1 1 .0 5 9 .1 3 0 .8 3 1 .0 2 1 .4 3 1 .1 2 4 .5 9 .8 1 0 .8 1 1 .0 3 2 .1 2 1 .0 2 1 0 .4 6 0 .8 0 1 .0 3 2 .0 8 1 .0 5 1 1 .8 5 0 .7 8 1 .0 3 1 .9 8 1 .1 3 f PC =3 5 .5 1 2 .1 5 0 .7 7 1 .0 3 2 .6 1 1 .0 2 1 2 .8 2 0 .7 6 1 .0 3 2 .5 6 1 .0 5 1 4 .2 7 0 .7 5 1 .0 3 2 .4 5 1 .1 4 3 1 0 .8 6 0 .8 6 1 .0 2 1 .5 7 1 .0 1 1 1 .4 5 0 .8 6 1 .0 2 1 .5 6 1 .0 2 1 2 .6 7 0 .8 5 1 .0 2 1 .5 2 1 .0 6 4 .5 1 5 .1 0 0 .8 2 1 .0 2 2 .1 6 1 .0 1 1 5 .7 2 0 .8 1 1 .0 2 2 .1 4 1 .0 3 1 7 .0 2 0 .8 0 1 .0 2 2 .0 9 1 .0 7 f PC =5 5 .5 1 8 .9 3 0 .7 8 1 .0 2 2 .6 5 1 .0 1 1 9 .5 8 0 .7 8 1 .0 2 2 .6 3 1 .0 3 2 0 .9 4 0 .7 7 1 .0 2 2 .5 8 1 .0 7 3 1 9 .9 8 0 .8 7 1 .0 1 1 .5 9 1 .0 0 2 0 .5 5 0 .8 7 1 .0 1 1 .5 8 1 .0 1 2 1 .7 3 0 .8 6 1 .0 1 1 .5 7 1 .0 2 4 .5 2 8 .3 5 0 .8 2 1 .0 1 2 .1 9 1 .0 0 2 8 .9 6 0 .8 2 1 .0 1 2 .1 8 1 .0 1 3 0 .2 0 0 .8 2 1 .0 1 2 .1 6 1 .0 3 f PC = 10 5 .5 3 5 .9 3 0 .7 9 1 .0 1 2 .6 8 1 .0 0 3 6 .5 7 0 .7 9 1 .0 1 2 .6 7 1 .0 1 3 7 .8 6 0 .7 8 1 .0 1 2 .6 5 1 .0 3 3 9 3 .0 3 0 .8 8 1 .0 0 1 .6 0 0 .9 9 9 3 .5 9 0 .8 8 1 .0 0 1 .6 0 0 .9 9 9 4 .7 3 0 .8 8 1 .0 0 1 .6 0 1 .0 0 4 .5 1 3 4 .5 0 .8 3 1 .0 0 2 .2 0 0 .9 9 1 3 5 .1 0 .8 3 1 .0 0 2 .2 0 0 .9 9 1 3 6 .3 0 .8 3 1 .0 0 2 .2 0 1 .0 0 f PC =5 0 5 .5 1 7 2 .1 0 .8 0 1 .0 0 2 .7 1 0 .9 9 1 7 2 .7 0 .7 9 1 .0 0 2 .7 0 0 .9 9 1 7 3 .9 0 .7 9 1 .0 0 2 .7 0 1 .0 0 307 T ab le C -2 5 a: A d ju st ed N o m ia n l R es is ta n ce F ac to rs f o r C ar b o n S te el a n d Ȗ ǹ = 1 .1 , Ȗ Ȃ = Ȗ P C = 1 .2 f o r g 1 3 C ar b o n S te el f M = 0 .5 f M = 1 .0 f M = 2 .0 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 3 1 .0 0 (1 ) 0 .9 6 0 .9 8 0 .9 2 0 .7 4 0 .9 8 (1 ) 0 .9 4 0 .9 7 0 .9 1 0 .7 3 0 .9 3 (1 ) 0 .9 1 0 .9 3 0 .8 8 0 .7 0 4 .5 0 .9 0 0 .8 2 0 .8 1 0 .7 6 0 .6 1 0 .8 9 0 .8 1 0 .8 0 0 .7 5 0 .6 0 0 .8 3 0 .7 6 0 .7 6 0 .7 1 0 .5 7 f PC =0 .5 5 .5 0 .8 0 0 .7 3 0 .7 1 0 .6 7 0 .5 3 0 .8 0 0 .7 3 0 .7 0 0 .6 6 0 .5 3 0 .7 3 0 .6 7 0 .6 5 0 .6 2 0 .4 9 3 0 .9 9 (1 ) 0 .9 6 0 .9 9 0 .9 3 0 .7 4 0 .9 9 (1 ) 0 .9 6 0 .9 9 0 .9 3 0 .7 5 0 .9 6 (1 ) 0 .9 4 0 .9 7 0 .9 1 0 .7 3 4 .5 0 .8 5 0 .7 8 0 .7 9 0 .7 4 0 .5 9 0 .8 8 0 .8 0 0 .8 0 0 .7 6 0 .6 0 0 .8 7 0 .7 9 0 .7 9 0 .7 4 0 .5 9 f PC =1 5 .5 0 .7 3 0 .6 7 0 .6 7 0 .6 3 0 .5 0 0 .7 7 0 .7 0 0 .6 9 0 .6 5 0 .5 2 0 .7 7 0 .7 1 0 .6 8 0 .6 4 0 .5 2 3 0 .9 3 (1 ) 0 .9 1 0 .9 6 0 .9 0 0 .7 2 0 .9 5 (1 ) 0 .9 3 0 .9 8 0 .9 2 0 .7 3 0 .9 6 (1 ) 0 .9 5 0 .9 9 0 .9 3 0 .7 5 4 .5 0 .7 5 0 .6 8 0 .7 1 0 .6 7 0 .5 3 0 .7 8 0 .7 1 0 .7 3 0 .6 9 0 .5 5 0 .8 2 0 .7 5 0 .7 6 0 .7 2 0 .5 7 f PC =3 5 .5 0 .6 1 0 .5 6 0 .5 7 0 .5 4 0 .4 3 0 .6 4 0 .5 9 0 .6 0 0 .5 6 0 .4 5 0 .6 9 0 .6 3 0 .6 3 0 .6 0 0 .4 8 3 0 .9 0 (1 ) 0 .8 9 0 .9 4 0 .8 9 0 .7 1 0 .9 2 (1 ) 0 .9 0 0 .9 6 0 .9 0 0 .7 2 0 .9 4 (1 ) 0 .9 3 0 .9 8 0 .9 2 0 .7 3 4 .5 0 .7 1 0 .6 5 0 .6 8 0 .6 4 0 .5 1 0 .7 3 0 .6 7 0 .7 0 0 .6 5 0 .5 2 0 .7 7 0 .7 0 0 .7 3 0 .6 8 0 .5 5 f PB =5 5 .5 0 .5 8 0 .5 3 0 .5 4 0 .5 1 0 .4 1 0 .6 0 0 .5 5 0 .5 6 0 .5 3 0 .4 2 0 .6 3 0 .5 8 0 .5 9 0 .5 5 0 .4 4 3 0 .8 8 (1 ) 0 .8 7 0 .9 3 0 .8 7 0 .7 0 0 .8 9 (1 ) 0 .8 8 0 .9 3 0 .8 8 0 .7 0 0 .9 0 (1 ) 0 .8 9 0 .9 5 0 .8 9 0 .7 1 4 .5 0 .6 8 0 .6 3 0 .6 5 0 .6 1 0 .4 9 0 .7 0 0 .6 4 0 .6 6 0 .6 2 0 .5 0 0 .7 2 0 .6 6 0 .6 8 0 .6 4 0 .5 1 f PC =1 0 5 .5 0 .5 5 0 .5 0 0 .5 2 0 .4 8 0 .3 9 0 .5 6 0 .5 1 0 .5 3 0 .4 9 0 .4 0 0 .5 8 0 .5 3 0 .5 4 0 .5 1 0 .4 1 3 0 .8 6 (1 ) 0 .8 5 0 .9 1 0 .8 5 0 .6 8 0 .8 6 (1 ) 0 .8 5 0 .9 1 0 .8 6 0 .6 9 0 .8 6 (1 ) 0 .8 6 0 .9 2 0 .8 6 0 .6 9 4 .5 0 .6 6 0 .6 0 0 .6 3 0 .5 9 0 .4 7 0 .6 6 0 .6 0 0 .6 3 0 .5 9 0 .4 7 0 .6 7 0 .6 1 0 .6 4 0 .6 0 0 .4 8 f PC =5 0 5 .5 0 .5 2 0 .4 8 0 .4 9 0 .4 6 0 .3 7 0 .5 2 0 .4 8 0 .4 9 0 .4 6 0 .3 7 0 .5 3 0 .4 8 0 .5 0 0 .4 7 0 .3 7 (1 ) F o r th es e fa ct o rs Ȗ ǹ = Ȗ Ȃ = Ȗ P C = 1 .1 308 T ab le C -2 5 b : A d ju st ed N o m ia n l R es is ta n ce F ac to rs f o r S ta in le ss S te el a n d Ȗ ǹ = 1 .1 , Ȗ Ȃ = Ȗ P C = 1 .2 f o r g 1 3 S ta in le ss S te el f M = 0 .5 f M = 1 .0 f M = 2 .0 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 3 0 .9 9 (1 ) 0 .9 0 0 .8 1 0 .8 1 0 .7 8 0 .9 6 (1 ) 0 .8 8 0 .7 9 0 .7 9 0 .7 6 0 .9 2 (1 ) 0 .8 5 0 .7 6 0 .7 6 0 .7 3 4 .5 0 .8 9 0 .7 7 0 .6 9 0 .6 9 0 .6 7 0 .8 7 0 .7 6 0 .6 8 0 .6 8 0 .6 6 0 .8 2 0 .7 1 0 .6 3 0 .6 3 0 .6 1 f PC =0 5 5 .5 0 .7 9 0 .6 9 0 .6 2 0 .6 2 0 .5 9 0 .7 9 0 .6 8 0 .6 1 0 .6 1 0 .5 9 0 .7 2 0 .6 3 0 .5 6 0 .5 6 0 .5 4 3 0 .9 8 (1 ) 0 .8 9 0 .8 0 0 .8 0 0 .7 8 0 .9 8 (1 ) 0 .9 0 0 .8 1 0 .8 1 0 .7 8 0 .9 5 (1 ) 0 .8 8 0 .7 9 0 .7 9 0 .7 6 4 .5 0 .8 4 0 .7 3 0 .6 5 0 .6 5 0 .6 3 0 .8 6 0 .7 5 0 .6 7 0 .6 7 0 .6 5 0 .8 5 0 .7 4 0 .6 6 0 .6 6 0 .6 4 f PC =1 5 .5 0 .7 2 0 .6 2 0 .5 6 0 .5 6 0 .5 4 0 .7 5 0 .6 5 0 .5 9 0 .5 9 0 .5 7 0 .7 6 0 .6 6 0 .5 9 0 .5 9 0 .5 7 3 0 .9 1 (1 ) 0 .8 5 0 .7 6 0 .7 6 0 .7 4 0 .9 3 (1 ) 0 .8 7 0 .7 8 0 .7 8 0 .7 5 0 .9 5 (1 ) 0 .8 9 0 .8 0 0 .8 0 0 .7 7 4 .5 0 .7 4 0 .6 4 0 .5 7 0 .5 7 0 .5 5 0 .7 6 0 .6 6 0 .5 9 0 .5 9 0 .5 7 0 .8 0 0 .7 0 0 .6 3 0 .6 3 0 .6 0 f PC =3 5 .5 0 .6 0 0 .5 2 0 .4 7 0 .4 7 0 .4 5 0 .6 3 0 .5 5 0 .4 9 0 .4 9 0 .4 7 0 .6 8 0 .5 9 0 .5 3 0 .5 3 0 .5 1 3 0 .8 9 (1 ) 0 .8 3 0 .7 5 0 .7 5 0 .7 2 0 .9 0 (1 ) 0 .8 4 0 .7 6 0 .7 6 0 .7 3 0 .9 2 (1 ) 0 .8 6 0 .7 8 0 .7 8 0 .7 5 4 .5 0 .7 0 0 .6 1 0 .5 5 0 .5 5 0 .5 3 0 .7 2 0 .6 3 0 .5 6 0 .5 6 0 .5 4 0 .7 6 0 .6 6 0 .5 9 0 .5 9 0 .. 5 7 f PC =5 5 .5 0 .5 7 0 .4 9 0 .4 4 0 .4 4 0 .4 3 0 .5 9 0 .5 1 0 .4 6 0 .4 6 0 .4 4 0 .6 2 0 .5 4 0 .4 9 0 .4 9 0 .4 7 3 0 .8 6 (1 ) 0 .8 1 0 .7 3 0 .7 3 0 .7 1 0 .8 7 (1 ) 0 .8 2 0 .7 4 0 .7 4 0 .7 1 0 .8 9 (1 ) 0 .8 4 0 .7 5 0 .7 5 0 .7 2 4 .5 0 .6 7 0 .5 8 0 .5 2 0 .5 2 0 .5 1 0 .6 8 0 .5 9 0 .5 3 0 .5 3 0 .5 1 0 .7 1 0 .6 1 0 .5 5 0 .5 5 0 .5 3 f PC = 10 5 .5 0 .5 4 0 .4 7 0 .4 2 0 .4 2 0 .4 1 0 .5 5 0 .4 8 0 .4 3 0 .4 3 0 .4 1 0 .5 7 0 .5 0 0 .4 4 0 .4 4 0 .4 3 3 0 .8 4 (1 ) 0 .8 0 0 .7 1 0 .7 1 0 .6 9 0 .8 4 (1 ) 0 .8 0 0 .7 2 0 .7 2 0 .6 9 0 .8 5 (1 ) 0 .8 0 0 .7 2 0 .7 2 0 .7 0 4 .5 0 .6 5 0 .5 6 0 .5 0 0 .5 0 0 .4 9 0 .6 5 0 .5 6 0 .5 1 0 .5 1 0 .4 9 0 .6 5 0 .5 7 0 .5 1 0 .5 1 0 .4 9 f PC =5 0 5 .5 0 .5 1 0 .4 4 0 .4 0 0 .4 0 0 .3 9 0 .5 2 0 .4 5 0 .4 0 0 .4 0 0 .3 9 0 .5 2 0 .4 5 0 .4 1 0 .4 1 0 .3 9 (1 ) F o r th es e fa ct o rs Ȗ ǹ = Ȗ Ȃ = Ȗ P C = 1 .1 309 310 C.12. Performance Function g14 Table C-26 gives the calculated mean load and resistance factors for performance function g14. In this table, ȝfu is the converged mean value of the ultimate strength of steel. Table C-27 shows the evaluated adjusted nominal resistance factors for nominal load factors Ȗǹ=1.1, ȖPO=1.2, ȖM=1.2, and ȖO=1.5. T ab le C -2 6 a: M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T ”2 0 0 o F f o r g 1 4 C ar b o n S te el T ”2 0 0 o F & S ta in le ss S te el ȕ= 2 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o 0 .5 4 .2 7 0 .9 8 1 .0 1 1 .0 0 1 .0 0 4 .3 5 4 .7 8 0 .9 8 1 .0 1 1 .0 0 1 .0 1 4 .3 2 5 .8 4 0 .9 7 1 .0 1 1 .0 0 1 .0 4 4 .1 8 1 6 .5 9 0 .9 6 1 .0 1 1 .0 0 1 .0 0 4 .3 3 7 .0 0 0 .9 8 1 .0 1 1 .0 0 1 .0 0 4 .3 9 8 .0 3 0 .9 8 1 .0 1 1 .0 0 1 .0 1 4 .3 6 2 1 0 .9 7 0 .9 9 1 .0 0 0 .9 9 0 .9 9 4 .4 2 1 1 .4 7 0 .9 9 1 .0 0 0 .9 9 1 .0 0 4 .4 1 1 2 .4 8 0 .9 9 1 .0 0 0 .9 9 1 .0 0 4 .4 0 0.5 2 .5 1 3 .2 1 0 .9 9 1 .0 0 0 .9 9 0 .9 9 4 .4 2 1 3 .7 1 0 .9 9 1 .0 0 0 .9 9 0 .9 9 4 .4 2 1 4 .7 2 0 .9 9 1 .0 0 0 .9 9 1 .0 0 4 .4 1 0 .5 4 .7 8 0 .9 8 1 .0 1 1 .0 1 1 .0 0 4 .3 2 5 .3 0 0 .9 8 1 .0 1 1 .0 1 1 .0 1 4 .2 8 6 .3 5 0 .9 7 1 .0 1 1 .0 1 1 .0 4 4 .1 3 1 7 .0 0 0 .9 8 1 .0 1 1 .0 0 1 .0 0 4 .3 9 7 .5 1 0 .9 8 1 .0 1 1 .0 0 1 .0 8 4 .3 8 8 .5 4 0 .9 8 1 .0 1 1 .0 0 1 .0 1 4 .3 4 2 1 1 .4 7 0 .9 9 1 .0 0 1 .0 0 0 .9 9 4 .4 1 1 1 .9 8 0 .9 9 1 .0 0 1 .0 0 1 .0 8 4 .4 1 1 2 .7 9 0 .9 9 1 .0 0 1 .0 0 1 .0 0 4 .4 0 1 2 .5 1 3 .7 1 0 .9 9 1 .0 0 1 .0 0 0 .9 9 4 .4 2 1 4 .2 1 0 .9 9 1 .0 0 1 .0 0 0 .9 9 4 .4 1 1 5 .2 2 0 .9 9 1 .0 0 1 .0 0 1 .0 0 4 .4 1 0 .5 6 .9 0 0 .9 7 1 .0 1 1 .0 6 1 .0 0 4 .0 1 7 .4 2 0 .9 7 1 .0 1 1 .0 6 1 .0 2 3 .9 5 8 .4 9 0 .9 6 1 .0 1 1 .0 6 1 .0 5 3 .7 2 1 9 .0 6 0 .9 8 1 .0 1 1 .0 2 1 .0 0 4 .3 2 9 .5 7 0 .9 8 1 .0 1 1 .0 2 1 .0 0 4 .3 0 1 0 .6 0 0 .9 8 1 .0 1 1 .0 2 1 .0 1 4 .2 6 2 1 3 .5 0 0 .9 9 1 .0 0 1 .0 1 0 .9 9 4 .3 9 1 4 .0 1 0 .9 8 1 .0 0 1 .0 1 1 .0 0 4 .3 9 1 5 .0 2 0 .9 8 1 .0 0 1 .0 1 1 .0 0 4 .3 8 3 2 .5 1 5 .7 3 0 .9 9 1 .0 0 1 .0 0 0 .9 9 4 .4 0 1 6 .2 4 0 .9 9 1 .0 0 1 .0 0 0 .9 9 4 .4 0 1 7 .2 5 0 .9 8 1 .0 0 1 .0 0 1 .0 0 4 .3 9 0 .5 9 .2 3 0 .9 3 1 .0 2 1 .2 8 1 .0 2 1 .1 9 9 .7 8 0 .9 2 1 .0 2 1 .2 8 1 .0 4 1 .1 7 1 0 .9 5 0 .9 2 1 .0 2 1 .2 6 1 .1 0 1 .1 1 1 1 1 .1 7 0 .9 8 1 .0 1 1 .0 4 1 .0 0 4 .1 8 1 1 .6 8 0 .9 7 1 .0 1 1 .0 4 1 .0 0 4 .1 6 1 2 .7 2 0 .9 7 1 .0 1 1 .0 4 1 .0 2 4 .1 1 2 1 5 .5 6 0 .9 8 1 .0 0 1 .0 2 0 .9 9 4 .3 6 1 6 .0 6 0 .9 8 1 .0 0 1 .0 2 1 .0 0 4 .3 5 1 7 .0 8 0 .9 8 1 .0 0 1 .0 2 1 .0 0 4 .3 4 5 2 .5 1 7 .7 8 0 .9 8 1 .0 0 1 .0 1 0 .9 9 4 .3 8 1 8 .2 8 0 .9 8 1 .0 0 1 .0 1 0 .9 9 4 .3 8 1 9 .3 0 0 .9 8 1 .0 0 1 .0 1 1 .0 0 4 .3 7 0 .5 1 6 .2 6 0 .9 3 1 .0 1 1 .3 2 1 .0 0 0 .9 1 1 6 .8 0 0 .9 3 1 .0 1 1 .3 1 1 .0 2 0 .9 1 1 7 .9 2 0 .9 3 1 .0 1 1 .3 0 1 .0 5 0 .9 1 1 1 6 .8 1 0 .9 3 1 .0 1 1 .2 9 1 .0 0 1 .2 3 1 7 .3 5 0 .9 3 1 .0 1 1 .2 9 1 .0 2 1 .2 2 1 8 .4 6 0 .9 3 1 .0 1 1 .2 8 1 .0 5 1 .1 9 2 2 0 .8 1 0 .9 8 1 .0 0 1 .0 4 0 .9 9 4 .2 0 2 1 .3 1 0 .9 8 1 .0 0 1 .0 4 1 .0 0 4 .1 9 2 2 .3 4 0 .9 8 1 .0 0 1 .0 4 1 .0 0 4 .1 8 10 2 .5 2 2 .9 7 0 .9 8 1 .0 0 1 .0 3 0 .9 9 4 .2 8 2 3 .4 8 0 .9 8 1 .0 0 1 .0 3 0 .9 9 4 .2 8 2 4 .5 0 0 .9 8 1 .0 0 1 .0 3 1 .0 0 4 .2 6 0 .5 7 2 .9 9 0 .9 4 1 .0 0 1 .3 3 0 .9 9 0 .8 3 7 3 .5 2 0 .9 4 1 .0 0 1 .3 3 0 .9 9 0 .8 3 7 4 .5 9 0 .9 3 1 .0 0 1 .3 3 1 .0 0 0 .8 3 1 7 3 .4 4 0 .9 4 1 .0 0 1 .3 3 0 .9 9 0 .8 5 7 3 .9 9 0 .9 4 1 .0 0 1 .3 3 0 .9 9 0 .8 5 7 5 .0 4 0 .9 3 1 .0 0 1 .3 3 1 .0 0 0 .8 5 2 7 4 .3 7 0 .9 4 1 .0 0 1 .3 3 0 .9 9 0 .8 9 7 4 .9 0 0 .9 3 1 .0 0 1 .3 2 0 .9 9 0 .8 9 7 5 .9 6 0 .9 3 1 .0 0 1 .3 2 1 .0 0 0 .8 9 f Po 50 2 .5 7 4 .8 5 0 .9 3 1 .0 0 1 .3 2 0 .9 9 0 .9 1 7 5 .3 8 0 .9 3 1 .0 0 1 .3 2 0 .9 9 0 .9 2 7 6 .4 5 0 .9 3 1 .0 0 1 .3 2 1 .0 0 0 .9 1 311 T ab le C -2 6 a: ( C o n ti n u ed ) C ar b o n S te el T ”2 0 0 o F & S ta in le ss S te el ȕ= 3 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo ȝ f y Iƍ fu Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f y Iƍ fu Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f y Iƍ fu Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o 0 .5 9 .3 5 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .4 6 9 .8 6 0 .9 9 1 .0 0 0 .9 9 1 .0 0 1 4 .4 4 1 0 .8 7 0 .9 8 1 .0 0 0 .9 9 1 .0 0 1 4 .3 9 1 1 6 .6 9 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .4 9 1 7 .1 9 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .4 8 1 8 .2 0 0 .9 9 1 .0 0 0 .9 9 1 .0 0 1 4 .4 6 2 3 1 .3 5 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .5 0 3 1 .8 5 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .5 0 3 2 .8 6 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .4 9 0.5 2 .5 3 8 .6 9 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .5 0 3 9 .1 9 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .5 0 4 0 1 9 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .4 9 0 .5 9 .8 6 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 4 1 0 .3 6 0 .9 9 1 .0 0 1 .0 0 1 .0 0 1 4 .4 2 1 1 .3 8 0 .9 8 1 .0 0 1 .0 0 1 .0 0 1 4 .3 7 1 1 7 .1 9 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .4 8 1 7 .6 9 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .4 7 1 8 .7 0 0 .9 9 1 .0 0 0 .9 9 1 .0 0 1 4 .4 6 2 3 1 .8 6 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .5 0 3 2 .3 6 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .4 9 3 3 .3 6 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .4 9 1 2 .5 3 9 .1 9 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .5 0 3 9 .6 9 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .5 0 4 0 .6 9 0 .9 9 1 .0 0 0 .9 9 0 .9 9 1 4 .4 9 0 .5 1 1 .9 0 0 .9 8 1 .0 0 1 .0 1 0 .9 9 1 4 .3 4 1 2 .4 0 0 .9 8 1 .0 0 1 .0 1 1 .0 0 1 4 .3 1 1 3 .4 2 0 .9 8 1 .0 0 1 .0 1 1 .0 1 1 4 .2 5 1 1 9 .2 1 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 5 1 9 .7 1 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 4 2 0 .7 2 0 .9 9 1 .0 0 1 .0 0 1 .0 0 1 4 .4 2 2 3 3 .8 7 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 8 3 4 .3 7 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 8 3 5 .3 8 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 7 3 2 .5 4 1 .2 0 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 9 4 1 .7 0 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 9 4 2 .7 1 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 8 0 .5 1 3 .9 7 0 .9 8 1 .0 0 1 .0 2 0 .9 9 1 4 .1 6 1 4 .4 8 0 .9 8 1 .0 0 1 .0 2 1 .0 0 1 4 .1 3 1 5 .5 0 0 .9 8 1 .0 0 1 .0 2 1 .0 1 1 4 .0 7 1 2 1 .2 5 0 .9 8 1 .0 0 1 .0 1 0 .9 9 1 4 .3 9 2 1 .7 5 0 .9 8 1 .0 0 1 .0 1 0 .9 9 1 4 .3 9 2 2 .7 6 0 .9 8 1 .0 0 1 .0 1 1 .0 0 1 4 .3 7 2 3 5 .8 9 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 7 3 6 .3 9 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 