ABSTRACT Title of Dissertation: Stress Intensity Factors for Structural Steel I-beams Daqing Feng, Doctor of Philosophy, 1996 Dissertation directed by: Pedro Albrecht Professor, Department of Civil Engineering Robert J. Sanford Professor, Department of Mechanical Engineering The application of fracture mechanics to highway steel bridges has been hampered by a lack of stress intensity factor (SIF) solutions for cracks in I-beams. Previous work cannot provide satisfactory solutions. In this study, the finite element analysis method was used to develop accurate SIFs for two- tip and three-tip cracks in I-beams under tension or bending. Cracked I-beams were modeled with eight-node shell elements, with the web and flanges being fully joined along the junction lines. The region around the crack tips, singularity quarter-point elements were used. To ensure accurate and converging solutions, mesh patterns around the crack tips were studied. Also, different methods of extracting SIFs from FEA results were discussed based on benchmark problem studies. Governing parameters for cracked I-beams were determined. For two-tip web cracks, the SIFs are functions of applied stress, crack length, eccentricity, and flange-to-web cross-sectional area ratio. For three-tip cracks in web and flange, the SIFs are functions of applied stress, web and flange crack lengths, and flange-to-web cross-sectional area ratio. The flange-to-web area ratio describes the constraining effect of the flange on the web crack of a two-tip cracked I-beam; the interaction forces between web and flanges greatly affect SIFs for a three-tip cracked I-beam. The SIFs were calculated based on a total of 2,106 FEAs performed for a wide range of the parameters. The results were fitted with equations for ready use by practicing engineers. An example illustrates the calculation of SIFs for a three-tip crack in a composite steel-concrete beam of a steel bridge. u STRESS INTENSITY FACTORS FOR STRUCTURAL STEEL I-BEAMS by Daqing Feng Dissertation submitted to the Faculty of the Graduate School of the University of Maryland at College Park in partial fulfillment of the requirement for the degree of Doctor of Philosophy 1996 C.1 (.I/DI.., :t oF 11} kt..) p ) ~ I ( ! LD "52~' ,, M-,.o,J Advisory Committee: t='?e,.1"f, lJ I Professor Robert J. Sanford, Chairman/Advisor Professor Pedro Albrecht, Advisor '/0L \ Professor William L. Fourney Professor Donald W. Vannoy Assistant Professor Ian Flood DEDICATION To my parents and wife! ii ACKNOWLEDGMENT The author would like to express his most grateful appreciation to Professor Pedro Albrecht and Professor Robert J. Sanford for their continuous encouragement, support and guidance during the time of this study. The author would like to express his sincere gratitude to Mr. William Wright of Federal Highway Administration for his technical support and helpful discussions. Thanks are extended to Dr. Ian Flood, Department of Civil Engineering, University of Maryland, Dr. Xiaoguang (Chester) Chen, ACTA Incorporation, and Dr. Phillip Yen, Federal Highway Administration, for their helpful discussions. Many thanks for their help are extended to the staff and colleagues of the Mechanical and Civil Engineering Departments, University of Maryland; and the staff of the Structures Division, Federal Highway Administration, McLean, Virginia. He also likes to acknowledge the Teaching and Research Assistantships from the Mechanical and Civil Engineering Departments as well as the Graduate Research Fellowship from the Federal Highway Administration. Finally, he wishes to express his appreciation to his wife, Yingnan, and his parents for their constant support. Without them this work would not have been possible. iii TABLE OF CONTENTS List of tables ................................................... x List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Crack Types ............................................... 3 1.3 Previous Work .............................................. 5 1.3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.2 Center-cracked Plate with Stiffened Edges ................. 6 Stress Functions ..................................... 7 Boundary Conditions .................................. 9 SIF Results ........................................ 11 1.3.3 Two-tip Cracked I-Beam .............................. 13 Weight Function for Central Crack ...................... 13 Reference Loading for Cantral Crack . . . . . . . . . . . . . . . . . . . . 14 Weight Functions for Eccentric Crack .................... 15 Self-consistency Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Reference Loading for Eccentric Crack . . . . . . . . . . . . . . . . . . . 17 SIFs for Two-tip Cracked I-beam ....................... 19 1.3.4 Three-Tip Cracked I-Beam ............................ 20 COD Assumptions for Central Three-tip Crack ............. 21 Reference Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 CODs of Web and Flange Cracks ....................... 23 Normalized Junction Point COD ........................ 25 Eccentric Three-tip Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Shortcomings ...................................... 28 1.4 Objective ................................................. 30 iv Chapter 2: Finite Element Models of Cracked I-beams ............... 32 2.1 Introduction ............................................... 32 2.2 Modeling of Cracked I-beam .................................. 32 2.3 Crack Tip Elements ......................................... 33 Quarter-point Singularity Element ....................... 34 ABAQUS' Singularity Element .......................... 35 2.4 Mesh Pattern Around Crack Tip ............................... 35 2.5 Extracting SIFs from FEM Output .............................. 37 2.5.1 Displacement-Based Methods .......................... 37 Nonlinear Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Linear Extrapolation ................................. 39 Quarter-point Displacement ........................... 39 Comparison ........................................ 40 2.5.2 Stress-Based Methods ................................ 41 2.5.3 Energy-Based Methods ............................... 42 Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 J-lntegral .......................................... 43 Stiffness Derivative .................................. 44 2.6 Two-dimensional Model ...................................... 46 2.6.1 2-D Approach ....................................... 46 2.6.2 Numerical Results and Discussion ....................... 47 Parametric Analyses ................................. 47 Two-tip Cracked I-beams ............................. 49 Three-tip Cracked I-beams ............................ 50 2.6.3 Benefits of 2-D Modeling .............................. 51 Chapter 3: Parameters for Cracked I-beams ....................... 52 3.1 Two-tip Cracked I-Beam ..................................... 52 V 3.2 Three-tip Cracked I-Beam .................................... 54 3.3 Validation of Parameter ~ .................................... 56 3.3.1 Two-tip Crack under Tension and Bending ................ 57 3.3.2 Three-tip Crack under Tension and Bending ............... 59 3.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Chapter 4: Non-interacting Three-tip Cracks ...................... 62 4.1 Introduction ............................................... 62 4.2 Joined Infinite Plates ........................................ 63 4.2.1 Centerline Displacements ............................. 63 4.2.2 Edge Displacements ................................. 64 Paris' Method ...................................... 64 Edge Displacement .................................. 66 4.2.3 Non-interacting Crack Lengths ......................... 67 4.3 Joined Finite Plates ......................................... 69 4.3.1 CMODs ........................................... 69 4.3.2 Non-interacting Crack Lengths ......................... 70 4.3.3 Comparison with Single Plate Solutions .................. 71 4.4 I-beams .................................................. 73 4.4.1 Non-interacting Crack Lengths From CMODs .............. 73 4.4.2 Non-interacting Crack Lengths From FEA ................. 74 4.4.3 Comparisons ....................................... 75 4.5 Conclusions ............................................... 76 Chapter 5: Stress Intensity Factors for Cracked I-beams ............ 78 5.1 Introduction ............................................... 78 5.2 Two-tip Cracked I-beams ..................................... 78 5.2.1 Variables in Analysis ................................. 79 5.2.2 Two-tip Cracked I-beams under Tension .................. 80 vi Effect of Crack Length Aw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Effect of Eccentricity e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Effect of Flanges-to-Web Ratio J3 . . . . . . . . . . . . . . . . . . . . . . . 82 Comparison of I-beam with Single Plates . . . . . . . . . . . . . . . . . 82 2-D Plots ......................................... 84 5.2.3 Two-tip Cracked I-beams under Bending ................. 85 Effect of Crack Length Aw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Effect of Eccentricity e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Effect of Flanges-to-Web Ratio J3 . . . . . . . . . . . . . . . . . . . . . . . 87 Comparison of I-beam and Single Plate .................. 87 5.3 Three-tip Cracked I-beams ................................... 89 5.3.1 Variables in Analysis ................................. 90 5.3.2 Three-tip Cracked I-beams under Tension ................ 91 Correction Factor for Flange Crack Tip ................... 91 Correction Factor for Web Crack Tip ..................... 92 Interaction Between Web and Flange .................... 93 5.3.2 Three-tip Cracked I-beams under Bending ................ 93 5 .4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Chapter 6: Equations for Stress Intensity Factors .................. 96 6.1 Fitting Procedures .......................................... 96 6.2 Two-tip Cracked I-beams ..................................... 98 6.2.1 Parameter Ranges .................................. 98 6.2.2 Equations for Two-tip Cracks under Tension ............... 99 Coefficients for Upper Crack Tip . . . . . . . . . . . . . . . . . . . . . . . 1 00 Coefficients for Lower Crack Tip . . . . . . . . . . . . . . . . . . . . . . . 101 6.2.3 Equations for Two-tip Cracks under Bending ............. 101 Coefficients for Upper Crack Tip . . . . . . . . . . . . . . . . . . . . . . . 102 Coefficients for Lower Crack Tip . . . . . . . . . . . . . . . . . . . . . . . 102 vii 6.2.4 Two-tip Cracks in Engineering Practice . . . . . . . . . . . . . . . . . . 103 Minimum Crack Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Maximum Crack Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Maximum Eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.3 Three-tip Cracked I-beams .................................. 105 6.3.1 Parameter Ranges .................................. 105 6.3.2 Equations for Three-tip Cracks under Tension ............ 106 Coefficients for Web Crack Tip ........................ 106 Coefficients for Flange Crack Tip . . . . . . . . . . . . . . . . . . . . . . 107 6.3.3 Equations for Three-tip Cracks under Bending ............ 108 Coefficients for Web Crack Tip . . . . . . . . . . . . . . . . . . . . . . . . 108 Coefficients for Flange Crack Tip . . . . . . . . . . . . . . . . . . . . . . 108 6.4 Fatigue and Fracture Analysis ................................ 109 6.4.1 lnputfor Analysis ................................... 109 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Crack Geometry and Dimension . . . . . . . . . . . . . . . . . . . . . . . 1 09 Loading .......................................... 110 Stress Intensity Factor .............................. 11 0 6.4.2 Analysis Methods ................................... 11 O Fracture Analysis .................................. 11 O Fatigue Analysis ................................... 111 6.5 Composite and Singly Symmetric Beam ........................ 112 6.5.1 Composite Rolled Beam ............................. 113 6.5.2 Composite Plate Girder with Doubly Symmetric Section ..... 114 6.5.3 Non composite Plate Girder with Singly Symmetric Section .. 114 6.5.4 Composite Plate Girder with Singly Symmetric Section ...... 116 6.6 40-Ft Simple-span Composite Beam ........................... 116 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 W-shape Geometry Properties ........................ 116 viii Crack Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Properties of Composite Section . . . . . . . . . . . . . . . . . . . . . . . 117 Loads and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Stress in Steel Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 SIFs for Web and Flange Crack Tips ............. . ..... 120 6. 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Chapter 7: Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Appendix A. Benchmark Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 A. 1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 A.2 Benchmark Problems ...... .. ...... . ....................... 126 A.3 Results and Discussions .................................... 127 A.3.1 Mesh Patterns in Inner Region .......... . ............. 128 Effect of Parameter m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Effect of Parameter n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 A.3.2 Calculation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 J-integral Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 COD Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Apparent SIF ........ . ............... . ............. 130 A.3.3 Additional Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 A.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Appendix B. Tables and Figures ................................ 134 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 ix LIST OF TABLES Chapter 1: 1.1. Methods of determining stress intensity factors ................. 135 Chapter 3: 3.1. W-shapes used in calculations of SIFs ....................... 136 Chapter 4: 4.1. Non-interacting crack lengths for T-section ... .. ............. . . 137 4.2. Non-interacting crack lengths for three-tip cracked I-beams . . . . . . . 138 Chapter 5: 5.1. Correction factors fA for two-tip cracked I-beam under tension; upper tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.2. Correction factors f 8 for two-tip cracked I-beam under tension; lower tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.3. Correction factors fA for two-tip cracked I-beam under bending; upper tip ............................................... 145 5.4. Correction factors f 8 for two-tip cracked I-beam under bending; lower tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.5. Correction factors fw for three-tip cracked I-beam under tension; web crack tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.6. Correction factors f' for three-tip cracked I-beam under tension; flange crack tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.7. Correction factors fw for three-tip cracked I-beam under bending; web crack tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.8. Correction factors f' for three-tip cracked I-beam under bending; flange crack tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 X Chapter 6: 6.1. Fitting coefficients for two-tip cracked I-beam .................. 163 6.2. Fitting coefficients for three-tip cracked I-beam ................. 164 6.3. Three-tip cracks under tension with fitting error for web crack tip jlij ~ 3% .. . .............................. 165 6.4. Three-tip cracks under tension with fitting error for flange crack tip !Lil ~ 3% ................................ 165 6.5. Three-tip cracks under bending with fitting error for web crack tip !Lil ~ 3% ................................. 166 6.6. Three-tip cracks under bending with fitting error for flange crack tip lli I ~ 3% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.7. Fitting errors for two-tip and three-tip cracked I-beams ........... 167 Appendix A: A.1. Comparison of SIF for center-cracked plate under tension ........ 168 A.2. Comparison of SIF for edge-cracked plate under bending ........ 169 A.3. Comparison of SIF for edge-cracked plate under tension ......... 170 A.4. CODs at quarter point B and corner point C calculated with FEA (m x n = 3 x 8) .......................... 171 xi LIST OF FIGURES Chapter 1: 1.1. Stages of crack growth for stiffener welded to web only ......... 172 1.2. Stages of crack growth for stiffener welded to web and flange .... 172 1.3. Stages of crack growth for attachment welded to flange . . . . . . . . . 173 1.4. Stages of crack growth for welded I-beam . . . . . . . . . . . . . . . . . . . 173 1.5. Centrally cracked strip with stiffened edges .................. 174 1.6. Equilibrium of stiffener element ............................ 175 1. 7. Plate with central or eccentric cracks . . . . . . . . . . . . . . . . . . . . . . . 176 1.8. Two-tip cracked I-beam .................................. 177 1. 9 . Three-tip cracked I-beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 8 1.10. Coordinate and symbols used in previous analysis of three-tip crack (Chen and Albrecht 1994) .................... 179 Chapter 2: 2.1. Dimensions used for modeling of I-beam . . . . . . . . . . . . . . . . . . . . 180 2.2. Typical three-dimensional I-beam mesh for three-tip crack: (a) before deformation ................................... 181 (b) after deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 2.3. Quarter-point element with 1! {r singularity ................... 183 2.4. Degenerated quarter-point element with 11{r singularity ......... 183 2.5. Element sizes in inner region generated by ABAQUS' * SINGULAR command .................................. 184 2.6. Mesh scheme for different crack lengths ..................... 185 2.7. Mesh patterns in inner region around crack tip ................ 186 2.8. Nodes used for calculating SIFs ........................... 187 2.9. SIFs calculated from displacement-based methods ............ 188 xii 2.10. Schematic of two-dimensional I-beam model ................. 189 2.11. Comparison of 2-D and 3-D model results for SIFs of two-tip cracked I-beam under tension; upper tip ............... 190 2.12. Comparison of 2-D and 3-D model results for SIFs of two-tip cracked I-beam under tension; lower tip ............... 191 2.13. Comparison of 2-D and 3-D model results for SIFs of two-tip cracked I-beam under bending; lower tip ............... 192 2.14. Comparison of 2-D and 3-D model results for SIFs of three-tip cracked I-beam under tension; web crack tip .......... 193 2.15. Comparison of 2-D and 3-D model results for SIFs of three-tip cracked I-beam under tension; flange crack tip ........ 194 2.16. Comparison of 2-D and 3-D model results for SIFs of three-tip cracked I-beam under bending; web crack tip ......... 195 2.17. Comparison of 2-D and 3-D model results for SIFs of three-tip cracked I-beam under bending; flange crack tip ........ 196 Chapter 3: 3.1. Dimensions of two-tip web crack ........................... 197 3.2. Dimensions of symmetric three-tip crack ..................... 197 3.3. Effect of parameter ~ on correction factor for two-tip center-cracked I-beams under tension ...................... 198 3.4. Selected W-shapes for validation of ~ parameter .............. 199 3.5. Effect of parameter~ on correction factor for two-tip center-cracked I-beams under bending ...................... 200 3.6. Effect of parameter~ on correction factor for three-tip cracked I-beams under tension; Aw = 0.1 ........................... 201 3.7. Effect of parameter~ on correction factor for three-tip cracked I-beams under tension; Aw = 0.5 ........................... 202 3.8. Effect of parameter ~ on correction factor for three-tip cracked xiii J I-beams under bending; 'Aw= 0.1 ........................... 203 3.9. Effect of parameter~ on correction factor for three-tip cracked I-beams under bending; 'Aw= 0.5 ........................... 204 3.1 O. Range of parameter~ in finite element analysis ............... 205 Chapter 4: 4.1. Center-cracked infinite plate joined with edge-cracked semi-infinite plate ...................................... 206 4.2. (a) Infinite plate with central crack under tension, and (b) semi-infinite plate with edge crack under tension ........... 207 4.3. Displacements along junction line of center-cracked infinite plate and edge-cracked semi-infinite plate under tension ............ 208 4.4. (a) Joined two finite plates; (b) T-flange and T-web under tension; (c) T-flange under tension and T-web under bending ............. 209 4.5. Relationship between non-interacting T-flange and T-web crack lengths .................................... 210 4.6. Comparison of SIFs for non-interacting T-section and single plates under tension ............................... 211 4.7. Comparison of CODs for non-interacting T-section and single plates under tension ............................... 