6 3 7 .4 0 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 5 5 2 .5 4 3 .2 2 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 8 4 3 .7 2 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 7 4 4 .7 3 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 7 0 .5 1 8 .5 0 0 .9 1 1 .0 2 1 .4 8 1 .0 1 0 .9 5 1 9 .0 7 0 .9 0 1 .0 2 1 .4 7 1 .0 3 0 .9 5 2 0 .2 2 0 .9 0 1 .0 2 1 .4 6 1 .0 7 0 .9 5 1 2 6 .4 1 0 .9 8 1 .0 0 1 .0 2 0 .9 9 1 4 .1 9 2 6 .9 1 0 .9 8 1 .0 0 1 .0 2 0 .9 9 1 4 .1 8 2 7 .9 3 0 .9 8 1 .0 0 1 .0 2 1 .0 0 1 4 .1 0 2 4 0 .9 8 0 .9 9 1 .0 0 1 .0 1 0 .9 9 1 4 .4 1 4 1 .4 8 0 .9 9 1 .0 0 1 .0 1 0 .9 9 1 4 .4 0 4 2 .4 9 0 .9 8 1 .0 0 1 .0 1 0 .9 9 1 4 .3 9 10 2 .5 4 8 .2 9 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 3 4 8 .7 9 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 3 4 9 .8 0 0 .9 9 1 .0 0 1 .0 0 0 .9 9 1 4 .4 3 0 .5 8 3 .9 3 0 .9 1 1 .0 0 1 .4 9 0 .9 9 0 .8 3 8 4 .4 7 0 .9 1 1 .0 0 1 .4 9 1 .0 0 0 .8 3 8 5 .5 7 0 .9 1 1 .0 0 1 .4 9 1 .0 0 0 .8 3 1 8 4 .3 9 0 .9 1 1 .0 0 1 .4 9 0 .9 9 0 .8 5 8 4 .9 4 0 .9 1 1 .0 0 1 .4 9 1 .0 0 0 .8 5 8 6 .0 3 0 .9 1 1 .0 0 1 .4 9 1 .0 0 0 .8 5 2 8 5 .3 6 0 .9 1 1 .0 0 1 .4 9 0 .9 9 0 .9 7 8 5 .9 8 0 .9 1 1 .0 0 1 .4 9 1 .0 0 0 .9 1 8 7 .0 0 0 .9 1 1 .0 0 1 .4 9 1 .0 0 0 .9 1 f Po 50 2 .5 8 5 .8 7 0 .9 1 1 .0 0 1 .4 9 0 .9 9 0 .9 6 8 6 .4 2 0 .9 1 1 .0 0 1 .4 9 1 .0 0 0 .9 6 8 7 .5 2 0 .9 1 1 .0 0 1 .4 8 1 .0 0 0 .9 6 312 T ab le C -2 6 b : M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T > 2 0 0 o F f o r g 1 4 C ar b o n S te el T > 2 0 0 o F ȕ= 2 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o 0 .5 4 .3 5 0 .9 5 1 .0 1 1 .0 0 1 .0 0 4 .2 3 4 .8 8 0 .9 4 1 .0 1 1 .0 0 1 .0 2 4 .1 5 5 .9 8 0 .9 3 1 .0 1 1 .0 0 1 .0 5 3 .9 0 1 6 .5 0 0 .9 9 1 .0 1 1 .0 0 1 .0 0 4 .4 0 7 .1 1 0 .9 6 1 .0 1 1 .0 0 1 .0 0 4 .3 1 8 .1 7 0 .9 5 1 .0 1 1 .0 0 1 .0 1 4 .2 4 2 1 1 .1 1 0 .9 7 1 .0 0 0 .9 9 0 .9 9 4 .3 7 1 1 .6 2 0 .9 6 1 .0 0 0 .9 9 1 .0 0 4 .3 6 1 2 .6 6 0 .9 6 1 .0 0 0 .9 9 1 .0 0 4 .3 4 0.5 2 .5 1 .3 3 7 0 .9 7 1 .0 0 0 .9 9 0 .9 9 4 .3 7 1 3 .8 8 0 .9 7 1 .0 0 0 .9 9 0 .9 9 9 4 .3 7 1 4 .9 2 0 .9 6 1 .0 0 0 .9 9 1 .0 0 4 .3 5 0 .5 4 .8 8 0 .9 4 1 .0 1 1 .0 1 1 .0 0 4 .1 5 5 .4 2 0 .9 4 1 .0 1 1 .0 1 1 .0 2 4 .0 7 6 .5 3 0 .9 2 1 .0 1 1 .0 1 1 .0 5 3 .7 8 1 7 .1 1 0 .9 6 1 .0 1 1 .0 0 1 .0 0 4 .3 1 7 .6 3 0 .9 5 1 .0 1 1 .0 0 1 .0 0 4 .2 8 8 .6 9 0 .9 5 1 .0 1 1 .0 0 1 .0 2 4 .2 1 2 1 1 .6 2 0 .9 6 1 .0 0 1 .0 0 0 .9 9 4 .3 6 1 2 .1 4 0 .9 6 1 .0 0 1 .0 0 1 .0 0 4 .3 5 1 3 .1 8 0 .9 6 1 .0 0 1 .0 0 1 .0 0 4 .3 3 1 2 .5 1 3 .8 8 0 .9 7 1 .0 0 1 .0 0 0 .9 9 4 .3 7 1 4 .4 0 0 .9 6 1 .0 0 1 .0 0 0 .9 9 4 .3 6 1 5 .4 3 0 .9 6 1 .0 0 1 .0 0 1 .0 0 4 .3 4 0 .5 7 .1 1 0 .9 1 1 .0 1 1 .0 7 1 .0 0 3 .5 4 7 .6 7 0 .9 0 1 .0 1 1 .0 7 1 .0 2 3 .3 5 8 .8 7 0 .8 3 1 .0 2 1 .1 4 1 .1 2 1 .3 3 1 9 .2 4 0 .9 5 1 .0 1 1 .0 2 1 .0 0 4 .1 7 9 .7 7 0 .9 4 1 .0 1 1 .0 2 1 .0 0 4 .1 3 1 0 .8 4 0 .9 4 1 .0 1 1 .0 2 1 .0 2 4 .0 5 2 1 3 .7 1 0 .9 6 1 .0 0 1 .0 1 0 .9 9 4 .3 1 1 4 .2 2 0 .9 6 1 .0 0 1 .0 1 1 .0 0 4 .3 0 1 5 .2 7 0 .9 5 1 .0 0 1 .0 1 1 .0 0 4 .2 8 3 2 .5 1 5 .9 6 0 .9 6 1 .0 0 1 .0 0 0 .9 9 4 .3 3 1 6 .4 7 0 .9 6 1 .0 0 1 .0 0 0 .9 9 4 .3 3 1 7 .5 1 0 .9 6 1 .0 0 1 .0 0 1 .0 0 4 .3 1 0 .5 9 .8 3 0 .8 4 1 .0 2 1 .2 3 1 .0 1 1 .0 6 1 0 .4 4 0 .8 3 1 .0 2 1 .2 2 1 .0 4 1 .0 4 1 1 .7 2 0 .8 3 1 .0 2 1 .2 0 1 .0 8 1 .0 1 1 1 1 .4 4 0 .9 3 1 .0 1 1 .0 5 1 .0 0 3 .9 2 1 1 .9 8 0 .9 3 1 .0 1 1 .0 5 1 .0 0 3 .8 8 1 3 .0 7 0 .9 2 1 .0 1 1 .0 5 1 .0 2 3 .7 5 2 1 5 .8 2 0 .9 5 1 .0 0 1 .0 2 0 .9 9 4 .2 5 1 6 .3 4 0 .9 5 1 .0 0 1 .0 2 1 .0 0 4 .2 4 1 7 .3 9 0 .9 5 1 .0 0 1 .0 2 1 .0 0 4 .2 1 5 2 .5 1 8 .0 6 0 .9 6 1 .0 0 1 .0 1 0 .9 9 4 .2 9 1 8 .5 8 0 .9 6 1 .0 0 1 .0 1 0 .9 9 4 .2 8 1 9 .6 2 0 .9 5 1 .0 0 1 .0 1 1 .0 0 4 .2 6 0 .5 1 7 .2 7 0 .8 4 1 .0 1 1 .2 6 1 .0 0 0 .9 0 1 7 .8 7 0 .8 4 1 .0 1 1 .2 6 1 .0 1 0 .9 0 1 9 .0 9 0 .8 4 1 .0 1 1 .2 4 1 .0 4 0 .8 9 1 1 7 .8 4 0 .8 4 1 .0 1 1 .2 5 1 .0 0 1 .0 8 1 8 .4 4 0 .8 4 1 .0 1 1 .2 4 1 .0 1 1 .0 7 1 9 .6 6 0 .8 4 1 .0 1 1 .2 3 1 .0 4 1 .0 5 2 2 1 .2 7 0 .9 4 1 .0 0 1 .0 5 0 .9 9 3 .9 9 2 1 .8 0 0 .9 4 1 .0 0 1 .0 5 1 .0 0 3 .9 7 2 2 .8 7 0 .9 3 1 .0 0 1 .0 5 1 .0 0 3 .9 2 10 2 .5 2 3 .4 3 0 .9 4 1 .0 0 1 .0 3 0 .9 9 4 .1 2 2 3 .9 6 0 .9 4 1 .0 0 1 .0 3 0 .9 9 4 .1 1 2 5 .0 1 0 .9 4 1 .0 0 1 .0 3 1 .0 0 4 .0 8 0 .5 7 7 .2 1 0 .8 5 1 .0 0 1 .2 8 0 .9 9 0 .8 3 7 7 .7 9 0 .8 5 1 .0 0 1 .2 8 0 .9 9 0 .8 3 7 8 .9 6 0 .8 5 1 .0 0 1 .2 7 1 .0 0 0 .8 3 1 7 7 .7 0 0 .8 5 1 .0 0 1 .2 8 0 .9 9 0 .8 4 7 8 .2 8 0 .8 5 1 .0 0 1 .2 8 0 .9 9 0 .8 4 7 9 .4 6 0 .8 5 1 .0 0 1 .2 7 1 .0 0 0 .8 4 2 7 8 .7 1 0 .8 5 1 .0 0 1 .2 7 0 .9 9 0 .8 8 7 9 .2 9 0 .8 5 1 .0 0 1 .2 7 0 .9 9 0 .8 8 8 0 .4 7 0 .8 5 1 .0 0 1 .2 7 1 .0 0 0 .8 8 f Po 50 2 .5 7 9 .2 3 0 .8 5 1 .0 0 1 .2 7 0 .9 9 0 .9 0 7 9 .8 2 0 .8 5 1 .0 0 1 .2 7 0 .9 9 0 .9 0 8 0 .9 9 0 .8 5 1 .0 0 1 .2 7 1 .0 0 0 .9 0 313 T ab le C -2 6 b : (C o n ti n u ed ) C ar b o n S te el T > 2 0 0 o F ȕ= 3 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o Iƍ fu Ȗƍ ǹ Ȗƍ P o Ȗƍ M Ȗƍ o Iƍ fu 0 .5 9 .4 8 0 .9 6 1 .0 0 0 .9 9 0 .9 9 1 4 .2 7 1 0 .0 0 0 .9 6 1 .0 0 0 .9 9 1 .0 0 1 4 .2 2 1 1 .0 4 0 .9 6 1 .0 0 0 .9 9 1 .0 1 1 4 .1 2 1 1 6 .9 0 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 3 1 7 .4 1 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 2 1 8 .4 4 0 .9 6 1 .0 0 0 .9 9 1 .0 0 1 4 .2 8 2 3 1 .7 3 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 6 3 2 .3 4 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 6 3 3 .2 7 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 4 0.5 2 .5 3 9 .1 5 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 7 3 9 .6 6 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 6 4 0 .6 9 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 5 0 .5 1 0 .0 0 0 .9 6 1 .0 0 1 .0 0 0 .9 9 1 4 .2 3 1 0 .5 2 0 .9 6 1 .0 0 1 .0 0 1 .0 0 1 4 .1 8 1 1 .5 6 0 .9 6 1 .0 0 1 .0 0 1 .0 1 1 4 .0 8 1 1 7 .4 1 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 2 1 7 .9 2 0 .9 6 1 .0 0 0 .9 9 0 .9 9 1 4 .3 0 1 8 .9 6 0 .9 6 1 .0 0 0 .9 9 1 .0 0 1 4 .2 6 2 3 2 .2 4 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 6 3 2 .7 6 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 5 3 3 .7 8 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 3 1 2 .5 3 9 .6 6 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 6 4 0 .1 7 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 6 4 1 .2 0 0 .9 7 1 .0 0 0 .9 9 0 .9 9 1 4 .3 4 0 .5 1 2 .1 0 0 .9 5 1 .0 0 1 .0 1 0 .9 9 1 4 .0 1 1 2 .6 2 0 .9 5 1 .0 0 1 .0 1 1 .0 0 1 3 .9 6 1 3 .6 7 0 .9 5 1 .0 0 1 .0 1 1 .0 1 1 3 .8 4 1 1 9 .4 8 0 .9 6 1 .0 0 1 .0 0 0 .9 9 1 4 .2 4 2 0 .0 0 0 .9 6 1 .0 0 1 .0 0 0 .9 9 1 4 .2 2 2 1 .0 3 0 .9 6 1 .0 0 1 .0 0 1 .0 0 1 4 .1 8 2 3 4 .3 0 0 .9 7 1 .0 0 1 .0 0 0 .9 9 1 4 .3 2 3 4 .8 1 0 .9 7 1 .0 0 1 .0 0 0 .9 9 1 4 .3 2 3 5 .8 4 0 .9 6 1 .0 0 1 .0 0 0 .9 9 1 4 .3 0 3 2 .5 4 1 .7 2 0 .9 7 1 .0 0 1 .0 0 0 .9 9 1 4 .3 4 4 2 .2 3 0 .9 7 1 .0 0 1 .0 0 0 .9 9 1 4 .3 3 4 3 .2 6 0 .9 7 1 .0 0 1 .0 0 0 .9 9 1 4 .3 2 0 .5 1 4 .2 4 0 .9 5 1 .0 0 1 .0 2 0 .9 9 1 3 .7 0 1 4 .7 7 0 .9 4 1 .0 0 1 .0 2 1 .0 0 1 3 .6 4 1 3 .4 2 0 .7 7 1 .0 2 1 .2 9 1 .1 2 1 .2 6 1 2 1 .5 7 0 .9 6 1 .0 0 1 .0 1 0 .9 9 1 4 .1 4 2 2 .0 9 0 .9 6 1 .0 0 1 .0 1 0 .9 9 1 4 .1 2 2 3 .1 3 0 .9 6 1 .0 0 1 .0 1 1 .0 0 1 4 .0 7 2 3 6 .3 7 0 .9 6 1 .0 0 1 .0 0 0 .9 9 1 4 .2 9 3 6 .8 8 0 .9 6 1 .0 0 1 .0 0 0 .9 9 1 4 .2 8 3 7 .9 1 0 .9 6 1 .0 0 1 .0 0 0 .9 9 1 4 .2 6 5 2 .5 4 3 .7 8 0 .9 7 1 .0 0 1 .0 0 0 .9 9 1 4 .3 1 4 4 .2 9 0 .9 6 1 .0 0 1 .0 0 0 .9 9 1 4 .3 0 4 5 .3 2 0 .9 6 1 .0 0 1 .0 0 0 .9 9 1 4 .2 9 0 .5 2 0 .0 9 0 .7 9 1 .0 1 1 .3 9 1 .0 1 0 .9 3 2 0 .7 3 0 .7 9 1 .0 1 1 .3 8 1 .0 2 0 .9 3 2 2 .0 5 0 .7 8 1 .0 1 1 .3 7 1 .0 6 0 .9 3 1 2 6 .9 0 0 .9 5 1 .0 0 1 .0 2 0 .9 9 1 3 .7 8 2 7 .4 2 0 .9 5 1 .0 0 1 .0 2 0 .9 9 1 3 .7 5 2 8 .4 7 0 .9 5 1 .0 0 1 .0 2 1 .0 0 1 3 .6 9 2 4 1 .5 8 0 .9 6 1 .0 0 1 .0 1 0 .9 9 1 4 .1 7 4 2 .1 0 0 .9 6 1 .0 0 1 .0 1 0 .9 9 1 4 .1 6 4 3 .1 3 0 .9 6 1 .0 0 1 .0 1 0 .9 9 1 4 .1 4 10 2 .5 4 8 .9 7 0 .9 6 1 .0 0 1 .0 0 .9 9 1 4 .2 3 4 9 .4 9 0 .9 6 1 .0 0 1 .0 0 0 .9 9 1 4 .2 2 5 0 .5 2 0 .9 6 1 .0 0 1 .0 0 0 .9 9 1 4 .2 0 0 .5 9 0 .6 7 0 .8 0 1 .0 0 1 .4 1 0 .9 9 0 .8 3 9 1 .2 9 0 .8 0 1 .0 0 1 .4 1 1 .0 0 0 .8 3 9 2 .5 3 0 .8 0 1 .0 0 1 .4 1 1 .0 0 0 .8 3 1 9 1 .1 9 0 .8 0 1 .0 0 1 .4 1 0 .9 9 0 .8 5 9 1 .8 1 0 .8 0 1 .0 0 1 .4 1 1 .0 0 0 .8 5 9 3 .0 6 0 .8 0 1 .0 0 1 .4 1 1 .0 0 0 .8 5 2 9 2 .2 8 0 .8 0 1 .0 0 1 .4 1 0 .9 9 0 .9 0 9 2 .9 1 0 .8 0 1 .0 0 1 .4 1 1 .0 0 0 .9 0 9 4 .1 6 0 .8 0 1 .0 0 1 .4 1 1 .0 0 0 .9 0 f Po 50 2 .5 9 2 .8 6 0 .8 0 1 .0 0 1 .4 1 0 .9 9 0 .9 4 9 3 .4 8 0 .8 0 1 .0 0 1 .4 1 1 .0 0 0 .9 4 9 4 .7 3 0 .8 0 1 .0 0 1 .4 0 1 .0 0 0 .9 4 314 T ab le C -2 7 a: A d ju st ed N o m in al R es is ta n ce F ac to rs f o r C ar b o n S te el a n d Ȗ ǹ = 1 .1 , Ȗ P O = Ȗ M = 1 .2 , Ȗ ȅ = 1 .5 f o r g 1 4 C ar b o n S te el ȕ= 2 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F 0 .5 0 .7 3 (1 ) 0 .7 2 0 .7 6 (1 ) 0 .8 4 0 .6 7 0 .7 7 (1 ) 0 .8 6 0 .7 7 (1 ) 0 .8 9 0 .7 1 0 .8 3 (1 ) 0 .9 2 0 .8 2 (1 ) 0 .9 4 0 .7 5 1 0 .5 8 (1 ) 0 .6 0 0 .6 5 (1 ) 0 .7 2 0 .5 8 0 .6 2 (1 ) 0 .7 4 0 .6 3 (1 ) 0 .7 6 0 .6 1 0 .7 0 (1 ) 0 .8 0 0 .7 0 (1 ) 0 .8 3 0 .6 6 2 0 .4 9 (1 ) 0 .4 9 0 .5 7 (1 ) 0 .6 3 0 .5 0 0 .5 2 (1 ) 0 .6 3 0 .5 7 (1 ) 0 .6 6 0 .5 2 0 .5 7 (1 ) 0 .6 8 0 .5 7 (1 ) 0 .7 1 0 .5 7 0.5 2 .5 0 .4 7 (1 ) 0 .4 7 0 .5 4 (1 ) 0 .6 0 0 .4 8 0 .4 9 (1 ) 0 .6 1 0 .4 9 (1 ) 0 .6 3 0 .5 0 0 .5 4 (1 ) 0 .6 5 0 .5 4 (1 ) 0 .6 7 0 .5 4 0 .5 0 .7 7 (1 ) 0 .7 7 0 .8 1 (1 ) 0 .8 9 0 .7 1 0 .8 1 (1 ) 0 .9 0 0 .8 0 (1 ) 0 .9 3 0 .7 4 0 .8 5 (1 ) 0 .9 5 0 .8 5 (1 ) 0 .9 7 0 .7 8 1 0 .6 4 (1 ) 0 .6 4 0 .6 9 (1 ) 0 .7 7 0 .6 1 0 .6 7 (1 ) 0 .7 8 0 .6 7 (1 ) 0 .8 0 0 .6 4 0 .7 3 (1 ) 0 .8 3 0 .7 2 (1 ) 0 .8 6 0 .6 8 2 0 .5 2 (1 ) 0 .5 3 0 .5 9 (1 ) 0 .6 6 0 .5 3 0 .5 5 (1 ) 0 .6 6 0 .5 5 (1 ) 0 .6 9 0 .5 5 0 .5 8 (1 ) 0 .7 1 0 .6 0 (1 ) 0 .7 3 0 .5 9 1 2 .5 0 .4 9 (1 ) 0 .5 0 0 .5 7 (1 ) 0 .6 3 0 .5 0 0 .5 1 (1 ) 0 .6 3 0 .5 2 (1 ) 0 .6 5 0 .5 2 0 .5 6 (1 ) 0 .6 7 0 .5 6 (1 ) 0 .7 0 0 .5 6 0 .5 0 .8 9 (1 ) 0 .8 8 0 .9 2 (1 ) 1 .0 0 0 .8 0 0 .9 0 (1 ) 1 .0 0 0 .8 9 (1 ) 1 .0 2 0 .8 1 0 .9 2 (1 ) 1 .0 2 0 .9 0 (1 ) 1 .0 3 0 .8 2 1 0 .7 6 (1 ) 0 .7 6 0 .8 1 (1 ) 0 .8 9 0 .7 1 0 .7 8 (1 ) 0 .8 9 0 .7 8 (1 ) 0 .9 1 0 .7 3 0 .8 1 (1 ) 0 .9 2 0 .8 1 (1 ) 0 .9 4 0 .7 5 2 0 .6 2 (1 ) 0 .6 3 0 .6 9 (1 ) 0 .7 6 0 .6 1 0 .6 4 (1 ) 0 .7 6 0 .6 4 (1 ) 0 .7 8 0 .6 2 0 .6 8 (1 ) 0 .7 9 0 .6 8 (1 ) 0 .8 1 0 .6 5 3 2 .5 0 .5 8 (1 ) 0 .5 9 0 .6 5 (1 ) 0 .7 2 0 .5 8 0 .6 0 (1 ) 0 .7 2 0 .6 0 (1 ) 0 .7 4 0 .5 9 0 .6 3 (1 ) 0 .7 5 0 .6 3 (1 ) 0 .7 7 0 .6 2 0 .5 0 .9 3 (1 ) 0 .8 8 0 .9 6 (1 ) 1 .0 1 0 .8 1 0 .9 3 (1 ) 1 .0 3 0 .8 9 (1 ) 1 .0 1 0 .8 1 0 .9 4 (1 ) 1 .0 4 0 .8 9 (1 ) 1 .0 2 0 .8 1 1 0 .8 3 (1 ) 0 .8 3 0 .8 6 (1 ) 0 .9 6 0 .7 7 0 .8 5 (1 ) 0 .9 5 0 .8 4 (1 ) 0 .. 9 8 0 .7 8 0 .8 7 (1 ) 0 .9 8 0 .8 6 (1 ) 0 .9 9 0 .8 0 2 0 .7 0 (1 ) 0 .7 0 0 .7 6 (1 ) 0 .8 4 0 .6 7 0 .7 1 (1 ) 0 .8 2 0 .7 1 (1 ) 0 .8 5 0 .6 8 0 .7 4 (1 ) 0 .8 1 0 .7 4 (1 ) 0 .8 7 0 .7 0 5 2 .5 0 .6 5 (1 ) 0 .6 5 0 .7 2 (1 ) 0 .7 9 0 .6 3 0 .6 7 (1 ) 0 .7 8 0 .6 7 (1 ) 0 .8 1 0 .6 4 0 .6 9 (1 ) 0 .8 5 0 .6 9 (1 ) 0 .8 3 0 .6 6 0 .5 0 .9 0 (1 ) 0 .8 6 0 .9 2 (1 ) 0 .9 8 0 .7 8 0 .9 0 (1 ) 1 .0 0 0 .8 6 (1 ) 0 .9 8 0 .7 8 0 .9 1 (1 ) 1 .0 0 0 .8 7 (1 ) 0 .9 9 0 .7 9 1 0 .9 1 (1 ) 0 .8 8 0 .9 5 (1 ) 1 .0 1 0 .8 1 0 .9 2 (1 ) 1 .0 2 0 .8 8 (1 ) 1 .0 1 0 .8 1 0 .9 3 (1 ) 1 .0 3 0 .8 8 (1 ) 1 .0 1 0 .8 1 2 0 .8 1 (1 ) 0 .8 1 0 .8 6 (1 ) 0 .9 5 0 .7 6 0 .8 2 (1 ) 0 .9 3 0 .8 2 (1 ) 0 .9 6 0 .7 6 0 .8 3 (1 ) 0 .9 5 0 .8 3 (1 ) 0 .9 7 0 .7 7 10 2 .5 0 .7 7 (1 ) 0 .7 7 0 .8 2 (1 ) 0 .9 1 0 .7 3 0 .7 8 (1 ) 0 .8 9 0 .7 7 (1 ) 0 .9 2 0 .7 3 0 .7 9 (1 ) 0 .9 1 0 .7 9 (1 ) 0 .9 3 0 .7 4 0 .5 0 .8 6 (1 ) 0 .8 3 0 .8 8 (1 ) 0 .9 4 0 .7 5 0 .8 6 (1 ) 0 .9 5 0 .8 3 (1 ) 0 .9 4 0 .7 5 0 .8 7 (1 ) 0 .9 5 0 .8 3 (1 ) 0 .9 4 0 .7 5 1 0 .8 7 (1 ) 0 .8 4 0 .8 9 (1 ) 0 .9 5 0 .7 6 0 .8 7 (1 ) 0 .9 6 0 .8 4 (1 ) 0 .9 5 0 .7 6 0 .8 7 (1 ) 0 .9 6 0 .8 4 (1 ) 0 .9 5 0 .7 6 2 0 .8 8 (1 ) 0 .8 4 0 .9 1 (1 ) 0 .9 6 0 .7 7 0 .8 8 (1 ) 0 .9 7 0 .8 5 (1 ) 0 .9 6 0 .7 7 0 .8 8 (1 ) 0 .9 8 0 .8 5 (1 ) 0 .9 7 0 .7 7 f Po 50 2 .5 0 .8 8 (1 ) 0 .8 5 0 .9 1 (1 ) 0 .9 7 0 .7 8 0 .8 7 (1 ) 0 .9 8 0 .8 5 (1 ) 0 .9 7 0 .7 8 0 .8 9 (1 ) 0 .9 8 0 .8 5 (1 ) 0 .9 7 0 .7 8 (1 ) Ȗ ǹ = Ȗ Ȃ = Ȗ P ȅ = Ȗ ȅ = 1 .0 0 315 T ab le C -2 7 a: ( C o n ti n u ed ) C ar b o n S te el ȕ= 3 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F 0 .5 0 .4 1 0 .3 7 0 .4 1 0 .3 9 0 .3 1 0 .4 6 0 .4 2 0 .4 6 0 .4 3 0 .3 5 0 .5 4 0 .5 0 0 .5 4 0 .5 1 0 .4 1 1 0 .3 0 0 .2 7 0 .3 0 0 .2 8 0 .2 3 0 .3 3 0 .3 0 0 .3 3 0 .3 1 0 .2 5 0 .3 9 0 .3 5 0 .3 9 0 .3 7 0 .2 9 2 0 .2 3 0 .2 1 0 .2 3 0 .2 2 0 .1 8 0 .2 5 0 .2 3 0 .2 5 0 .2 4 0 .1 9 0 .2 8 0 .2 6 0 .2 9 0 .2 7 0 .2 2 0.5 2 .5 0 .2 2 0 .2 0 0 .2 2 0 .2 1 0 .1 0 0 .2 3 0 .2 1 0 .2 3 0 .2 2 0 .1 8 0 .2 6 0 .2 4 0 .2 6 0 .2 5 0 .2 0 0 .5 0 .4 6 0 .4 2 0 .4 6 0 .4 4 0 .3 5 0 .5 1 0 .4 6 0 .5 1 0 .4 8 0 .3 8 0 .5 8 0 .5 3 0 .5 8 0 .5 5 0 .4 4 1 0 .3 0 .3 0 0 .3 3 0 .3 1 0 .2 5 0 .3 6 0 .3 3 0 .3 6 0 .3 4 0 .2 7 0 .4 2 0 .3 8 0 .4 2 0 .3 9 0 .3 1 2 0 .2 5 0 .2 3 0 .2 5 0 .2 4 0 .1 9 0 .2 7 0 .2 5 0 .2 7 0 .2 5 0 .2 0 0 .3 0 0 .2 8 0 .3 0 0 .2 9 0 .2 3 1 2 .5 0 .2 3 0 .2 1 0 .2 3 0 .2 2 0 .1 8 0 .2 5 0 .2 3 0 .2 5 0 .2 3 0 .1 9 0 .2 8 0 .2 5 0 .2 8 0 .2 6 0 .2 1 0 .5 0 .6 3 0 .5 7 0 .6 3 0 .5 9 0 .4 7 0 .6 6 0 .6 0 0 .6 6 0 .6 2 0 .4 9 0 .7 1 0 .7 1 0 .7 1 0 .6 7 0 .5 3 1 0 .4 5 0 .4 1 0 .4 5 0 .4 2 0 .3 4 0 .4 7 0 .4 3 0 .4 7 0 .4 4 0 .3 6 0 .5 2 0 .5 2 0 .5 2 0 .4 9 0 .3 9 2 0 .3 2 0 .2 9 0 .3 2 0 .3 0 0 .2 4 0 .3 4 0 .3 1 0 .3 4 0 .3 2 0 .2 6 0 .3 7 0 .3 7 0 .3 7 0 .3 5 0 .2 8 3 2 .5 0 .2 9 0 .2 7 0 .2 9 0 .2 8 0 .2 2 0 .3 1 0 .2 8 0 .3 1 0 .2 9 0 .2 3 0 .3 3 0 .3 3 0 .3 3 0 .3 1 0 .2 5 0 .5 0 .7 4 0 .6 8 0 .7 4 0 .7 0 0 .5 6 0 .7 6 0 .7 0 0 .7 6 0 .7 2 0 .5 7 0 .8 0 0 .9 4 0 .9 4 0 .8 9 0 .7 1 1 0 .5 4 0 .4 9 0 .5 4 0 .5 1 0 .4 1 0 .5 6 0 .5 1 0 .5 6 0 .5 8 0 .4 2 0 .6 0 0 .6 0 0 .6 0 0 .5 6 0 .4 5 2 0 .3 9 0 .3 5 0 .3 9 0 .3 6 0 .2 9 0 .4 0 0 .3 6 0 .4 0 0 .3 8 0 .3 0 0 .4 2 0 .4 3 0 .4 3 0 .4 0 0 .3 2 5 2 .5 0 .3 5 0 .3 2 0 .3 5 0 .3 3 0 .2 6 0 .3 6 0 .3 3 0 .3 6 0 .3 4 0 .2 7 0 .3 8 0 .3 8 0 .3 8 0 .3 6 0 .2 9 0 .5 0 .9 5 0 .8 7 0 .8 9 0 .8 4 0 .6 7 0 .9 6 0 .8 8 0 .9 0 0 .8 5 0 .6 8 0 .9 7 0 .9 1 0 .9 1 0 .8 5 0 .6 8 1 0 .7 1 0 .6 5 0 .7 0 .6 7 0 .5 3 0 .7 2 0 .6 6 0 .7 2 0 .6 8 0 .5 4 0 .7 5 0 .7 5 0 .7 5 0 .7 0 0 .5 6 2 0 .5 1 0 .4 7 0 .5 2 0 .4 8 0 .3 9 0 .5 2 0 .4 8 0 .5 3 0 .4 9 0 .4 0 0 .5 4 0 .5 5 0 .5 5 0 .5 1 0 .4 1 10 2 .5 0 .4 6 0 .4 2 0 .4 6 0 .4 3 0 .3 5 0 .4 7 0 .4 3 0 .4 7 0 .4 4 0 .3 5 0 .4 9 0 .4 9 0 .4 9 0 .4 6 0 .3 7 0 .5 0 .9 0 0 .8 2 0 .8 5 0 .8 0 0 .6 4 0 .9 0 0 .8 3 0 .8 5 0 .8 0 0 .6 4 0 .9 1 0 .8 6 0 .8 6 0 .8 1 0 .6 5 1 0 .9 1 0 .8 3 0 .8 6 0 .8 1 0 .6 5 0 .9 1 0 .8 3 0 .8 6 0 .8 1 0 .6 5 0 .9 2 0 .8 6 0 .8 6 0 .8 1 0 .6 5 2 0 .9 3 0 .8 5 0 .8 7 0 .8 2 0 .6 6 0 .9 3 0 .8 5 0 .8 7 0 .8 2 0 .6 6 0 .9 3 0 .8 8 0 .8 8 0 .8 3 0 .6 6 f Po 50 2 .5 0 .9 4 0 .8 5 0 .8 8 0 .8 3 0 .6 6 0 .9 4 0 .8 6 0 .8 8 0 .8 3 0 .6 6 0 .9 4 0 .8 9 0 .8 9 0 .8 3 0 .6 7 316 T ab le C -2 7 b : A d ju st ed N o m in al R es is ta n ce F ac to rs f o r S ta in le ss S te el a n d Ȗ ǹ = 1 .1 , Ȗ P O = Ȗ M = 1 .2 , Ȗ ȅ = 1 .5 f o r g 1 4 S ta in le ss S te el ȕ= 2 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F 0 .5 0 .7 1 (1 ) 0 .7 6 0 .6 9 0 .6 9 0 .6 6 0 .7 5 ( 1 ) 0 .8 1 0 .7 2 0 .7 2 0 .7 0 0 .8 1 (1 ) 0 .8 6 0 .7 7 0 .7 7 0 .7 5 1 0 .5 9 (1 ) 0 .6 5 0 .5 9 0 .5 9 0 .5 7 0 .6 3 (1 ) 0 .6 9 0 .6 2 0 .6 2 0 .6 0 0 .6 8 (1 ) 0 .7 5 0 .6 7 0 .6 7 0 .6 5 2 0 .4 8 (1 ) 0 .5 7 0 .5 1 0 .5 1 0 .4 9 0 .5 1 (1 ) 0 .5 9 0 .5 3 0 .5 3 0 .5 1 0 .5 6 (1 ) 0 .6 4 0 .5 7 0 .5 7 0 .5 5 0.5 2 .5 0 .4 6 (1 ) 0 .5 4 0 .4 9 0 .4 9 0 .4 7 0 .4 8 (1 ) 0 .5 7 0 .5 1 0 .5 1 0 .4 9 0 .5 3 (1 ) 0 .6 1 0 .5 5 0 .5 5 0 .5 3 0 .5 0 .7 6 (1 ) 0 .8 1 0 .7 3 0 .7 3 0 .7 0 0 .7 9 (1 ) 0 .8 4 0 .7 6 0 .7 6 0 .7 3 0 .8 4 (1 ) 0 .8 9 0 .8 0 0 .8 0 0 .7 7 1 0 .6 3 (1 ) 0 .6 9 0 .6 2 0 .6 2 0 .6 0 0 .6 6 (1 ) 0 .7 5 0 .6 5 0 .6 5 0 .6 3 0 .7 1 (1 ) 0 .7 8 0 .7 0 0 .7 0 0 .6 7 2 0 .5 1 (1 ) 0 .5 9 0 .5 3 0 .5 3 0 .5 2 0 .5 4 (1 ) 0 .6 2 0 .5 6 0 .5 6 0 .5 4 0 .5 9 (1 ) 0 .6 6 0 .5 9 0 .5 9 0 .5 7 1 2 .5 0 .4 9 (1 ) 0 .5 7 0 .5 1 0 .5 1 0 .4 9 0 .5 2 (1 ) 0 .5 9 0 .5 3 0 .5 3 0 .5 1 0 .5 5 (1 ) 0 .6 3 0 .5 6 0 .5 6 0 .5 5 0 .5 0 .8 7 (1 ) 0 .9 2 0 .8 3 0 .8 3 0 .8 0 0 .8 9 (1 ) 0 .9 4 0 .8 4 0 .8 4 0 .8 1 0 .9 1 (1 ) 0 .9 6 0 .8 6 0 .8 6 0 .8 3 1 0 .7 5 (1 ) 0 .8 1 0 .7 3 0 .7 3 0 .7 0 0 .7 7 (1 ) 0 .8 3 0 .7 4 0 .7 4 0 .7 2 0 .8 0 (1 ) 0 .8 6 0 .7 7 0 .7 7 0 .9 4 2 0 .6 1 (1 ) 0 .6 9 0 .6 2 0 .6 2 0 .6 0 0 .6 3 (1 ) 0 .7 1 0 .6 3 0 .6 3 0 .6 1 0 .6 6 (1 ) 0 .7 4 0 .6 6 0 .6 6 0 .6 4 3 2 .5 0 .5 7 (1 ) 0 .6 5 0 .5 9 0 .5 9 0 .5 7 0 .5 9 (1 ) 0 .6 7 0 .6 0 0 .6 0 0 .5 8 0 .6 2 (1 ) 0 .7 0 0 .6 3 0 .6 3 0 .6 1 0 .5 0 .9 1 (1 ) 0 .9 6 0 .8 6 0 .8 6 0 .8 3 0 .9 2 (1 ) 0 .9 6 0 .8 7 0 .8 7 0 .8 4 0 .9 2 (1 ) 0 .9 7 0 .8 7 0 .8 7 0 .8 4 1 0 .8 2 (1 ) 0 .8 8 0 .7 9 0 .7 9 0 .7 6 0 .8 3 (1 ) 0 .8 9 0 .8 0 0 .8 0 0 .7 7 0 .8 5 (1 ) 0 .9 1 0 .8 2 0 .8 2 0 .7 9 2 0 .6 8 (1 ) 0 .7 6 0 .6 8 0 .6 8 0 .6 6 0 .7 0 (1 ) 0 .7 7 0 .6 9 0 .6 9 0 .6 7 0 .7 2 (1 ) 0 .7 9 0 .7 1 0 .7 1 0 .6 9 5 2 .5 0 .6 4 (1 ) 0 .7 2 0 .6 4 0 .6 4 0 .6 2 0 .6 5 (1 ) 0 .7 3 0 .6 6 0 .6 6 0 .6 3 0 .6 8 (1 ) 0 .7 5 0 .6 8 0 .6 8 0 .6 5 0 .5 0 .8 8 (1 ) 0 .9 2 0 .8 3 0 .8 3 0 .8 0 0 .8 9 (1 ) 0 .9 3 0 .8 3 0 .8 3 0 .8 1 0 .9 0 (1 ) 0 .9 4 0 .8 4 0 .8 4 0 .8 1 1 0 .9 0 (1 ) 0 .9 5 0 .8 5 0 .8 5 0 .8 3 0 .9 0 (1 ) 0 .9 6 0 .8 6 0 .8 6 0 .8 3 0 .9 1 (1 ) 0 .9 6 0 .8 6 0 .8 6 0 .8 3 2 0 .8 0 (1 ) 0 .8 6 0 .7 8 0 .7 8 0 .7 5 0 .8 1 (1 ) 0 .8 7 0 .7 8 0 .7 8 0 .7 5 0 .8 2 (1 ) 0 .8 8 0 .7 9 0 .7 9 0 .7 7 10 2 .5 0 .7 6 (1 ) 0 .8 2 0 .7 4 0 .7 4 0 .7 2 0 .7 6 (1 ) 0 .8 3 0 .7 5 0 .7 5 0 .7 2 0 .7 8 (1 ) 0 .8 5 0 .7 6 0 .7 6 0 .7 3 0 .5 0 .8 5 (1 ) 0 .8 8 0 .7 9 0 .7 9 0 .7 7 0 .8 5 (1 ) 0 .8 9 0 .8 0 0 .8 0 0 .7 7 0 .8 5 (1 ) 0 .8 9 0 .8 0 0 .8 0 0 .7 7 1 0 .8 5 (1 ) 0 .8 9 0 .8 0 0 .8 0 0 .7 7 0 .8 5 (1 ) 0 .8 9 0 .8 0 0 .8 0 0 .7 8 0 .8 6 (1 ) 0 .9 0 0 .8 1 0 .8 1 0 .7 8 2 0 .8 6 (1 ) 0 .9 1 0 .8 1 0 .8 1 0 .7 9 0 .8 6 (1 ) 0 .9 1 0 .8 2 0 .8 2 0 .7 9 0 .8 7 (1 ) 0 .9 1 0 .8 2 0 .8 2 0 .7 9 f Po 50 2 .5 0 .8 7 (1 ) 0 .9 1 0 .8 2 0 .8 2 0 .7 9 0 .8 8 (1 ) 0 .9 2 0 .8 2 0 .8 2 0 .7 9 0 .8 7 (1 ) 0 .9 2 0 .8 2 0 .8 2 0 .8 0 (1 ) Ȗ ǹ = Ȗ Ȃ = Ȗ P ȅ = Ȗ ȅ = 1 317 T ab le C -2 7 a: ( C o n ti n u ed ) C ar b o n S te el ȕ= 3 .5 f M = 0 .5 f M = 1 .0 f M = 2 .0 fo R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F 0 .5 0 .4 1 0 .3 7 0 .4 1 0 .3 9 0 .3 1 0 .4 6 0 .4 2 0 .4 6 0 .4 3 0 .3 5 0 .5 4 0 .5 0 0 .5 4 0 .5 1 0 .4 1 1 0 .3 0 0 .2 7 0 .3 0 0 .2 8 0 .2 3 0 .3 3 0 .3 0 0 .3 3 0 .3 1 0 .2 5 0 .3 9 0 .3 5 0 .3 9 0 .3 7 0 .2 9 2 0 .2 3 0 .2 1 0 .2 3 0 .2 2 0 .1 8 0 .2 5 0 .2 3 0 .2 5 0 .2 4 0 .1 9 0 .2 8 0 .2 6 0 .2 9 0 .2 7 0 .2 2 0.5 2 .5 0 .2 2 0 .2 0 0 .2 2 0 .2 1 0 .1 0 0 .2 3 0 .2 1 0 .2 3 0 .2 2 0 .1 8 0 .2 6 0 .2 4 0 .2 6 0 .2 5 0 .2 0 0 .5 0 .4 6 0 .4 2 0 .4 6 0 .4 4 0 .3 5 0 .5 1 0 .4 6 0 .5 1 0 .4 8 0 .3 8 0 .5 8 0 .5 3 0 .5 8 0 .5 5 0 .4 4 1 0 .3 0 .3 0 0 .3 3 0 .3 1 0 .2 5 0 .3 6 0 .3 3 0 .3 6 0 .3 4 0 .2 7 0 .4 2 0 .3 8 0 .4 2 0 .3 9 0 .3 1 2 0 .2 5 0 .2 3 0 .2 5 0 .2 4 0 .1 9 0 .2 7 0 .2 5 0 .2 7 0 .2 5 0 .2 0 0 .3 0 0 .2 8 0 .3 0 0 .2 9 0 .2 3 1 2 .5 0 .2 3 0 .2 1 0 .2 3 0 .2 2 0 .1 8 0 .2 5 0 .2 3 0 .2 5 0 .2 3 0 .1 9 0 .2 8 0 .2 5 0 .2 8 0 .2 6 0 .2 1 0 .5 0 .6 3 0 .5 7 0 .6 3 0 .5 9 0 .4 7 0 .6 6 0 .6 0 0 .6 6 0 .6 2 0 .4 9 0 .7 1 0 .7 1 0 .7 1 0 .6 7 0 .5 3 1 0 .4 5 0 .4 1 0 .4 5 0 .4 2 0 .3 4 0 .4 7 0 .4 3 0 .4 7 0 .4 4 0 .3 6 0 .5 2 0 .5 2 0 .5 2 0 .4 9 0 .3 9 2 0 .3 2 0 .2 9 0 .3 2 0 .3 0 0 .2 4 0 .3 4 0 .3 1 0 .3 4 0 .3 2 0 .2 6 0 .3 7 0 .3 7 0 .3 7 0 .3 5 0 .2 8 3 2 .5 0 .2 9 0 .2 7 0 .2 9 0 .2 8 0 .2 2 0 .3 1 0 .2 8 0 .3 1 0 .2 9 0 .2 3 0 .3 3 0 .3 3 0 .3 3 0 .3 1 0 .2 5 0 .5 0 .7 4 0 .6 8 0 .7 4 0 .7 0 0 .5 6 0 .7 6 0 .7 0 0 .7 6 0 .7 2 0 .5 7 0 .8 0 0 .9 4 0 .9 4 0 .8 9 0 .7 1 1 0 .5 4 0 .4 9 0 .5 4 0 .5 1 0 .4 1 0 .5 6 0 .5 1 0 .5 6 0 .5 8 0 .4 2 0 .6 0 0 .6 0 0 .6 0 0 .5 6 0 .4 5 2 0 .3 9 0 .3 5 0 .3 9 0 .3 6 0 .2 9 0 .4 0 0 .3 6 0 .4 0 0 .3 8 0 .3 0 0 .4 2 0 .4 3 0 .4 3 0 .4 0 0 .3 2 5 2 .5 0 .3 5 0 .3 2 0 .3 5 0 .3 3 0 .2 6 0 .3 6 0 .3 3 0 .3 6 0 .3 4 0 .2 7 0 .3 8 0 .3 8 0 .3 8 0 .3 6 0 .2 9 0 .5 0 .9 5 0 .8 7 0 .8 9 0 .8 4 0 .6 7 0 .9 6 0 .8 8 0 .9 0 0 .8 5 0 .6 8 0 .9 7 0 .9 1 0 .9 1 0 .8 5 0 .6 8 1 0 .7 1 0 .6 5 0 .7 0 .6 7 0 .5 3 0 .7 2 0 .6 6 0 .7 2 0 .6 8 0 .5 4 0 .7 5 0 .7 5 0 .7 5 0 .7 0 0 .5 6 2 0 .5 1 0 .4 7 0 .5 2 0 .4 8 0 .3 9 0 .5 2 0 .4 8 0 .5 3 0 .4 9 0 .4 0 0 .5 4 0 .5 5 0 .5 5 0 .5 1 0 .4 1 10 2 .5 0 .4 6 0 .4 2 0 .4 6 0 .4 3 0 .3 5 0 .4 7 0 .4 3 0 .4 7 0 .4 4 0 .3 5 0 .4 9 0 .4 9 0 .4 9 0 .4 6 0 .3 7 0 .5 0 .9 0 0 .8 2 0 .8 5 0 .8 0 0 .6 4 0 .9 0 0 .8 3 0 .8 5 0 .8 0 0 .6 4 0 .9 1 0 .8 6 0 .8 6 0 .8 1 0 .6 5 1 0 .9 1 0 .8 3 0 .8 6 0 .8 1 0 .6 5 0 .9 1 0 .8 3 0 .8 6 0 .8 1 0 .6 5 0 .9 2 0 .8 6 0 .8 6 0 .8 1 0 .6 5 2 0 .9 3 0 .8 5 0 .8 7 0 .8 2 0 .6 6 0 .9 3 0 .8 5 0 .8 7 0 .8 2 0 .6 6 0 .9 3 0 .8 8 0 .8 8 0 .8 3 0 .6 6 f Po 50 2 .5 0 .9 4 0 .8 5 0 .8 8 0 .8 3 0 .6 6 0 .9 4 0 .8 6 0 .8 8 0 .8 3 0 .6 6 0 .9 4 0 .8 9 0 .8 9 0 .8 3 0 .6 7 318 319 C.13. Performance Function g16 Table C-28 gives the calculated mean load and resistance factors for performance function g16. In this table, ȝfu is the converged mean value of the ultimate strength of steel. Table C-29 shows the evaluated adjusted nominal resistance factors for nominal load factors Ȗǹ=1.1 and ȖS=1.5. Table C-28: Computed Mean Partial Safety Factors for g16 Carbon Steel (T”200oF) & Stainless Steel Carbon Steel (T>200oF) ȕ ȝfu I'fu Ȗ'ǹ Ȗ'S ȝfu I'fu Ȗ'ǹ Ȗ'S 1.5 4.77 0.989 1.006 1.853 4.82 0.969 1.006 1.834 2 6.60 0.988 1.004 2.758 6.68 0.968 1.004 2.730 f S= 2 3 16.56 0.989 1.002 7.684 16.75 0.969 1.002 7.614 1.5 6.64 0.989 1.004 1.855 6.71 0.971 1.004 1.839 2 9.39 0.989 1.003 2.761 9.50 0.970 1.003 2.735 f S= 3 3 24.33 0.989 1.001 7.686 24.62 0.969 1.001 7.619 1.5 8.52 0.990 1.003 1.856 8.61 0.972 1.003 1.841 2 12.18 0.989 1.002 2.761 12.32 0.970 1.002 2.738 f S= 4 3 32.11 0.989 1.001 7.687 32.48 0.970 1.001 7.621 Table C-29a: Adjusted Nominal Resistance Factor for Ȗǹ=1.1 and ȖS=1.5 and Carbon Steel for g16 Carbon Steel fS ȕ R.T. 200 o F 400 o F 600 o F 800 o F 1.5 0.97 (1) 0.89 (1) 0.98 (1) 0.92 (1) 0.93 2 0.89 0.81 0.89 0.84 0.67 2 3 0.35 0.32 0.36 0.33 0.27 1.5 0.95 (1) 0.87 (1) 0.96 (1) 0.90 (1) 0.93 2 0.87 0.79 0.87 0.82 0.66 3 3 0.34 0.31 0.34 0.32 0.25 1.5 0.94 (1) 0.86 (1) 0.95 (1) 0.89 (1) 0.93 2 0.86 0.78 0.86 0.81 0.65 4 3 0.33 0.30 0.33 0.31 0.25 (1) Ȗǹ=1.1 and ȖS=1.1, R.T.=Room Temperature 320 Table C-29b: Adjusted Nominal Resistance Factors for Ȗǹ=1.1 and ȖS=1.5 and Stainless Steel for g16 Stainless Steel fS ȕ R.T. 200 o F 400 o F 600 o F 800 o F 1.5 0.96 (1) 0.83 (1) 0.94 0.94 0.91 2 0.87 0.76 0.68 0.68 0.66 2 3 0.35 0.30 0.27 0.27 0.26 1.5 0.94 (1) 0.81 (1) 0.94 0.94 0.91 2 0.85 0.74 0.67 0.67 0.64 3 3 0.33 0.29 0.26 0.26 0.25 1.5 0.92 (1) 0.80 (1) 0.94 0.94 0.91 2 0.84 0.73 0.66 0.66 0.64 4 3 0.32 0.28 0.25 0.25 0.24 (1) Ȗǹ=1.1 and ȖS=1.1, R.T.=Room Temperature C.14. Performance Function g17 Table C-30 gives the calculated mean load and resistance factors for performance function g17. In this table, ȝfu is the converged mean value of the ultimate strength of steel. Table C-31 shows the evaluated adjusted nominal resistance factors for nominal load factors Ȗǹ=1.1, ȖPS=1.2, and ȖS=1.5. T ab le C -3 0 a: M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T ”2 0 0 o F a n d S ta in le ss S te el f o r g 1 7 C ar b o n S te el f o r T” 2 0 0 o F a n d S ta in le ss S te el f S= 2 f S= 3 f S= 4 ȕ ȝ f u I' f u Ȗƍ ǹ Ȗƍ S Ȗƍ P S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ S Ȗƍ P S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ S Ȗƍ P S 2 7 .1 0 0 .9 8 8 1 .0 0 4 2 .7 5 5 0 .9 9 5 9 .8 9 0 .9 9 1 .0 0 2 .7 6 0 .9 9 1 2 .6 8 0 .9 9 1 .0 0 2 .7 6 0 .9 9 2 .5 1 0 .4 7 0 .9 9 1 .0 0 4 .4 2 0 .9 9 1 4 .9 4 0 .9 9 1 .0 0 4 .4 2 0 .9 9 1 9 .4 2 0 .9 9 1 .0 0 4 .4 3 0 .9 9 f PS = 0.5 3 1 7 .0 6 0 .9 8 8 1 .0 0 2 7 .6 8 2 0 .9 9 3 2 4 .8 3 0 .9 9 1 .0 0 7 .6 8 0 .9 9 3 2 .6 1 0 .9 9 1 .0 0 7 .6 9 0 .9 9 2 7 .6 0 0 .9 8 7 1 .0 0 4 2 .7 5 6 0 .9 9 9 1 0 .3 9 0 .9 9 1 .0 0 2 .7 6 1 .0 0 1 3 .1 8 0 .9 9 1 .0 0 2 .7 6 1 .0 0 2 .5 1 0 .9 7 0 .9 9 1 .0 0 4 .4 2 1 .0 0 1 5 .4 5 0 .9 9 1 .0 0 4 .4 2 0 .9 9 1 9 .9 2 0 .9 9 1 .0 0 4 .4 2 0 .9 9 f PS =1 3 1 7 .5 6 0 .9 8 8 1 .0 0 2 7 .6 7 9 0 .9 9 4 2 5 .3 4 0 .9 9 1 .0 0 7 .6 8 0 .9 9 3 3 .1 1 0 .9 9 1 .0 0 7 .6 8 0 .9 9 2 9 .6 5 0 .9 8 4 1 .0 0 4 2 .7 2 3 1 .0 1 3 1 2 .4 2 0 .9 9 1 .0 0 2 7 4 1 .0 1 1 5 .2 1 0 .9 9 1 .0 0 2 .7 5 1 .0 0 2 .5 1 3 .0 0 0 .9 9 1 .0 0 4 .4 0 1 .0 1 1 7 .4 7 0 .9 9 1 .0 0 4 .4 1 1 .0 0 2 1 .9 4 0 .9 9 1 .0 0 4 .4 2 1 .0 0 f PS =3 3 1 9 .5 8 0 .9 8 7 1 .0 0 2 7 .6 6 2 1 .0 0 0 2 7 .3 5 0 .9 9 1 .0 0 7 .6 7 1 .0 0 3 5 .1 2 0 .9 9 1 .0 0 7 .6 8 1 .0 0 2 1 1 .7 2 0 .9 8 1 1 .0 0 4 2 .6 7 4 1 .0 2 9 1 4 .4 7 0 .9 8 1 .0 0 2 .7 2 1 .0 2 1 7 .2 5 0 .9 9 1 .0 0 2 .7 4 1 .0 1 2 .5 1 5 .0 5 0 .9 8 1 .0 0 4 .3 6 1 .0 2 1 9 .5 0 0 .9 9 1 .0 0 4 .3 9 1 .0 1 2 3 .9 7 0 .9 9 1 .0 0 4 .4 1 1 .0 0 f PS =5 3 2 1 .6 2 0 .9 8 6 1 .0 0 2 7 .6 3 8 1 .0 0 6 2 9 .3 7 0 .9 9 1 .0 0 7 .6 6 1 .0 0 3 7 .1 4 0 .9 9 1 .0 0 7 .6 7 1 .0 0 2 1 7 .1 1 0 .9 7 1 1 .0 0 5 2 .4 1 3 1 .0 7 9 1 9 .7 2 0 .9 8 1 .0 0 2 .6 2 1 .0 4 2 2 .4 3 0 .9 8 1 .0 0 2 .6 3 1 .0 3 2 .5 2 0 .3 0 0 .9 8 1 .0 0 4 .2 1 1 .0 4 2 4 .6 6 0 .9 8 1 .0 0 4 .3 3 1 .0 2 2 9 .0 9 0 .9 8 1 .0 0 4 .3 7 1 .0 2 f PS = 10 3 2 6 .7 6 0 .9 8 3 1 .0 0 2 7 .5 4 5 1 .0 2 0 3 4 .4 8 0 .9 8 1 .0 0 7 .6 2 1 .0 1 4 2 .2 2 0 .9 9 1 .0 0 7 .6 4 1 .0 1 2 6 8 .9 0 0 .9 4 8 1 .0 0 2 0 .8 7 3 1 .2 5 1 6 9 .8 5 0 .9 5 1 .0 0 0 .9 2 1 .2 5 7 0 .8 4 0 .9 5 1 .0 0 0 .9 7 1 .2 4 2 .5 7 3 .8 4 0 .9 4 1 .0 0 0 .8 9 1 .3 3 7 4 .8 2 0 .9 4 1 .0 0 0 .9 5 1 .3 2 7 5 .8 7 0 .9 4 1 .0 0 1 .0 4 1 .3 2 f PS = 50 3 7 9 .1 3 0 .9 2 4 1 .0 0 3 0 .9 0 2 1 .4 0 5 8 0 .1 5 0 .9 2 1 .0 0 0 .9 8 1 .4 0 8 1 .2 8 0 .9 2 1 .0 0 1 .1 4 1 .3 9 321 T ab le C -3 0 b : M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T > 2 0 0 o F f o r g 1 7 C ar b o n S te el f o r T> 2 0 0 o F f S= 2 f S= 3 f S= 4 ȕ ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ S Ȗƍ P S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ S Ȗƍ P S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ S Ȗƍ P S 2 7 .1 9 0 .9 6 6 1 .0 0 4 2 .7 2 3 0 .9 9 5 1 0 .0 1 0 .9 7 1 .0 0 2 .7 3 0 .9 9 1 2 .8 3 0 .9 7 1 .0 0 2 .7 3 0 .9 9 2 .5 1 0 .6 0 0 .9 7 1 .0 0 4 .3 7 0 .9 9 1 5 .1 2 0 .9 7 1 .0 0 4 .3 8 0 .9 9 1 9 .6 4 0 .9 7 1 .0 0 4 .3 9 0 .9 9 f PS = 0.5 3 1 7 .2 7 0 .9 6 8 1 .0 0 2 7 .6 0 7 0 .9 9 3 2 5 .1 3 0 .9 7 1 .0 0 7 .6 1 0 .9 9 3 2 .9 9 0 .9 7 1 .0 0 7 .6 2 0 .9 9 2 7 .7 1 0 .9 6 4 1 .0 0 4 2 .7 1 4 0 .9 9 9 1 0 .5 2 0 .9 7 1 .0 0 2 .7 3 1 .0 0 1 3 .3 4 0 .9 7 1 .0 0 2 .7 3 1 .0 0 2 .5 1 1 .1 1 0 .9 7 1 .0 0 4 .3 7 1 .0 0 1 5 .6 3 0 .9 7 1 .0 0 4 .3 8 0 .9 9 2 0 .1 6 0 .9 7 1 .0 0 4 .3 8 0 .9 9 f PS =1 3 1 7 .7 8 0 .9 6 7 1 .0 0 2 7 .6 0 0 0 .9 9 4 2 5 .6 4 0 .9 7 1 .0 0 7 .6 1 0 .9 9 3 3 .5 0 0 .9 7 1 .0 0 7 .6 1 0 .9 9 2 9 .8 0 0 .9 5 6 1 .0 0 4 2 .6 6 2 1 .0 1 4 1 2 .6 0 0 .9 6 1 .0 0 2 .7 0 1 .0 1 1 5 .4 1 0 .9 6 1 .0 0 2 .7 1 1 .0 0 2 .5 1 3 .1 9 0 .9 6 1 .0 0 4 .3 2 1 .0 1 7 .7 0 0 .9 6 1 .0 0 4 .3 5 1 .0 0 2 2 .2 2 0 .9 7 1 .0 0 4 .3 6 1 .0 0 f PS =3 3 1 9 .8 5 0 .9 6 4 1 .0 0 2 7 .5 6 4 1 .0 0 0 2 4 .7 0 0 .9 7 1 .0 0 7 .5 9 1 .0 0 3 5 .5 5 0 .9 7 1 .0 0 7 .6 0 1 .0 0 2 1 1 .9 5 0 .9 4 7 1 .0 0 4 2 .5 7 9 1 .0 3 0 1 4 .7 1 0 .9 6 1 .0 0 2 .6 6 1 .0 2 1 7 .5 0 0 .9 6 1 .0 0 2 .6 9 1 .0 1 2 .5 1 5 .3 0 0 .9 5 1 .0 0 4 .2 6 1 .0 2 1 9 .7 9 0 .9 6 1 .0 0 4 .3 2 1 .0 1 2 4 .2 9 0 .9 6 1 .0 0 4 .3 4 1 .0 0 f PS =5 3 2 1 .9 3 0 .9 6 1 1 .0 0 2 7 .5 1 8 1 .0 0 6 2 9 .7 7 0 .9 6 1 .0 0 7 .5 6 1 .0 0 3 7 .6 2 0 .9 7 1 .0 0 7 .5 8 1 .0 0 2 1 7 .6 1 0 .9 1 7 1 .0 0 5 2 .1 2 8 1 .0 8 8 2 0 .1 4 0 .9 4 1 .0 0 2 .4 9 1 .0 5 2 2 .8 4 0 .9 5 1 .0 0 2 .5 9 1 .0 3 2 .5 2 0 .7 4 0 .9 4 1 .0 0 4 .0 0 1 .0 5 2 5 .1 1 0 .9 5 1 .0 0 4 .2 0 1 .0 3 2 9 .5 5 0 .9 6 1 .0 0 4 .2 7 1 .0 2 f PS = 10 3 2 7 .2 3 0 .9 5 2 1 .0 0 2 7 .3 5 4 1 .0 2 1 3 5 .0 0 0 .9 6 1 .0 0 7 .4 8 1 .0 1 4 2 .8 2 0 .9 6 1 .0 0 7 .5 3 1 .0 1 2 7 2 .1 4 0 .8 7 8 1 .0 0 2 0 .8 6 6 1 .2 1 3 7 3 .1 4 0 .8 8 1 .0 0 0 .9 0 1 .2 1 7 4 .1 9 0 .8 8 1 .0 0 0 .9 5 1 .2 1 2 .5 7 8 .1 3 0 .8 5 1 .0 0 0 .8 8 1 .2 8 7 9 .1 9 0 .8 5 1 .0 0 0 .9 3 1 .2 7 8 0 .3 1 0 .8 5 1 .0 0 0 .9 9 1 .2 7 f PS = 50 3 8 4 .6 2 0 .8 2 5 1 .0 0 3 0 .8 9 1 1 .3 4 1 8 5 .7 4 0 .8 2 1 .0 0 0 .9 5 1 .3 4 8 6 .9 5 0 .8 2 1 .0 0 1 .0 5 1 .3 3 322 T ab le C -3 1 a: A d ju st ed N o m in al R es is ta n ce F ac to r fo r g 1 7 a n d C ar b o n S te el f o r Ȗ ǹ = 1 .1 , Ȗ P S= 1 .2 a n d Ȗ S = 1 .5 C ar b o n S te el f S= 2 f S= 3 f S= 4 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 2 0 .6 4 (1 ) 0 .5 8 (1 ) 0 .6 4 (1 ) 0 .6 0 (1 ) 0 .7 0 0 .6 0 (1 ) 0 .5 4 (1 ) 0 .6 0 (1 ) 0 .5 6 (1 ) 0 .6 8 0 .5 7 (1 ) 0 .5 2 (1 ) 0 .5 8 (1 ) 0 .5 4 (1 ) 0 .6 7 2 .5 0 .6 3 0 .5 7 0 .6 3 0 .5 9 0 .4 8 0 .6 0 0 .5 4 0 .6 0 0 .5 6 0 .4 5 0 .5 8 0 .5 3 0 .5 8 0 .5 4 0 .4 4 f PS = 05 3 0 .3 9 0 .3 5 0 .3 9 0 .3 7 0 .2 9 0 .3 6 0 .3 3 0 .3 6 0 .3 4 0 .2 7 0 .3 4 0 .3 1 0 .3 5 0 .3 2 0 .2 6 2 0 .6 7 (1 ) 0 .6 2 (1 ) 0 .6 8 (1 ) 0 .6 4 (1 ) 0 .7 3 0 .6 3 (1 ) 0 .5 7 (1 ) 0 .6 3 (1 ) 0 .5 9 (1 ) 0 .7 0 0 .6 0 (1 ) 0 .5 5 (1 ) 0 .6 0 (1 ) 0 .5 7 (1 ) 0 .6 8 2 .5 0 .6 7 0 .6 1 0 .6 7 0 .6 3 0 .5 0 0 .6 2 0 .5 7 0 .6 3 0 .5 9 0 .4 7 0 .6 0 0 .5 5 0 .6 0 0 .5 7 0 .4 5 f PS =1 3 0 .4 2 0 .3 8 0 .4 2 0 .3 9 0 .3 1 0 .3 8 0 .3 5 0 .3 8 0 .3 6 0 .2 9 0 .3 6 0 .3 3 0 .3 6 0 .3 4 0 .2 7 2 0 .7 8 (1 ) 0 .7 1 (1 ) 0 .7 8 (1 ) 0 .7 4 (1 ) 0 .8 0 0 .7 2 (1 ) 0 .6 6 (1 ) 0 .7 2 (1 ) 0 .6 8 (1 ) 0 .7 6 0 .6 8 (1 ) 0 .6 2 (1 ) 0 .6 8 (1 ) 0 .6 4 (1 ) 0 .7 4 2 .5 0 .7 9 0 .7 2 0 .7 9 0 .7 4 0 .5 9 0 .7 2 0 .6 5 0 .7 2 0 .6 8 0 .5 4 0 .6 8 0 .6 2 0 .6 8 0 .6 4 0 .5 1 f PS =3 3 0 .5 2 0 .4 8 0 .5 2 0 .4 9 0 .3 9 0 .4 6 0 .4 2 0 .4 6 0 .4 3 0 .3 5 0 .4 2 0 .3 9 0 .4 2 0 .4 0 0 .3 2 2 0 .8 5 (1 ) 0 .7 8 (1 ) 0 .8 5 (1 ) 0 .8 0 (1 ) 0 .8 4 0 .7 8 (1 ) 0 .7 2 (1 ) 0 .7 8 (1 ) 0 .7 4 (1 ) 0 .8 0 0 .7 4 (1 ) 0 .6 7 (1 ) 0 .7 4 (1 ) 0 .7 0 (1 ) 0 .7 8 2 .5 0 .8 7 0 .8 0 0 .8 7 0 .8 2 0 .6 6 0 .7 9 0 .7 2 0 .7 9 0 .7 5 0 .6 0 0 .7 4 0 .6 8 0 .7 4 0 .7 0 0 .5 6 f PS =5 3 0 .6 1 0 .5 5 0 .6 1 0 .5 7 0 .4 6 0 .5 3 0 .4 8 0 .5 3 0 .5 0 0 .4 0 0 .4 8 0 .4 4 0 .4 8 0 .4 5 0 .3 6 2 0 .9 4 (1 ) 0 .8 5 (1 ) 0 .9 3 (1 ) 0 .8 7 (1 ) 0 .8 9 0 .8 8 (1 ) 0 .8 1 (1 ) 0 .8 8 (1 ) 0 .8 3 (1 ) 0 .8 6 0 .8 4 (1 ) 0 .7 6 (1 ) 0 .8 4 (1 ) 0 .7 9 (1 ) 0 .8 4 2 .5 1 .0 0 0 .9 2 1 .0 0 0 .9 4 0 .7 5 0 .9 2 0 .8 4 0 .9 2 0 .8 6 0 .6 9 0 .8 6 0 .7 8 0 .8 6 0 .8 1 0 .6 5 f PS = 10 3 0 .7 6 0 .7 0 0 .7 6 0 .7 2 0 .5 7 0 .6 6 0 .6 0 0 .6 6 0 .6 2 0 .5 0 0 .5 9 0 .5 4 0 .5 9 0 .5 6 0 .4 5 2 0 .9 4 (1 ) 0 .8 5 (1 ) 0 .9 1 (1 ) 0 .8 5 (1 ) 0 .8 3 0 .9 4 (1 ) 0 .8 6 (1 ) 0 .9 2 (1 ) 0 .8 6 (1 ) 0 .8 5 0 .9 5 (1 ) 0 .8 7 (1 ) 0 .9 2 (1 ) 0 .8 7 (1 ) 0 .8 6 2 .5 1 .0 6 0 .9 7 1 .0 2 0 .9 6 0 .7 7 1 .0 8 0 .9 9 1 .0 4 0 .9 8 0 .7 8 1 .1 0 1 .0 0 1 .0 5 0 .9 9 0 .7 9 f PS = 50 3 0 .9 9 0 .9 1 0 .9 4 0 .8 9 0 .7 1 1 .0 1 0 .9 2 0 .9 6 0 .9 0 0 .7 2 1 .0 2 0 .9 3 0 .9 7 0 .9 1 0 .7 3 (1 ) Ȗ ǹ = Ȗ P S = 1 a n d Ȗ P S = 0 .9 323 T ab le C -3 1 b : A d ju st ed N o m in al R es is ta n ce F ac to r fo r g 1 7 a n d S ta in le ss S te el f o r Ȗ ǹ = 1 .1 , Ȗ P S= 1 .2 a n d Ȗ S = 1 .5 S ta in le ss S te el f S= 2 f S= 3 f S= 4 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 2 0 .6 2 (1 ) 0 .5 4 (1 ) 0 .7 1 0 .7 1 0 .6 9 0 .5 9 (1 0 .5 1 (1 0 .6 9 0 .6 9 0 .6 6 0 .5 6 0 .5 2 0 .6 8 0 .6 8 0 .6 5 2 .5 0 .6 2 0 .5 4 0 .4 8 0 .4 8 0 .4 7 0 .5 8 0 .5 1 0 .4 6 0 .4 6 0 .4 4 0 .5 7 0 .4 9 0 .4 4 0 .4 4 0 .4 3 f PS = 0.5 3 0 .3 8 0 .3 3 0 .3 0 0 .3 0 0 .2 9 0 .3 5 0 .3 1 0 .2 7 0 .2 7 0 .2 6 0 .3 4 0 .2 9 0 .2 6 0 .2 6 0 .2 5 2 0 .6 6 (1 ) 0 .5 7 (1 ) 0 .7 4 0 .7 4 0 .7 1 0 .6 1 (1 0 .5 3 (1 0 .7 1 0 .7 1 0 .6 9 0 .5 9 0 .5 1 0 .6 9 0 .6 9 0 .6 7 2 .5 0 .6 6 0 .5 7 0 .5 1 0 .5 1 0 .4 9 0 .6 1 0 .5 3 0 .4 8 0 .4 8 0 .4 6 0 .5 9 0 .5 1 0 .4 6 0 .4 6 0 .4 4 f PS =1 3 0 .4 1 0 .6 0 .3 2 0 .3 2 0 .3 1 0 .3 7 0 .3 2 0 .2 9 0 .2 9 0 .2 8 0 .3 5 0 .3 1 0 .2 8 0 .2 8 0 .2 7 2 0 .7 7 (1 ) 0 .6 7 (1 ) 0 .8 1 0 .8 1 0 .7 8 0 .7 1 (1 0 .6 1 (1 0 .7 7 0 .7 7 0 .7 5 0 .6 7 0 .5 8 0 .7 5 0 .7 5 0 .7 2 2 .5 0 .7 7 0 .6 7 0 .6 0 0 .6 0 0 .5 0 0 .7 0 0 .6 1 0 .5 5 0 .5 5 0 .5 3 0 .6 6 0 .5 8 0 .5 2 0 .5 2 0 .5 0 f PS =3 3 0 .5 7 0 .5 0 0 .4 0 0 .4 0 0 .3 9 0 .4 5 0 .3 9 0 .3 5 0 .3 5 0 .3 4 0 .4 1 0 .3 6 0 .3 2 0 .3 2 0 .3 1 2 0 .8 4 (1 ) 0 .7 2 (1 ) 0 .8 6 0 .8 6 0 .8 3 0 .7 7 (1 0 .6 7 (1 0 .8 2 0 .8 2 0 .7 9 0 .7 2 0 .6 3 0 .7 9 0 .7 9 0 .7 6 2 .5 0 .8 6 0 .7 4 0 .6 7 0 .6 7 0 .6 4 0 .7 8 0 .6 7 0 .6 1 0 .6 1 0 .5 8 0 .7 3 0 .6 3 0 .5 7 0 .5 7 0 .5 5 f PS =5 3 0 .6 0 0 .5 2 0 .4 6 0 .4 6 0 .4 5 0 .5 2 0 .4 5 0 .4 0 0 .4 0 0 .3 9 0 .4 7 0 .4 1 0 .3 7 0 .3 7 0 .3 5 2 0 .9 2 (1 ) 0 .8 0 (1 ) 0 .9 1 0 .9 1 0 .8 8 0 .8 8 (1 0 .8 1 (1 0 .8 8 0 .8 8 0 .8 5 0 .8 2 0 .7 1 0 .8 5 0 .8 5 0 .8 1 2 .5 0 .9 9 0 .8 6 0 .7 7 0 .7 7 0 .7 4 0 .9 0 0 .7 8 0 .7 0 0 .7 0 0 .7 0 0 .8 4 0 .7 3 0 .6 6 0 .6 6 0 .6 4 f PS = 10 3 0 .7 5 0 .6 5 0 .5 8 0 .5 8 0 .5 6 0 .6 5 0 .5 6 0 .5 0 0 .5 0 0 .4 9 0 .5 8 0 .5 0 0 .4 5 0 .4 5 0 .4 4 2 0 .9 2 (1 ) 0 .8 0 (1 ) 0 .8 7 0 .8 7 0 .8 4 0 .9 3 (1 0 .8 0 (1 0 .8 9 0 .8 9 0 .8 6 0 .9 3 0 .8 1 0 .9 0 0 .9 0 0 .8 7 2 .5 1 .0 4 0 .9 1 0 .8 1 0 .8 1 0 .7 9 1 .0 6 0 .9 2 0 .8 3 0 .8 3 0 .8 0 1 .0 8 0 .9 3 0 .8 4 0 .8 4 0 .8 1 f PS = 50 3 0 .9 7 0 .8 5 0 .7 6 0 .7 6 0 .7 3 0 .9 9 0 .8 6 0 .7 7 0 .7 7 0 .7 5 1 .0 0 0 .8 7 0 .7 8 0 .7 8 0 .7 6 (1 ) F o r th es e fa ct o rs Ȗ ǹ = Ȗ P S= 1 a n d Ȗ S= 0 .9 324 325 C.15. Performance Function g18 Table C-32 gives the calculated mean load and resistance factors for performance function g18. In this table, ȝfu is the converged mean value of the ultimate strength of steel. Table C-33 shows the evaluated adjusted nominal resistance factors for nominal load factors Ȗǹ=1.1, ȖPD=1.2, and ȖL=1.3. T ab le C -3 2 a: M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T ”2 0 0 o F a n d S ta in le ss S te el f o r g 1 8 C ar b o n S te el T ”2 0 0 o F & S ta in le ss S te el f L= 0 .5 f L= 1 .0 f L= 2 .0 ȕ ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P D Ȗƍ L ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P D Ȗƍ L ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P D Ȗƍ L 3 2 .7 3 0 .9 1 1 .1 0 1 .3 9 1 .3 9 3 .6 1 0 .9 3 1 .0 6 1 .1 0 1 .7 6 5 .5 5 0 .9 5 1 .0 3 1 .0 2 1 .8 5 4 3 .0 9 0 .8 9 1 .1 1 1 .6 5 1 .6 5 4 .2 6 0 .9 2 1 .0 6 1 .1 1 2 .2 9 6 .8 1 0 .9 3 1 .0 3 1 .0 2 2 .4 0 f PD =0.5 5 3 .7 9 0 .8 7 1 .1 2 2 .1 6 2 .1 6 5 .5 4 0 .8 9 1 .0 6 1 .1 2 3 .3 4 9 .3 3 0 .9 1 1 .0 3 1 .0 3 3 .4 7 3 3 .6 1 0 .9 3 1 .0 6 1 .1 0 1 .1 0 4 .2 9 0 .9 3 1 .0 5 1 .4 6 1 .4 6 6 .1 2 0 .9 4 1 .0 3 1 .1 0 1 .8 1 4 4 .2 6 0 .9 2 1 .0 6 1 .1 1 1 .1 1 4 .9 8 0 .9 1 1 .0 6 1 .7 4 1 .7 4 7 .3 8 0 .9 3 1 .0 3 1 .1 1 2 .3 5 f PD =1 5 5 .5 4 0 .8 9 1 .0 6 1 .1 2 1 .1 2 6 .3 3 0 .8 8 1 .0 6 2 .2 7 2 .2 7 9 .9 2 0 .9 0 1 .0 3 1 .1 2 3 .4 1 3 7 .5 2 0 .9 5 1 .0 2 1 .0 0 1 .0 0 8 .0 6 0 .9 5 1 .0 2 1 .8 6 1 .0 4 9 .2 3 0 .9 4 1 .0 2 1 .7 7 1 .1 8 4 9 .4 0 0 .9 4 1 .0 2 1 .0 0 1 .0 0 9 .9 4 0 .9 3 1 .0 2 2 .4 0 2 .4 0 1 1 .1 4 0 .9 3 1 .0 2 2 .3 0 1 .2 0 f PD =3 5 1 3 .1 6 0 .9 1 1 .0 2 1 .0 0 1 .0 0 1 3 .7 2 0 .9 1 1 .0 2 3 .4 7 1 .0 5 1 4 .9 6 0 .9 0 1 .0 2 3 .3 5 1 .2 2 3 1 1 .4 8 0 .9 5 1 .0 1 0 .9 9 0 .9 9 1 2 .0 0 0 .9 5 1 .0 1 1 .8 8 1 .0 1 1 3 .0 9 0 .9 5 1 .0 1 1 .8 5 1 .0 7 4 1 4 .5 8 0 .9 4 1 .0 1 0 .9 9 0 .9 9 1 5 .1 1 0 .9 4 1 .0 1 2 .4 3 1 .0 1 1 6 .2 2 0 .9 3 1 .0 1 2 .4 0 1 .0 7 f PD =5 5 2 0 .8 3 0 .9 2 1 .0 1 0 .9 9 0 .9 9 2 1 .3 8 0 .9 1 1 .0 1 3 .5 1 1 .0 1 2 2 .5 2 0 .9 1 1 .0 1 3 .4 7 1 .0 7 3 2 1 .3 9 0 .9 6 1 .0 1 0 .9 8 0 .9 8 2 1 .9 0 0 .9 6 1 .0 1 1 .8 9 0 .9 9 2 2 .9 5 0 .9 5 1 .0 1 1 .8 9 1 .6 1 4 2 7 .5 7 0 .9 4 1 .0 1 0 .9 8 0 .9 8 2 8 .0 9 0 .9 4 1 .0 1 2 .4 5 0 .9 9 2 9 .1 6 0 .9 4 1 .0 1 2 .4 4 1 .0 1 f PD =10 5 4 0 .0 3 0 .9 2 1 .0 1 0 .9 8 0 .9 8 4 0 .5 6 0 .9 2 1 .0 1 3 .5 3 0 .9 9 4 1 .6 5 0 .9 2 1 .0 1 3 .5 2 1 .0 1 3 1 0 0 .7 0 .9 6 1 .0 0 0 .9 7 0 .9 7 1 0 1 .2 0 .9 6 1 .0 0 1 .9 0 0 .9 9 1 0 2 .2 0 .9 6 1 .0 0 1 .9 0 0 .9 7 4 1 3 1 .5 0 .9 5 1 .0 0 0 .9 7 0 .9 7 1 3 2 .1 0 .9 5 1 .0 0 2 .4 6 0 .9 7 1 3 3 .1 0 .9 5 1 .0 0 2 .4 6 0 .9 8 f PD =50 5 1 9 3 .6 0 .9 2 1 .0 0 0 .9 7 0 .9 7 1 4 4 .2 0 .9 2 1 .0 0 3 .5 4 0 .9 7 1 9 5 .2 0 .9 2 1 .0 0 3 .5 4 0 .9 8 326 T ab le C -3 2 b : M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T > 2 0 0 o F f o r g 1 8 C ar b o n S te el T > 2 0 0 o F f L= 0 .5 f L= 1 .0 f L= 2 .0 ȕ ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P D Ȗƍ L ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P D Ȗƍ L ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P D Ȗƍ L 3 2 .9 5 0 .8 0 1 .0 9 1 .2 8 1 .2 8 3 .8 5 0 .8 3 1 .0 6 1 .0 9 1 .6 1 5 .8 4 0 .8 6 1 .0 3 1 .0 2 1 .7 5 4 3 .4 0 0 .7 6 1 .1 1 1 .4 7 1 .4 7 4 .5 9 0 .8 0 1 .0 6 1 .1 1 2 .0 6 7 .2 5 0 .8 3 1 .0 3 1 .0 2 2 .2 4 f PD =0.5 5 4 .2 7 0 .7 0 1 .1 2 1 .8 9 1 .8 9 6 .1 1 0 .7 5 1 .0 6 1 .1 1 2 .9 7 1 0 .1 5 0 .7 8 1 .0 3 1 .0 3 3 .1 9 3 3 .8 5 0 .8 3 1 .0 6 1 .6 1 1 .0 9 4 .5 8 0 .8 3 1 .0 5 1 .3 7 1 .3 7 6 .4 6 0 .8 5 1 .0 3 1 .1 0 1 .6 9 4 4 .5 9 0 .8 0 1 .0 6 2 .0 6 1 .1 1 5 .4 0 0 .7 9 1 .0 6 1 .6 0 1 .6 0 7 .8 9 0 .8 2 1 .0 3 1 .1 1 2 .1 7 f PD =1 5 6 .1 1 0 .7 5 1 .0 6 2 .9 7 1 .1 1 7 .0 3 0 .7 3 1 .0 6 2 .0 5 2 .0 5 1 0 .8 4 0 .7 7 1 .0 3 1 .1 1 3 .1 0 3 7 .8 8 0 .8 7 1 .0 2 1 .7 9 1 .0 0 8 .4 7 0 .8 7 1 .0 2 1 .7 6 1 .0 4 9 .7 5 0 .8 5 1 .0 2 1 .6 5 1 .1 7 4 9 .9 6 0 .8 4 1 .0 2 2 .2 9 1 .0 0 1 0 .5 7 0 .8 3 1 .0 2 2 .2 5 1 .0 5 1 1 .9 1 0 .8 2 1 .0 2 2 .1 2 1 .2 0 f PD =3 5 1 4 .2 5 0 .7 9 1 .0 2 3 .2 4 1 .0 0 1 4 .9 0 0 .7 8 1 .0 2 3 .2 0 1 .0 5 1 6 .3 4 0 .7 7 1 .0 2 3 .0 4 1 .2 1 3 1 1 .9 8 0 .8 8 1 .0 1 1 .8 1 0 .9 9 1 2 .5 5 0 .8 8 1 .0 1 1 .8 0 1 .0 1 1 3 .7 3 0 .8 7 1 .0 1 1 .7 6 1 .0 6 4 1 5 .4 0 0 .8 5 1 .0 1 2 .3 2 0 .9 9 1 5 .9 9 0 .8 5 1 .0 1 2 .3 0 1 .0 1 1 7 .2 2 0 .8 4 1 .0 1 2 .2 6 1 .0 7 f PD =5 5 2 2 .4 7 0 .8 0 1 .0 1 3 .2 9 0 .9 9 2 3 .1 0 0 .7 9 1 .0 1 3 .2 6 1 .0 1 2 4 .4 2 0 .7 9 1 .0 1 3 .2 1 1 .0 7 3 2 2 .2 7 0 .8 9 1 .0 1 1 .8 3 0 .9 8 2 2 .8 2 0 .8 9 1 .0 1 1 .8 2 0 .9 9 2 3 .9 5 0 .8 8 1 .0 1 1 .8 1 1 .0 1 4 2 9 .0 4 0 .8 6 1 .0 1 2 .3 4 0 .9 8 2 9 .6 2 0 .8 5 1 .0 1 2 .3 3 0 .9 9 3 0 .7 9 0 .8 5 1 .0 1 2 .3 2 1 .0 1 f PD =10 5 4 3 .0 8 0 .8 0 1 .0 1 3 .3 1 0 .9 8 4 3 .6 9 0 .8 0 1 .0 1 3 .3 0 0 .9 9 4 4 .9 4 0 .8 0 1 .0 1 3 .2 8 1 .0 1 3 1 0 4 .7 0 .8 9 1 .0 0 1 .8 4 0 .9 7 1 0 5 .2 0 .8 9 1 .0 0 1 .8 4 0 .9 7 1 0 6 .3 0 .8 9 1 .0 0 1 .8 4 0 .9 7 4 1 3 8 .3 0 .8 6 1 .0 0 2 .3 6 0 .9 7 1 3 8 .9 0 .8 6 1 .0 0 2 .3 5 0 .9 7 1 4 0 0 .8 6 1 .0 0 2 .3 5 0 .9 8 f PD =50 5 2 0 8 0 .8 1 1 .0 0 3 .3 3 0 .9 7 2 0 8 .6 0 .8 1 1 .0 0 3 .3 3 0 .9 7 2 0 2 .7 0 .8 1 1 .0 0 3 .3 3 0 .9 8 327 T ab le C -3 3 a: A d ju st ed N o m ia n l R es is ta