212 4.8. Comparison of displacements along junction line of non-interacting T-section and single plates under tension ....... 213 4.9. Comparison of CODs for T-section and single plates under tension ......................................... 214 4.10. Comparison of displacements along junction line of T-section and single plates under tension .................... 215 4.11. Non-interacting flange crack length determined by equating Kt; W33 x 201, 'Aw= 0.1, under tension ........................ 216 4.12. Non-interacting flange crack length determined by equating Kw; xiv W33 x 201, Aw= 0.1, under tension ........................ 217 4.13. Comparison of non-interacting flange crack lengths determined from CMOD and SIFs; Aw = 0.1, under tension ................ 218 4.14. Comparison of non-interacting flange crack lengths determined from CMOD and SIFs; Aw= 0.1, under bending ............... 219 Chapter 5: 5.1. Correction factor for two-tip cracked I-beam under tension; central crack .......................................... 220 5.2. Correction factor for two-tip cracked I-beam under tension; eccentric crack, upper tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 5.3. Correction factor for two-tip cracked I-beam under tension; eccentric crack, lower tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 5.4. Comparison of single plate and I-beam under tension; upper tip, ~ = 0.83 ...................................... 223 5.5. Comparison of single plate and I-beam under tension; lower tip, ~ = 0.83 ...................................... 224 5.6. Correction factors for two-tip cracked I-beam under tension; ~ = 0.83 .............................................. 225 5.7. Correction factors for two-tip cracked I-beam under tension; ~ = 1.37 .............................................. 226 5.8. Correction factors for two-tip cracked I-beam under tension; ~ = 2.05 .............................................. 227 5.9. Correction factors for two-tip cracked I-beam under tension; upper tip, ~ = 0.83 ...................................... 228 5.10. Correction factors for two-tip cracked I-beam under tension; lower tip, ~ = 0.83 ...................................... 229 5.11. Correction factors for two-tip cracked I-beam under tension; upper tip, ~ = 2.05 ...................................... 230 xv 5.12. Correction factors for two-tip cracked I-beam under tension; lower tip, J3 = 2.05 ...................................... 231 5.13. Correction factors for two-tip cracked I-beam under tension; upper tip, & = 0.5 ....................................... 232 5.14. Correction factors for two-tip cracked I-beam under tension; lower tip, e = 0.5 ....................................... 233 5.15. Correction factor for two-tip cracked I-beam under bending; upper tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 5.16. Correction factor for two-tip cracked I-beam under bending; lower tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ? . . . . . . . . . . . . 235 5.17. Comparison of single web plate and I-beam under bending; upper tip, J3 = 0.83 ...................................... 236 5.18. Comparison of single web plate and I-beam under bending; lower tip, J3 = 0.83 ...................................... 237 5.19. Correction factors for two-tip cracked I-beam under bending; J3 = 0.83 .............................................. 238 5.20. Correction factors for two-tip cracked I-beam under bending; J3 = 1.37 .............................................. 239 5.21. Correction factors for two-tip cracked I-beam under bending; J3 = 2.05 .............................................. 240 5.22. Correction factors for two-tip cracked I-beam under bending; upper tip, ~ = 0.83 ...................................... 241 5.23. Correction factors for two-tip cracked I-beam under bending; lower tip, ~ = 0.83 ...................................... 242 5.24. Correction factors for two-tip cracked I-beam under bending; upper tip, ~ = 2.05 ...................................... 243 5.25. Correction factors for two-tip cracked I-beam under bending; lower tip, ~ = 2.05 ...................................... 244 5.26. Correction factors for two-tip cracked I-beam under bending; xvi upper tip, e = 0.5 ....................................... 245 5.27. Correction factors for two-tip cracked I-beam under bending; lower tip, e = 0.5 ....................................... 246 5.28. Correction factor for three-tip cracked I-beam under tension; flange crack tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 7 5.29. Correction factor for three-tip cracked I-beam under tension; web crack tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 5.30. Correction factor for flange crack of three-tip cracked I-beam under tension; P= 0.83 ?................................. 249 5.31. Correction factor for flange crack of three-tip cracked I-beam under tension; P= 2.05 .................................. 250 5.32. Correction factor for three-tip cracked I-beam under bending; flange crack tip ........................................ 251 5.33. Correction factor for three-tip cracked I-beam under bending; web crack tip .......................................... 252 Chapter 6: 6.1. Crack tip positions for central crack in two-tip cracked I-beam .... 253 6.2. Crack tip positions for eccentric crack in two-tip cracked I-beam .. 254 6.3. Preliminary fit for two-tip cracked I-beam under tension; W40 x 149, upper tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.4. Preliminary fit for two-tip cracked I-beam under tension; W18 x 97, upper tip ..................................... 256 6.5. Error in predicting correction factor for two-tip cracked I-beam under tension; upper crack tip ............................. 257 6.6. Variation in prediction error with area ratio; two-tip cracked I-beam under tension, upper crack tip ....................... 258 6.7. Variation in prediction error with eccentricity; two-tip cracked I-beam under tension, upper crack tip ....................... 259 xvii 6.8. Variation in prediction error with web crack length; two-tip cracked I-beam under tension, upper crack tip ....................... 260 6.9. Comparison of predicted and calculated correction factors for two-tip cracked I-beam under tension; W40 x 149, upper tip ..... 261 6.1 0. Comparison of predicted and calculated correction factors for two-tip cracked I-beam under tension; W18 x 97, upper tip ...... 262 6.11. Error in predicting correction factor for two-tip cracked I-beam under tension; lower crack tip ............................. 263 6.12. Variation in prediction error with area ratio; two-tip cracked I-beam under tension, lower crack tip ............................. 264 6.13. Variation in prediction error with eccentricity; two-tip cracked I-beam under tension, lower crack tip ............................. 265 6.14. Variation in prediction error with web crack length; two-tip cracked I-beam under tension, lower crack tip ....................... 266 6.15. Comparison of predicted and calculated correction factors for two-tip cracked I-beam under tension; W40 x 149, lower tip ...... 267 6.16. Comparison of predicted and calculated correction factors for two-tip cracked I-beam under tension; W18 x 97, lower tip ....... 268 6.17. Error in predicting correction factor for two-tip cracked I-beam under bending; upper crack tip ............................ 269 6.18. Variation in prediction error with area ratio; two-tip cracked I-beam under bending, upper crack tip ............................ 270 6.19. Variation in prediction error with eccentricity; two-tip cracked I-beam under bending, upper crack tip ............................ 271 6.20. Variation in prediction error with web crack length; two-tip cracked I-beam under bending, upper crack tip ...................... 272 6.21. Comparison of predicted and calculated correction factors for two-tip cracked I-beam under bending; W40 x 149, upper tip ..... 273 6.22. Comparison of predicted and calculated correction factors for xviii two-tip cracked I-beam under bending; W18 x 97, upper tip ...... 274 6.23. Error in predicting correction factor for two-tip cracked I-beam under bending; lower crack tip ............................ 275 6.24. Variation in prediction error with area ratio; two-tip cracked I-beam under bending, lower crack tip ............................ 276 6.25. Variation in prediction error with eccentricity; two-tip cracked I-beam under bending, lower crack tip ............................ 277 6.26. Variation in prediction error with web crack length; two-tip cracked I-beam under bending, lower crack tip ...................... 278 6.27. Comparison of predicted and calculated correction factors for two-tip cracked I-beam under bending; W40 x 149, lower tip ..... 279 6.28. Comparison of predicted and calculated correction factors for two-tip cracked I-beam under bending; W18 x 97, lower tip ...... 280 6.29. Maximum and minimum web crack lengths for two-tip cracked I-beam ........................................ 281 6.30. Maximum eccentricity in two-tip cracked I-beam ............... 282 6.31. Maximum eccentricity for rolled I-beams used in engineering ..... 283 6.32. Maximum and minimum crack lengths for two-tip eccentrically cracked I-beams; e = 0.1 ................................. 284 6.33. Maximum and minimum crack lengths for two-tip eccentrically cracked I-beams; e = 0. 7 ................................. 285 6.34. Preliminary fit for three-tip cracked I-beam under tension; W40 x 149, flange crack tip ............................... 286 6.35. Preliminary fit for three-tip cracked I-beam under tension; W18 x 97, flange crack tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 6.36. Error in predicting correction factor for three-tip cracked I-beam under tension; web crack tip .............................. 288 6.37. Variation in prediction error with area ratio; three-tip cracked I-beam under tension, web crack tip ........................ 289 xix 6.38. Variation in prediction error with web crack length; three-tip cracked I-beam under tension, web crack tip ................. 290 6.39. Variation in prediction error with flange crack length; three-tip cracked I-beam under tension, web crack tip .......... 291 6.40. Comparison of predicted and calculated correction factors for three-tip cracked I-beam under tension; W40 x 149, web crack tip. 292 6.41. Comparison of predicted and calculated correction factors for three-tip cracked I-beam under tension; W18 x 97, web crack tip .. 293 6.42. Error in predicting correction factor for three-tip cracked I-beam under tension; flange crack tip ...................... 294 6.43. Variation in prediction error with area ratio; three-tip cracked I-beam under tension, flange crack tip ...................... 295 6.44. Variation in prediction error with web crack length; three-tip cracked I-beam under tension, flange crack tip ........ 296 6.45. Variation in prediction error with flange crack length; three-tip cracked I-beam under tension, flange crack tip ........ 297 6.46. Comparison of predicted and calculated correction factors for three-tip cracked I-beam under tension; W40 x 149, flange crack tip ........................................ 298 6.47. Comparison of predicted and calculated correction factors for three-tip cracked I-beam under tension; W18 x 97, flange crack tip ........................................ 299 6.48. Error in predicting correction factor for three-tip cracked I-beam under bending; web crack tip ....................... 300 6.49. Variation in prediction error with area ratio; three-tip cracked I-beam under bending, web crack tip ....................... 301 6.50. Variation in prediction error with web crack length; three-tip cracked I-beam under bending, web crack tip ......... 302 6.51. Variation in prediction error with flange crack length; xx three-tip cracked I-beam under bending, web crack tip ......... 303 6.52. Comparison of predicted and calculated correction factors for three-tip cracked I-beam under bending; W40 x 149, web crack tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 6.53. Comparison of predicted and calculated correction factors for three-tip cracked I-beam under bending; W18 x 97, web crack tip .......................................... 305 6.54 Error in predicting correction factor for three-tip cracked I-beam under bending; flange crack tip ...................... 306 6.55 Variation in prediction error with area ratio; three-tip cracked I-beam under bending, flange crack tip ...................... 307 6.56 Variation in prediction error with web crack length; three-tip cracked I-beam under bending, flange crack tip ........ 308 6.57 Variation in prediction error with flange crack length; three-tip cracked I-beam under bending, flange crack tip ........ 309 6.58. Comparison of predicted and calculated correction factors for three-tip cracked I-beam under bending; W40 x 149, flange crack tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 0 6.59. Comparison of predicted and calculated correction factors for three-tip cracked I-beam under bending; W18 x 97, flange crack tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 6.60. (a) Composite rolled beam; (b) composite plate girder with doubly symmetric section; (c) noncomposite singly symmetric section; (d) composite plate girder with singly symmetric section ........... 312 6.61. (a) Noncomposite singly symmetric section; (b) stress distribution; and? equivalent doubly symmetric section .................. 313 6.62. Composite, simple supported, rolled beam ................... 314 6.63. Composite section ...................................... 315 6.64. Bending moment diagrams ............................... 316 xxi 6.65. Decomposition of linearly distributed loading into axial tension and pure bending ................................ . ............. 317 Appendix A: A.1. (a) center-cracked plate; (b) edge-cracked plate .............. 318 A.2. J-integral versus contour distance from crack tip within inner region for center-cracked plate under tension; a/w = 0.5 ........ 319 A.3. COD profiles within crack tip element for center-cracked plate under tension; a/w = 0.5 ............................. 320 A.4. Normalized apparent SIFs within crack tip element for center-cracked plate under tension; a/W = 0.5 ............... . 321 A.5. Normalized apparent SIFs within crack tip element for edge-cracked plate under bending; a/W = 0.1 .... . ........... 322 A.6. Normalized apparent SIFs within crack tip element for edge-cracked plate under bending; a/W = 0.9 ................ 323 A. 7. Plate with eccentric crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 A.8. Comparison of FEA and existing solutions for center-cracked plate under tension or bending ................ 325 A.9. Comparison of FEA and existing solutions for edge-cracked plate under tension or bending ................. 326 A.1 O. Comparison of FEA and lsida's (1965) solutions for plate with eccentric crack under tension ..................... 327 A.11. Comparison of FEA and Chen and Albrecht (1994) solutions for plate with eccentric crack under bending ....... . .. ... ....... 328 xxii Chapter 1: Introduction 1.1 Problem Fatigue and fracture have caused steel bridges to fail for many years. Over one-half of the 577,710 bridges in the federal highway system are more than 30 years old. Many of them are made of steel. Because of such factors as type of material, design details and quality of fabrication, initial cracks or crack- like flaws in structural components of steel bridges sometimes cannot be avoided. Cracks propagate as a result of stress ranges induced by trucks crossing the bridges. When a crack reaches the critical size, the girder may fracture. Fatigue cracking of a steel bridge is costly. Besides the tangible repair cost, traffic delays when one or more lanes have to be closed inflict intangible cost that may greatly exceed the repair cost. The discovery of a crack in a steel bridge immediately places the state engineer who is responsible for public safety in a dilemma. He must decide whether to close the entire bridge or just an individual lane while the bridge is being retrofitted. To help him decide, the engineer must determine if the calculated stress intensity factor (SIF), K, is smaller or greater than the fracture toughness, Kc, of the steel. Fracture mechanics analysis of crack extension provides a systematic, 1 scientific approach to characterizing the severity of cracks and predicting when they may become unstable during the structure's service life. A complete fracture study involves both stress analysis to determine the SIF and material testing to determine the material's resistance to crack extension. This study focuses on the former, that is, the calculation of SIFs in cracked steel bridge girders. Cracks in steel bridges can usually be analyzed with linear-elastic fracture mechanics (LEFM). The fundamental postulate is that crack extension is governed solely by the value of the SIF. Since bridges typically fracture during the winter months when the temperature is low, the critical value of the SIF, called the fracture toughness Kc, lies on the lower shelf and the lower portion of the transition region. Therefore, plastic deformation is limited to a small crack-tip region and the linear-elastic fracture mechanics approach is valid. Knowing the SIF, one can determine the static strength of a cracked structure (residual strength) and the crack growth rate under cyclic loading (fatigue). Type of loading and the size, shape and orientation of the crack play a major role in determining the applicable K values. Solutions for simple structural configurations and loadings are available in several handbooks (Tada, et al. 1973, Sih 1973, Rooke and Cartwright 1976, and Murakami 1986). But few, if any, solutions exist for cracks in rolled I-beams, welded I-beams and plate 2 girders that are commonly used in steel bridges. This has greatly hindered the application of fracture mechanics to fatigue and fracture analysis of steel highway bridges. 1.2 Crack Types Previous studies on fatigue strength of steel bridge details, including stiffeners and attachments, showed that all cracks initiate from flaws at stress raisers, typically at weld toes on web or flange surfaces (Fisher, et al. 1970; Fisher, et al. 1974). When a transverse stiffener is welded to the web alone, the crack propagates in three stages. After initiating at points along the toe of the stiffener to web weld, the crack grows first as a surface crack through the web thickness, then as a two-tip through crack with one front moving up and the other down the web. Finally it grows as a three-tip crack with one front moving further up the web and the other two fronts extending across the flange width (Figure 1.1 ). When the stiffener is welded to the web and flanges, shown in Figure 1.2, the crack initiates at points along the toe of the fillet weld connecting the stiffener to the tension flange. In this case the crack grows in two or three stages depending on where the crack initiates across the flange width. During the first stage, one or more cracks propagate in a semi-elliptical shape and 3 coalesce. After this part-through crack breaks through the extreme fiber of the tension flange, it grows in the second stage as a two-tip crack across the tension flange. Upon further propagation, a three-tip crack forms if the inner tip enters the web before the outer tip reaches the flange edge. But if the tip closest to the flange edge breaks through, the crack then grows with one front moving up the web and the other across the flange width. Cracks at attachments welded eccentrically to the flange grow like those at stiffeners. The attachment in Figure 1.3 is welded to the flange along its length but not the width. The crack initiates at the end of a weld and then propagates in two or three stages. Cracks in welded beams, shown in Figure 1.4, initiate at a porosity, weld repair, tack weld, or stop-start position in the longitudinal flange to web fillet weld. When the crack initiates at an internal porosity, it grows approximately in a circular shape, with the initial defect at its center, until the crack front reaches the extreme fiber of the tension flange. The crack then changes into a three-tip crack with two fronts propagating across the flange and one front advancing upward into the web. These examples illustrate that typical cracks in steel beams have either two tips or three tips. SIFs for such cracks under an applied moment or axial force are essential to the application of fracture mechanics in highway bridges. 