n ce F ac to rs f o r C ar b o n S te el a n d Ȗ ǹ = 1 .1 , Ȗ L = 1 .3 , Ȗ P D = 1 .2 f o r g 1 8 C ar b o n S te el f L= 0 .5 f L= 1 .0 f L= 2 .0 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 3 1 .0 3 (1 ) 0 .9 4 (1 ) 0 .9 7 (1 ) 0 .9 1 (1 ) 0 .7 3 (1 ) 0 .9 8 (1 ) 0 .9 0 (1 ) 0 .9 4 (1 ) 0 .8 8 (1 ) 0 .7 1 (1 ) 0 .9 1 (1 ) 0 .8 3 (1 ) 0 .8 8 (1 ) 0 .8 2 (1 ) 0 .6 6 (1 ) 4 0 .9 7 0 .8 9 0 .9 0 0 .8 5 0 .6 8 0 .9 7 0 .8 3 0 .8 6 0 .8 1 0 .6 5 0 .8 3 0 .7 6 0 .7 9 0 .7 5 0 .6 0 f PD =0.5 5 0 .7 9 0 .7 2 0 .7 2 0 .6 7 0 .5 4 0 .7 0 0 .6 4 0 .6 5 0 .6 1 0 .4 9 0 .6 0 0 .5 5 0 .5 7 0 .5 3 0 .4 3 3 0 .8 1 (1 ) 0 .8 5 (1 ) 0 .9 5 (1 ) 0 .8 9 (1 ) 0 .7 1 (1 ) 1 .0 1 (1 ) 0 .9 2 (1 ) 0 .9 6 (1 ) 0 .9 1 (1 ) 0 .7 2 (1 ) 0 .9 5 (1 ) 0 .8 7 (1 ) 0 .9 2 (1 ) 0 .8 6 (1 ) 0 .6 9 (1 ) 4 0 .9 1 0 .8 3 0 .8 6 0 .8 1 0 .6 5 0 .9 5 0 .8 7 0 .8 9 0 .8 4 0 .6 7 0 .8 7 0 .7 5 0 .8 4 0 .7 9 0 .6 3 f PD =1 5 0 .7 0 0 .6 4 0 .6 4 0 .6 1 0 .4 8 0 .7 5 0 .6 8 0 .6 9 0 .6 5 0 .5 2 0 .6 6 0 .6 0 0 .6 1 0 .5 7 0 .4 6 3 0 .9 0 (1 ) 0 .8 2 (1 ) 0 .8 7 (1 ) 0 .8 2 (1 ) 0 .6 6 (1 ) 0 .9 3 (1 ) 0 .8 5 (1 ) 0 .9 0 (1 ) 0 .8 5 (1 ) 0 .6 8 (1 ) 0 .9 7 (1 ) 0 .8 9 (1 ) 0 .9 4 (1 ) 0 .8 8 (1 ) 0 .7 1 (1 ) 4 0 .7 8 0 .7 1 0 .7 5 0 .7 0 0 .5 6 0 .8 2 0 .7 5 0 .7 9 0 .7 4 0 .5 9 0 .8 9 0 .8 2 0 .8 5 0 .8 0 0 .6 4 f PD =3 5 0 .5 6 0 .5 1 0 .5 2 0 .4 9 0 .3 9 0 .6 0 0 .5 5 0 .5 6 0 .5 3 0 .4 2 0 .6 5 0 .5 7 0 .6 2 0 .5 8 0 .4 7 3 0 .8 6 (1 ) 0 .7 9 (1 ) 0 .8 4 (1 ) 0 .7 9 (1 ) 0 .6 3 (1 ) 0 .8 9 (1 ) 0 .8 1 (1 ) 0 .8 6 (1 ) 0 .8 1 (1 0 .6 5 (1 ) 0 .9 3 (1 ) 0 .8 5 (1 ) 0 .9 0 (1 ) 0 .8 5 (1 ) 0 .6 8 (1 ) 4 0 .7 4 0 .6 7 0 .7 1 0 .6 7 0 .5 4 0 .7 7 0 .7 0 0 .7 4 0 .7 0 0 .5 6 0 .8 2 0 .7 5 0 .7 9 0 .7 4 0 .6 0 f PD =5 5 0 .5 2 0 .4 7 0 .4 9 0 .4 6 0 .3 7 0 .5 4 0 .5 0 0 .5 1 0 .4 8 0 .3 9 0 .6 0 0 .5 4 0 .5 6 0 .5 3 0 .4 2 3 0 .8 3 (1 ) 0 .7 6 (1 ) 0 .8 1 (1 ) 0 .7 7 (1 ) 0 .6 1 (1 ) 0 .8 5 (1 ) 0 .7 7 (1 ) 0 .8 3 (1 ) 0 .7 8 (1 ) 0 .6 2 (1 ) 0 .8 7 (1 ) 0 .8 0 (1 ) 0 .8 5 (1 ) 0 .8 0 (1 ) 0 .6 4 (1 ) 4 0 .7 0 0 .6 4 0 .6 8 0 .6 4 0 .5 1 0 .7 2 0 .6 6 0 .7 0 0 .6 5 0 .5 2 0 .7 6 0 .6 9 0 .7 3 0 .6 8 0 .5 5 f PD =10 5 0 .4 8 0 .4 4 0 .4 6 0 .4 3 0 .3 4 0 .5 0 0 .4 6 0 .4 7 0 .4 4 0 .3 6 0 .5 3 0 .4 8 0 .5 0 0 .4 7 0 .3 8 3 0 .8 0 (1 ) 0 .7 3 (1 ) 0 .7 9 (1 ) 0 .7 4 (1 ) 0 .5 9 (1 ) 0 .8 1 (1 ) 0 .7 4 (1 ) 0 .7 9 (1 ) 0 .7 4 (1 ) 0 .6 0 (1 ) 0 .8 1 (1 ) 0 .7 4 (1 ) 0 .8 0 (1 ) 0 .7 5 (1 ) 0 .6 0 (1 ) 4 0 .6 7 0 .6 1 0 .6 5 0 .6 1 0 .4 9 0 .6 8 0 .6 2 0 .6 5 0 .6 1 0 .4 9 0 .6 8 0 .6 2 0 .6 6 0 .6 2 0 .5 0 f PD =50 5 0 .4 6 0 .4 2 0 .4 3 0 .4 1 0 .3 3 0 .4 4 0 .4 2 0 .4 4 0 .4 1 0 .3 3 0 .4 7 0 .4 3 0 .4 4 0 .4 2 0 .3 3 (1 ) F o r th es e fa ct o rs Ȗ ǹ = Ȗ L = Ȗ P D = 1 .1 328 T ab le C -3 3 b : A d ju st ed N o m ia n l R es is ta n ce F ac to rs f o r S ta in le ss S te el a n d Ȗ ǹ = 1 .1 , Ȗ L = 1 .3 , Ȗ P D = 1 .2 f o r g 1 8 S ta in le ss S te el f L= 0 .5 f L= 1 .0 f L= 2 .0 ȕ R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F R .T . 2 0 0 o F 4 0 0 o F 6 0 0 o F 8 0 0 o F 3 1 .0 1 (1 ) 0 .8 7 (1 ) 0 .7 8 (1 ) 0 .7 8 (1 ) 0 .7 6 (1 ) 0 .9 6 (1 ) 0 .8 4 (1 ) 0 .7 5 (1 ) 0 .7 5 (1 ) 0 .7 2 (1 ) 0 .8 9 (1 ) 0 .7 7 (1 ) 0 .6 9 (1 ) 0 .6 9 (1 ) 0 .6 7 (1 ) 4 0 .9 6 0 .6 3 0 .7 4 0 .7 4 0 .7 2 0 .9 0 0 .7 8 0 .7 0 0 .7 0 0 .6 7 0 .8 1 0 .7 1 0 .6 3 0 .6 3 0 .6 1 f PD =0.5 5 0 .7 8 0 .6 8 0 .6 1 0 .6 1 0 .5 9 0 .6 9 0 .6 0 0 .5 4 0 .5 4 0 .5 2 0 .5 9 0 .5 2 0 .4 6 0 .4 6 0 .4 5 3 0 .9 8 (1 ) 0 .8 5 (1 ) 0 .7 6 (1 ) 0 .7 6 (1 ) 0 .7 3 (1 ) 0 .9 9 (1 ) 0 .8 6 (1 ) 0 .7 7 (1 ) 0 .7 7 (1 ) 0 .7 5 (1 ) 0 .9 4 (1 ) 0 .8 1 (1 ) 0 .7 3 (1 ) 0 .7 3 (1 ) 0 .7 0 (1 ) 4 0 .8 9 0 .7 7 0 .7 0 0 .7 0 0 .6 7 0 .9 4 0 .8 1 0 .7 3 0 .7 3 0 .7 0 0 .8 7 0 .7 5 0 .6 7 0 .6 7 0 .6 5 f PD =1 5 0 .6 9 0 .5 9 0 .5 3 0 .5 3 0 .5 2 0 .7 4 0 .6 4 0 .5 7 0 .5 7 0 .5 5 0 .6 4 0 .5 6 0 .5 0 0 .5 0 0 .4 8 3 0 .8 8 (1 ) 0 .7 6 (1 ) 0 .6 9 (1 ) 0 .6 9 (1 ) 0 .6 6 (1 ) 0 .9 1 (1 ) 0 .7 9 (1 ) 0 .7 1 (1 ) 0 .7 1 (1 ) 0 .6 9 (1 ) 0 .9 6 (1 ) 0 .8 3 (1 ) 0 .7 4 (1 ) 0 .7 4 (1 ) 0 .7 2 (1 ) 4 0 .7 7 0 .6 6 0 .6 0 0 .6 0 0 .5 8 0 .8 1 0 .7 0 0 .6 3 0 .6 3 0 .6 1 0 .8 8 0 .7 6 0 .6 8 0 .6 8 0 .6 6 f PD =3 5 0 .5 5 0 .4 7 0 .4 3 0 .4 3 0 .4 1 0 .5 9 0 .5 1 0 .4 6 0 .4 6 0 .4 4 0 .6 5 0 .5 7 0 .5 1 0 .5 1 0 .4 9 3 0 .8 5 (1 ) 0 .7 4 (1 ) 0 .6 6 (1 ) 0 .6 6 (1 ) 0 .6 4 (1 ) 0 .8 7 (1 ) 0 .7 6 (1 ) 0 .6 8 (1 ) 0 .6 8 (1 ) 0 .6 6 (1 ) 0 .9 1 (1 ) 0 .7 9 (1 ) 0 .7 1 (1 ) 0 .7 1 (1 ) 0 .6 9 (1 ) 4 0 .7 3 0 .6 3 0 .5 7 0 .5 7 0 .5 5 0 .7 6 0 .6 6 0 .5 9 0 .5 9 0 .5 7 0 .8 1 0 .7 0 0 .6 3 0 .6 3 0 .6 1 f PD =5 5 0 .5 1 0 .4 4 0 .4 0 0 .4 0 0 .3 8 0 .5 4 0 .4 6 0 .4 2 0 .4 2 0 .4 0 0 .5 9 0 .5 1 0 .4 6 0 .4 6 0 .4 4 3 0 .8 2 (1 ) 0 .7 7 (1 ) 0 .6 4 (1 ) 0 .6 4 (1 ) 0 .6 2 (1 ) 0 .8 3 (1 ) 0 .7 2 (1 ) 0 .6 5 (1 ) 0 .6 5 (1 ) 0 .6 3 (1 ) 0 .8 6 (1 ) 0 .7 4 (1 ) 0 .6 7 (1 ) 0 .6 7 (1 ) 0 .6 5 (1 ) 4 0 .6 1 0 .6 0 0 .5 4 0 .5 4 0 .5 2 0 .7 1 0 .6 1 0 .5 5 0 .5 5 0 .5 3 0 .7 4 0 .6 4 0 .5 8 0 .5 8 0 .5 6 f PD =10 5 0 .4 8 0 .4 1 0 .3 7 0 .3 7 0 .3 6 0 .4 9 0 .4 3 0 .3 8 0 .3 8 0 .3 7 0 .5 2 0 .4 5 0 .4 0 0 .4 0 0 .3 9 3 0 .7 9 (1 ) 0 .6 9 (1 ) 0 .6 2 (1 ) 0 .6 2 (1 ) 0 .6 5 (1 ) 0 .7 9 (1 ) 0 .6 9 (1 ) 0 .6 2 (1 ) 0 .6 2 (1 ) 0 .6 0 (1 ) 0 .8 9 (1 ) 0 .6 9 (1 ) 0 .6 2 (1 ) 0 .6 2 (1 ) 0 .6 0 (1 ) 4 0 .6 6 0 .5 7 0 .5 1 0 .5 1 0 .5 0 0 .6 6 0 .5 8 0 .5 2 0 .5 2 0 .5 0 0 .6 7 0 .5 8 0 .5 2 0 .5 2 0 .5 1 f PD =50 5 0 .4 5 0 .3 9 0 .3 5 0 .3 5 0 .3 4 0 .4 5 0 .3 9 0 .3 5 0 .3 5 0 .3 4 0 .4 6 0 .4 0 0 .3 6 0 .3 6 0 .3 4 (1 ) F o r th es e fa ct o rs Ȗ ǹ = Ȗ L = Ȗ P D = 1 .1 329 330 C.16. Performance Function g19 Table C-34 gives the calculated mean load and resistance factors for performance function g19. In this table, ȝfu is the converged mean value of the ultimate strength of steel. Table C-35 shows the evaluated adjusted nominal resistance factors for nominal load factors Ȗǹ=1.1, ȖPS=1.2, ȖL=1.3, and ȖS=1.5. T ab le C -3 4 a: M ea n L o ad a n d R es is ta n ce F ac to rs f o r S ta in le ss S te el a n d C ar b o n S te el f o r T” 2 0 0 o F f o r g 1 9 C ar b o n S te el T ”2 0 0 o F & S ta in le ss S te el ȕ= 2 .5 f L= 0 .5 f L= 1 .0 f L= 2 .0 f S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P S Ȗƍ L Ȗƍ S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P S Ȗƍ L Ȗƍ S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P S Ȗƍ L Ȗƍ S 1 6 .4 9 0 .9 9 1 .0 1 1 .0 0 0 .9 8 4 .4 0 6 .9 8 0 .9 8 1 .0 1 1 .0 0 0 .9 9 4 .3 9 8 .0 0 0 .9 8 1 .0 1 1 .0 0 1 .0 1 4 .3 4 2 1 0 .9 6 0 .9 9 1 .0 0 0 .9 9 0 .9 7 4 .4 2 1 1 .4 5 0 .9 9 1 .0 0 0 .9 9 0 .9 8 4 4 1 1 2 .4 5 0 .9 9 1 .0 0 0 .9 9 0 .9 9 4 .4 0 0.5 3 1 5 .4 3 0 .9 9 1 .0 0 0 .9 9 0 .9 7 4 .4 2 1 5 .9 2 0 .9 9 1 .0 0 0 .9 9 0 .9 7 4 .4 2 1 6 .9 1 0 .9 9 1 .0 0 0 .9 9 0 .9 8 4 .4 1 1 6 .9 9 0 .9 8 1 .0 1 1 .0 0 0 .9 8 4 .3 9 7 .4 9 0 .9 8 1 .0 1 1 .0 0 0 .9 9 4 .3 8 8 .5 1 0 .9 8 1 .0 1 1 .0 0 1 .0 1 4 .3 3 2 1 1 .4 6 0 .9 9 1 .0 0 1 .0 0 0 .9 7 4 .4 1 1 1 .9 6 0 .9 9 1 .0 0 1 .0 0 0 .9 8 4 .4 1 1 2 .9 5 0 .9 9 1 .0 0 1 .0 0 0 .9 9 4 .4 0 1 3 1 5 .9 4 0 .9 9 1 .0 0 0 .9 9 0 .9 7 4 .4 2 1 6 .4 3 0 .9 9 1 .0 0 0 .9 9 0 .9 7 4 .4 2 1 7 .4 2 0 .9 9 1 .0 0 0 .9 9 0 .9 8 4 .4 1 1 9 .0 5 0 .9 8 1 .0 1 1 .0 2 0 .9 8 4 .3 2 9 .5 5 0 .9 8 1 .0 1 1 .0 2 0 .9 9 4 .3 0 1 0 .5 7 0 .9 8 1 .0 1 1 .0 2 1 .0 1 4 .2 4 2 1 3 .4 9 0 .9 9 1 .0 0 1 .0 1 0 .9 7 4 .3 9 1 3 .9 9 0 .9 8 1 .0 0 1 .0 1 0 .9 8 4 .3 9 1 4 .9 8 0 .9 8 1 .0 0 1 .0 1 0 .9 9 4 .3 7 3 3 1 7 .9 6 0 .9 9 1 .0 0 1 .0 0 0 .9 7 4 .4 1 1 8 .4 5 0 .9 9 1 .0 0 1 .0 0 0 .9 7 4 .4 1 1 9 .4 4 0 .9 9 1 .0 0 1 .0 0 0 .9 8 4 .4 0 1 1 1 .1 6 0 .9 8 1 .0 1 1 .0 4 0 .9 8 4 .1 8 1 1 .6 6 0 .9 7 1 .0 1 1 .0 4 0 .9 9 4 .1 6 1 2 .6 9 0 .9 7 1 .0 1 1 .0 4 1 .0 1 4 .0 9 2 1 5 .5 5 0 .9 8 1 .0 0 1 .0 2 0 .9 7 4 .3 6 1 6 .0 4 0 .9 8 1 .0 0 1 .0 2 0 .9 8 4 .3 5 1 7 .0 4 0 .9 8 1 .0 0 1 .0 2 0 .9 9 4 .3 4 5 3 1 9 .9 9 0 .9 9 1 .0 0 1 .0 1 0 .9 7 4 .3 9 2 0 .4 9 0 .9 8 1 .0 0 1 .0 1 0 .9 7 4 .3 9 2 1 .4 8 0 .9 8 1 .0 0 1 .0 1 0 .9 8 4 .3 8 1 1 6 .8 0 0 .9 3 1 .0 1 1 .2 9 0 .9 9 1 .2 3 1 7 .3 4 0 .9 3 1 .0 1 1 .2 9 1 .0 1 1 .2 2 1 8 .4 6 0 .9 3 1 .0 1 1 .2 8 1 .0 7 1 .1 7 2 2 0 .7 9 0 .9 8 1 .0 0 1 .0 4 0 .9 7 4 .2 0 2 1 .2 9 0 .9 8 1 .0 0 1 .0 4 0 .9 8 4 .1 9 2 2 .3 0 0 .9 8 1 .0 0 1 .0 4 0 .9 9 4 .1 7 10 3 2 5 .1 6 0 .9 8 1 .0 0 1 .0 2 0 .9 7 4 .3 2 2 5 .6 5 0 .9 8 1 .0 0 1 .0 2 0 .9 7 4 .3 2 2 6 .6 5 0 .9 8 1 .0 0 1 .0 2 0 .9 8 4 .3 1 1 7 3 .4 3 0 .9 4 1 .0 0 1 .3 3 0 .9 7 0 .8 5 7 3 .9 5 0 .9 4 1 .0 0 1 .3 3 0 .9 8 0 .8 5 7 5 .0 0 0 .9 3 1 .0 0 1 .3 3 0 .9 8 0 .8 5 2 7 4 .3 5 0 .9 4 1 .0 0 1 .3 3 0 .9 7 0 .8 9 7 4 .8 8 0 .9 3 1 .0 0 1 .3 2 0 .9 8 0 .8 9 7 5 .9 2 0 .9 3 1 .0 0 1 .3 2 0 .9 8 0 .8 9 f PS 50 3 7 5 .3 3 0 .9 3 1 .0 0 1 .3 2 0 .9 7 0 .9 5 7 5 .8 5 0 .9 3 1 .0 0 1 .3 2 0 .9 8 0 .9 5 7 6 .9 0 0 .9 3 1 .0 0 1 .3 2 0 .9 8 0 .9 5 331 T ab le C -3 4 a: ( C o n ti n u ed ) C ar b o n S te el T ”2 0 0 o F & S ta in le ss S te el ȕ= 3 .0 f L= 0 .5 f L= 1 .0 f L= 2 .0 f S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P S Ȗƍ L Ȗƍ S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P S Ȗƍ L Ȗƍ S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P S Ȗƍ L Ȗƍ S 1 9 .9 1 0 .9 6 1 .0 0 0 .9 9 0 .9 7 7 .5 7 1 0 .4 2 0 .9 6 1 .0 0 0 .9 9 0 .9 8 7 .5 5 1 1 .4 4 0 .9 6 1 .0 0 0 .9 9 0 .9 9 7 .4 9 2 1 7 .7 7 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .6 0 1 8 .2 7 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .5 9 1 9 .2 8 0 .9 6 1 .0 0 0 .9 9 0 .9 8 7 .5 7 0.5 3 2 5 .6 3 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .6 1 2 6 .1 3 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .6 1 2 7 .1 3 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .5 9 1 1 0 .4 3 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 5 1 0 .9 4 0 .9 6 1 .0 0 1 .0 0 0 .9 8 7 .5 3 1 1 .9 6 0 .9 6 1 .0 0 1 .0 0 0 .9 9 7 .4 7 2 1 8 .2 8 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .5 9 1 8 .7 8 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .5 8 1 9 .8 0 0 .9 6 1 .0 0 0 .9 9 0 .9 8 7 .5 6 1 3 2 6 .1 4 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .6 1 2 6 .6 4 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .6 0 2 7 .6 5 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .5 9 1 1 2 .5 2 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .4 5 1 3 .0 3 0 .9 5 1 .0 0 1 .0 1 0 .9 8 7 .4 2 1 4 .0 6 0 .9 5 1 .0 0 1 .0 1 0 .9 9 7 .3 6 2 2 0 .3 5 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 6 2 0 .8 5 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 5 2 1 .8 7 0 .9 6 1 .0 0 1 .0 0 0 .9 8 7 .5 3 3 3 2 8 .2 0 0 .9 7 1 .0 0 1 .0 0 0 .9 7 7 .5 8 2 8 .7 0 0 .9 7 1 .0 0 1 .0 0 0 .9 7 7 .5 8 2 9 .7 1 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 7 1 1 4 .6 5 0 .9 5 1 .0 0 1 .0 2 0 .9 7 7 .3 0 1 5 .1 7 0 .9 5 1 .0 0 1 .0 2 0 .9 8 7 .2 7 1 6 .2 1 0 .9 4 1 .0 0 1 .0 2 0 .9 9 7 .2 0 2 2 2 .4 4 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .5 1 2 2 .9 4 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .5 0 2 3 .9 6 0 .9 6 1 .0 0 1 .0 1 0 .9 8 7 .4 8 5 3 3 0 .2 7 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 6 3 0 .7 7 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 5 3 1 .7 9 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 4 1 1 9 .2 4 0 .8 2 1 .0 1 1 .3 0 0 .9 9 1 .2 9 1 9 .8 6 0 .8 2 1 .0 1 1 .2 9 1 .0 1 1 .2 5 2 1 .1 3 0 .8 1 1 .0 1 1 .2 8 1 .0 7 1 .1 8 2 2 7 .7 4 0 .9 5 1 .0 0 1 .0 2 0 .9 7 7 .3 4 2 8 .2 5 0 .9 5 1 .0 0 1 .0 2 0 .9 7 7 .3 3 2 9 .2 8 / 0 .9 5 1 .0 0 1 .0 2 0 .9 8 7 .3 0 10 3 3 5 .5 1 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .4 7 3 6 .0 1 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .4 6 3 7 .0 3 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .4 5 1 8 4 .1 6 0 .8 3 1 .0 0 1 .3 4 0 .9 7 0 .8 5 8 4 .. 