4 1.3 Previous Work 1.3.1 Summary Many methods for obtaining SIFs have been developed. Aliabadi and Rooke (1991) divided them into three categories depending on degree of sophistication and time required to obtain a solution (Table 1.1 ). For a simple geometrical configuration, or where a structural component can be easily modeled, SIF solutions may be obtained directly from handbooks (Tada, et al. 1973, Sih 1973, Rooke and Cartwright 1976, and Murakami 1986). When handbook solutions do not exist or only an estimate is required, a relatively simple method from category 2 may be adequate. Finally, when a highly accurate SIF is required, or the structural shape is complex, the numerical methods listed in category 3 must be used. For a two-tip or three-tip cracked I-beam, the complex geometry and interaction between the web and flanges make it exceedingly difficult to calculate SIFs. An I-beam consists of one web plate and two flange plates in 3- D space. As the crack extends into both the web and flange, the crack opening displacements (CODs) must be equal at the flange-web junction. Likewise the web and flange displacements along the junction line away from the intersection of the two cracks must also be equal. These displacement continuity conditions induce interaction forces whose magnitude and distribution depend on the applied load, I-beam geometry, and crack lengths. 5 Greif and Sanders (1965), lsida (1973), Nishimura (1991) and others analyzed a cracked plate with stringers or edge stiffeners. This geometry is commonly found in both aircraft fuselages and ship hulls. Although the boundary conditions are different from those of an I-beam, the effect of edge stiffeners on the cracked plate is similar to that of flanges on a two-tip cracked web. Chen (1992) obtained SIF solutions for two-tip central and eccentric cracks in the web with the weight function method (WFM). For three-tip symmetric or non-symmetric cracks, he used the energy release rate method. lsida's (1973) solution for center-cracked plate with edge stiffener and Chen's (1992) solutions for two-tip and three-tip cracked I-beams are discussed in the following three subsections. 1.3.2 Center-cracked Plate with Stiffened Edges The flanges in an I-beam with a two-tip web crack act like edge stiffeners that effectively restrain the CODs and thus reduce the SIF. Similar geometries exist in many other structures. For example, sheets in aircraft fuselages and ship hulls are usually reinforced with stiffeners that may arrest a crack or alleviate stresses. The effects of these stiffeners on the SIF have been studied by Greif and Sanders (1965), lsida (1973). In actual airplane fuselages and ship hulls, the stiffeners are usually riveted or bolted to the plate, but in the studies 6 of Greif and Sanders (1965) and lsida (1973) the stiffeners were assumed to be integral with the plate. In rolled or welded I-beams used in steel bridges, the flanges and web are also connected integrally. Greif and Sanders (1965) were the first to study the problem of a plate with longitudinal stiffeners. The stiffener was assumed to have zero flexural stiffness. lsida (1973) studied the same problem by including both the axial and flexural rigidities. In his study, lsida used the method of Laurent expansion of the complex potentials and determined the expansion coefficients from the boundary conditions. He performed the numerical calculations with the perturbation technique and obtained the SIFs from the 36-term power series of '/..2, where>.. is the ratio of crack length to plate width. Stress Functions Figure 1.5 shows a center-cracked plate reinforced with stiffeners along both edges. Only the plate is subjected to axial tension. The stress function x of this problem can be expressed in terms of complex potentials cp(~) and 41(~): (1.1) where Tis the applied tension stress and b is the half width of the web. The corresponding stress and displacement components for plane stress are given by: a" = T Re{2q>1(0 - ~q> 11(l) - 4J 11(l)} o = T Re {2cp'(0 + tcp''(l) + 41 11(0} 1 (1.2) Txy = T Im {~q>11(l) + 4,111(l)} E(u - iv) = Tb ((3 -v)q>(~) - (1 + v){~q>'(l) + ttJ1(l)}] 7 where~ and r, are dimensionless coordinates, and vis Poisson's ratio. In the complex plane: Z = X + iy ~ = ~ I r, = 1. I ~ = ~ + ;17 z (1.3) b b b the stress function can be rewritten in the more convenient form: (1.4) where x0 corresponds to the axial tension stress at infinity and is written as: Xo = Tb 2 Re{~q>0 (Q + 'Pi~} with c.p (~) = l (1.5) and 4J (Q = 'l!- 0 4 0 4 x1 is a stress function with singularities at the origin. Because stresses are symmetric about the x and y axes, it is expressed as: X1 = Tb 2 Re{~q>1(0 + 411(0} L"' q>1 (0 = E2n ~-(2n ? 1) n =O (1.6) ... 4'1(0 = -DO log~+ L D2n~-2n n = 1 where the unknown coefficients D2n and E2n are determined from the boundary conditions. The third stress function X2 is added to Xo and X1 to satisfy the boundary conditions along the crack surface and the outer edges where the plate meets the stiffeners. Stress function X2 is assumed in the integral form: X2 = Tb 2 [tA (m) ~ sinh m~ + B (m) cosh m~] cos m~ dm 0 (1.7) ? T b'R?{[[ A ~m) (~ + i;>slnhm~ + B (m )cash m~]dm} 8 Again the unknown functions A(m) and B(m) are determined from the boundary conditions. Inserting these stress functions into Equation 1.2 yields stresses and displacements. Boundary Conditions To determine the unknown coefficients and functions, the boundary conditions along the plate-stiffener junction lines x = ? b (Figure 1.5) and the crack surface (-a :s: x :s: a, y = 0) are examined. From the equilibrium of the stiffener element shown in Figure 1.6, the following relations are obtained: (1.8) where subscripts denotes the stiffener. The continuity condition along the joint line gives a y -va X (1.9) E Therefore, replacing us by (u)f.= 1, Oys by (E/E)(ay - vax)f.= 1, and noting that Oys :::: = ~[_?__ (Vax - or>] ar, ~ = 1 -__ -a( (1.10) 4-a E-u] b ar,4 ~ = 1 where the flexural and axial rigidity ratios are 9 E I a= _s ? _s_ and ~ = Es ? As (1.11) E b 3 t E bt The stresses and displacements derived from stress functions Xo, x1 and x2 must satisfy the boundary condition Equations 1.10. Since Xo itself satisfies these relations, only stresses and displacements derived from X1 and X2 are involved in Equations 1.10. By using Equations 1.2, 1.5, 1.6 and 1.10, the unknown functions A(m) and B(m) of the stress function X2 can be written in terms of D2n and E2n which are the unknown coefficients of x1? Expanding Equation 1.7 into a power series in~ yields the following form of X2 which, together with x1, satisfies the boundary conditions along the junction lines: X = Tb 2 2 Re [~cp2(~ + 4,1 (~)] .. 2 -= L L2nl':n 2 + n ?O in Equation 1.12, L2n and M2n are expressed in linear forms of D2P and E2p: ... L2n = L ca:: 02p + ~:: E2p> p .?.O ( 1.13) M2n = L .. Poisson's ratio was assumed to be 0.3. Retaining the first 36 terms of the series, gives K 1 = T{rraF(a,~,A) 35 L (1.17) F(a,~,A) = 1 + c2n;..2n n ~1 where the coefficient c2n is a function of a and ~- The function F(a, ~. A) represents the correction on K1 due to the effects of relative rigidities of plate and stiffeners. Values of the correction factor F (a, ~. >.), plotted as a function of A for 11 different values of a and ~ (lsida 1973), show that increasing flexural and axial rigidities a and~ reduces the value of SIF. This trend becomes more pronounced the longer the crack is. Of the two parameters, the axial rigidity ~. which is the cross-sectional area ratio of stiffener-to-plate, has the greater effect on SIF. The flexural rigidity a becomes appreciable only when the crack tip approaches the flanges. lsida checked the accuracy of his solution by calculating the value of F for several specially chosen cases of a and ~ and then comparing them with the existing solutions. ? a =~ =O ; the extreme case of a finite width plate with a central crack. lsida's solution yields values of F accurate to less than 0.1% for>-.~ 0.9. ? a =0 11 , ~ =O ; wide plate with an infinite row of collinear cracks. lsida's solution yields values of F accurate to less than 0.1% for>-. !i: 0.9. ? a = '3 = 00; the other extreme case of a center-cracked plate with clamped edges. lsida's solution yields values varying from F = 1.0 at>-. ... 0 to F = 0.6 >-. = 0.9. These results are clearly wrong. Infinitely stiff flanges do not deform and the stress applied at the remote end of the plate cannot be transferred along the plate to the cracked section. Theoretically, the SIF for this case should be zero. lsida's solution can not be applied to a two-tip cracked I-beam for the following reasons: ? In his analysis, the plate (which is similar to the web of an I-beam) is 12 under remote axial tension but the stiffeners (flanges) are not, whereas both the web and flanges of an I-beam are stressed. ? lsida's solution is for a central crack, a symmetric problem. Two-tip cracks in the web of an I-beam are always eccentric, making the geometry non-symmetric. 1.3.3 Two-tip Cracked I-Beam Chen and Albrecht (1994) calculated SIFs for central and eccentric cracks in a finite-width plate using the WFM (Figure 1.7). Weight functions are powerful in the sense that knowing the solution for one loading condition enables one to determine the solutions for the same cracked body under any other loading (Bueckner 1970, Rice 1972, and Wu and Carlsson 1983). Weight Function for Central Crack SIFs can be calculated from the weight function m(a, x) and the stress distribution o(x) in the crack-free body along the line of the prospective crack: (1.18) m (a, x) = -5._ avo(a, x) (1.19) K aa 0 where E = Young's modulus K0 = SIF for arbitrary reference loading 13 v0 = COD for reference loading To obtain the weight function from Equation 1.19, both the SIF and the corresponding COD are needed for the reference loading. Reference Loading for Central Crack For a central crack in a finite-width plate, Chen and Albrecht chose Tada's (1973) solution for axial tension as the reference. This solution is given by: K, = o,friar,( ;) (1.20) with (1.21) It is accurate to 0.1 % for any crack length. With the reference SIF determined, the accuracy of the weight function depends on the accuracy of COD for the reference loading. Wu (1984) assumed an elliptical COD profile of a center- cracked plate under axial tension: (1.22) where and f (-a) = _Ko_(a_) 0 w oo./na (1.23) Equations 1.22 and 1.23 give good results for a/V'-/ ~ 0.7. To obtain a more accurate weight function that is valid over a wider range of crack lengths, Chen added a second term to the expression for the COD profile: 14 v (a,x) = ~Ert2 f (_!_) ?a2-x2 + G(a/W)(a2 -x2)312] (1.24) o o W a2 where G is a function of f0 (1 .25) (1.26) Inserting Equations 1.20 and 1.24 through 1.26 into Equation 1.19 yields a closed-form weight function for a finite-width plate with a central crack. Chen and Albrecht applied his weight function to three known cases (single concentrated load, partial uniform pressure acting on arbitrary part of crack, and central crack in a rotating circular disk) and obtained good results. Weight Functions for Eccentric Crack For an eccentrically cracked plate, shown in Figure 1. 7 (b ), the weight functions for each tip are (1.27) (1 .28) where K/ = SIF at crack tip A under reference loading K 8 0 = SIF at crack tip B under reference loading 15 v = COD for reference loading 0 The SIFs are then given by: (1.29) crack surface and (1 .30) crack surface Self-consistency Conditions Chen and Albrecht developed self-consistency conditions for determining the accuracy of SIFs and hence weight functions for an eccentric crack. For virtual crack extensions, the energy balance condition requires that aA (K/)2 f daA = ao VO dx (1.31) E f - aB crack surface aB (K/)2 f da 8 = f ao VO dx E ( 1. 32) -aA crack surface and (1.33) 0 crack surface where aA and a8 are positions of crack tips A and B, and a is the crack length as shown in Figure 1.7 (b). Equation 1.31 was obtained by assuming crack tip A virtually extends by ~aA while crack tip B does not extend . Similarly, Equation 16 1.32 was obtained by assuming crack tip B virtually extends by 5a8 while crack tip A does not extend. Equation 1.33 was obtained by assuming both crack tips A and B extend by equal amounts, 5aA = 5a8 = 5a, without changing the eccentricity 5e = 0. The above three equations are not independent because the right sides of the equations are the same. Let (1.34) (1.35) ( 1. 36) Two so-called self-consistency conditions were then obtained by equating 1.31 to 1.33 and 1.32 to 1.33: QA = Q and Q B =Q (1.37) The close the ratios of QAIQ and QafQ are to unity, the more accurate is the weight function . Reference Loading for Eccentric Crack Chen and Albrecht chose lsida's (1965) solution for an eccentric crack in a finite-width plate under axial tension as the reference, which is given by the following equations: KoA = Oo ,/na foA (1.38) KoB = OoVTTafoB 17 where 19 foA =: 1 + L Ci (8) >,/ i = 2 (1.39) 19 8 f == 1 + L (-1)iC/8)t./ 0 i = 2 and e == e/Wand ).. == al(W-e) are the normalized eccentricity and crack length respectively, with e being the eccentricity and W being the half width of the plate. The numerical coefficients Ci (e) are given in tabular form for different values of normalized eccentricity. To obtain a closed-form weight function , Chen and Albrecht fitted the tabulated values with a 10th order polynomial: 10 C;(e) = L vijei (1 .40) j 0 The fitting error is less than 1% for C2 to C14 , and less than 2.5% for C15 to C19? The self-consistency conditions, Equation 1.37, were checked and the error was found to be less than 2% (Chen and Albrecht 1994). To obtain the weight functions for the two crack tips, a reasonable COD profile is needed. Chen and Albrecht assumed an expression for the COD of an eccentric crack under axial tension that is similar to Equation 1.24: (1.41) where for crack tips A and B located at aA and a8 from the origin (Figure 1.7): 18 a -1 = (a A - a B ) 2 e 1 = -(a A + a B ) 2 (1.42) rA = BA - X 's=x-as In Equation 1.41: KB and fB = __o_ 0 (1.43) and G is obtained by satisfying the energy balance condition of Equation 1.33. To further check the accuracy of the developed weight functions, Chen J and Albrecht (1994) applied them to solve several special problems. ? Central crack in an infinite plate subjected to a pair of symmetric point loads; The result is the same as the theoretical one. ? Central crack in a finite-width plate subjected to bending; The WFM solution is much closer to Benthem's (1972) result than to that of lsida (1956). ? Eccentric crack in a plate under remote tension; this is the reference loading case from which the weight functions were derived. As expected, the results compared well with the error being less than 1% for a!W ~ 0.9. SIFs for Two-tip Cracked I-beam Chen and Albrecht used the weight function for the finite-width plate to 19 calculate SIFs for a two-tip cracked I-beam, as shown in Figure 1.8 (if not specifically noted, the two-tip crack is on the web). The underlying idea is to apply on the plate the stress distribution o(x) that is acting on the web of an 1- beam. For example, the stress is uniformly distributed for axial loading: O=I._= T (1.44) A Aw+ 2A, where Tis the axial force and A is the cross-sectional area of the I-beam. For the case of pure bending, the stress is linearly distributed: a= My (1.45) I where Mis the applied moment, y is the distance from the neutral axis, and / is the moment inertia of the I-beam. Chen and Albrecht calculated the SIFs for a two-tip cracked I-beam by simply multiplying the single plate solution with a correction factor of Aw'A for axial tension, or lw'I for pure bending. In so doing, the weight function becomes an approximation because the geometries of the plate and I-beam are different. Although the stresses are the same, the web of an I-beam is constrained by the flanges along the junction line while the single plate is not constrained. As crack length increases and the crack tips move closer to the flange, the weight function becomes increasingly inaccurate. 1.3.4 Three-Tip Cracked I-Beam A solution for a three-tip cracked I-beam (Figure 1.9) was first developed 20 by Chen (1992). The COD expression and the SIFs were determined with the energy release rate method. The three-tip cracked I-beam was modeled as an edge-cracked web plate combined with a flange plate having a central crack or an eccentric crack, as shown in Figure 1.10. COD Assumptions for Central Three-tip Crack Chen made some several assumptions about the COD of the central three-tip crack: v = v (x, y, ax, ay) (1.46) where the coordinates x, y and crack lengths ax, ay are defined in Figure 1.10 (a). ? The COD vanishes at the flange crack tips: v(0, ? ay, ax, ay) = 0 ( 1.4 7) and at the web crack tip: v( ax, 0, ax, ay) = 0 (1.48) ? If any one of the crack lengths is zero, the crack does not open at the junction point where the web and the flange center lines intersect. This is a special case of the three-tip crack. If the web or flange crack length is zero, the COD at the junction point closes as well the entire crack length: v(x, y, 0, ay) = O (1.49) v(x, Y, ax, 0) = 0 (1.50) ? The COD profile of the flange crack is similar to that of a center-cracked plate of the same width and crack length as the flange; the COD profile of the web crack is similar to that of an edge-cracked plate of the same 21 width and crack length as the web. ? Web and flange cracks interact only through the displacement at the junction point. ? A long web (flange) crack has a small effect on the COD of the flange (web) crack. ? For any point along the web and flange cracks, the COD across the thickness of the element is the same as the COD at mid-thickness. Based on the assumption cited above, Chen chose the following expression for the COD of a three-tip crack subjected to an opening stress a(x, y) applied on the crack faces: V2(y, a ) 1 (1.51) v2(0, a ) E 1 where E= Young's modulus v1 = COD expression for edge crack in web v2 = COD expression for central or eccentric crack in flange v0 = normalized displacement at the junction 0 0 = reference stress Reference Stress The reference stress 0 0 is defined differently for the cases of axial load and pure bending. When the I-beam is under axial tension, the value of the uniformly distributed stress across the web and flanges is taken as the 22 reference stress: o(x, O) = o" (O:f.X:f.dw) b b (1.52) o(0,y) = 0 (--':f.ys-') 0 2 2 When the I-beam is under pure bending, the stresses over the web and flanges are given by: o(x, 0) = o (1 - 2~) 0 d w (1.53) b, b, o(0,y) = 0 (--sys-) 0 2 2 and the reference stress 0 0 here is taken as the value of the bending stress at the flange-web junction. CODs of Web and Flange Cracks As assumed by Chen, the COD profiles of the web and flange cracks are equal to those of the corresponding 2-D edged-cracked and center-cracked plate respectively. If the stress a(x, y) applied on the crack faces is uniformly or linearly distributed, 2-D COD expressions take the following forms for the flange crack ( IYI s By): v,(y,a,) i a,,~ o,[1 -(:Jr (1.54) and the web crack (0 s x s Bx) (1.55) where the non-dimensional coefficients C;(B/dw) and O;(B/bJ can be obtained from two-dimensional analysis. 23 In Chen's analysis, COD of the flange crack always takes the form of Equation 1.24 which is a special case of the Equation 1.54 with m = 2. It makes no difference whether the I-beam is under axial tension or pure bending. The bottom flange in an I-beam is subjected to axial tension. For the web crack, Chen used Petroski and Achenbach's (1978) edge crack solution: (1.56) where the normalized COD v1 is given by: (1.57) If the I-beam is under axial tension, coefficients C1 and C2 become: c1 : 2 /2 fXO (1.58) C2 - /G2x = SIT -~f 2 X 312 XO where fxo and x are defined as: KXO f XO [~d ) = w ooJnax 3 (1.59) 0.752 + 2.02( ::l + 0.37 1 - sin -::-: 2d na =-- -----------~ _w tan-x na nax 2dw cos--x 2dw and 24 (1.60) If the I-beam is under pure bending, the coefficients C1 and C2 are given by: c1 : 2./ifKO c2 = GK == [snK -~,KO( 1 - i~11 1 (1.61) {2. 2 3{2. 5 dw 1 - i~ 7 dw where cJ>x is defined as in Equation 1.60, and fxo given by KKO 00?TT8K na 4 0.923 +0.199 1 -sin-K (1.62) == _____ __.__ _____2_d_w; . ..i_ na cos-K 2dw Normalized Junction Point COD The only unknown in Equation 1.51 is the normalized COD at junction point, v0 , which is calculated with the energy release rate method. The crack closure work is given by the potential energy n: (1.63) crack surface or, for a three-tip cracked I-beam: 25 (1.64) Normalizing the integrals by the web and flange areas yields 02 n = -~E v (H +H) = -- a!v H (1.65) O X y E o where (1.66) (1.67) = A, Jo (O,y) v,(Y, a,) d ( 1..J = A, Ry 0 0 v1(0, ay) W 0 Using Irwin's method, the energy release rate is K2 X an Gx = - = E aAx (1.68) K2 an 2Gy = 2-1.. = E aAr here Ax= ax fw and Ay = ayt, are the surface areas of the web and flange cracks respectively. So that the change of the total energy release rate is then obtained by: 26 K2 K2 dG = _x dA + _Y 2dA = -d n (1.69) E X E y Inserting n from Equation 1.65 into 1.69 leads to: 1.a,( ::r +21,a,( :J = a~~H) (1 .70) where -3( -) -_a -3( -) +a2-( -) 3g x3a y3a (1.71) X y Furthermore, from the near-field equation, the COD vis related to K by: v z IBK[r (1.72) ~~E Equating the two COD expressions, Equations 1.72 and 1.51 , near the web crack tip and flange crack tip, the following equations are obtained: _Kx = ~-C fa_Vo = 'na f _Vo = ~K -oV 0 8 1vax V"Clx XO 0 v1o v1o 0 o v1o (1.73) where (1.74) is the normalized crack mouth COD of single web crack, and (1.75) is the normalized crack mouth COD of single flange crack. Equation 1.73 also shows that the v scales SIFs also. 0 Inserting Equation 1. 