7 5 0 .8 3 1 .0 0 1 .3 4 0 .9 8 0 .8 5 8 5 .9 4 0 .8 2 1 .0 0 1 .3 4 0 .9 9 0 .8 5 2 8 5 .2 1 0 .8 2 1 .0 0 1 .3 4 0 .9 7 0 .8 9 8 5 .8 0 0 .8 2 1 .0 0 1 .3 4 0 .9 8 0 .8 9 8 6 .9 9 0 .8 2 1 .0 0 1 .3 4 0 .9 9 0 .8 9 f PS 50 3 8 6 .3 3 0 .8 2 1 .0 0 1 .3 4 0 .9 7 0 .9 5 8 6 .9 2 0 .8 2 1 .0 0 1 .3 4 0 .9 8 0 .9 5 8 8 .1 1 0 .8 2 1 .0 0 1 .3 4 0 .9 9 0 .9 5 332 T ab le C -3 4 b : M ea n L o ad a n d R es is ta n ce F ac to rs f o r C ar b o n S te el a n d T > 2 0 0 o F f o r g 1 9 C ar b o n S te el , T> 2 0 0 o F ȕ= 2 .5 f L= 0 .5 f L= 1 .0 f L= 2 .0 f S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P S Ȗƍ L Ȗƍ S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P S Ȗƍ L Ȗƍ S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P S Ȗƍ L Ȗƍ S 1 6 .5 8 0 .9 6 1 .0 1 1 .0 0 0 .9 8 4 .3 3 1 0 .4 2 0 .9 6 1 .0 0 0 .9 9 0 .9 8 7 .5 5 1 1 .4 4 0 .9 6 1 .0 0 0 .9 9 0 .9 9 7 .4 9 2 1 1 .1 0 0 .9 7 1 .0 0 0 .9 9 0 .9 7 .3 7 1 8 .2 7 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .5 9 1 9 .2 8 0 .9 6 1 .0 0 0 .9 9 0 .9 8 7 .5 7 0.5 3 1 5 .6 2 0 .9 7 1 .0 0 0 .9 9 0 .9 7 4 .3 8 2 6 .1 3 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .6 1 2 7 .1 3 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .5 9 1 7 .1 0 0 .9 6 1 .0 1 1 .0 0 0 .9 8 4 .3 1 1 0 .9 4 0 .9 6 1 .0 0 1 .0 0 0 .9 8 7 .5 3 1 1 .9 6 0 .9 6 1 .0 0 1 .0 0 0 .9 9 7 .4 7 2 1 1 .6 1 0 .9 6 1 .0 0 1 .0 0 0 .9 7 4 .3 6 1 8 .7 8 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .5 8 1 9 .8 0 0 .9 6 1 .0 0 0 .9 9 0 .9 8 7 .5 6 1 3 1 6 .1 3 0 .9 7 1 .0 0 0 .9 9 0 .9 7 4 .3 7 2 6 .6 4 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .6 0 2 7 .6 5 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .5 9 1 9 .2 3 0 .9 5 1 .0 1 1 .0 2 0 .9 8 4 .1 7 1 3 .0 3 0 .9 5 1 .0 0 1 .0 1 0 .9 8 7 .4 2 1 4 .0 6 0 .9 5 1 .0 0 1 .0 1 0 .9 9 7 .3 6 2 1 3 .7 0 0 .9 6 1 .0 0 1 .0 1 0 .9 7 4 .3 1 2 0 .8 5 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 5 2 1 .8 7 0 .9 6 1 .0 0 1 .0 0 0 .9 8 7 .5 3 3 3 1 8 .2 0 0 .9 6 1 .0 0 1 .0 0 0 .9 7 4 .3 5 2 8 .7 0 0 .9 7 1 .0 0 1 .0 0 0 .9 7 7 .5 8 2 9 .7 1 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 7 1 1 1 .4 3 0 .9 3 1 .0 1 1 .0 5 0 .9 8 3 .9 2 1 5 .1 7 0 .9 5 1 .0 0 1 .0 2 0 .9 8 7 .2 7 1 6 .2 1 0 .9 4 1 .0 0 1 .0 2 0 .9 9 7 .2 0 2 1 5 .8 1 0 .9 5 1 .0 0 1 .0 2 0 .9 7 4 .2 5 2 2 .9 4 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .5 0 2 3 .9 6 0 .9 6 1 .0 0 1 .0 1 0 .9 8 7 .4 8 5 3 2 0 .2 9 0 .9 6 1 .0 0 1 .0 1 0 .9 7 4 .3 1 3 0 .7 7 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 5 3 1 .7 9 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 4 1 1 7 .8 3 0 .8 4 1 .0 1 1 .2 5 0 .9 9 1 .0 8 1 9 .8 6 0 .8 2 1 .0 1 1 .2 9 1 .0 1 1 .2 5 2 1 .1 3 0 .8 1 1 .0 1 1 .2 8 1 .0 7 1 .1 8 2 2 1 .2 6 0 .9 4 1 .0 0 1 .0 5 0 .9 7 3 .9 9 2 8 .2 5 0 .9 5 1 .0 0 1 .0 2 0 .9 7 7 .3 3 2 9 .2 8 / 0 .9 5 1 .0 0 1 .0 2 0 .9 8 7 .3 0 10 3 2 5 .6 2 0 .9 5 1 .0 0 1 .0 3 0 .9 7 4 .1 9 3 6 .0 1 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .4 6 3 7 .0 3 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .4 5 1 7 7 .6 9 0 .8 5 1 .0 0 1 .2 8 0 .9 7 0 .8 4 8 4 .. 7 5 0 .8 3 1 .0 0 1 .3 4 0 .9 8 0 .8 5 8 5 .9 4 0 .8 2 1 .0 0 1 .3 4 0 .9 9 0 .8 5 2 7 8 .7 0 0 .8 5 1 .0 0 1 .2 7 0 .9 7 0 .8 8 8 5 .8 0 0 .8 2 1 .0 0 1 .3 4 0 .9 8 0 .8 9 8 6 .9 9 0 .8 2 1 .0 0 1 .3 4 0 .9 9 0 .8 9 f PS 50 3 7 9 .7 6 0 .8 5 1 .0 0 1 .2 7 0 .9 7 0 .9 3 8 6 .9 2 0 .8 2 1 .0 0 1 .3 4 0 .9 8 0 .9 5 8 8 .1 1 0 .8 2 1 .0 0 1 .3 4 0 .9 9 0 .9 5 333 T ab le C -3 4 b : (C o n ti n u ed ) C ar b o n S te el , T> 2 0 0 o F ȕ= 3 .0 f L= 0 .5 f L= 1 .0 f L= 2 .0 f S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P S Ȗƍ L Ȗƍ S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P S Ȗƍ L Ȗƍ S ȝ f u Iƍ fu Ȗƍ ǹ Ȗƍ P S Ȗƍ L Ȗƍ S 1 9 .9 1 0 .9 6 1 .0 0 0 .9 9 0 .9 7 7 .5 7 1 0 .4 2 0 .9 6 1 .0 0 0 .9 9 0 .9 8 7 .5 5 1 1 .4 4 0 .9 6 1 .0 0 0 .9 9 0 .9 9 7 .4 9 2 1 7 .7 7 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .6 0 1 8 .2 7 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .5 9 1 9 .2 8 0 .9 6 1 .0 0 0 .9 9 0 .9 8 7 .5 7 0.5 3 2 5 .6 3 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .6 1 2 6 .1 3 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .6 1 2 7 .1 3 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .5 9 1 1 0 .4 3 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 5 1 0 .9 4 0 .9 6 1 .0 0 1 .0 0 0 .9 8 7 .5 3 1 1 .9 6 0 .9 6 1 .0 0 1 .0 0 0 .9 9 7 .4 7 2 1 8 .2 8 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .5 9 1 8 .7 8 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .5 8 1 9 .8 0 0 .9 6 1 .0 0 0 .9 9 0 .9 8 7 .5 6 1 3 2 6 .1 4 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .6 1 2 6 .6 4 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .6 0 2 7 .6 5 0 .9 7 1 .0 0 0 .9 9 0 .9 7 7 .5 9 1 1 2 .5 2 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .4 5 1 3 .0 3 0 .9 5 1 .0 0 1 .0 1 0 .9 8 7 .4 2 1 4 .0 6 0 .9 5 1 .0 0 1 .0 1 0 .9 9 7 .3 6 2 2 0 .3 5 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 6 2 0 .8 5 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 5 2 1 .8 7 0 .9 6 1 .0 0 1 .0 0 0 .9 8 7 .5 3 3 3 2 8 .2 0 0 .9 7 1 .0 0 1 .0 0 0 .9 7 7 .5 8 2 8 .7 0 0 .9 7 1 .0 0 1 .0 0 0 .9 7 7 .5 8 2 9 .7 1 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 7 1 1 4 .6 5 0 .9 5 1 .0 0 1 .0 2 0 .9 7 7 .3 0 1 5 .1 7 0 .9 5 1 .0 0 1 .0 2 0 .9 8 7 .2 7 1 6 .2 1 0 .9 4 1 .0 0 1 .0 2 0 .9 9 7 .2 0 2 2 2 .4 4 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .5 1 2 2 .9 4 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .5 0 2 3 .9 6 0 .9 6 1 .0 0 1 .0 1 0 .9 8 7 .4 8 5 3 3 0 .2 7 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 6 3 0 .7 7 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 5 3 1 .7 9 0 .9 6 1 .0 0 1 .0 0 0 .9 7 7 .5 4 1 1 9 .2 4 0 .8 2 1 .0 0 1 .3 0 0 .9 7 1 .2 9 1 9 .8 6 0 .8 2 1 .0 0 1 .2 9 1 .0 1 1 .2 5 2 1 .1 3 0 .8 1 1 .0 0 1 .2 8 1 .0 7 1 .1 8 2 2 7 .7 4 0 .9 5 1 .0 0 1 .0 2 0 .9 7 7 .3 4 2 8 .2 5 0 .9 5 1 .0 0 1 .0 2 0 .9 7 7 .3 3 2 9 .2 8 0 .9 5 1 .0 0 1 .0 2 0 .9 8 7 .3 0 10 3 3 5 .5 1 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .4 7 3 6 .0 1 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .4 6 3 7 .0 3 0 .9 6 1 .0 0 1 .0 1 0 .9 7 7 .4 5 1 8 4 .1 6 0 .8 3 1 .0 0 1 .3 4 0 .9 7 0 .8 5 8 4 .7 5 0 .8 3 1 .0 0 1 .3 4 0 .9 8 0 .8 5 8 5 .9 4 0 .8 2 1 .0 0 1 .3 4 0 .9 9 0 .8 5 2 8 5 .2 1 0 .8 2 1 .0 0 1 .3 4 0 .9 7 0 .8 9 8 5 .8 0 0 .8 2 1 .0 0 1 .3 4 0 .9 8 0 .8 9 8 6 .9 9 0 .8 2 1 .0 0 1 .3 4 0 .9 9 0 .8 9 f PS 50 3 8 6 .3 3 0 .8 2 1 .0 0 1 .3 4 0 .9 7 0 .9 5 8 6 .9 2 0 .8 2 1 .0 0 1 .3 3 0 .9 8 0 .9 5 8 8 .1 1 0 .8 2 1 .0 0 1 .3 3 0 .9 9 0 .9 5 334 T ab le C -3 5 a: A d ju st ed N o m in al R es is ta n ce F ac to rs f o r C ar b o n S te el a n d Ȗ ǹ = 1 .1 , Ȗ P S= 1 .0 , Ȗ L = 1 .3 , Ȗ S = 1 .5 f o r g 1 9 C ar b o n S te el ȕ= 2 .5 f L= 0 .5 f L= 1 .0 f L= 2 .0 f S R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F 1 0 .7 8 0 .7 1 0 .7 8 0 .7 3 0 .5 9 0 .8 5 0 .7 8 0 .8 5 0 .8 0 0 .6 4 0 .9 6 0 .8 8 0 .9 6 0 .9 0 0 .7 2 2 0 .6 7 0 .6 1 0 .6 7 0 .6 3 0 .5 1 0 .7 2 0 .6 6 0 .7 2 0 .6 8 0 .5 4 0 .8 0 0 .7 3 0 .8 1 0 .7 6 0 .6 1 0.5 3 0 .6 3 0 .5 7 0 .6 3 0 .5 9 0 .4 7 0 .6 6 0 .6 0 0 .6 6 0 .6 2 0 .5 0 0 .7 3 0 .6 6 0 .7 3 0 .6 9 0 .5 5 1 0 .8 1 0 .7 4 0 .8 1 0 .7 6 0 .6 1 0 .8 7 0 .8 0 0 .8 7 0 .8 2 0 .6 6 0 .9 7 0 .8 9 0 .9 7 0 .9 2 0 .7 3 2 0 .6 9 0 .6 3 0 .7 0 0 .6 6 0 .5 2 0 .7 4 0 .6 7 0 .7 4 0 .7 0 0 .5 6 0 .8 2 0 .7 5 0 .8 2 0 .7 7 0 .6 2 1 3 0 .6 4 0 .5 9 0 .6 5 0 .6 1 0 .4 9 0 .6 8 0 .6 2 0 .6 8 0 .6 4 0 .5 1 0 .7 4 0 .6 8 0 .7 4 0 .7 0 0 .5 6 1 0 .8 9 0 .8 1 0 .8 9 0 .8 4 0 .6 7 0 .9 4 0 .8 6 0 .9 3 0 .8 8 0 .7 0 1 .0 1 0 .9 3 1 .0 1 0 .9 5 0 .7 6 2 0 .7 7 0 .7 0 0 .7 7 0 .7 2 0 .5 8 0 .8 0 0 .7 3 0 .8 1 0 .7 6 0 .6 1 0 .8 7 0 .7 9 0 .8 7 0 .8 2 0 .6 5 3 3 0 .7 1 0 .6 4 0 .7 1 0 .6 7 0 .5 3 0 .7 3 0 .6 7 0 .7 4 0 .6 9 0 .5 5 0 .7 9 0 .7 2 0 .7 9 0 .7 4 0 .5 9 1 0 .9 4 0 .8 6 0 .9 3 0 .8 8 0 .7 0 0 .9 8 0 .8 9 0 .9 7 0 .9 1 0 .7 3 1 .0 4 0 .9 5 1 .0 2 0 .9 6 0 .7 7 2 0 .8 2 0 .7 5 0 .8 2 0 .7 7 0 .6 2 0 .8 5 0 .7 8 0 .8 5 0 .8 0 0 .6 4 0 .9 1 0 .8 3 0 .9 0 0 .8 5 0 .6 8 5 3 0 .7 6 0 .6 9 0 .7 6 0 .7 1 0 .5 7 0 .7 8 0 .7 1 0 .7 8 0 .7 3 0 .5 9 0 .8 3 0 .7 5 0 .8 3 0 .7 8 0 .6 2 1 0 .9 8 0 .9 0 0 .9 4 0 .8 9 0 .7 1 1 .0 1 0 .9 2 0 .9 6 0 .9 0 0 .7 2 1 .0 4 0 .9 5 0 .9 9 0 .9 3 0 .7 5 2 0 .9 1 0 .8 3 0 .9 0 0 .8 5 0 .6 8 0 .9 3 0 .8 5 0 .9 2 0 .8 7 0 .6 9 0 .9 6 0 .8 8 0 .9 6 0 .9 0 0 .7 2 10 3 0 .8 4 0 .7 7 0 .8 4 0 .7 9 0 .6 3 0 .8 6 0 .7 8 0 .8 6 0 .8 1 0 .6 5 0 .8 9 0 .8 2 0 .8 9 0 .8 4 0 .6 7 1 0 .8 8 0 .8 1 0 .8 5 0 .8 0 0 .6 4 0 .8 9 0 .8 1 0 .8 6 0 .8 0 0 .6 4 0 .9 0 0 .8 2 0 .8 7 0 .8 1 0 .6 5 2 0 .9 0 0 .8 3 0 .8 7 0 .8 2 0 .6 5 0 .9 1 0 .8 3 0 .8 7 0 .8 2 0 .6 6 0 .9 2 0 .8 4 0 .8 8 0 .8 3 0 .6 7 f PS 50 3 0 .9 2 0 .8 4 0 .8 9 0 .8 3 0 .6 7 0 .9 3 0 .8 5 0 .8 9 0 .8 4 0 .6 7 0 .9 4 0 .8 6 0 .9 0 0 .8 5 0 .6 8 335 T ab le C -3 5 a: ( C o n ti n u ed ) C o n si d er ed L o ad F ac to rs Ȗ ǹ = 1 .1 , Ȗ P S= 1 .2 , Ȗ L = 1 .3 , Ȗ S = 1 .5 C ar b o n S te el ȕ= 3 .0 f L= 0 .5 f L= 1 .0 f L= 2 .0 f S R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F 1 0 .5 3 0 .4 8 0 .5 3 0 .5 0 0 .4 0 0 .5 9 0 .5 4 0 .5 9 0 .5 6 .4 4 0 .6 9 0 .6 3 0 .6 9 0 .6 5 0 .5 2 2 0 .4 3 0 .3 9 0 .4 3 0 .4 0 0 .3 2 0 .4 6 0 .4 2 0 .4 6 0 .4 4 0 .3 5 0 .5 3 0 .4 9 0 .5 3 0 .5 0 0 .4 0 0.5 3 0 .3 9 0 .3 5 0 .3 9 0 .3 6 0 .2 9 0 .4 1 0 .3 8 0 .4 1 0 .3 9 0 .3 1 0 .4 6 0 .4 2 0 .4 7 0 .4 4 0 .3 5 1 0 .5 7 0 .5 2 0 .5 8 0 .5 4 0 .4 3 0 .6 3 0 .5 9 0 .6 3 0 .5 9 0 .4 7 0 .7 2 0 .6 6 0 .7 3 0 .6 8 0 .5 5 2 0 .4 5 0 .4 1 0 .4 6 0 .4 3 0 .3 4 0 .4 9 0 .4 5 0 .4 9 0 .4 6 0 .3 7 0 .5 5 0 .5 1 0 .5 6 0 .5 3 0 .4 2 1 3 0 .4 1 0 .3 7 0 .4 1 0 .3 8 0 .3 1 0 .4 3 0 .3 9 0 .4 3 0 .4 1 0 .3 3 0 .4 8 0 .4 4 0 .4 8 0 .4 5 0 .3 6 1 0 .7 1 0 .6 5 0 .7 2 0 .6 7 0 .5 4 0 .7 6 0 .6 9 0 .7 6 0 .7 1 0 .5 7 0 .8 3 0 .7 6 0 .8 3 0 .7 8 0 .6 2 2 0 .5 5 0 .5 0 0 .5 6 0 .5 2 0 .4 2 0 .5 8 0 .5 3 0 .5 8 0 .5 5 0 .4 4 0 .6 4 0 .5 8 0 .6 4 0 .6 0 0 .4 8 3 3 0 .4 8 0 .4 4 0 .4 8 0 .4 5 0 .3 6 0 .5 0 0 .4 6 0 .5 1 0 .4 8 0 .. 3 8 0 .5 5 0 .5 0 0 .5 5 0 .5 2 0 .4 1 1 0 .8 1 0 .7 4 0 .8 1 0 .7 6 0 .6 1 0 .8 5 0 .7 7 0 .8 4 0 .7 9 0 .6 4 0 .9 0 0 .8 2 0 .9 0 0 .8 5 0 .6 8 2 0 .6 3 0 .5 8 0 .6 4 0 .6 0 0 .4 8 0 .6 6 0 .6 0 0 .6 6 0 .6 2 0 .5 0 0 .7 1 0 .6 4 0 .7 1 0 .6 6 0 .5 3 5 3 0 .5 5 0 .5 0 0 .5 5 0 .5 2 0 .4 1 0 .5 7 0 .5 2 0 .5 7 0 .5 3 0 .4 3 0 .6 6 0 .5 5 0 .6 1 0 .5 7 0 .4 6 1 0 .9 6 0 .8 8 1 .0 0 0 .9 4 0 .7 5 0 .9 8 0 .9 0 1 .0 2 0 .9 6 0 .7 6 1 .0 2 0 .9 3 1 .0 4 0 .9 8 0 .7 8 2 0 .7 8 0 .7 1 0 .7 8 0 .7 3 0 .5 9 0 .8 0 0 .7 3 0 .8 0 0 .7 5 0 .6 0 0 .8 3 0 .7 6 0 .8 3 0 .7 8 0 .6 2 10 3 0 .6 7 0 .6 2 0 .6 8 0 .6 3 0 .5 1 0 .6 9 0 .6 3 0 .6 9 0 .6 5 0 .5 2 0 .7 2 0 .6 6 0 .7 2 0 .6 8 0 .5 4 1 0 .9 8 0 .8 1 0 .9 3 0 .8 8 0 .7 0 0 .9 8 0 .9 0 0 .9 4 0 .8 8 0 .7 0 0 .9 9 0 .9 1 0 .9 4 0 .8 9 0 .7 1 2 1 .0 0 0 .9 1 0 .9 5 0 .8 9 0 .7 1 1 .0 0 .9 1 0 .9 5 0 .8 9 0 .7 2 1 .0 1 0 .9 2 0 .9 6 0 .9 0 0 .7 2 f PS 50 3 1 .0 1 0 .9 2 0 .9 6 0 .9 1 0 .7 2 1 .0 2 0 .9 3 0 .9 7 0 .9 1 0 .7 3 1 .0 2 0 .9 4 0 .9 7 0 .9 2 0 .7 3 336 T ab le C -3 5 a: A d ju st ed N o m in al R es is ta n ce F ac to rs f o r S ta in le ss S te el a n d Ȗ ǹ = 1 .1 , Ȗ P S= 1 .0 , Ȗ L = 1 .3 , Ȗ S = 1 .5 f o r g 1 9 S ta in le ss S te el ȕ= 2 .5 f L= 0 .5 f L= 1 .0 f L= 2 .0 f S R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F 1 0 .7 6 0 .6 6 0 .6 0 0 .6 0 0 .5 8 0 .8 3 0 .7 2 0 .6 5 0 .6 5 0 .6 3 0 .9 4 0 .8 2 0 .7 4 0 .7 4 0 .7 1 2 0 .6 0 .5 7 0 .5 1 0 .5 1 0 .5 0 0 .7 1 0 .6 1 0 .5 5 0 .5 5 0 .5 3 0 .7 9 0 .6 8 0 .6 1 0 .6 1 0 .5 9 0.5 3 0 .6 1 0 .5 3 0 .4 8 0 .4 8 0 .4 6 0 .6 5 0 .5 6 0 .5 1 0 .5 1 0 .4 9 0 .7 1 0 .6 2 0 .5 6 0 .5 6 0 .5 4 1 0 .7 9 0 .6 9 0 .6 2 0 .6 2 0 .6 0 0 .8 6 0 .7 4 0 .6 7 0 .6 7 0 .6 4 0 .9 6 0 .8 3 0 .7 5 0 .7 5 0 .7 2 2 0 .6 8 0 .5 9 0 .5 3 0 .5 3 0 .5 1 0 .7 3 0 .6 3 0 .5 7 0 .5 7 0 .5 5 0 .8 0 0 .7 0 0 .6 3 0 .6 3 0 .6 0 1 3 0 .6 3 0 .5 5 0 .4 9 0 .4 9 0 .4 8 0 .6 7 0 .5 8 0 .5 2 0 .5 2 0 .5 0 0 .7 3 0 .6 3 0 .5 7 0 .5 7 0 .5 5 1 0 .8 8 0 .7 6 0 .6 8 0 .6 8 0 .6 6 0 .9 2 0 .8 0 0 .7 2 0 .7 2 0 .6 9 1 .0 0 0 .8 6 0 .7 8 0 .7 8 0 .7 4 2 0 .7 6 0 .6 6 0 .5 9 0 .5 9 0 .5 7 0 .7 9 0 .6 9 0 .6 2 0 .6 2 0 .5 9 0 .8 5 0 .7 4 0 .6 6 0 .6 6 0 .6 4 3 3 0 .6 9 0 .6 0 0 .5 4 0 .5 4 0 .5 2 0 .7 2 0 .6 3 0 .5 6 0 .5 6 0 .5 4 0 .7 7 0 .6 7 0 .6 0 0 .6 0 0 .5 8 1 0 .9 2 0 .8 0 0 .7 2 0 .7 2 0 .7 0 0 .9 6 0 .7 5 0 .7 5 0 .7 2 1 .0 2 0 .8 8 0 .7 9 0 .7 9 0 .7 7 2 0 .8 1 0 .7 0 0 .6 3 0 .6 3 0 .6 1 0 .8 4 0 .8 3 0 .6 5 0 .6 5 0 .6 3 0 .8 9 0 .7 7 0 .6 9 0 .6 9 0 .6 7 5 3 0 .7 4 0 .6 4 0 .5 8 0 .5 8 0 .5 6 0 .7 7 0 .7 3 0 .6 0 0 .6 0 0 .5 8 0 .8 1 0 .7 0 0 .6 3 0 .6 3 0 .6 1 1 0 .9 7 0 .8 4 0 .7 5 0 .7 5 0 .7 3 0 .9 9 0 .6 6 0 .7 7 0 .7 7 0 .7 4 1 .0 2 0 .8 9 0 .8 0 0 .8 0 0 .7 7 2 0 .8 9 0 .7 7 0 .6 9 0 .6 9 0 .6 7 0 .9 1 0 .8 6 0 .7 1 0 .7 1 0 .6 8 0 .9 5 0 .8 2 0 .7 4 0 .7 4 0 .7 1 10 3 0 .8 3 0 .7 2 0 .6 4 0 .6 4 0 .6 2 0 .8 4 0 .7 9 0 .6 6 0 .6 6 0 .6 3 0 .8 8 0 .7 6 0 .6 8 0 .6 8 0 .6 6 1 0 .8 7 0 .7 5 0 .6 8 0 .6 8 0 .6 5 0 .8 7 0 .7 3 0 .6 8 0 .6 8 0 .6 6 0 .8 9 0 .7 7 0 .6 9 0 .6 9 0 .6 7 2 0 .8 9 0 .7 7 0 .6 9 0 .6 9 0 .6 7 0 .8 9 0 .7 6 0 .7 0 0 .7 0 0 .6 7 0 .9 0 0 .7 8 0 .7 0 0 .7 0 0 .6 8 f PS 50 3 0 .9 1 0 .7 9 0 .7 1 0 .7 1 0 .6 8 0 .9 1 0 .7 8 0 .7 1 0 .7 1 0 .6 9 0 .9 2 0 .8 0 0 .7 2 0 .7 2 0 .6 9 337 T ab le C -3 5 b : (C o n ti n u ed ) C o n si d er ed L o ad F ac to rs Ȗ ǹ = 1 .1 , Ȗ P S= 1 .2 , Ȗ L = 1 .3 , Ȗ S = 1 .5 S ta in le ss S te el ȕ= 3 .0 f L= 0 .5 f L= 1 .0 f L= 2 .0 f S R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F R .T .  R F 4 0 0 R F 6 0 0 R F 8 0 0 R F 1 0 .5 2 0 .4 5 0 .4 0 0 .4 0 0 .3 9 0 .5 8 0 .5 0 0 .4 5 0 .4 5 0 .4 4 0 .6 8 0 .5 9 0 .5 3 0 .5 3 0 .5 1 2 0 .4 2 0 .3 6 0 .3 3 0 .3 3 0 .3 1 0 .4 5 0 .3 9 0 .3 5 0 .3 5 0 .3 4 0 .5 2 0 .4 5 0 .4 1 0 .4 1 0 .. 