73 into 1.7 0, and making some necessary 27 numerical assumptions for the integral calculation yields the normalized COD v0 H v10 v20 Vo:::::----- (1.76) H" v10 + Hy v20 It is clear from the above equations that the accuracy of determining the normalized COD at the junction point v and therefore the SIF depends solely 0 on the accuracy of the assumed three-tip crack opening profile expression. Eccentric Three-tip Crack For the eccentric three-tip crack shown in Figure 1.1 Ob, Chen adopted an approach similar to that for the central three-tip crack. In this case, the I- beam was modeled as an edge-cracked web plate combined with an eccentrically cracked flange plate. The two are joined, having the same COD, at the junction point. The COD expressions are given by Equation 1. 51 for the three-tip cracked I-beam, Equations 1.56 through 1.62 for the edge crack in the web under tension and bending, and Equation 1.41 for the eccentric crack in the flange under tension. Shortcomings The COD expression for a three-tip cracked I-beam, Equation 1.51, is mainly based on the assumptions that the edge crack in the web and central crack in the flange take the same profiles as 2-D cracks, and the cracked web and flange are joined and interact only at the junction point. Those two assumptions are arguable. 28 In a real I-beam, the cracked web plate and flange plate are not only joined at the junction point but are also stitched along the common junction line. Single cracked webs and flanges deform independently in their own plane. But when they are joined to form an I-beam, the displacements along the junction line should be compatible; that is, the displacement at every point along the centerline of the flange should equal the displacement at the matching point along the edge of the web. If the I-beam is not cracked, the deformations of the flange and web along the junction line under applied loading (axial tension and pure bending) are the same, and no longitudinal shear is transferred between the web and flange. But when the I-beam has a three-tip crack, the flange and web deformations are disturbed along the junction line, inducing longitudinal shear between the web and the flanges in the vicinity of the cracked section. If the displacements along the center line of the flange are larger than those of l the corresponding points along the web edge, the web tends to close the flange crack while the flange tends to open the web crack. Conversely, if the displacements along the center line of the flange are smaller than those of the corresponding points along the web edge, the web tends to open the flange crack while the flange tends to close the web crack. The interaction forces between the web and flange will therefore alter the COD and SIFs of the web and flange cracks relative to those of the corresponding single plate solutions. While web and flange interact along the entire junction line, the interaction is greatest at the junction point and rapidly diminishes with distance 29 from the junction point, alter the COD profiles of the edge-cracked web plate and center-cracked flange plate under axial tension or pure bending. An interesting special case is worth noting. With varying crack lengths in web and flange, the interaction may change from the flange pulling open the web crack to the flange restraining the opening of the web crack. This change must be physically continuous as a function of relative crack lengths. So the two behaviors are separated by a pair of matching web and flange crack lengths for which there is no interaction. In this case, called no-interaction, the cracked web and flange can be parted freely, without altering significant the displacement at the junction point. The assumption here is that the junction line displacements mismatching away from the junction point have a much lesser effect and may be neglected. 1.4 Objective Stress intensity factors are needed in fatigue and fracture analysis of steel bridge girders. But no satisfactory SIF solutions exist for two-tip web cracks and three-tip web and flange cracks. Therefore, to develop SIF solutions for I-beams is the foremost objective of the present study. SIFs will be determined from finite element analysis of I-beams with different flange-web area ratios, crack eccentricities, web crack lengths, and flange crack lengths. 30 The numerical results will be fitted with equations suitable for ready use by engineers. 31 Chapter 2: Finite Element Models of Cracked I-Beam 2.1 Introduction I-beams with two-tip and three-tip cracks are difficult to model. The interaction forces between the cracked web and flanges invalidate single-plate solutions. Accurate closed-form solutions of the SIF are not available in the literature. After careful evaluation of several options for calculating SIFs, the finite element method was chosen for its powerful ability to treat complex geometric configurations and boundary conditions. The present chapter discusses the modeling of cracked I-beams. Finite elements and meshes are selected, methods of extracting SIFs are examined, and a simple 2-D method of modeling a cracked I-beam is evaluated. FEAs throughout this study were performed with the ABAQUS program, a general finite element code. 2.2 Modeling of Cracked I-beam Rolled and welded I-beams are treated in this study as a combination of three plates: one web and two flanges, each represented by its mid-plane, width, and thickness. They are joined along the junction lines defined as the intersection of the web and flange mid-planes shown in Figure 2.1. 32 The web and flanges are modeled in 3-D with eight-node shell elements -ABAQUS designation S8R5 (ABAQUS/Standard User's Manual 1993). At most nodes, there are five active degrees of freedom: three displacements and two in-surface rotations. But when the node is part of a multi-point constraint (MPG), such as the nodes along the junction lines, the sixth DOF - the out-of- surface rotation - is activated. Figure 2.2 shows typical mesh for a three-tip cracked I-beam. The fillets at the transition between the web and flanges are neglected. This is compensated by counting twice the area where web and flanges overlap as shown in Figure 2.1. The errors from the two approximations tend to cancel out. 2.3 Crack Tip Elements A major problem in applying FEM to fracture analysis arises from the square-root singularity of stresses and strains around the crack tip. Several crack tip elements have been developed to overcome this difficulty. Among them, the most common are conventional elements with a special shape function that produces a 1/{r singularity, the quarter-point element, and the hybrid approach. All have strong supporters and have been applied with success to a variety of problems. However, no method has been established as optimal for all problems. 33 The quarter-point element is the most widely used because it describes the desired singularity, is simple to program, and has other advantages such as convergence and continuity. For these reasons it was adopted in the present study. Quarter-point Singularity Element Barsoum (1974) and Henshell and Shaw (1975) independently showed that placing two mid-side nodes of an eight-node isoparametric quadrilateral element at the quarter point yields a 11{r singularity (Figure 2.3a). The crack tip is located at node 1. The element exhibits 1 t{r singularity along the two sides containing the quarter-point nodes but not along other rays emanating from the crack tip. Barsoum (197 4) found that the six-node isoparametric triangular element with quarter-point nodes on two sides also has 1t {r singularity (Figure 2.3b). This triangular element better represents the stress field because the 11{r singularity exists along all rays emanating from the crack tip. It turns out that the six-node triangular element can be obtained by degenerating the eight-node quadrilateral element as shown in Figure 2.4; nodes 1, 8 and 4 are collapsed into one node that is placed at the crack tip; and nodes 5 and 7 are moved from mid-sides to quarter points. According to Freese and Tracy (1976), this degenerated quadrilateral element yields the same stresses and strains as the triangular element (Figure 2.3b). Since this element comes from an isoparametric element, even in its singular form, it still satisfies 34 the requirements for convergence and passes the patch test. It also possesses rigid body motion, constant strain modes, inter-element compatibility, and continuity of displacements. From here on, the two elements shown in Figure 2.3 are simply called the quarter-point quadrilateral and triangular elements, while the one in Figure 2.4 is called the degenerated quarter-point element. ABAQUS' Singularity Element In this study, the degenerated quarter-point elements come from the ABAQUS library: - CPS8R for plane stress problems and S8R5 for shells. Command *SINGULAR creates the mesh in the region surrounding the crack tip, with quarter-point spacing for the first element and progressive increase in element size with distance from the crack tip. Figure 2.5 shows how element size is increased along any line emanating from the crack tip. 2.4 Mesh Pattern Around Crack Tip An important problem in modeling a cracked geometry is to choose suitable mesh density, size of crack tip element, and element aspect ratio in a region surrounding the crack tip. The cracked web or flange was divided into a so-called inner region at the crack tip surrounded by an outer region. Figure 2.6 shows the shaded inner region and nonshaded outer region for crack lengths of O < a/W < 0.5, a/W = 35 0.5, and 0.5 < a/W < 1.0. The vertical line on the left of each sub-figure represents the centerline of a center-cracked flange, or the edge of an edge- J cracked web, or the centerline of an eccentric crack in the web. All three are symmetric about the line of crack extension, meaning that Figure 2.6 shows only the upper half of the cracked web or flange. Using standard terminology, the width is defined as W = b,12 for the center-cracked flange, W = di for the edge-cracked web, and W = d112 - e for the eccentrically cracked web. The inner region consists of two squares of sides/ placed symmetrically about the vertical line through the crack tip, with the left square trailing the crack tip and the right square leading it. The characteristic length/ is: a (0 < a/W ~ 0.5) I - (2.1) { W-a (0.5 < a/W < 1) Parameter m defines the number of elements around a concentric square in the mesh, and parameter n defines the number of concentric squares (Figure 2.7). The sides of all elements lying on the perimeter of a concentric square have equal lengths. The sides of all elements on a radial line emanating from the crack tip increase in length as specified in the ABAQUS command *SINGULAR. Typical meshes in the outer region are shown in Figure 2.2. To determine how mesh density affects the accuracy of calculating SIFs, benchmark analyses were performed for three geometries for which reliable solutions are available in the literature: 36 ? Center-cracked finite-width plate under tension ? Edge-cracked finite-width plate under tension ? Edge-cracked finite-width plate under bending The calculations were repeated for the ten combinations of parameters m and n shown in Figure 2.7. Mesh patterns m x n = 2 x 16 and m x n =4 x 2 were not included because they resulted in large element aspect ratios. The results, presented in Appendix A, lead to the following conclusions: ? Increasing the parameter n improves the accuracy of SIFs more than increasing the parameter m. ? When n = 8, the mesh is fine enough to give convergent results. Since a fine mesh is needed when the crack tip approaches the web- flange junction, it was decided to use the mesh pattern m x n = 3 x 16 for all analyses of cracked I-beams (center-right in Figure 2.7). 2.5 Extracting SIFs from FEA Output FEA yields nodal displacements, nodal forces, and stresses and strains at selected positions - but not SIFs. Methods of extracting SIFs are classified as being displacement-based, stress-based, and energy-based. 2.5.1 Displacement-Based Methods The first method involves examining the crack opening displacements 37 (COD) of the crack tip elements. The COD around the crack tip is given by: u = o + o(r) v = (2.2) K ."...:.!.~ ' + o(r) 2G 2TT where K =( 3 - v)/(1 + v) for plane stress, K =3 - 4v for plane strain, v is Poisson's ratio, G is the shear modulus, K, is the SIF, and r is the distance from the crack tip. The so-called apparent SIF K,* can then be calculated from: K ? = v ~~ 21T (2.3) K + 1 r The three common methods of calculating SIFs from COD are discussed in the following. Nonlinear Extrapolation Tracy (1977) proposed to calculate SIFs from the displacements of nodes Band C of the quarter-point element trailing the crack tip, a= TT in Figure 2.8. According to Barsoum (1976), the COD of the quarter-point element is given by: (2.4) where v8 and Ve are the quarter-point and corner node displacements at r8 = L/4 and re = L respectively, and L is the length of the trailing element along the crack surface shown in Figure 2.8. Inserting Equation 2.4 into 2.3 gives the apparent SIF: K . -_ _2.G...,/.2,_I_T ( -2v-c --4v-8H - + -4v8 ---Ve) (2.5) (K + 1) {f. L {[. which amounts to nonlinear extrapolation of the apparent SIFs within the crack 38 tip element: K ? = A{r + B (2.6) The SIF at the crack tip is then obtained from Equation 2.5 as r ... O: K = 2Gt./2.n (4v8 - Ve) (2.7) (K + 1) /f. Equation 2.7 can also be obtained by equating the coefficients of the {r terms in Equations 2.2 and 2.4. Linear Extrapolation The linear extrapolation technique (Chan, et al. 1970) was applied to the quarter-point singularity element by Chen and Kuang (1992). The SIF was extrapolated linearly: K1" = Ar+ B (2.8) Inserting the apparent SIFs at the quarter point B and corner point C yields K. = 2Gt./2.n( 4 Ve-2VB !_ +..! Bva-Ve) (2.9) K+1 3 fl. L 3 /f. The SIF at the crack tip, which is the intercept of the linear equation, is therefore: K = 2Gt./2.n (BvB - Ve) (2.10) K + 1 3/L. Inserting Equation 2.9 into Equation 2.2 gives the corresponding COD: 3 2 v = -1( 8v8 -vc)H !_+ -4( ve-2v8 ) ( !_ ) ' (2.11) 3 L 3 L Quarter-point Displacement In this method, the COD at the quarter point (r8 = L/4) trailing the crack 39 tip is inserted in Equation 2.3, yielding directly the value of the SIF: K = 2Gy'2n 2va (K + 1) ./L. (2.12) Comparison Equations 2. 7, 2.10 and 2.12 are all based on the CO Os of the element trailing the crack tip. Shih, et al. 1976, Saouma and Schwemmer 1984, Yehia and Shephard 1985, Lim, et al. 1992, Chen and Kuang 1992, and Pang 1993 studied the relative accuracies of the three equations but reached no consistent conclusions. The three equations yield equal values of SIF only if: (2.13) To satisfy this condition, the crack tip element must be very small. For example, considering the case of an infinite plate with a central crack, the analytical solution for the COD profile is given by: V = (K + 1) 0 ? rf.2a - r) (2.14) 4G where a is the half crack length. For a crack tip element of size L, CO Os at the quarter point B and corner point C can be calculated as: v " (K + 1) a~ L (2a _ L) v " (K + 1 )a ?L(2a _ L) (2.15) 8 4G 4 4 c 4G Inserting these values in Equation 2.13 yields JL(2a - L) = 2 I~ (2a ~ ~) (2.16) ~ 4 4 Equation 2.16 results in L = 0. However, in FEA the size of the crack tip element cannot be zero. Inserting Equation 2.15 into Equations 2.7, 2.10 and 2.12 yields, for 40 nonlinear extrapolation: (2.17) linear extrapolation: K ? ~ ( 2~ a - ~ - ~ 2 - ~ ) o ?na (2.18) and quarter-point displacement: (2.19) Equations 2.17, 2.18, and 2.19 are normalized and plotted in Figure 2.9 as a function of the dimensionless element length L la at the crack tip. The exact solution is Kio ?rr a = 1 . As can be seen, linear extrapolation (Equation 2.18) gives the most accurate result and is largely insensitive to element length. Its error for L = 0.5a is about 0.2%. As the crack tip element becomes longer, the quarter-point displacement (Equation 2.19) increasingly underestimates SIF. Nonlinear extrapolation (Equation 2.17), the least accurate of the three, increasingly overestimates SIF. Detailed results of relative accuracy are given in appendix A for finite- width plates under tension and bending. 2.5.2 Stress-Based Methods SIFs can also be calculated from the normal stress in elements leading the crack tip, a = 0.0 in Figure 2.8. From LEFM, this stress in given by: 41 K, a= -- (2.20) Y ./2rrr resulting in the following equation for the SIF: K1 = Lim 1,.0 aY ?2rr, (2.21) Since Equation 2.21 contains only the leading asymptotic term, the result is an apparent SIF at a discrete distance ahead of the crack tip. SIFs are then extrapolated to the crack tip using linear regression. Since in FEA displacements are more accurate than stresses, K1 should be calculated from displacements of nodes trailing the crack rather than stresses in nodes leading the crack. 2.5.3 Energy-Based Methods Energy-based computation of SIF is based on the energy release rate, J- integral, and stiffness derivative. These three approaches are discussed next. Energy Release Rate Irwin (1957) showed that the SIF is related to the strain energy release rate G for a mode I crack: G = (1 -v2) K 2 (for plane strain) E (2.22) G = _!_ K 2 (for plane stress) E The strain energy released at the crack tip per unit area of newly formed crack surfaces is: 42 ] dU D..U G = ?-:::: (2.23) dA M where U is the elastic strain energy of the body containing the crack and A is the crack surface. The plus and minus signs refer to constant-load and constant-displacement conditions respectively. In FEA the strain energy release rate is calculated for two slightly different crack lengths, the difference being of order D..a/a = 1/50 of the original crack length. For each cracked body, the stored strain energy is calculated from the sum of the strain energies of all elements. The SIF can then be determined from Equations 2.22 and 2.23. The major weakness of this method is that two FEAs are needed. J-lntegral Rice (1968) defined the J-integral as: J = ( [wn 1 -T aaux '] ds (2.24) Jr / where r is any path beginning at the lower crack surface, encircling the crack tip, and ending at the upper surface. The strain energy density is given by W = ? o .. e .. for linear elasticity. n1 is the x-component of the outward unit normal to I) I) r T- = a -n is the traction vector, and u-1 is the displacement vector. The J- ' I I} J integral, one of seven conservation energy integrals in continuum mechanics, is path independent for any elastic material (Rice 1968) . For a linear elastic material, the J-integral and the energy release rate are equivalent: 43 J = G (2.25) To calculate the J-integral with FEA, stresses, strains and displacements are needed along a path surrounding the crack tip for use in Equation 2.24. But the components comprising the integrand, as determined from FEA, are likely to be inaccurate near the crack tip. Fortunately, path independence allows the contour to be chosen away from the crack tip. With the J-integral value determined, the SIF is then obtained from Equations 2.25 and 2.22: K = ./EJ (2.26) By using this method, only one FEA is needed per crack length. Being an energy approach, the J-integral method has the advantage of not requiring an elaborate representation of the crack tip stress and strain fields. The J value is based on the strain energy of the mesh rather than individual local values of stresses, so that the result is still accurate even when the mesh is coarse. Stiffness Derivative A third way of computing SIFs from energy is the stiffness derivative method (Parks 1974), sometimes called the virtual crack extension method; it is a variant of the energy release rate method. The change in strain energy liU is associated with elements surrounding the crack tip and is evaluated by displacing the nodal points laying on a contour around the crack tip by an incremental distance, tiL. This method, as employed widely today, is efficient and accurate. 44 ? Parks (1974) and Banks-Sills and Sherman (1992) investigated the equivalency of the J-integral and stiffness derivative methods. The contour J- integral of Equation 2.24 can also be converted into an area integral through Green's divergence theorem. The J-integral (contour or area) and stiffness derivative methods are theoretically equivalent. In reality, the equivalence depends on element type and computation method when the contour and area J-integrals are obtained from FEA. Parks (1974) showed that the contour J-integral method is the same as the stiffness derivative method if constant-strain triangular elements are used. Banks-Sills and Sherman (1992) pointed out that for triangular and four-node isoparametric elements, stiffness derivative, area J-integral and contour J- integral are equivalent. For eight-node and eight-node quarter-point isoparametric elements, with three-point Gaussian integration, only the area J- integral method is equivalent to the stiffness derivative method. However, if the two-point Gaussian (reduced) integration rule is used, the three results are equivalent. Saouma and Schwemmer (1984) recommended the two-point Gaussian integration rule for fracture mechanics analysis. The J-integral method is used in all cracked I-beam analyses performed in this study because it yields accurate results and can be obtained directly from the ABAQUS output through its special procedure * J-INTEGRAL. 45 2.6 Two-dimensional Model The cracked I-beams were modeled in section 2.2 with shell elements in the actual 3-D configuration of web and flanges. Since the model is three- dimensional and the shell elements have at /east five DOFs per node, the FEA is time consuming. 2.6.1 2-D Approach This section explores the possibility of modeling a aacked I-beam in two dimensions. Figure 2.10 shows the schematic of the 2-D I-beam model, with the mid-planes of the flanges rotated by 90 ? onto a common plane with the web. The edges of the web mid-planes are joined to the centerlines of the flange mid-planes. This reduces the problem from 3-D to 2-D, making it possible to model the I-beam with eight-node plane-stress elements that have only two DOFs per node and reduced integration - ABAQUS designation CPSBR. In this method the web and flanges should not be fully joined. Only the y- direction displacements of corresponding nodes along the junction line should be equated, thus allowing web and flanges to deform freely in the x-direction. If the x-direction displacements were also equated, the web and flanges would be over constrained, especially for long cracks. Theoretically, the flanges could be at any angle. The only requirement is 46 that corresponding nodes along the junction line of the web and flanges have equal y-direction displacements. Practically, the flanges must rotate 90 ? because the plane-stress element has only two DOFs; therefore, the flanges and web must lie in the same plane. Because the problem is now planar and the plane-stress element has only two DOFs per node, meshes are easily generated and software runs faster than in the 3-D I-beam model. 2.6.2 Numerical Results and Discussion Parametric Analyses Two-tip and three-tip cracked I-beams were analyzed for the following combinations of models, W-shapes, and loading: ? Models: 2-D and 3-0 ? W-shapes: W40x149, W40x199, and W18x97; with cross-sectional area ratios of flanges and web~= 2A,IAw = 0.83, 1.37, and 2.05 respectively ? Loading: tension and bending Besides the variables listed above, the two-tip cracked I-beams were analyzed for the following 27 combinations of: ? Crack eccentricity: e = e/(d/2) = 0.0, 0.3, 0. 7 ? Web crack length: >aw= awl(d/2 - e) = 0.1 to 0.9, in steps of 0.1 The three-tip cracked I-beams were analyzed for the following 18 combinations 47 of: ? Web crack length: Aw = awl~= 0.1 and 0.5 ? Flange crack length: A,= a,l(b,12) = 0. 1 to 0.9, in steps of 0. 1 Altogether, 12(27+18) = 540 FEAs were performed to determine whether a simple 2-D model could replace a complex 3-D model without significant loss of accuracy in calculating SlFs. A two-tip crack in an I-beam web is defined by the non-dimensional eccentricity of the crack midpoint about the major axis of the I-beam, e = el(~ 12), and the non-dimensional crack length, Aw= aw l(d/2 - e). All dimensions are shown in Figure 1.8. Defining the coordinates as in Figure 2.1, the upper and .. . j lower crack tips are located at: d. x = (e - 1 + e A ) ..J. upper "w w 2 (2.27) d x = (e + 1 - e A ) -21 . lower "w w For a three-tip cracked I-beam, the crack lengths are given by the two non-dimensional parameters Aw= awlt:4 for the web crack and A,= a,l(b,12) for the flange crack. All dimensions are shown in Figure 1.9. With these definitions, the web and flange crack tips are located at: d x web = (1 - 2 Aw ) -1. 2 (2.28) b, Z ffange = ? A, 2 SlFs computed with Equation 2.26 are compared next in terms of their ratio K IK where K and K are the SlFs from the 2-D and 3-D models. ' 2D 3D, 2D 3D 48 n Two-tip Cracked I-beams Figures 2.11 and 2.12 compare SIFs for the upper and lower crack tips of an I-beam under tension and Figure 2.13 compares the SIFs for the lower crack tip of an I-beam under bending. Results for the upper crack tip of an I- beam under bending are not presented because the crack tip is near the neutral axes of the section and the SIF values are small. The trends are very similar in all three figures. Specifically: ? Relative to the 3-D model, the 2-D model consistently overestimates the SIFs. The error is always on the safe side. 2-D results exceed 3-D results by less than 1% for >-w ~ 0.6 in Figure 2.11 and >-w ~ 0.5 in Figures 2.12 and 2.13. The difference is largest at Aw= 0.9, with values of 5% for the upper tip and 7% for the lower tip. ? For same eccentricity e and crack length >-w, the difference increases with cross-sectional area ratio of flanges to web. In another words, as ~ increases, the 2-D model underestimates the beneficial constraint provided by a heavier flange. ? For same area ratio ~ and crack length >-w, the difference increases with eccentricity e. That is, the closer a tip of a constant length crack gets to the flange (Equation 2.27), the more 2-D analysis underestimates the beneficial constraint provided by the flange. Judging by these conclusions, one might expect that SIFs calculated With the 2-D model would approach those calculated with the 3-D model if both 49 .a the x- and y-direction displacements along the junction lines of web and flanges were equated - not just they-direction displacements. To check this hypothesis, the following I-beams were reanalyzed with junction-line displacements equated in both directions: ? W-shapes: W40x149 (~ =0 .83), W18x97 (~ =2 .05) ? Loading: tension and bending ? Eccentricity: e = 0.0 and 0. 7 ? Web crack length: Aw= 0.1 and 0.9 It was found that a fully joined 2-D model always yields smaller SIF values than the 3-D model. The error is on the unsafe side. The maximum difference is about 20% for the W18X97 shape with e = 0. 7 and Aw= 0.9. Clearly, equating also the x-direction displacements gives less accurate and unsafe SIF values. Three-tip Cracked I-beams SlFs for the web and flange crack tips are compared in Figures 2.14 and 2-15 for I-beams under tension and in Figures 2.16 and 2.17 for I-beams under bending. Good agreement between 2-D and 3-D models was found: ? For the web crack tip, the SIFs calculated with 2-D and 3-D models differ by at most ?1% (Figures 2.14 and 2.16). The only exception is I-beam W18x97 with Aw= 0.5 and >.., = 0.1 for which the maximum difference is 1.5% (Figure 2.16). ? For the flange crack tip, the difference is at most ?1 % for flange cracks >.., 50 ~ 0.3 (Figures 2.15 and 2.17). The maximum difference is about 4% for I-beam W18x97 with a web crack (Aw= 0.5) significantly longer than the flange crack (A,= 0.1) (Figure 2.15). 2.6.J Benefits of 2-D Modeling Based on the results presented above, cracked I-beams can be conservatively modeled in 2-D, with only small losses in accuracy of determining SIFs. In a 2-D model, mesh preparation is simplified and FEA software runs faster. For example, it took only 5 minutes to execute a FEA of a three-tip ..., . cracked I-beam in 2-0, versus 30 minutes in 3-0. The small difference between the 2-0 and 3-0 analysis results strongly suggests that the interaction between the cracked web and flange is controlled mostly by the compatibility of y-direction displacements along the junction line. 51 Chapter 3: Parameters for Cracked I-beams Most cracks found in I-beams of steel bridges have either two tips or three tips as shown in Figures 1.8 and 1.9. Two-tip web cracks are always eccentrically and, therefore the SIFs have different values for the upper and lower crack tips. Three-tip cracks are usually, but not always symmetric. As a result, two SIFs are needed; one for the web crack tip, and the other for the two flange crack tips. 3.1 Two-tip Cracked I-Beam ,. I ~ ;, I ,, I jl The SIF for a two-tip crack in the web of an I-beam, shown in Figure 3.1, is expressed as: KA,B = ,A,B a /naw (3.1) Where f is the correction factor, aw is one-half of the web crack length, and the superscripts A and a represent the upper and lower crack tips, respectively. The reference stress a is defined as the remote uniform stress in an I-beam under tension and the stress at the flange-web junction in an I-beam under bending. The correction factors for each crack tip: 52 fA = fA(Aw, e, (3) ,a = fa(Aw, e, f3) (3.2) are functions of normalized web crack length: A = aw w (3.3) d/2 -e normalized eccentricity: e e = (3.4) d/2 and cross-sectional area ratio of flanges to web: /3 = 2 A, (3.5) Aw In the above equations, 0 = d - t, is the web depth between the upper and lower junction points, e is the eccentricity, and A, and Aw are the cross-sectional areas of the flange and web (Figure 3.1) . Parameters Aw and e describe the length and eccentricity of the crack. Parameter p accounts for the constraint imposed by the flanges on the Web crack. It is basically an axial rigidity. lsida (1973a) also used the area ratio to account for the interaction between two joined half planes of different th icknesses under tension, and when he analyzed the problem of a series of joined strips under tension, the same parameter was used. Cartwright and Miller (1975) analyzed two uniformly stressed infinite sheets, each containing a central crack. The two sheets bisect each other at a right angle. The cracks lie in the same plane and are centered about a common point on the junction line. 53 .. Again the interaction between the two intersecting cracked sheets was expressed in terms of their area ratio. The plates analyzed by lsida {1973a) and Cartwright and Miller {1975) are infinitely wide, and all joined parts are stressed. Therefore, only the axial rigidity needed to be modeled. lsida (1973) calculated the SIF for a central crack in a plate with edge stiffeners under axial stress applied only on the plate. Both the plate and stiffener widths were finite. He chose two parameters a and ~. where a is the ratio of flexural rigidities, El, and f3 is the ratio of axial rigidities, EA, of stiffeners and plate. While he accounted for the effect of bending rigidity, lsida pointed out that the axial rigidity affects the SIF most. The flexural rigidity is appreciable only when the crack tip moves close to the stiffeners. The two-tip cracked I-beam differs from the problems of lsida (1973a) and Cartwright and Miller {1975) in that the web and flange widths are finite. It also differs from lsida's {1973) problem in that both the web and flange plates are stressed. 3.2 Three-tip Cracked I-Beam The SIF for a symmetric three-tip cracked I-beam, shown in Figure 3.2, is expressed as: K w.f = f w.f a ? 11 awJ {3.6) Where f is the correction factor, and aw and a, are the web and flange crack 54 lengths. The subscripts and superscripts wand f represent the web and flange crack tips, respectively. As for the two-tip crack, the reference stress a is defined as the remote uniform stress for an I-beam under tension and the stress at the flange-web junction for an I-beam under bending. Here, the correction factors for the web and flange crack tips: fW=fW(A w, A, , ~A ) (3.7) t'=t'(Aw ' A, , ~A ) are functions of normalized web crack length: (3.8) normalized flange crack length: a, A=- (3.9) ' b 12 ' and cross-sectional area ratio of flanges and web: 2 A, p =- (3.10) Aw In the above equations, b, is the flange width. Parameters >.wand>., describe the Web and flange crack lengths (Figure 3.2). The web is under uniform or linear stress depending on the loading conditions. The flange is always under uniform stress. . area ratio of flanges and web. Parameter~. again, is the cross-sect 1ona 1 Ch . d I b ms used the parameter en (1992), in his analysis of three-tip cracke - ea ' 55 A, /Aw. Here the more common ratio of both flange areas divided by the web area is used, 13 = 2A,IAw. 3.3 Validation of Parameter p The parameters in the correction factor Equation 3.2 for the two-tip crack and Equation 3. 7 for the three-tip crack account for two major effects: (1) the Web crack length and eccentricity of the two-tip crack, and the web and flange crack lengths of the three-tip crack; and (2) cross-sectional dimensions of the Web and flanges. The validity of 13 as an independent parameter is checked numerically ,, by comparing correction factors for W-shapes that have significantly different geometries but equal 13 values. For example, Table 3.1 lists seven pairs of . - rolled W-shapes with equal 13 values. All were taken from the Manual of Steel Construction (Manual 1986). If the correction factors can be shown to be nearly equal for each pair of W-shapes, then 13 can be used as an independent Parameter. Finite element analyses were perfonned for the seven pairs of rolled W- Shapes identified by the footnote (a) in Table 3.1. These pairs were selected because: (1) their 13 values of 0.83 to 1.91 cover most of the range of 13 values for W-shapes listed in the Manual, and (2) the width-thickness ratios of the web and flanges satisfy the minimum requirements for using shell elements. Typical 56 finite element meshes for two-tip and three-tip cracked I-beams are shown in Figures 2.7 and 2.9. Parameter '3 is validated next for both two-tip and three-tip cracked ,_ beams under tension or bending. 3-3.1 Two-tip Crack under Tension and Bending Correction factors were calculated for two-tip cracked I-beams under tension, with a central crack in the web varying in length from Aw= awl(~ /2) = 0-1 to 0.9 in steps of 0.2. The material properties for both the web and flanges Were Young's modulus, E =2 00 GPa, and Poisson's ratio, v =0 .3. Figure 3.3 .?? I i compares results for each pair of W-shapes. The ordinate is the ratio of the ,j correction factors for shapes 1 and 2, t<1>tf '2' . As Figure 3.3 shows, the correction factors for two shapes of equal J3 values differ by less than 1% for crack lengths up to Aw = 2aJ0 ~ 0. 7 and less than 2% for crack lengths up to Aw< 0.9. The sole exception is shape 2 with J3 = 1-11 (W36X 170 in Table 3. 1) whose correction factor is 3. 5 % larger than its shape 1 counterpart (W40X192). Differences smaller than 3.5% would be Preferable. On the other hand, linear-elastic fracture mechanics would no longer be valid at crack lengths Aw= 0.9 since the net ligament would yield. Clearly, '3 is a valid parameter Another parameter, besides 13, that may affect the SIF is the depth-width 57 - ratio V = d i lb, , wh ere di i.s th ewe b depth between J? uncti?o n poi?n ts and b, is the flange width. In an analogy to the structure of a mammal, the parameter y is to the skeleton as the parameter S is to the flesh. Values of y for all selected W-shapes are listed in Table 3.1 . For each Pair having equal area ratios, y is always smaller for shape 1 than shape 2; meaning that, for same area ratio (3, shape 2 is relatively deeper and narrower than shape 1. A narrower and thicker flange (shape 2) constrains the web crack more than does a wider and thinner flange (shape 1). This explains why the ratio of correction factors t<1l!f<2J is greater than unity in Figure 3.3. Ideally, both parameters Sandy should be used to improve the accuracy of calculating SIFs. But y is about a linear function of f3 as Figure 3.4 shows for all shapes listed in the Manual. Each data point corresponds to one W-shape. The sections used in validating f3 are identified with solid symbols. Shapes 1 lie near the bottom of the bandwidth and shapes 2 near the middle. The variation in Y values accounts for the slight differences in correction factors for W-shapes With equal f3 values - less than 1% for crack lengths up to A. w = 2alq s: O. 7 (Figure 3.3). Since Y is approximately a linear function of (3, and the difference between the correction factors for shapes 1 and 2 is small, the W-shapes are characterized with the parameter f3 alone in the subsequent finite element calculations. T ? F' s 3 3 and 3.4 is the W40X192 he exception to the trend shown m igure ? 58 (shape 1) section with P = 1.11 and y = 2.11, which has a wider flange relative to the depth than is common in W-shapes used as flexural members for bridges. Because it is an outlier, the W40X 192 is omitted in subsequent calculations. Parameter p was also validated for the two-tip center-cracked I-beam under bending, but on a smaller scale than was done for I-beams in tension. Instead of all seven pairs of W-shapes, only the three pairs with P= 0.83, 1.37 and 1.91 were analyzed. As Figure 3.5 shows, the maximum difference between the correction factors for two shapes with equal ~ values is about 1% . 3-3.2 Three-tip Crack under Tension and Bending I t 'I' -':~j Interaction between the web and flanges of a three-tip cracked I-beam differs from that of a two-tip cracked I-beam. In a three-tip cracked I-beam, the cracks bisect the line joining the web and flange, thus complicating the interaction. Correction factors for both the web and flange cracks were calculated for the following combinations: flange-web area ratio, ~ = 0.83 and 1-91 (smalleSt and largest values); web crack length, Aw= awld1 = 0.1 and 0.5; and flange crack length, >., = 28, lb,= 0_1 to 0_9 in steps of 0.2. The results are shown in Figures 3.6 and 3. 7 for tension, and Figures 3.8 and 3.9 for bending. Differences between correction factors were found to be less than 2%, 59 - suggesting that parameter p characterizes cracked W-shapes well. 3.4 Summary and Conclusions Correction factors for two-tip and three-tip cracked I-beams were found to be functions of crack length, crack position, and area ratio of web and flanges. The validation of pas an independent parameter was checked for both two-tip and three-tip cracked I-beams under tension and bending. The following is concluded: ? Two different W-shapes of equal p values have nearly equal correction . factors, the maximum difference being less than 2%. Thus, parameter p ' accounts for the interaction between the web and flanges. ? The depth-width ratio of a W-shape, y = d/b,, is not a significant parameter in the calculation of S/Fs. It becomes noticeable only for long cracks. For all I-beams fisted in the Manual, f3 and y are related about finearly within a bandwidth. Since the w-shapes listed in the Manual can be grouped according to Parameter f3 alone, the number of FEAs in Chapter 5 can be reduced. Analyses are needed only for typical W-shapes, and fitted functions for correction factors become simpler. The Manual lists 297 W-shapes, of which about 170 are typically used as beams. To create a database of correction factors, it is not necessary to 60 -- analyze all W-shapes. Instead, in this study, nine W-shapes are retained for finite element analyses in Chapter 5. Besides the seven W-shape 1 sections used for f3 validation, two more were selected - W33X201 with p = 1.53 and VV18X97 with f3 = 2.05. All nine are identified with a superscript bin Table 3.1. The values of f3 = 0.83 to 2.05 for the nine W-shapes to be analyzed in Chapter 5 cover the range of p = O. 77 to 2.25 for all WB to W40 I-beams listed in the Manual (Figure 3.1 O). Such selected calculation range also covers part of welded girders. In civil engineering, the size of the top flange and the bottom flange of a welded girder may not be the same. For positive bending girders, AttfAw = 0.24 to 0.41 and Ab,!A w = 0.33 to 1.11 (Schilling, private communication), where At, is the top flange area, Abt is the bottom flange area, and Aw is the web area. For negative bending girders, A,,IAw = 0.50 to 1.51 and Ab,IAw = 0.61 to 1.71 (Schilling, Private communication). Parameter f3 for a welded girder is defined as P= 2Ab/Aw, thus f3 = 0.66 to 2.22 for positive bending girders and P = 1.22 to 3.42 for negative bending girders. Therefore, parameter Ps elected for finite element analysis covers 77% of positive bending girders and 42% of negative bending girders. However, for welded girders, to use the results of th is study, the relationship between skelton ratio y and cross-sectional area ratio Ps hould lie Within the fuzzy band as shown in Figure 3.4. 61 Chapter 4: Non-interacting Three-tip Cracks 4.1 Introduction A three-tip cracked I-beam is modeled as a combination of an edge- cracked web and a center-cracked bottom flange, with the top flange remaining uncracked. Starting from the junction point, the cracked web and flange are joined along the full length of the junction line. When separated from each 0ther, the cracked web and flange deform independently in their own planes. However, once joined to form an I-beam, corresponding points on the junction . line must displace by equal amounts. In other words, displacements of nodes f on the junction line common to both the web and flange must be compatible. To enforce displacement compatibility, pairs of equal and opposite interaction forces must be applied at each node on the junction line. The values of the interaction forces depend on the relative lengths of web and flange cracks. If the displacements along the flange centerline are larger than those of the corresponding nodes along the web edge, the flange magnifies the opening of the Web crack while the web restrains the opening of the flange crack. Conversely, if the displacements along the flange centerline are smaller than those of the corresponding points along the web, the flange restraints the OPening of the web crack while the web magnifies the opening of the flange crack ? s,?nc e th e process reverses depend,?ng on the relative crack lengths of 62 Web and flange, there should be pairs of matching crack lengths that induce compatible displacements. For each such pair, the interaction forces would be zero at the junction point and small - perhaps even negligible - at points along the junction line away from the junction point. This condition is hereafter referred to as one of non-interacting cracks. 