3 9 0.5 3 0 .3 8 0 .3 3 0 .3 0 0 .3 0 0 .2 9 0 .4 1 0 .3 5 0 .3 2 0 .3 2 0 .3 0 0 .4 5 0 .3 9 0 .3 5 0 .3 5 0 .3 4 1 0 .5 6 0 .4 9 0 .4 4 0 .4 4 0 .4 2 0 .6 2 0 .5 4 0 .4 8 0 .4 8 0 .4 6 0 .7 1 0 .6 2 0 .5 5 0 .5 5 0 .5 4 2 0 .4 5 0 .3 9 0 .3 5 0 .3 5 0 .3 4 0 .4 8 0 .4 2 0 .3 7 0 .3 7 0 .3 6 0 .5 5 0 .4 7 0 .4 2 0 .4 2 0 .4 1 1 3 0 .4 0 0 .3 5 0 .3 1 0 .3 1 0 .3 0 0 .4 2 0 .3 7 0 .3 3 0 .3 3 0 .3 2 0 .4 7 0 .4 1 0 .3 7 0 .3 7 0 .3 6 1 0 .7 0 0 .6 1 0 .5 5 0 .5 5 0 .5 3 0 .7 4 0 .6 4 0 .5 8 0 .5 8 0 .5 6 0 .8 1 0 .7 1 0 .6 3 0 .6 3 0 .6 1 2 0 .5 4 0 .4 7 0 .4 2 0 .4 2 0 .4 1 0 .5 7 0 .5 0 0 .4 5 0 .4 5 0 .4 3 0 .6 3 0 .5 4 0 .4 9 0 .4 9 0 .4 7 3 3 0 .4 7 0 .4 1 0 .3 7 0 .3 7 0 .3 6 0 .5 0 0 .4 3 0 .3 9 0 .3 9 0 .3 7 0 .5 4 0 .4 7 0 .4 2 0 .4 2 0 .4 0 1 0 .8 0 0 .6 9 0 .6 2 0 .6 2 0 .6 0 0 .8 3 0 .7 2 0 .6 5 0 .6 5 0 .6 3 0 .8 9 0 .7 7 0 .6 9 0 .6 9 0 .6 7 2 0 .6 2 0 .5 4 0 .4 8 0 .4 8 0 .4 7 0 .6 5 0 .5 6 0 .5 0 0 .5 0 0 .4 9 0 .6 9 0 .6 0 0 .5 4 0 .5 4 0 .5 2 5 3 0 .5 4 0 .4 7 0 .4 2 0 .4 2 0 .4 0 0 .5 6 0 .4 8 0 .4 3 0 .4 3 0 .4 2 0 .5 9 0 .5 2 0 .4 6 0 .4 6 0 .4 5 1 0 .9 5 0 .8 2 0 .7 4 0 .7 4 0 .7 1 0 .9 7 0 .8 4 0 .7 5 0 .7 5 0 .7 3 1 .0 0 0 .8 7 0 .7 8 0 .7 8 0 .7 5 2 0 .7 7 0 .6 7 0 .6 0 0 .6 0 0 .5 8 0 .7 8 0 .6 8 0 .6 1 0 .6 1 0 .5 9 0 .8 2 0 .7 1 0 .6 4 0 .6 4 0 .6 1 10 3 0 .6 6 0 .5 7 0 .5 2 0 .5 2 0 .5 0 0 .6 8 0 .5 9 0 .5 3 0 .5 3 0 .5 1 0 .7 1 0 .6 1 0 .5 5 0 .5 5 0 .5 3 1 0 .9 6 0 .8 3 0 .7 5 0 .7 5 0 .7 2 0 .9 7 0 .8 4 0 .7 5 0 .7 5 0 .7 3 0 .9 8 0 .8 5 0 .7 6 0 .7 6 0 .7 3 2 0 .9 8 0 .8 5 0 .7 6 0 .7 6 0 .7 4 0 .9 8 0 .8 5 0 .7 7 0 .7 7 0 .7 4 0 .9 9 0 .8 6 0 .7 7 0 .7 7 0 .7 5 f PS 50 3 0 .9 9 0 .8 6 0 .7 7 0 .7 7 0 .7 5 1 .0 0 0 .8 7 0 .7 8 0 .7 8 0 .7 5 1 .0 1 0 .8 7 0 .7 8 0 .7 8 0 .7 6 338 339 REFERENCES: 1. American Association of State Highway and Transportation Officials, 1994, “LRFD Bridge Design and Construction Specifications. AASHTO,” Washington, DC. 2. American Concrete Institute International, 2001, “Standard Code Requirements for Nuclear Safety Related Concrete Structures,” ACI 349-01. 3. American Concrete Institute, 1977, “Building Code Requirements for Reinforced Concrete,” ACI 318-77, Detroit Michigan. 4. American Institute for Steel Construction, 2003, “Load and Resistance Factor Design Specification for Safety–related Steel Structures for Nuclear Facilities,” ANSI/AISC N690L-03. 5. American Institute for Steel Construction, 1986, “Manual for Steel Construction, Load and Resistance Factor Design,” Chicago, Illinois, AISC. 6. American Petroleum Institute, 1989, “Draft Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms Load and Resistance Factor Design,” API RP2A-LRFD, API, Dallas, TX. 7. American Petroleum Institute, 1999, “Design, Construction, Operation and Maintenance of Offshore Hydrocarbon Pipelines (Limit State Design),” Third Edition, July. 8. American Society of Civil Engineers, Structural Division Committee on Nuclear Structures and Materials, SMiRT-4, 1977, “International Seminar on Probabilistic and Extreme Load Design of Nuclear Plant Facilities,’’ San Francisco, California, August 22-24, p. 302. 9. American Society of Civil Engineers, 1992, “Load and Resistance Factor Design: Specification for Engineered Wood Construction,” New York, ASCE. 10. American Society of Mechanical Engineers, 1969, “Criteria of the ASME Boiler and Pressure Vessel Code for Design by Analysis in Sections III and VIII, Division 2,” ASME United Engineering Center, New York. 11. American Society of Mechanical Engineers, 2001, “Rules for Construction of Nuclear Facility Components,” Boiler and Pressure Vessel Code, Section III, ASME. 340 12. American Society of Mechanical Engineers, 1985, “An American National Standard : Stainless Steel Pipe,” ANSI/ASME B36.19M-1985, ASME, New York. 13. American Society of Mechanical Engineers, 2004 “An American National Standard: Factory-Made Wrought Buttwelding Fittings,” ASME B16.9-2003, ASME. 14. American Society for Metals, 1961, “Metals Handbook,” 8th Edition, Vol. 1, Metals Handbook Committee, ASM, Metals Park, Novelty, Ohio. 15. American Water Work Association, 2004, “Steel Pipe: A Guide for Design and Installation,” 4th Edition, AWWA Manual, Denver, Colorado. 16. Ang, A. H-S, and Tang, W.H., 1984, “Probability Concepts in Engineering Planning and Design, Volume II: Decision, Risk, and Reliability,” John Wiley and Sons, New York. 17. Antaki, G. A., 1999, “Design and In-service Margins of Power Piping Systems: A Comparative Study of U.S., Canadian and European Codes and Standards,” Bulletin, Vol. 434, pp. 1-21. 18. Antaki, G. A., 1993, “Failure and Factors of Safety in Piping System Design (U),” PVP-Vol. 264, Piping, Supports, and Structural Dynamics, ASME, pp. 59-64. 19. Armenákas, A. E., 2006, “Advanced Mechanics of Materials and Applied Elasticity,” CRC Press. 20. Assakkaf, I. A., 1998, “Reliability-Based Design of Panels and Fatigue Details of Ship Structures,” Dissertation Submitted to the Faculty of the Graduate School of the Univ. of Maryland, Coll. Park in Partial Fulfillment of the Requirements for the PhD Degree. 21. Ayyub, B. M, Assakkaf I. A., Kihl, D. P., and Siev, M. S., 2002, “Reliability-Based Design Guidelines for Fatigue of Ship Structures,” J. Naval Engineering, Vol. 114, No. 2. 22. Ayyub, B. M. and McCuen, R. H., 2003, “Probability, Statistics and Reliability for Engineers and Scientists,” CRC Press, FL. 23. Ayyub, B. M. and Klir, G. J., 2006, “Uncertainty Modeling and Analysis for Engineers and Scientists,” Chapman & Hall/CRC, Press Boca Raton, FL. 24. Ayyub, B. M., Gupta, A., Assakkaf, I., Shah, N., Kotwicki, P., and Avrithi, K., 2005, “Development of Reliability-Based Load and Resistance Factor Design (LRFD) Methods for Piping,” Preliminary Interim Report, ASME, for the U. S. Nuclear Regulatory Commission, Washington, DC, and the Government of Japan. 341 25. Ayyub, B. M., Assakkaf, I. A., Avrithi, K., Gupta, A., Shah, N., Kotwicki, P., and Hill, R. S. III, 2005, “Development of Reliability–Based Load and Resistance Factor Design (LRFD) Methods for Piping,” 13th International Conference on Nuclear Engineering, Beijing, China, May 16-20, ICONE13-50486. 26. Bai, Y. and Song, R., 1997, “Fracture Assessment of Dented Pipes with Cracks and Reliability-Based Calibration of Safety Factors,” Int. J. Pres. Ves. and Piping,” Vol. 74, pp. 221-229. 27. Barnes, R. W., Harris, D. O., Hill, R. S., and Stevenson, J. D., 2000, “Demonstration of Risk-Informed Design Procedures for the ASME Nuclear Code,” ICONE8-8256, April 2-6, 2000, Baltimore, MD. 28. Becht, C., IV, 2004, “Process Piping: The Complete Guide to ASME B31.3,” Second Edition, ASME, Three Park Avenue, New York. 29. Belke, L., 1984, “A simple Approach for Failure Bending Moments of Straight Pipes,” Nucl. Eng. and Des., 77, pp. 1-5. 30. Benham, P. P, Crawford, R. J., and Armstrong, C. G., 1996, “Mechanics of engineering materials,” Second Edition, Longman Group Lim. 31. Benjamin, M., 1983, “Nuclear Reactor Materials and Applications,” New York, Van Nostrand Reinhold Co. 32. Brust, F. W., Scott, P., Rahman, S., Ghadiali, N., Kilinski, T., Francini, B., Marschall, C. W., Miura, N., Krishnaswamy, P., and Wilkowski, G. M., 1994, “Assessment of Short Through-Wall Circumferential Cracks in Pipes,” 33. Burrows, W. R., Michel, R., and Rankin, A. W., 1954, “A Wall-Thickness Formula for High-Temperature Piping,” ASME Trans., 76 (3), pp. 427-444. 34. Canonico, D. A., 2000, “The Origins of ASME’s Boiler and Pressure Vessel Code,” Mechanical Engineering, 122, No 2, ASME, New York, February. 35. Cardarelli, F., 1999, “Materials Handbook: A Concise Desktop Reference,” Springer-Verlag, London. 36. Casciati, F.and Faravelli, L., 1982, “Load Combination by Partial Safety Factors,” Nuclear Engineering and Design, Vol. 75, pp. 439-452. 37. Casciati, F., 1983, “Partial Safety Factors for Combined Loading,” Transactions of the International Conference on Structural Mechanics in Reactor Technology, pp. 57-64. 342 38. Lamb, S., Editor, 2002, “CASTI Handbook of Stainless Steel and Nickel Alloys,” Second Edition, CASTI Publishing Inc. 39. CEC, 1984, “Common Unified Rules for Steel Structures. Eurocode 3,” Brussels, Belgium: Commission of the European Communities. 40. Chopra, O.K. and Shack, W. J., 1998, “Low-cycle Fatigue of Piping and Pressure Vessel Steels in LWR Environments,” Nuclear Engineering and Design, 184, pp. 49-76. 41. Chopra, O.K. and Shack, W. J., 1999, “Overwiew of Fatigue Crack Initiation in Carbon and Low-Alloy Steels in Light Water Reactors Environments,” Journal of Pressure Vessel Technology, Vol. 121, pp. 49-60. 42. CISC, 1974, “Steel Structures for Buildings – Limit State Design. Standard CSA- 516.1-1974,” Rexdale, Ontario, Canada: Canadian Institute of Steel Construction. 43. Cornell, C. A., 1968, “Engineering Seismic Risk Analysis,” Bulletin of the Seismological Society of America, pp. 1583-1605. 44. Crocker, S., King, R. C., 1967, “Piping Handbook,” 5th edition, McGraw-Hill, New York. 45. Cullen, T. M. and Davis, M. W., 1973, “Influence of Nitrogen on the Creep- Rupture Properties of Type 316 Steel,” ASTM STP 522, American Society for Testing and Materials, pp. 60-78. 46. Davis, J. R., 1996, “Carbon and Alloy Steels,” ASM Specialty Handbook. 47. Davis, J. R., 2000, “Alloy Digest Sourcebook, Stainless Steels,” ASM International. 48. Ellingwood, B. R., 1994. “Validation of Seismic Probabilistic Risk Assessments of Nuclear Power Plants,” NUREG/GR-0008. 49. Ellingwood, B. R., 1995 “Event Combination Analysis for Design and Rehabilitation of U. S. Army Corps of Engineers Navigation Structures,” WES ITL-95-2, U. S. Army Corps of Engineers. 50. Ellingwood, B. R., 1994 “Probability-based Codified Design for Earthquakes,” engineering Structures, Volume 16, Issue 7, pp. 498-506. 51. Ellingwood, B. R. and Batts, M.E., 1982 “Characterization of Earthquake Forces for Probability-Based Design of Nuclear Structures,” NUREG/CR-2945. 52. Ellingwood, B., Galambos, T. V., MacGregor, J. G., and Cornell, C. A., 1980, “Development of a Probability Based Load Criterion for American National 343 Standard A58,” U.S. Department of Commerce, Washington, DC, Special Publication No. 577. 53. Ellyin, F., 1976, “Experimental investigation on Limit Loads of Nozzles in Cylindrical Vessels,” Weld. Res. Counc., Bull. No. 219, Sept. 54. Ellyin, F., 1977, “An Experimental Study of Elastic-plastic Response of Branch- Pipe Tee Connections Subjected to Internal Pressure, External Couples and Combined Loadings,” Weld. Res. Council, Bull. No. 230, Sept. 55. Eurocode 1, “Basis of Design and Actions on Structures,” ENV 1991-1:1994, Part I. 56. Franzen, W. E., Stokey, W. F., 1972, “The elastic-plastic Behavior of Stainless Steel Tubing Subjected to Bending, Pressure and Torsion,” Second Int. Conf. on Pressure Vessel Technology, San Antonio, ASME. 57. Galambos, T. V. and Ravindra, M. K., 1973, “Tentative Load and Resistance Criteria for Steel Buildings,” Res. Rep. No 18, Structural Division, CEE, Washington Univ., Sept. 58. Gerber, T. L, “Plastic Deformation of Piping Due to Pipe-whip Loading,” ASME Paper No. 74-NE-1. 59. Gerdeen, J. C., 1979, “Critical Evaluation of Plastic Behavior Data and a United Definition of Plastic Loads for Pressure Components,” Weld. Res. Counc. Bull., n. 254, pp. 1-64. 60. GP COURSEWARE, (Firm), 1982, ‘‘Reactor Plant Materials,” Division of GP Publishing, Inc. 61. Gupta, A., Choi, B., 2003, “Reliability Based Load and Resistance Factor Design for Piping: an Exploratory Case Study,” Nuclear Engineering and Design, Vol. 224, pp. 161-178. 62. Gupta, A, Gupta, A. K., 1995. ‘‘Application of new developments in coupled seismic analysis of piping systems,’’ Transactions of the 13th International Conference of Structural Mechanics in Reactor Technology, Porto Allegre, Brazil, August. 63. Haldar, A. and Mahadevan, S., 2000, “Probability, Reliability and Statistical Methods in Engineering Design,” John Wiley & Sons, Inc., New York 64. Hasofer, A. M. and Lind, N. C., 1974, “Exact and Invariant Second Moment Code Format,” Journal of Engineering Mechanics, ASCE, Vol. 100, No. EM1, pp. 111- 121. 344 65. Helguero, V. M., 1983, “Piping Stress Handbook,” Houston, TX: Distributed by Gulf Pub. Co. 66. Higuchi, M., Iida, K., and Asada, Y., 1995, “Fatigue and Crack Growth; Environmental Effects, Modeling Studies, and design considerations,” Yukawa, S., Editor, Vol . PVP-306, ASME, N.Y., pp. 111-116. 67. Higuchi, M. and Iida, K., 1991, “Fatigue Strength Correction Factors for Carbon and Low-alloy Steels in Oxyben-Containing High-Temperature Water,” Nuclear Engineering and Design, 129, pp. 293-306. 68. Hwang, H., Wang, P. C., Shooman, M. and Reich, M., 1983, “A Consensus Estimation Study of Nuclear Power Plant Structural Loads,” NUREG/CR-3315. 69. Hwang, H., Wang, P. C. and Reich, M., 1983, “Probabilistic Models for Operational and Accidental Loads on Seismic Category I Structures,” NUREG/CR- 3342. 70. Hwang, H., Ellingwood, B., Shinozuka, M., and Reich, M., 1987 “Probability- Based Design Criteria for Nuclear Plant Structures,” Journal of Structural Engineering, Vol. 113, No 5, pp. 925-942. 71. International Atomic Energy Agency, 2003, “Assessment and Management of Ageing of Major Nuclear Power Plant Components Important to Safety, Primary Piping in PWRs,” IAEA-TECDOC-1361, July. 72. Jirsa, J. O., Lee, F. H., and Wilheit, J. C., 1972, “Ovaling of Pipelines under Pure Bending,” Proc. Offshore Technology Conf., Dallas, OTC Paper 1569. 