4.2 Joined Infinite Plates The first problem analyzed in this chapter is that of a center-cracked infinite plate bisected perpendicularly with an edge-cracked semi-infinite plate as shown in Figure 4.1. The plates, joined along the centerline of the infinite plate and the edge of the semi-infinite plate, are under uniform tension. 4.2.1 Centerline Displacements The exact displacement solutions for a center-cracked infinite plate subject to uniform tension stress a, shown in Figure 4.2(a), can be found with the Musklishvili stress function method. In two-dimensional elaSticity, displacements u in the x direction and v in the y direction can be expressed in terms of a complex function ct>(z): 2?u = (K -1)Ref{f>(z)J-2ylm[cf>'(z)J +Ax (4.1) 2?v = (K+1)1m[cf>(z)J-2yRe[ct>'(z)J-AY Where z = x + iy, K = 3 - 4v for plane strain, and K = (3 - v)/(1 + v) for plane 63 stress. The material constants are the shear modulus ? = E/2(1 + v), Poisson's ratio v, and Young's modulus E. The boundary conditions for an infinite-width Plate With a central crack: a = Txy = 0, (y = 0, -a< x (z) = a lz z - a z - a z (4.3) 2 4 Inserting the constant A and stress function cf>(z) into Equation 4.1, and assuming plane stress, the displacement along the centerline, on x = 0, is obtained as: 2 2 . V = -a t ." Vy + ,(.:.1.:, __-_..v.!.).~- y_ :::-+. =2. :a:: :-1 (4.4) I ? IY2+ 8 2 At the crack mouth, x =o and y =o , the displacement is reduced to the simple form: a V = 2 -a (4.5) E Therefore, the crack mouth opening displacement (CMOD) is aa c5=2v=4- (4.6) E 4.2.2 Edge Displacements . . . th displacement along the For an edge-cracked sem1-mfimte plate, e edge can be obtained with Paris' (1957) method. Paris' Method . t in the direction of a force From Castigliano's theorem, the d1spfacemen 64 - Fis /J. = BUT F BF (4.7) Where the t ot a I s t ram? energy U ? ? ? ? 7 ,s the sum of the strain energy m the noncracked plate and the strain energy generated as the crack is introduced Wh1'I e the load is held constant: rA aur UT = Uno crack + Jo BA dA (4.8) According to Equations 4. 7 and 4.8, the displacement is then: /J. = auno crack + _g__f A auT dA (4.9) F aF aF o aA Where the first addend on the right side of Equation 4.9 is the load-point d' ,.. tsplacement of the noncracked plate: ~,~, BUno ,. crack F /J.F (4.10) I = no crack ., .. The second term on the right side of Equation 4.9 is calculated as " follows. The crack extension force, G, is equal to the rate at which the total strain energy, Ur, increases with crack area, dA, under constant-force loading: au7 (4.11) G = aA I constant force For a mode I crack, G is related to the SIF by: 2 (4.12) EG = K1 Now in a body loaded by a force, P, and virtual force, F, the SIF is additive: (4.13) and Eq ua t?i on 4.12 becomes: 2 (4.14) EG, = (K,p + K IF ) Inserting Equations 4_1 0, 4_1 1 and 4.14 into Equation 4.9 and letting the 65 -- virtual force approach zero (F. .. Q) yields the displacement of the virtual force: 2 rA aK,F /).F = /).F no crack + E JO K,P aF dA (4.15) Therefore, I). F may be computed from the equations for the displacement with no crack present and the SIFs. Edge Displacement According to Equation 4.15, the displacement along the edge of an edge-cracked semi-infinite plate under uniform tension is determined as follows. A pair of virtual forces Fis applied at two symmetric points along the edge where the displacement is to be determined, as shown in Figure 4.2(b). The SIF is for uniform tension: K,P = 1. 12 15 a {iii (4.16) and for the virtual force pair F (Tada 1973): K = 2 F (1 +asin28)cos8F(8) (4.17) ,, , IF r.:;:: ?Jf vna ;,,:I' Where 8 is defined in Figure 4.2(b): I~ ,!;,~,.- 9 = arctan Y (4.19) a The function F(8) is given by: F(B) = 1.12 +0.18sech(tan8) (4-18) and Parameter a= ?(1 + v). Inserting Equations 4.16 through 4? 19 into the second term on the right side of Equation 4.15 yields 66 fl - 2f ? aK,F F crack - - K1p -da E o aF = ?. ? 21.1215 afria ? - - (1 + asin28) cos8 F(8) ?a (4.20) E {na 40 = a ? 1.121 s (1 + a sin28) cos8 F (8) E The first term on the right side of Equation 4.15 is calculated as the change in length of a noncracked plate of length, 2a tan8: A 20a e u.F no crack = -- fan (4.21) E The total displacement between the virtual forces Fis /:1F = /:1 F crack + /:1 F no crack (4.22) Inserting Equations 4.20 and 4.21 into Equation 4.22 yields fl = 20a[.!.tan8+1.1215(1 +asin28)F(8)coseJ (4.23) F E 2 At the crack mouth, where a= 't, 0, flF becomes the CMOD: j flF = 2V (4.24) and the CMOD is ,fl' :j 0 ?:# ~ = = a a 2 V 4 - [1.458] (4.25) ~ E .~ 4?2-3 Non-interacting Crack Lengths Non-interacting (matching) crack lengths are obtained by equating the CMoo of the center-cracked infinite plate (Equation 4-6>t o that of the edge- cracked semi-infinite plate (Equation 4.25): (4.26) a = 1.45Ba C 8 Where 8 c is the central crack length and Be is the edge crack length. T . f ng crack lengths have 0 check whether the calculated non-interac 1 67 compatible displacements along the junction line, the centerline displacements of an infin 1?t e plate and the edge displacements of a semi-infinite plate are compared. Inserting Equation 4.26 into displacement Equations 4.4 yields the non-dimensional displacement of the center-cracked plate: vcla9 = vL + (1 -v)(y/a 2 8 ) +2?1.4582 (4.27) atE a,, ?(y1a.)2 + 1.4592 While the non-dimensional displacement of the edge-cracked plate is: 2 v,,ta,, = L+ 2 24 [ 1 +a(L] } F[arctan (Yta.)] (4.28) O/E a,, ? a. cos farctan (yla,,)] Where Ve and V8 are the displacements along the junction line of the center- and edge-cracked plates respectively, and y is the coordinate of the point where the displacement is being calculated. In Figure 4.3, Equations 4.27 and 4.28 are Plotted in terms of normalized displacement, (via. )l(a!E), versus normalized distan ce, yla,,. Equation 4.28 (for edge crack) is plotted as a so I1'd 1,? ne, an d Equation 4.27 (for central crack) as a dashed line. The CODs are equal at the junction Point, but differ somewhat along the junction line for O < Y < 38e ? and are nearly equal for y > 3a ?. The question arises as to how joining the two plates at the junction point only_ versus joining them along the entire junction line- affects the SIFs. Unfortunately, an analytical and numerical solutions are difficult to obtain for foined infinite plates. So, in the following section two joined finite plates, one Center-cracked and the other edge-cracked, are analyzed with FEA. 68 4?3 Joined Finite Plates As was done in section 4.2, when an edge-cracked finite-width plate (called T-web) is joined with a center-cracked finite-width plate (called T-flange) to form a T-section, non-interacting is defined as both cracks having equal CMoos at the junction point. As an example, the T-flange and T-web studied in th is section have the same thickness t= 10 mm, width W= 200 mm, and Young's modulus E = 207 GPa. The T-flange is under tension and the T-web is under tension or bending with reference stress of a = 100 MPa, as shown in figure 4.4. 4.3.1 CMODs For a center-cracked finite-width plate under tension, the T-flange, the CMoo?1 s given by Tada (1973): l5=4aav(2a) (4.29) 1 E W Where th e d1' splacement correction factor 1. s v.(:) ?" -0.071 -0.535(:) +o.169{: )' ,.Ji-~) (4.30) +0.020(: 1-E )'-1.071 (: j and ? d ? th reference a is the half crack length, Wis the plate width, an ? ,s e rern 0 t st fi any a/W (Tada e ress. This formula has better than 0.6% accuracy or 69 1973). The CMOD for an edge-cracked finite?width plate under tension, the T? Web' ?, s also given by Tada (1973): Ii = 4 ~ ? v,( ; ) (4.31) Where v,( ~) = 1.46 + 3.42( 1 - cosf;) (4.32) w ( cosf; r Here ? a ?rs the edge crack length. When the edge-cracked plate is under bending, the CMOD takes the samefo rm as Equation 4.31, but the reference stress o becomes the remote bending st r ess at the extreme fiber of the plate, and the di.s placement correct'i on function is (Tada 1973): 2 a ) ( a ) ( (4.33) V , w = o.a-1.1 W +2.4 Wa ) ?r,-0.-66; )' ( As a/Wapproaches zero, Equations 4.29 and 4.31 reduce to Equations 4.6 and 4 ?2 5 for infinite plates. 4.3.2 N on-i.n teracting Crack Lengths the For T-flange crack lengths of 2a,tW= 0.1 to 0.9 in steps O.OS, corresponding non-interacting T-web crack lengths were calculated by equating 4.29 and 4.31 70 The results are listed in Table 4.1, with the first pair of columns listing the T-flange crac k length and the second and third pairs of columns listing the corresponding non-interacting T-web crack lengths for tension and bending. As shown in Figure 4.5, the non-interacting T-web crack length aw is a/ways shorter than th e corresponding T-flange crack length a,. 4?3-3 Comparison with Single Plate Solutions To validate the calculated non-interacting crack lengths, the SIFs and Coo profiles of the non-interacting cracks in the T-section are compared with those 0 f si?n gle plate solutions. This is done first tor S/Fs of three non-interacting crack lengths selected I n, from Table 4.1: a,= 30.0 mm, aw= 20.55 mm; a,= 50.0 mm, aw= 34.79 mm; ,,.. and a,== 70.00 mm, aw= 50.98 mm. SIFs for single plates are given by Tada (1973) While those for the T-section were calculated with FEA As shown in Figure 4-6, the S/Fs of T-section cracks are about equal to th0se of the single Plate cracks. The maximum difference is less than 3.5%. Compared next, in Figure 4. 7, are the COD profiles of the non- interacf - 34 79 mm Because there ing flange and web cracks, a,= 50.0 mm aw - ? ? are no analytical solutions for the COD profiles of finite-width plates, the comparison is limited to those calculated with FEA As can be seen, the COD Profile f . . th corresponding profiles of s O the T-section cracks are very s1m1lar to e 71 the single plates. F' . rnally, Figure 4.8 compares the displacements along the junction line of the T?secrio n wi?t h those of the corresponding single plates. Again, lacking analYtical solutions, the comparison is limited to displacements calculated with FEA. The results agree well, meaning that, for non-interacting crack lengths, displacements along the junction line differ by a small amount, irrespective of Whether the plates are joined or separated. The above examples show that SIFs and coos for the T-section with non.;nteia ct?1 ng crack lengths can be calculated with good accuracy from si?n gle- Plate solutions. On the other hand, when crack lengths are mismatched, differences betwee . n the COD profiles of the T-section and single plates become large. This .I. . ? is shown '?" Fi?g ure 4.9 for a T-section with a,= 50.0 mm and aw= 1o o ?o mm, th e :,.,. ,. latter b e,.n g about three times the matching crack length of 34. 79 mm (t a bl e :":";' 4 ' 1) . When separated, the single edge-cracked plate has a CMOD of 3-29 ?m and th e s,.n g/e center-cracked plate has 0.39 ?m. When the two PIa te s are ioinec1 to form a T-section, the interaction forces reduce the CMOD of the edge craek and enlarge the CMOD of the central crack to a common value of 0-56 IJrn. Figure 4. 10 shows big differences in displacements of the T-section and th e single plates along the junction line. Obviously, mismatched crack lengths induce I . ? h SIF and coos for the T- arge interaction forces in a T-sect1on, and t e s 72 Web and T-fl ange cracks cannot be replaced by single-plate solutions. 4?4 I-beams Thre e-t?1 p cracked I-beams are modeled as an edge-cracked web plate ioined to a center-cracked bottom flange and a noncracked top flange. The web and flan . . . . ges are represented by their mid-planes; the web 1s Joined to the flanges I . a ong the intersection lines of the mid-planes (Figure 2.2). Non- ~~~~ . . ng crack lengths for the eight representative I-beams lrsted rn Table 4.2 Were calculated with two methods: one is based on CMODs, the other on a cornbinaf ion of FEA and single plate solutions. ,~ 4 4 , . .1 No n-,?n teracting Crack Lengths From CM OD s In th is method, an I-beam is replaced by a T-section whose web and flange are equal to the cracked web and bottom flange of the I-beam. The 1- ~rn stress r.s applied on the T-section. The non-interacti.n g crack lengths are deterrn? . tned by equating the CMODs of the flange and web cracks, Equations 4.29 and 4.31 . In the process, since the T-section has only one flange, the effect of the I-beam? s top flange was neglected. The error should be small because t he we b c~ck . h of a non-interacting three-tip cracked I-beam 1s often short and hence t e crack r Ip does not approach the top flange. 73 N . . on-interacting flange crack lengths were calculated for web crack lengths b ? ? . egrnning wrth Aw= awl0 = 0.05 and increasing in steps of 0.05. The increm t . en atron was stopped when the corresponding flange crack would have become longer than the flange width, which is not physically possible. The results for the eight I-beams are listed in Table 4.2. Crack lengths, aw and a,, net Web depth 0, and flange width b, are shown in Figure 3.2. 4 ?4?2 Non-interacting Crack Lengths From FEA Non-interacting crack lengths were also obtained from FEA of the eight f- beams rr sted Table 4.2, with the following crack lengths: ? Web crack lengths: Aw= awl0 = 0. 1 to 0. 7, in steps of 0.1 I, I ? Flange crack lengths: A,= 2a,lb, = 0.1 to 0.9, in steps of 0. 1 ihe Web and flange of an I-beam with non-interacting cracks can be separated :: It :1 and the SfFs of the web and flange cracks are equal to those of the ;~ !I I Cor responding edge-and center-cracked plates. Therefore, the non-r?n t erac f, ng crack lengths are determined here as the value at the intersection point of the I- beam . and srngfe-plate solutions. For example, in Figure 4.11, the normalized SIF for the flange crack tip of I-beam W33X201 is plotted with a dashed curve as a function of flange crack length A Id - o 1 fn the same figure, , == 2a,lb,. The web crack length is Aw= aw 'i - ? ? the c . tt d with a solid curve. At enter-cracked plate solution (Tada 1973) rs P1O e 74 the interse t? . c ion point of the two curves lies the non-interading flange crack length With value of A,= 2a,lb, = 0.50. The same non-interacting flange crack length can also be determined by equar ing the SIF for the web crack to Tada's (1973) solution for an edge- cracked plate. According to Figure 4.12, the non-interacting flange crack length is A,== 2a,Jb, = 0.50 for a web crack length Aw= a.,,I~ = 0.1. Pairs of non-interacting web and flange crack lengths are listed in Table 4 ?2 for eight I-beams and two loadings - tension and bending, with the results being determined in the left column by equating K, values and in the right COiumn by equating Kw values. 4,4 ? 3 Com par1.s on I , Figures 4.13 and 4.14 compare non-interacting flange crack lengths ?' ! J determined from CMOD and FEA for I-beams are under tension or bending. The former Were determined from matching CMODs from single-plate solutions. The latter Were determined from equating SIFs from FEAs aod single-plate Solutions. No n-,?n teracting flange crack lengths are PI o t ted as a function of depth- Width r t? 0 . n was done in Chapter a ? V instead of area ratio of flanges to web, ,.,, as 3 s; f ? ot important in ? nee the Web and flange do not interact, the area ra 10 ,s n this case. 75 Figures 4.13 and 4.14 show that the non-interacting crack lengths determined fr om FEA SIFs for flange or web cracks (dashed lines) agree \<(6 eccentricities)x(9 W-shapes)x(2 loadings)= 972 ? I-beam modeling method: 3-D model, web and flanges represented by their I' ' mid-planes, with all DOFs joined along the junction lines of web and flanges u' K? ?ft ??I I ? Elements: eight-node shell element, reduced integration, five degrees of freed om Per node, ABAQUS designation S8R5 ? Crack tip elements: degenerated quarter-point elements from SBRS ? Mesh Pattern around crack tip: m x n = 3 x 16 (Figure 2.4) ? M Ode I SIFs obtained from J-integral of ABAQUS output The Positions of the upper and lower crack tips are described by the Crack I ength Aw . Id and eccentricity e. Different combinations of Aw and e yie 79 increases , th e f A curves become gradually stratified, with the higher curves corresponding to small eccentricities and the lower curves to large eccentr,c ,? t?i es. The only deviation from this trend is the curve for the central crack (e = 0.0) which first rises and then falls at large crack length, Aw= 0.8. The fall occurs because the upper tip of a long central crack approaches the Uppe r J? uncti?o n point and becomes constrained by the upper flange. ? Forthe upper ti?p and stocky sections such asp= 2.05: when the eccentn ?c ,?t y 1? s small, the fA curves first rise slowly for short cracks and then fall for long cracks; when the eccentricity is large, the crack tip approaches th e lower junction point and the f A curves always fall. ? For the lower crack tip, the trends of the f 8 curves are similar to those of the Upper crack tip. The major difference occurs in the slender sections with srnall eccentricities: instead of rising, the f 8 curves fall for long cracks. This is bec ause the lower tip of a long crack approaches the 1o wer 1? unc t?, on po,?n t ?? ? and becomes constrained. ' ? ~ t r slender sections such as p = 0.83, when both crack tips are ar away fron, the junction points (small eccentricities and short cracks), the co rrection factors of the lower tip are larger than those of the upper t,' p. A s the crack tip moves closer to the junction point (large eccentricity and10r long c ck tip are smaller than rack), the correction factors of the lower era those 0 f the upper tip. ? For stocky sections such as 13 = 2.05, the correction factors of the lower tip 81 are always smaller than those of corresponding upper crack tip. Effect of Eccentricity e ? As the eccentricity increases and, as a result, the crack lies farther down the Web, the correction factors for both the upper and lower crack tips decrease. The reduction is greater for the lower crack tip because it is much closer to th e bottom flange than the upper tip is to the top flange. ? For h s ort cracks, the correction factors of the . upper and lower crack tips Change little with eccentricity. But as the crack length increases, the correction factors become more sensitive to the eccentricity. For example, When Aw= 0.9, the correction factor of the lower crack tip of an I-beam with f3 = 0-83 drops from f 8 = 1.056 fore= 0.0 to f 8 = 0.815 fore= 0.9. Effect 0 f F langes-to-Web Ratio f3 ? As the f3 value increases, the correction factors for both the upper and lower crack tips decrease. A high f3 value means the flange is larger and I' I I Const rai?n s the web crack more. ? For a short crack such as Aw = 0.1, the flange constraint is small and both the fA and fB curves are flat. For a long crack, Aw= 0.9, the flange constraint becom . . . es s1gnif1cant, and the correcti.o n factors e more sensitive to ar Parameter f3. For example, the correction factor of the lower crack tip drops from 0.94 to 0.82 as f3 changes from 0.83 to 2.0S. Cornp . anson of I-beam with Single Plates 149 Figures 5.4 and 5.5 compare the solutions for I-beam W40 x (f3 = 82 O.B3) With th ? ose of single plates of same dimensions and crack lengths as the Web and fl ange of the I-beam. Similar trends can be found for other I-beams. Figure 5 4 ? ? is for the upper crack tip while Figure 5.5 is for the lower crack tip. The corre r c ron factors are plotted as a function of crack length, with the solid and dashed curves representing the I-beam and single plate solutions respecf . ?ve/y. The single plate solutions are given by Tada (1973) for a central crack and lsida (1965) for an eccentric crack. Each curve is for a different eCCentricity. ? For any crack length and eccentricity, the correction factors of the I-beam are always smaller than those of corresponding single web plate. Web crack opening in an /-beam is constrained by the flanges, causing stresses to be reduced. ? The d'"""e rence between the I-beam and single plate soluti?o ns i?n creases "th WI Web crack length. A long crack deforms more and the crack tip is closer to ~e?J tmcti?o n points. Therefore, the crack tip is constrai?n ed more '?" an l-b earn than? . in a single plate. 1 ? n a sing le Plate, the correction factor of the lower crack tip is always larger th th ? 't' an at of the upper tip. In an I-beam, depending on crack tip pos, ion (determined by parameters Aw and e) and flange size (parameter (3), the correction factor of the lower crack tip may be smaller than that of the upper crack tip. ? For a ? single Plate, . es with crack length. the correction factor always 1ncreas 83 But for an I- be arn, the correction factor increases slightly for a short crack but the n d ecreases as the crack becomes longer. 2-0 Plots Figures 5.1 to 5.3 were replotted in 2-D as Figures 5.6 through 5.14 to more c/ea I . r Y illustrate the effect of parameters Aw, e, and p. Figures 5.6 through 5.8 show the correction factor as a function of crack length 1 '"" for values of f3 = 0.83, 1.37, and 2.05 respectively. These figures represent th ree vertical planes normal to the p axis in Figures 5. 1 to 5.3. There are two . sets of curves in each figure: solid lines for the upper crack tip and dashec1,- . ines for the lower crack tip, each line corresponding to a given E!ccentricity. F? gures 5.9 through 5.12 show the correction factors for the upper and low er crack tips as a function of eccentricity. The first two figures are for the most 1 sender section ('3 = 0.83) and the other two figures are for the moSt stacky ('3 nds ck section = 2.05). Each curve in the figures correspo to a web era length Aw. Finally, Figures 5.13 and 5.14 show the correction factor as a function of Paramet '3 ? 1y These figures er for the upper and lower crack tips respective ? represent h ack length axis in a vertical cut of thee= 0.5 surfaces normal tot e er F"1?9 Ures 5 ?2 d to a given crack and 5.3. Each curve in the figures correspon s lera9th -'w- 84 5.2.3 T wo ...t ;p Cracked f. .. beams under Bending tn an 1?beam under bending, the web is subjected to a linear stress distribution . ' with reference stress a of equal values and opposite signs at the Upper and t . . . ower Junction points. The tower and upper Ranges are subjected to equat Unito rm reference stresses in tension and compression. Compressive stresses act on the portion of the crack length lying above the centroidal axis, causing P . art,al penetration of the crack surfaces and resulting in a negative SIF for the u . . . Pper crack tip. Such behavior is not physicatly possible. So the bending solution h . . as meaning only when applied in combination with an ax,at stress distributi on of a magnitude that prevents crack surfaces from penetrating each Other. C . . . . omb,ning tension and bending shifts the neutral axis from the centroid ~~~ . eam upward, a condition commonly found in composite steel-concrete beams I . . ? n th is way, the separate solutions for tension and bending can be Usec1 . ' in combination, to calculate SIFs for composite beams. Crack cfosure, without penetration, on the compressive side of a center? cracked ? ? ? b single plate increases the SIF at the tip on the tension side by a out 10% (8 . ow,e and Freese 1976, Woo et al. 1988). The correction factors for the upper and lower crack tips of a two-tip cracked J F' 5 15 ?beam under bending are listed in Tables 5.3 and 5.4. ,gures ? and 51 ? 6 Show surface plots of the correction factors as functions of parameters \.and 13 ? Ea ch surface corresponds to an eccentri?c i?t y. F,' gure 5? 15 is for the 85 Upper tip h ? . w rle Figure 5.16 is for the lower tip. The effect of individual parameters Aw, e and ~ are discussed in following. Effect of Crack Length Aw ? The cor~ t? ec ion factors for both the upper and lower crack tips have values of fA:: fB - . . . - e, m the llmrt as Aw ? 0.0, which is theoretically expected. ? The cor t? rec ion factors for the upper crack tip decreases as the crack length increases. For small eccentricities of e = 0.0 and 0.1, the correction factors are negative, because the upper crack tip lies in the compressive stress re ? 9lon (except fore= 0.1 and Aw= 0.1). ? For an eccentricity of e = 0.3 and crack length A..,~ 0.5, the upper crack tip st ill lies in the compressive stress region, but the correction factors are Positive ? sm ? ce the portion of the crack in the compressi?v e st ress regi?o n 1? s Shorter th an that in the tensile stress region, the era ck "c Io s,.n g " displacements are overcome by the crack "opening" displacements. As a result ' t he whole crack is still open and the correcti.o n factors are posr' f,v e. ? l=or large eccentricities of e ~ 0.5, the upper crack tip lies in the lower portion ~~ Web and the entire crack is open. Therefore, the correct ?i on ia~ rs~ Positive. ? For th e lower crack tip which always lies in the tensile ? th stress region, e correction factor curves of f 8 first rise with increasing crack length A~ then, 86 as the crack tip approaches the lower junction point, the cu1Ves become flat for small eccentricities or fall for larger eccentricities e ~ 0. 7. ? For same crack length and eccentricity, the correction factor of the lower crack tip is always greater than that of the upper crack tip, except fore= 0.9 and "-w ~ 0. 8. The correction factor cu1Ves for the upper crack tip are more st raufied than those for the lower crack tip. Effect of Eccentricity e As the eccentricity increases, so do the correction factors for both the lipper ad n lower crack tips. Effect 0 f F langes to Web Ratio ~ The correction factors for both the upper and tower crack tips are Practica11 Y not affected by parameter ~ for short cracks, but change s 1i? g htl Y io r long er acks. As an example of the latter, when 'Aw = . ~ 0.9 and '3 increases rom 0.83 to 2 ? oS , . o 293 t o2 62 the correction 0 factor of the upper tip drops from ? ? and that 0 f the lower tip drops from 0.684 to 0.615. CornPar? Ison of I-beam and Single Plate The constraining effect that the flange has on the web crack is evident by comparing the I-beam and single plate solutions. The correction factors for cracks in the web of I-beam W40X149 ('3 = 0.83) ands single plate of same Sile as th d as a function of crack e Web are compared in Figures 5.17 an 5? 18 length S. . The curves correspond ? ,rn,tar results can be found for other I-beams. to ec08 . . . d 5 18 are for the ntnc,t,es e = 0.0, 0.1, 0.5 and 0.9. Figures 5-17 an ? 87 Upper and I . ower crack trps respectively. The solutions for a single plate with a central or eccentric crack under bending . were given by Chen and Albrecht (1994). Their solutions were checked With FEA ' and the error was found to be less than 0.5% for all eccentricities and crack le h ngt s Aw s 0. 7 (Appendix A). The fol low? ? tng is observed from the figures: ? The corr ect,.o n factors for the upper and lower crack tips of the I-beam are a~~ smaller than those of the single plate because the flange constrai.n s the Web crack opening. ? D~e . rences between correction factors of the I-beam and single plate increase With crack length. A long crack opens more and its tips are closer to the junction points, thus increasing constraint. A short crack, on the 0ther hand ' opens much less and its tips are farther from the J?u ne t?i on po,?n t s? rem?. arn,ng largely unaffected by the flange. ? Differences between the I-beam and single plate solutions for the upper crack tip are smaller than those for the lower crack tip, meaning the upper crack tip is constrained less than the lower tip. For example, when e = O.S, th k f . e crack lies in the lower portion of the web and the upper crac ?Pis l0cated near the major axis of the I-beam. The upper crack tip is subjected to sm II fo little flange constraint. a stresses and deformations and, there re, The I ? ? f the I-beam and the ower crack tip is located far from the ma1or axis 0 stresses and deformations around the crack tip are large. Also, the lower 88 crack tip Ii . . . es near the lower Junction point. Therefore, the lower crack tip is highly constrained by the flange. ? The cor t? rec ion factor trends differ greatly for the I-beam and the corresp d' . on mg single plates. For example, the lower tip correction factor of a single Pl t . a e increase with crack length; that of an I-beam decreases, espe . I cia ly for long cracks. Add1't i?o nal 2-D plots of correction factors are presented I?n Fi?g ures 5.19 through 5 2 . - 7 for other I-beams. The findings are similar to those described above. 5 ?3 Three-tip Cracked I-beams The SIFs for a symmetric three-tip crack in an I-beam is expressed as: Kw.,= rw,,0 V~ (5.2) ,,c;,w,' Where 8 b ? t d -w and a, are the web and flange crack lengths, and the su scrip s an supers ? cripts wand f refer to ? I s? ?1 the web and flange crack tips, respective y. imi ar to the.... . t ?i LVVO-tip crack, the reference stress a is defined as the remo e uni orm st ress for an 1- beam under tensi.o n and the stress a t th e flange-web 1?unction for ar, I-beam under bending. The correction factor f is a function of web crack length .1 fl crack length >.,, to b cro ss-sectional area ratio ''-w, ange and flanges we !3. 89 5.3.1 V ariables in Analysis The variables used .1 n FEAs of three-tip cracked I-beams are: ree-t1p crack, with center-cracked bottom flange ? Symmetric th . rma ,zed web crack lengths: A. = 0 .1 to O.7 , in steps of 0.1 ? Seven no 1? ma ,zed flange crack lengths: A,= 0.1 to 0.9, in steps of 0.1 ? N'i ne nor 1? ? N'i ne W-shapes listed in Table 3.1 ? Twot ypes of loading: tension and bending ? Nu mber of FEAs: rack lengths) x (9 flange crack lengths) x (9 W-shapes) x(2 (7 web c load?i ngs) = 1,134 modeling method: 3-D model, web and flanges represented by their ? I-beam ? mid-pla nes, wi.t h all DOFs joined along the junction 1.1 nes of web an d fl anges ? Elements?? ei?g ht-node shell element reduced integrati?o n, f', ve de gree s of I freedom per node, ABAQUS designation S8R5 Crack tip d f rn 1ernent S8R5 ? e ements: quarter-point element degenerate ro e I 2 ? Mesh pattern around crack tip: m x n = 3 x 16 (Fi.g ure .4) ? Mode IS IFs obtained from J-integral of ABAQUS output. Them ? A = o 7 because, in aximum web crack length was chosen as w ? bridge gi.r ders, the flange crack has extended across the full flange wi.d th when the Web er ac k length exceeds 70% of the depth. 90 5.3.2 Th . ree-t,p Cracked I-beams under Tension The calculated correction factors for the web and flange crack tips are listed in T ab/es 5.5 and 5.6, respectively, as a function of Aw, A,, and~- These same Va/u es are plotted in Figures 5.28 and Figure 5.29 as functions of Parameter s A, and f3. Each surface corresponds to a given web crack length Aw- Smooth and . . . regular relat1onsh1ps exist between the correction factors and the Paramete rs. Correctfo " Factor for Flange Crack Tip F'rom Fi . gure 5.28, the following is observed for the flange crack tip: ? As the flange crack length A, increases, the f' curves first drop slightly and then ris e as A, ... 0.9. This trend is uniform for all web crac k /e ng th s. ? The correction factor f' also increases with web crack /ength Aw? Clearly, When th e Web crack is long and the flange crack is short, the m? terac f, on forces along the junction line tend to open the flange crack more than would be the . case if the flange were separated from the web. ? F'i nauy f3 . In a two-tip cracked ' as increases, the correction factor decreases. '?bean, f3 ? t on the web crack ' Parameter accounts for the flange constrain 0Pening? . . meter ~ accounts for ? s,m,lar/y in a three-tip cracked I-beam, para interact? The interaction effect Ion between the cracked web and flange. decre th heavier the flange is ases With increasing f3 value. In other w0rds, e relative to the web, the less the CMOD of the flange crack is affected by the 91 Web crack. Correct; on Factor for Web Crack Tip .F? '9 tJre 5,29 sho ws the following effects on the web crack tip , As the flan e g crack fength >., increases, the correction factor for the web crack tip tw ,?n creases. ? Contra,y t0 . the find,ng for f ~ the correction factor fw decreases with lncreas;n g of Web crack length Aw. The interaction forces acting along the Web ed . ge are equal 1n magnitude and opposite in direction to those acting along th e flange. Hence, if the interaction forces tend to open the flange crack int ' urn to close the web crack. ? The cor . . rect,on factor for the web crack generally increases with parameter 13, the inc . rease being greatest for combination of long flange and short web cracks. However, by comparing the values of correction factor for the web and flan9e crack t1? PS, Figures 5.28 and 5.29, it can be seen that the responses of web craCk s to the interaction forces are much greater than those of the flange cracks ,= ? or example, for I-beam W40 x 149 with web and flange crack lengths lf .1 ._ ,:::: 0.7 and >.,= 0.9 (the interaction forces open the flange crack and close he Web crack), the correction factor of the flange crack is increased from t' = -576"> ' to 4? 8 3 46, White the correction factor of the we b crac k i?s reduced from r 6 37 ? 55 to 1? 0 38 7. Thi.s ,.s so because, for equa f ~,o rces and crack lengths, the ,too is ? ked plate greater for a single edge crack than for a single cen,er-crac ? 92 Interact; on Between Web and Flange The inte rac ti'o n between the web and flange depends on the relative crack length . s. Figures 5.30 and 5.31 show the correction factor of the flange crack tips a s a function of flange crack length. These two figures represent two Planes no rmal to the~ axis at 0.83 and 2.05 in Figure 5.28. Each dashed line in the figures corresponds to a web crack length>._. To investigate the interaction between th e edge-cracked web and the center-cracked flange, a solid curve Was added to the figures representing Tada's (1973) solution for a center- cracked I Pate. It can be seen that: ? p. airs of fl ange and web crack lengths, A, and >.,,, are free of ,?n terac t?i on Where Ta da ,s solid curve for a single center-cracked plate 1. ntersect s th e dashed d & curves for the I-beam. SIFs for such pairs may be calculate ,rom single-plate solutions. ? Tada's s 0 Iu t,?o n divides the space in Figure 5.30 and 5 ?3 1 ,?n t? t wo fegions of behavior. The opening of the flange crack is restrained by th8 web crack at Points b e Io w the solid curve and augmented by the we b era ck at points ,a, bove the solid curve. Accordingly, the correction factor for the flange crack . is sma11 er and larger, respectively, than Tada ,s soIu t,?on would predict. ?3?2 Th ree-tip Cracked I-beams under Ben d", ng VVh d' the stress is constant en a three-tip cracked I-beam is under ben ,ng, 93 in the flange . . s and linear ,n the web. The interaction behavior between the cracked Web . . . and flange 1s s1m1lar to that tor I-beams under tension. The correction factors are listed in Tables 5. 7 and 5.8 tor the web and flan 9e era k ? c tips respectively. They are plotted in Figures 5.32 and 5.33 as a function of A . , and (3. Each surface in the figures corresponds to a given web crack fen th . . . g Aw, Smooth and regular relationships exist between the correction factors and . . . the parameters. Also, the trends in these figures are very s1m1lar to those for 1-b earns under tension (Figures 5.28 and 5.29). ? As the ti ange crack length A, increases, the correcti?o n factor for the ti ange crack tips first drops and then rises; the correction factor for the web crack tip in creases gradually. ? As the Web crack length Aw increases, the correction factor increases tor the flange . . crack tips and decreases tor the web crack tip. ? As the fl ratio the t1' o n factor anges-to-web area (3 increases, correc decreas t es or the flange . ~ the web crack tip. crack tip and increases ,or 5.4 C0 nclusions Tw . ? n or bending were o-t,p and three-tip cracked I-beams under tensio analyzed T t ?ned from FEAs. The ? he correction factors for the SIFs were ob al ~elected I lded girders used in -beams cover most rolled W-sections and we lighway b . d for a wide range of r,dges. For each I-beam, FEAs were performe 94 eccentricities w ' eb crack lengths and flange crack lengths. Based on the results th ' e following conclusions were reached: ? The calcu1 t . a ed SIFs (correction factors) are smooth and regular functions of eccentri ?t er Y, crack lengths, and flanges-to-web area ratio. ? Ti1.1, vO-tip C k . rac s rn I-beam webs are constrained by the flanges on each side. As a res u It , the SIF for the web crack is smaller than that for a single plate. The flange co ns t rar?n t increases as crack tip, whose pos1?t 1? on 1?s d e te rmi.n e d by Paramet ers Aw and e, approaches the juncti.o n poi.n ts of web and ~,,, anges. As eccent n?c 1?t y, web crack length, and flange area increase, t he constrai?n i?n g effect of th e flange on the web crack becomes larger. ? SIFs to r th . . ree-tip cracks are affected by the interaction forces between the Web and flange. These forces depend on the web and flange displacements along th . . . e Junction line. If the displacements of a single flange P1a te are greater th an those of the corresponding points of a si?n g 1e we b PI a te ' enfor ? crng displacement compatibility reduces the CMOD of the fl k ange crac nd a augments the CMOO of the web crack. Conversely, if the displac h th se of the ements of a single flange plate are smaller t an ? corresp0 d' b ugments the CMOD of n rng points of a single web plate, the we a th e flange crack and the flange reduces the CMOD of the web crack. 95 Chapter 6.? Equations for Stress Intensity Factors Correctio n ta ctors for two-tip and three-tip cracked I-beams under ten s1?o n or be nd. mg were determined with FEAs in Chapter 5. The results were Presented. in tables and curves. In this chapter, the numerical values were fitted With empirical ? equations for ready use by engineers. 6.1 F1?t t? ?ng Procedures The fitting was carried out in two steps. First, the correction factors were frtt8d as a function of two variables - web crack length Aw and eccentricity e for ta fwo-t;p c rack, and web and flange crack lengths Aw and A, for a three-tip crack. he third Parameter, the flanges-to-web area ratio, was kept constant. The Purp0 Se of this Preliminary step was to explore the functional form of the fitting equation fi d I or each I-beam. Software TableCurve3D- a product of Jan e Scienr1t ; ? ic for surface fitting -was used. TableCurve3D can hand1e two 1nct epend ? d . kl ent variables and has the advantage of automatically an qu,c Y finc:1? Ing the b t . . ,nong a large family of es fitting functions for complex data from a llossibfe ~ unctions (TableCurve3D User's Manual 1994), In th fitted with a three-variable e second step, the correction factors were 1 equation th . . th third variable. The at included the flanges-to-web area ratio as e best fittin . . with different functional 9 equation was determined after several tria 1s 96 forn, s selected b ased on the results from the first step. The Unknown coefficients were determined with another Jandel Scientific 80ftware _ s. igmaPlot - which can handle as many as 15 dependent variables and25 coeffj ? cients. SigmaPlot uses the Marquardt-Levenberg algorithm (SisrnaPJot , . Users Manual 1994) to find the coefficients of the independent Van? ables th . . . at give the best fit between an equation and the data. This algorithm finds the co . . efficients that minimize the sum of the squared differences between the Values . . of the observed and predicted values of the dependent variable. n ss = E Where Y . , = 1 ' is the observed and i, is the predicted value of the dependent Va. riab1e Th . . .. ? e fitting process is iterative. SigmaPLot 1n1t1ally guesses the COefficient ? I . s, checks to see how well the equation fits, and then continuous Y 'lllProve th f s e guesses until the difference between the residual sum O squares no longe ffi ? t r decreases significantly. Curve fitting stops when: (1) all coe ,c,en Values stop h . th bsolute value of the c anging in all significant places; and (2) e a diffi erence b ... _ ~ ~ ne iteration to the eu-n,en the norm of the residuals, ySS, ,rom 0 next is les h ed in the present st an the tolerance value. The tolerance value us Sfudy Was 0.0001. Co . f n are compared with rrect,on factors predicted with the fitted equa '0 the F'~ r 0 f small values off, the esults. To exclude meaningless large errors fltt?? ng erro ? r is defined as: 97 f - f. t:,_ ::: pre~ FEA 1O O (6.2) Where f . 'FEA ~, thPred is e predicted value of the empirical equation ?~f. is the FEA or culated result - ' and fFEA is the average value of all FEA results . 6.2 Tw 0 -tip Cracked I-beam The SIF fi . . s or two-tip cracked I-beams are given by equation 3. 1, and the Correct? ion fa t c ors are defined by equations 3.2. The parameters are the web crack length . . . Aw, eccentricity e, and cross-sectional area ratio of flanges-to-web S. Bee ause th . e Web crack is eccentric, the correction factors differ for the upper and lo Wer crack t?l ps, 6-2.1 p aram eter Ranges In Chapter 5, FEAs were performed for eight selected I-beams /iSted in 'abte 3 1 . ? ? Curves Were fitted for the following combinations of crack t,ps and load1? ngs: ? Two cr ack ti.p s: correction factors fA and f 8 for the upp er and lower crack tips ? 1Wo/ ? Oad,ngs: tension and bending f: ? Numb d. s) = 4 er of curve fittings : (2 crack tips) x (2 /oa ,ng or each curve fitting , the parameter ranges were: 98 ? Eight er . oss-sect,onat area ratios: f3 = 0.83, 1.00, 1.25, 1.37, 1.53, 1.69, 1.91 and 2 0 ' ? 