73. Kannappan, S., 1986, “Introduction to Pipe Stress Analysis,” John Wiley & Sons Inc. 74. Kecioglu, D., 1991, “Reliability Engineering Handbook,” Vol. 2, Prentice Hall Inc. 75. King, R. C, Crocker, S., and Walker, J. H., 1967, “Piping Handbook,” 5th Edition, McGraw-Hill, New York. 76. Kirkermo, F., 2001, “Burst and Gross Plastic Deformation Limit State Equations for Pipes: Part 1-Theory,” Proc. of the Eleventh International Offshore and Polar Engineering Conf., Stavanger, Norway, June 17-22. 77. Kirkermo, F. and Holden, H., 2001, “Burst and Gross Plastic Deformation Limit State Equations of Pipes: Part 2-Apllication,” Proc. of the Eleventh International Offshore and Polar Engineering Conf., Stavanger, Norway, June 17-22. 345 78. Kumar, R. and Saleem, M. A., 2002, “B2 and C2 Stress Indices for Large-Angle Bends,” Journal of Pressure Vessel Technology, Vol. 124, May. 79. Lamit, Louis Gary, 1981, “Piping Systems, Drafting and Design,” Prentice Hall, Inc., Englewood Cliffs, N.J. 07632. 80. Langer, B. F., 1962, “Design of Pressure Vessels for Low-Cycle Fatigue,” Journal of Basic Engineering, September, pp. 389-402. 81. Ling, J., 2000, “The Evolution of the ASME Boiler AND Pressure Vessel Code,” Transactions of the ASME, Vol. 122, August. 82. Lynch, C. T., 1989, “Practical Handbook of Materials Science,’’ CRC Press. 83. MacDonald, D. D., Cragnolino, G. A., 1989 “Corrosion of Steam Cycle Materials,” The ASME Handbook on Water Technology for Thermal Power Systems, Cohen, P., Editor-in-Chief, p. 673. 84. MacGregor, J. G., 1976, “Safety and Limit States Design for Reinforced Concrete,” Can. J. Civil Eng., 3 (4), pp. 484-513. 85. Madsen, H. O., Krenk, S., and Lind, N. C., 1986, “Methods of Structural Safety,” Prentice-Hall, Englewood Cliffs, New Jersey. 86. Mansour, A. E., Wirsching, P. H., White, G. J., and Ayyub, B. M., 1996, “Probability-Based Ship Design: Implementation of Design Guidelines,” SSC 392, NTIS, Washington D.C. 87. Markl, A. R. C. and Louisville, K. Y., 1955, “Piping-Flexibility Analysis,” ASME Transactions, Vol. 77. 88. Markl, A. R. C. and Louisville, K. Y.,1952, “Fatigue Tests of Piping Components,” Journal Transactions of the ASME, Vol. 74, pp. 287-303. 89. Marschall, C. W., Landow, M. P. and Wilkowski, G. M., 1993, “Loading Rate Effects on Strength and Fracture Toughness of Pipe Steels Used in Task 1 of the IPIRG Program,” NUREG/CR-6098. 90. Matzen, V. C., and Tan Y., 2002, “The History of the B2 Stress Index,” Transactions of the ASME, Vol. 124, May. 91. Maxey, W. A., 1986, “Y/T Significance in Line Pipe,” 7th Symposium on Line Pipe Research, Houston, Texas. 346 92. Mello, R. M. and Griffin, D. S., 1974, “Plastic Collapse Loads for Pipe Elbows Using Inelastic Analysis,” Journal of Pressure Vessel Technology, ASME, No.74- PVP-16, pp. 177-183. 93. Mikita, R. W., Reedy, R. F., 1988, “Guidelines for Piping System Reconciliation (NCIG-05, Revision 1): Final Report,” EPRI-NP-5639. 94. Miner, A. M., 1945, “Cumulative Damage in Fatigue,” Journal of Applied Mechanics, September, pp. A159-A164. 95. Nakai, Y., Kurahasi, H., and Totsuka, N., 1982, “Hydrostatic Burst Test of Pipe with HIC,” Intern. Corrosion Forum, March 22-26, Houston, Texas. 96. Nayyar, M. L., 2000, “Piping Handbook,” 7th edition, McGraw – Hill, New York. 97. Nowak, A. S. and Collins, K. R., 2000, “Reliability of Structures,” McGraw-Hill Comp. Inc. 98. Payne, M. L. and Swanson, J. D., 1989, “Application of Probabilistic Reliability Methods to Tubular Design,” Society of Petroleum Engineers, ARCO Oil and Gas Co. 99. Pretorius, J., Van Der Merwe, P., Van Der Berg, P., 1996, “Burst Strength of Type 304L Stainless Steel Tubes Subjected to Internal Pressure and External Forces,” Thirteen International Specialty Conference on Cold-Formed Steel Structures, St. Louis, Missouri, U.S.A., October 17-18. 100. Prost, J. P., Taupin P. and Delidais, M., 1983, “Experimental Study of Austenitic Stainless Steel Pipes and Elbows under Pressure and Moment Loadings,” Transactions of the International Conference on Structural Mechanics in Reactor Technology, pp. 381-385. 101. Rajdeep Metals, 526 Duncan Road, 2nd Floor, Office #36, near Gulalwadi Circle, Mumbai-400004, Tel: 00-91-22-23898428, http://www.rajdeepmetals.com/generally_piping.htm. 102. Rao, S. S., 1992, “Reliability-Based Design,” McGraw-Hill, Inc., New York, pp. 10-15. 103. Ravindra, M. K., Su, T. Y., Won, D. J., and Schwartz, M. W., 1981, “Development of Load Combinations for Design of Nuclear Components: Applications of Probabilistic Methodology,” Trans. of the Intern. Conf. on Struct. Mech. in Reactor Tech., V(b). 104. Regulatory Guide 1.60, 1973, “Design Response Spectra for Seismic Design of Nuclear Power Plants,” Rev. 1, U.S. Nuclear Regulatory Commission, December. 347 105. Reich, M. and Hwang, H., 1984 “Probability-Based Load Combinations for Design of Category I Structures–Overview of Research Program and Recent Results,” Nuclear Engineering and Design, V. 79, pp. 129-135. 106. Rodabaugh, E. G. and Moore, S. E., 1978, “Evaluation of the Plastic Characteristics of Piping Products in Relation to ASME Code Criteria,” Battelle-Columbus labs, Ohio, USA. 107. Rodabaugh, E. G., 1984, “Sources of Uncertainty in the Calculation of Loads on Supports of Piping Systems,” Work performed for U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research, NUREG/CR–3599. 108. Rodabaugh, E. G. and Moore, S. E., 1987, “Preparation of Design Specifications and Design Reports for Pumps, Valves, Piping,” NUREG/CR-4943, Oak Ridge National Lab, TN (USA), 75p. 109. Royer, C. P. and Rolfe, S. T., 1974, “Effect of Strain-Hardening Exponent and Strain Concentrations on the Bursting Behavior of Pressure Vessels,” Trans. of the ASME, J. of Eng. Mat. and Tech., Octob., pp. 292-298. 110. Pretorius, J., Van Der Merwe, P., and Van Der Berg, P., 1996, “Burst Strength of Type 304L Stainless Steel Tubes Subjected to Internal Pressure and External Forces,” Thirteen Intern. Spec. Conf. on Cold-formed Steel Structures, St. Louis, Missouri, Octob. 17-18. 111. Saigal, R. K, 2005, “Seismic Analysis and Reliability-Based Design of Secondary Systems,” Dissertation Submitted to the Faculty of the Graduate School of North Carolina State Univ. in Partial Fulfillment of the Requirements for the PhD Degree. 112. Sbazó, J., 1972, “Hoehere Technische Mechanik nach Vorlesungen,” Fifth Edition, Springler-Verlag, Berlin. 113. Scavuzzo, R. J., 2006, “Effect of Loading on Stress Intensification Factors,” Journal of Pressure Vessel Technology, Vol. 128, pp. 33-38. 114. Schroeder, J., Srinivasaiah, K., and Graham, P., 1974, “Analysis of Test Data on Branch Connections Exposed to Internal Pressure and/or External Couples,” Weld. Res. Counc. Bull. No. 200, Nov. 115. Schwartz, M. W., Ravindra, M. K., Cornell, C. A., Chou, C. K., 1981, “Load Combination Methodology Development. Load Combination Program,” Project II, Final Report, NUREG/CR-2087. 116. Schuëler, G. I., Ang, A. H-S, 1992, “Advances in Structural Reliability,” Nuclear Engineering and Design, V. 134, pp. 121-140. 348 117. Sherman, D. R., 1976, “Tests of Circular Steel Tubes in Bending,” Journal of Struct. Div., ASCE, pp. 2181-2195. 118. Sherman, D. R. and Glass, A.M., 1974, “Ultimate Bending Capacity of Circular Tubes,” Proc. Offshore Technology Conf., Dallas, OTC Paper No 2119. 119. Sikka, V. K and Booker, M. K., 1977, “Assessment of Tensile and Creep Data for Types 304 and 316 Stainless Steel,” ASME J. of Press. Ves. Tech., pp. 298-213. 120. Simmons, W. F. and Cross, H. C., 1955, “Elevated-Temperature Properties of Carbon Steels,” ASTM Spec. Tech. Publ. No 180. 121. Simmons, W. F., and Van Echo, J. A., 1965, “Elevated-Temperature Properties of Stainless Steels,” ASTM, DS 5-S1, Philadelphia. 122. Smith, G. V., 1969 “An Evaluation of the Yield, Tensile, Creep, and Rupture Strengths of Wrought 304, 316, 321, and 347 Stainless Steels at Elevated Temperatures,” ASTM, DS 5S2, Philadelphia. 123. Sorenson, J. E., Mesloh, R. E., Rybicki, E., Hopper, A. T., and Atterbury, T. J., 1970, “Buckling Strength of Offshore Pipelines,” Battelle-Columbus Labs, Report to the Offshore Pipeline Group, July 13. 124. Sotberg, T., and Leira B. J., 1994, “Reliability-Based Pipeline Design and Code Calibration,” Pipeline Technology, OMAE, Vol. V., pp. 351-363. 125. Sotberg, T., Mørk, K. J., Barbas S., 1999, “ISO Standard Pipeline Transportation Systems: Reliability-Based Limit State Methods,” Proceedings of the Ninth International Offshore and Polar Engineering Conference, Brest, France, May 30- June 4. 126. Staat, M., 2004, “Plastic Collapse Analysis of Longitudinally Flawed Pipes and Vessels,” Nuclear Engineering and Design, Vol. 234, pp. 25-43. 127. Stancampiano, P. A. and Zemanick, P. P.,1976, “Estimates of the Burst Reliability of Thin-walled Cylinders Designed to Meet the ASME Code Allowables,” International Joint Pressure Vessels and Piping and Petreleum mechanical Engineering Conference, Mexico City, Mexico, September. 128. Steele, R. Jr., Nitzel, M. E., 1992, “Piping System Response During High – Level Simulated Seismic Tests at the Heissdampfreaktor Facility, (SHAM Test Facility),” Prepared for the Division of Engineering Office of Nuclear Regulatory Research, Contract No. DE-AC07–761D01570, NUREG/CR–5646. 349 129. Stevenson, J. D., 1995, “Application of Bounding Spectra to Seismic Design of Piping Based on the Performance of above ground piping in Power Plants Subjected to Strong Motion Earthquakes,” NUREG/CR-6240, ORNL/Sub/94-SD427/1. 130. Stevenson, J. D., Harris, D. O., Hill, R. S, 1999, “Analysis of the Reliability of Piping Designed to ASME Boiler and Pressure Vessel Code Allowables,” Report Submitted to ASME Working Group on Piping Design, ASME. 131. Stevenson, J. D., 2003, “Historical Development of the Seismic Requirements for Construction of Nuclear Power Plants in the U.S. and Worldwide and their Current Impact on Cost and Safety,” Trans. SMIRT 17, Czech Republic, August 17-22. 132. Stewart, G., Klever, F. J., and Ritchie, D., 1994, “An Analytical Model to Predict the Burst Capacity of Pipelines,” OMAE, Pipeline Technology, Vol. V. 133. Stoner, K. J., Sindelar, R. L., Caskey, G. R., Jr., 1991,“Reactor Materials Program- Baseline Material Property Handbook-Mechanical Properties Of 1950’s Vintage Stainless Steel Weldment Components (U),” Task Number: 89-023-A-1, Savannah River Laboratory, Aiken, SC 29808. 134. Stubbe, E. J., VanHoenacker, L., Otero, R., 1994, “RELAP5/MOD3 Assessment for Calculation of Safety and Relief Valve Discharge Piping Hydrodynamic Loads,” International Agreement Report, NUREG/IA-0093. 135. Sudret, B. and Guédé, Z., 2005, “Probabilistic Assessment of Thermal Fatigue in Nuclesr Components,” Nuclear Engineering and Design, Vol. 235, pp.1819-1835. 136. Timoshenko, S., 1930, “Strength of Materials, Part II: Advanced Theory and Problems,” D. Van Nostrand Company, Inc., New York. 137. Touboul, F., Sollogoub, P., and Blay, N., 1999, “Seismic behavior of piping systems with and without defects: experimental and numerical evaluations,” Nuclear Engineering and Design, Vol. 192, pp. 243-260. 138. Turkstra, C. J., 1970, “Theory of Structural Design Decisions Study No. 2,” Solid Mechanics Division, University of Waterloo, Waterloo, Ontario. 139. Ukrainian Industrial Energetic Company, Ukraine, Kiev, UK Fax: + 44870160- 6954, www.geocities.com/ferroslav/fother.html, e-mail: tubing@usa.com 140. Venkataramana, K., Bhasin, V., Vaze, K. K., and Kushwaha, H. S., 2004, “B′2 Stress Indices for Elbows Using Non-Linear Finite Element Analysis,” PVP- Vol. 480. 350 141. Ware, A. G., 1995, “Estimates of Margins in ASME Code Strength Values for Stainless Steel Nuclear Piping,” Idaho Nat. Eng. Lab, INEL-95/00197, CONF- 950740-102. 142. Weiner, P. D. and Smith, S. A. Jr., 1976, “Maximum Moment Capability of Pipe with Various D/T Ratios,” J. of Eng. for Industry, Trans. of the ASME, 98 B (3), pp. 1107-1111. 143. Wellinger, K. and Sturm, D., 1971, “Festigkeits Verhalten von Zylindrischen Hohlkoerpern,” Fortschr. Ber. VDI-Z, Reiche 5, Nr. 13, VDI – Verl., Düsseldorf. 144. Wesley, D. A., 1993, “Interfacing Systems LOCA (ISLOCA) Component Pressure Capacity Methodology and Typical Plant Results,” Nuclear Engineering and Design Vol. 142, pp. 209-224. 145. Woods, G. E., Baguley, R. B., 2000, “CASTI Guidebook to ASME B31.3,” CASTI Guidebook Series, Vol. 3, Second Edition. 146. Zhao, J., Tang, J., and Wu, H. C., 2000, “A Generalized Random Variable Approach for Strain-Based Fatigue Reliability Analysis,” Journal of Pressure Vessel Technology, Vol. 122, pp. 156-161. 147. Zhu, X.-K. and Leis, B. N., 2005, “Analytic Prediction of Plastic Collapse Failure Pressure of Line Pipes,” Proc. of Press. Ves. and Piping Div. Conf., PVP 2005- 71204, Denver, Colorado, July 17-21.