5. The range of f3 = 0.83 to 2.05 covers most I-beams used in civil ? . engmeenng (Figure 3. 10 ). ? s?I X eccent ? ? ? nc,t,es: e = 0.0, and 0.1 to 0.9, in steps of 0.2 ? Nin e Web crack lengths: Aw = O.O to 0.8, in steps of 0.1. FEA data for the longest crack length were not used because Aw = 0. 9 is rarely found in Practice ? Fo r example, a central crack (e = 0.0) of length Aw= h 0.9, s own in Figure 6 ? 1, extends through most of the web depth, causi?n g si?g n,' fi, can t Yielding 0 f the net section that would invalidate linear-elasti?c ca tc u ta t?i ons of Sf Fs t ? ft th "1 ? n another example, a greatly eccentric crack O eng w = 0? 9 ' show ? ? n In Figure 6.2, would extend into the bottom flange whereas, ,n reat;t Y, the analysis should be limited t? to cracks whose lower ?P 1? ,es above the upper surface of the bottom flange. ? iotats . . ample points for each curve fitting: (B (3 Values) x (6 eccentricities) x (9 crack lengths)= 432 ihe limit case of A = 0.0 which corresponds to no crack, yields ad w , ditionat . d fA - ,s = e for Values of fA =, s =1 for an I-beam under tension an - an l~bea rn under bending. s.~.~ e . Quations for Two-tip Cracks under Tension Preti ? and 6 4 for the upper minary fitting results, shown in Figures 6 ?3 ? 99 crack tip of t wo I-beams W40x149 (/3 =0 .83) and W18x97 (/3 =2 .05) under tension ' suggested a polynomial functional form. After several trials, the fottowing 15-t . erm equation 6.3 was found to best fit the calculated correction fact ors for both h t e upper and tower crack tips: f - ao + a, e + a fl.g + a e2 + a A + 2~ 3 4 w as !3A.., + a6 eAw + a7 Aw 2 +a ~e>._, + a9 /3 2 e + (6.3) 8 a,o !3e2 + a,, ~132 + a,2~e2 + Bu ~A/+ a, .. eA/ The con st ant . ,n the equation is 80 = 1 because . fA. 8 = 1.0 for the special case of e - - O.O and Aw= 0.0. The other 14 coefficients were determined with the s? '9fl1aPfot softw are. Coefficia nts for Upper Crack Tip The coefficients of equation 6.3 for the upper crack tip of a two-tip cracked l~b eam under tension are given in Table 6.1. F'igure 6-5 shows the differences between the predicted (fitted) and Cafcut ated (F EA) correction factors. The points in the figure are arrange d ? ,n e? '9ht 9ro th ups, one for each f3 value of 0.83 to 2.05; within each such group, e eecentricit t ? ?ty the ye Was incremented from 0.0 to 0.9; and for each eccen nc, ' crack I ength ~ Was incremented from 0.0 to 0.8. The average value of the Corre . ct,on factors for at/ the 432 data points is f = 1.0031. Good agreement ~~b ~ . etween predicted and calculated values. The maximum poSitive error is 1,41? 0 ? and th e . . t rrors lie within ?1 %. maximum negative error ,s -3.0%. Mos e To id . t ? td the /east accurate entity the combinations of parameters tha y,e res u1 ts , the t? . . . 6 8 versus parameters 1tt1ng errors were replotted ,n Figures 6.B to ? 100 a, e, and A_, respectively. Clearly errors are largest for the long central crack, e = o.o and A...- ,. - D.B, a case that rarely occurs in practice and would, in all ?kelihood ' Produce net section yielding in violation of LEFM. Figures 6 9 f ? and 6. 10 compare the predicted and calculated correction actors for I b - eams W40X149 (13 = 0.83j and W18X97 (13 = 2.05), further Confirm? rng the goodness of fit. Co,e,,c?,.e nta for Lower Crack Tip ihe coeffi, c,?e nts of equation 6.3 for predicting correcti?o n factors ~,o r th e Iowa r~~ti .. P of a two-tip cracked I-beam under tension are grven ,n Table 6.1. ihe errors between the predicted and calculated correction factors are Shown. ... in Figures 6.11 to 6.14. The average value of all the correction factors is ', ~ ::: 0 9 ? 933? All errors fall within the range of 1.29% to -1.93%. Figures 6? 15 ar,d 6. 18 show comparative surface plots for I-beams W40X149 (13 = O.B3) and W1ax97 (l3 == 2-0S). Again, the fitted equation produces smooth surfaces and 9ood agree ment with the calculated values. s.2.3 e Quations for Two-tip Cracks under Sending ihe functional form of equation 6.3 was also selected to fit the correction fact 0rs for .... . h' e the constant is ,vvo-t,p cracked I-beams under bending. In t is cas ' 80:::::: 0 becau se, . Aw= 0 the correction factors t rn the limit, when e = 0.0 and 0. , JlJ.>roach fA ? a ==o.o. 101 The other 14 . C coefficients were determined with SigmaPlot software. C>efffcienta fi or Upper Crack Tip The COeffi . th cients of equation 6.3 for predicting the correction factors for e Up Per crack f T: 1P of a two-tip cracked I-beam under bending are given in ab/e 6.3. Figures 6 1 ? 7 to 6.20 show the errors between the predicted and Ca/cutatec1 . r: correction factors and their distribution as a function of fJ, e, and A.. he avera e 9 e of all correction factors is f = o.2979. The maximum positive rror is 2 FEA ?25%' an d t h e maximum negative error is -2.7 3%. The largest fitt'm g errors are th t ose for the central crack, e = o.o, and shortest and longest crack engths ,\., == o . . o a ? 1 and 0.9. For most data , the errors fall within ?1 %. Good 9reernent b . . etween the predicted and calculated correction factors is further collfirrned in . b Figures 6.21 and 6.22, where the values are compared for l- earns W4 0 C " 149 (13 = 0.83) and W18x97 (13 = 2.05). Oeffic?t ents ti or lower Crack Tip The kt? . coefficients of the fitting equation 6. 3 for the lower crac ?P are 91~e11 1?1 1 Table 6 ?3 ? The fitti.n g errors and thei.r d1. stn.b .u duio n are shown in Figures 6.23 t 0 ll, 6.25 ? Th . ? - = o 5247. e average value of all the correction factors is fFEA ? en,~ . . fa .... x1rnum . . . d I ulated correction Pos,t,ve error between the predicted an ca c ~~- is 1 950/ . B2? . The largest fitting 7 , ? o, and the maximum negative error ,s -2? 0 ? ?tro rs are fi ? 6 3 tor I-beams h ore == 0.9 and Aw= 0.9. Surface plots of equation ? ,40 ? ~149 (n . F 627and ? == 0.83) and W18x97 (13 = 2.05) are shown ,n ,gures ? 102 6,28. Good a re 9 ement between the predicted and calculated correction factors can b e observed. s.2.4 r Wo-tip Cracks in Engineering Practice In the ab . . ove sections, the correction factor for two-tip cracked I-beam under tension . . or bending were fitted with equation 6.3 for a wide range of Pararnet . nc crack lengths and eccentricities. In engineering practice, there are some limitar ions on crack position and crack length. '1iriirnu rn Crack Length r In engineering practice , two-tip web cracks found in service are at least 4 'Il'les as/ . ong as the thickness of the web That is the minimum crack length ,s (I=? . , 1 9ure 6_ 29): (6 .4) Crac 8 w,min = 2 tw k Shorter th hav an a . are virtually not detectable an d wo uld tv,.p ically not w, min e 9rown k en much less Yet through the web thickness. Part-through crac sop than .... 'VVO~tip th rough.cracks. ~&Jcirnurn Crack Length The rn . th lowest position of ax,mum web crack length is determined by e ~~ ' Wer era k . t possible crack t,p c t,p. For a given eccentricity e, the lowes Posir1 0n is ri h fl nge which is a . 9 t above the upper surface of the bottom a ' dist ance of t I2 . Th refore the maximum ' away from the lower junction point. e ' crack length . for a given eccentricity is 103 ""a~imum aw, max = di 12 - e - t, 12 (6.S ) Eccentricity As th e eccentricity i and the . ncreases, the crack is located farther down the web, d1fferenc e b sn-iaue etween the maxi?m um and minimum crack lengths is r. When a Sholl, . w, max = aw, min, the eccentricity reaches its maximum value. As ryn 1n r-? rigure 6.30 . ' the maximum eccentricity is Ba 8 max = d/2 - 2tw - 1,12 (6.6) lenllth seci on the above analyses, limiting values of eccentricities and crack s Were c I a culated for all I-beams. w l=igure 6.31 . shows the maximum eccentricity for a/I I-beams of sizes 40 towa the . The Circ u Ia r symbols in each group of nominal size are plotted in orde r ~,r om hea . . sn-iau viest to lightest section. Typically heavy I-beams have eece er. m ax; mum eccentricities, and light I-beams have larger maximum ntncities the ? l=or all I-beams combined, the maximum eccentricity falls within range e - - O.S6 to 0.91. ,=i? gures 6 .. for ?3 2 and 6.33 show the maximum and minimum crack lengths eccentricit' Ieng ies e == 0. 1 and 0. 7 respectively. Fore = 0.1, practical crack fhs are b for hea ound by A... m" and A.. m1,? The bounds in each group are narrower 0 7 "Y section s and wi.d er for light sections. For all I-beams, >w. ?min is ab o u t ? and w ~- rnax about 0.98. Some heavy sections with large eccentricity e = 0.7 OlJld h a\te Val lirtipiy ues of A... . .,, greater than A., 6 33 Th' m"" as shown in Figure - - ,s means th 0 at the geometry of these sections limits the eccentri.c ,? ty t e < 104 0.7. 6.3 Th ree-tip Cracked I-beams SIFs for th . e Web and flange crack tips of a three-tip cracked I-beam are 9tven b Yequar . Web ion 3.6. The correction factors ' equation 3. 7, are functions of crack I ength Aw, flange crack length A,, and flanges-to-web area ratio ~- 6.J.1 p ara,- .-. eter Ranges SIFs . ? a calculated With FEAs (Table 3.1) were fitted for the following cracks nd load? ings: ? 'Wo c k rack tips: correction factors 1w and f' for the web and flange crac tips ? 'Wo lo d. a mgs: tension and bending ? Number O f . . . ? - 4 'h curve fitting: (2 crack tips) x (2 toadmg) - e Pararnet er for each curve fitting were: ? Et ,ght cross-sectional area ratios: (3 = 0.83, 1.00, 1.25, 1?37? 1?53? 1?69' 1.91 a ' nd 2.05 ? Seven Web crack lengths:,\ w = 0.1 to 0. 7, in steps of 0? 1 ? N? ine flange crack lengths: A,= 0.1 to 0.9, in steps of 0? 1 ? iotat s ampte points for each curve fitting: (B 13 1 k lengths) = 504 va ues) x (7 web crack lengths) x (9 flange crac 105 6.J.2 E Quation s f or Three-ti.p Cracks under Tensi.o n b ihe Preliminary results suggest that a polynomial ratio equation is the est fun . ct1ona1 i . . orm for f1tt1ng SIFs of three-tip cracks. Figures 6.34 and 6.35 Con,p are Pred? Vv. icted and calculated SIFs for the flange crack tips of I-beams 40x149 (f3 - - 0.83) and W18x97 (/3 = 2.05) under tension. Based on th . t? e preliminary results, different polynomial ratio equa 10ns Were tried. Th . . . e foltow,ng 26-term equation 6. 7 yielded the best fit. ( a 1 + a2 f3 + a {3 2 3 +a 4A w +a 5 A~ +a A~ +a 7 AF +a s A; + 6 f "' ~ A~ + a10 f3AwAF +a 11 /3A~AF +a12 f3AwA; +a13 f3A~A;) (6.7) (1 +a14 f3 +a1sf32 +a1aAw +a17A~ +a1sA~ +B19AF +a20A; + a A3 2 2 nj.2 A2) 'he 21 F+a22f3AwAF +a23f3AwAF+a24f3AwAF+a25I-' w F Unknow ? ? Pl t n coefficients of equation 6. 7 were determined with Sigma O ? Coek! ?ricients for Web Crack Tip 1 able 6 ?2 1i? sts the 25 unknown coefficients of equa t?i on 6 ? 7 for predicting Corre . ction fact b m under tension . .,.. ors for the web crack of a three-tip cracked I- ea ?he avera ge value of all correction factors is-; = 0.9517? FEA F'igure 6 3 . t the calculated ? 6 shows the errors in fitting equation 6? 7 0 Corre . Chon fact . ed first by f3 values t ors. The points in the figure are agatn arrang ron-, 0 -83 to 2 5 t o 7 in steps of 0.1 , .o , then by web crack lengths Aw from 0-1 0 ? ana f'1 '1a//y b . st s of o 1 Y flange crack lengths A from 0.1 to 0.9 ,n ep ? ? I In Pig tt d versus~. Aw, and A, re ures 6.37 to 6.39, the fitting errors are plo e Spective1 . o/c and the maximum ri Y. The maximum positive fitting error ,s 3-34 0 ' e9ative err . . . 2% for most sample or is -3.58%. But the fitting errors lie with in :1: 106 _.. ... Points t ? able 6 ,- .3 ists the 9 three-tip cracks whose absolute fitting errors are Qreater th 3 an %. Evidently, fitting errors are largest when the web and flange crack Ie ngth s are greatly mismatched, that is, a short web crack joined with a 1O ng flange crack or a Io ng web crack 1. 0.,n 6 ed wi .t h a short flange crack. Fi.g ures -40 and 6 41 th ? are surface plots for I-beams W40x149 and W18x97 showing at the Predict' . ion equation 6. 7 yields good agreement with the calculated correct? ion f.act ors. COefficients for Flange Crack Tip The~k . f +&. nown coefficients of equation 6. 7 for the flange crack tips O a "'''ee .. r,p cracked I-beam under tension are given in Table 6.2. The average value of au co,~e ct ?i on factors for the flange crack tips is -; = 1? 8 703 ? FEA 8 The fittin errors and ? F 6 42 to g their distributions are shown 1n igures ? -45. Th . e maximum positive fitting error is 4.42%, and the maximum negative erro,. is --4. 76?0 F . ? 2o/c0 Table 6 4 lists th ? or most data the fitting errors are w1th1n :t ? ? e15 I three t' th 3% These - 'P cracks whose absolute fitting errors are greater an ? errors v? Occur in th I flange crack or ree-tip cracks with short web crack and ong 'Ce v be ersa ? F' f uation 6 7 for I-igures 6.41 and 6.42, which are surface plots O eq ? 8 1lls \/V40 . . tion fits well the ~ >c 14 9 and W18x97, indicate that the prediction equa ~ICIJ/at edco rrection factors. 107 6.J.J e quations to r Three-tip Cracks under Bending The sa Pred; me 26-term po Iy nom,.a l rati.o of equation 6 7 was also used to ct the ? correction fact cracked ors for the web and flange crack tips of three-tip I-beam underbend~g CO effj . ?cients fi or Web Crack Tip The . In th; coefficients 0 f equati.o n 6. 7 for the web crack are given in Table 6.2. s case negar 'the maxim um posi.t i.v e fi.t ti.n g error is 6.31%, and the maximum tve er . 6.4 t ror is -6 ? SOo/c wi. 0 ? th most falling within ?2% (Figures 6.48). Figures 9 0 6 51 lists th . show the f'It t'm g error versus /3, Aw and A, respectively. Table 6.5 e 37 th large ree-ti P cracks whose absolute fitting errors exceed 3%. These errors . crack . 'in most cases, occur when the web crack is short and the flange IS long. Th anQ Vv e surface plots in Figures 6.52 and 6.53 for I-beams W40X149 1BX97 ? Co fr, . iftustrate th e good prediction accuracy. 8 ?c?ents fi or Flange Crack Tip I ihe coeff' . able ,c,ents of equation 6. 7 for the Range crack tips are given in 6.2. The . flegat? maximum positive fitting error is 4.14%, and the maximum IVe 1 error is -3 o . 6 7 9 ?66 1/o. The average of all the correcti .o n . - -factors is f - r- FEA 8 . ror betw most sample points, the fitting error lies within ?2%. The errors een th are e Pred?ic te d and calculated correction factors and thei?r d"i s tr ,' b uf , on s Sho 6 th tip Wn in F?i gures 6.54 to 6.57. As listed in Table 6.6, there are only ree- crack s Whose ab IO t of equation solute fitting errors exceed 3%. Surface P s 108 6,7for, b - eams W4o e X149 and W18X97 are shown in Figures 6.58 and 6.59. As XPected, the ob . . . . . tamed emprncal equation produces good f1tt1ng results. 6.4 Far ?Que and Fracture Analysis 6,4, 1 I "Put for Analysis Before a . . . ~ cracked bridge girder is analyzed for fatigue and fracture, the 10110,.,. rvlOg infor . ?? mation must be obtained: '"lfer;a , Properties ? M . easure the fracture toughness, K,c, at the loweSt service temperature, or estimate the fracture toughness from the Charpy If far V-notch energy (CVN) tests. '9ue anal .. Ysis is needed, then: ? Estimate the threshold value of the SIF range, llK,h ? ? Obtain the parameters C and n of the crack growth rate equation C (Paris' Law). rackG eornetry and Dimension ? T . h nd the eccentricity, Wo-t,p Web crack: measure the crack lengt a th . ck to the centroid of at is, the distance from mid-length of the era the girder section. ? Th n e crack lengths. The ree-tip crack: measure the web and fla g ck tip to the junction Web crack length is the distance from the era 109 Point between web and flange. loading ? Obtain the dead load and live toad moments at the cracked If section fro m th e be ndi.n g moment di.a gram. fatigue a natysis is needed, then: ? Count the number of trucks that cross the bridge every day (ADTT S or average daily truck traffic) tress I ntens;ty Factor ? Calculate the SIF, K, by choosing Equation 6.3 or 6. 7 according to crack type and loading condition, or using the software FACI - a Windows application software for Fracture Analysis of Cracked I- beams Written by the author. 8.4., ?< A nalys;s Methods a With th e above m. format ,.o n as i.n put, pe rfo rm ",'r a cture and fatigue na,Yses as follows? f:rac... . ""'9 An afys,.s ? Ca t b' ed dead toad cu/ate the maximum SIF K under com ,n ? max' and live load stresses . ? Compare K with the fracture toughnesS, K,, ' reduced by a mu ti mmend to the sa ety factor (S. F.): if Kmax :? K1, I (S.F.)' reco 110 bridge en gm? eer the closi.n g of the traffic lane above the girder, or th e closing of the entire bridge if the member is fracture critical; if Km,x < K,c I (S.F.), perform fatigue crack propagation analysis to determine the remaining safe life and recommend a crack arrest treatment. l=au 9ue Analysis ? Calculate SIF range, llK, corresponding to stress ranges induced by the fatigue truck as defined in the Specifications for Highway Bridges (1994) . ? Compare thellK value with the crack growth threshold, AK,,,: if ~"K > llK,h for one of the crack tips, perform &,.a t?, gue an alysis?' if AK s AK,h for all crack tips, then the cracks will not propagate and no fatigue analysis is needed. Fatigue analysis for both two-tip (on the web) and three-tip ( ation equations. symmetric) cracked I-beams involves two propa g ? Two-tip crack: da,ower - C ('AK ) n - u 1ow?r ? dN Three-tip crack: daw - (AV )n -- - C '-"'w dN 111 da, - = C(IJ.K )n ? dN ' Sub f s I t . ute /),,K mto the crack growth rate equation and calculate th e number of cycles, IJ.N, needed for the crack to grow from its current length to the critical length where K,,,. = K,1:l(S.F.) ? ? Convert the calculated number of truck crossings, N, into a number of days: days = N / ADTT. ? Advise the bridge engineer how much time he has to repair the cracked bridge girders. 6.5 C omposite and Singly Symmetric Beams Ii ihe above equations 6.3 and 6. 7 were deve/Oped for rolled w..srnipes. be Owe\ler, in en . . . etric beams may 9tneer,ng practice, composite and stngly symm Used. As . . . shown tn Figure 6.60, such sections constst of. ? Con,po . site rolled beam ? Con,p . stee I ection os,te Plate girder with doubly symmetric s ? Noncom t. ? tee/ section PS te plate girder with singly symmetric s ? C0n,Po8 n?8 ? t el section t Plate girder with singly symmetnc s e r, br;d 9e gird _,.ted to bending. ers, these sections are always sub}""'' the e . . for cracks in doubly a Quat,ons for calculating correction factors Y,n'lletr;c b ection types listed eams are not directly applicable to the tour 5 112 - abo11 e. In the foll . owmg, procedures are outlined for obtaining approximate Correct' ton factors. 6.s.1 C omposite Rolled Beam the folio ? . wmg Procedure is recommended for calculating SIFs for a cracked compo s . ite rolled beam (Figure 6.60a): ? Ca/cu/at e the position of the elastic neutral axis and the stress distribut' ion through the depth of the composite rolled beam. ? Separate the stress distribution through the depth of the steel section into an . ax,aI tensile stress component: ab, + a,, a = --- a 2 and ab ending stress component ab, - a,, ab =-- -2 0 br - Stress at mid-thickness of bottom flange or, "' stress at mid-thickness of top flange ? Deter . I section with a two-tip mine the values of A e and (3 for the stee W ' crack defined for rolled beams and Aw , A, and (3 for the three-tip crack as Witho . 3 3 to 3.5 and 3.8 to ut a composite concrete deck (equations ? 3, 10). ? ~or tw . /cu/ate the correction o~t,p and three-tip cracked I-beams, ca 113 a factors With . equations 6.3 and 6.7 respectively. ? Ca/cu/ate th S . e IFs with equations 3.1 and 3.6, using aa as the reference stress for th . e axial component and ab for the bending component. The fina1 S/F values are obtained by superposing the SIF values for axial tension and bending. 6.s.2 C 0 mp os,?t e Plate Girder with Doubly Symmetr,?c s ec t?i on The Pro cedure for calculating S/Fs for a composite plate gm?d e r rWI n doubly symmetric steel section is the same as that for a composite rolled beam. Sin,.. ..... ~ equaf tons 6.3 and 6. 7 were developed for rolled W-shapeS, the Correct?1 0n fa ct o rs are strictly valid only for sections whose flanges-to -web area ratio 13 nd a skeleton ratio y lie within the band shown in Figure 3.4? If the st . 3 4 equations 6.3 ee/ section falls inside the ~-y band of Figure ? ' ane16 7 . ? Yield accurate 'rd r: S/Fs for doubly symmetric welded gi e ? But th tive) if the section f. ' e calculated S/Fs are overestimated (conserva ans b . a ove th ative) if the section f; e /3-y band and underestimated (non-conserv ans b eJaw th d be determined from ,:~ e /3-y band. In this latter case, SIFs shoul 4s of the s .. Pecif1c steel section. 6.s.3 N . metric Section oncompsite Plate Girder with Singly Sym F'or a . mmetric steel section noncomposite plate girder with a singly sy 114 (top flan ge smaller th ,n.. ,roeed an bottom flange) as shown in Figure 6.60c, the following Ure of ca/c I . u atmg S/Fs is recommended: ? Ca/cu/ate th .. e Pos1t10n of the elastic neutral axis. ? Ca/cu/ate th e stress distribution through the depth of singly symmetric stee/ section. ? Retain th . e stress at mid-thickness of the bottom flange, abt ' Figure 6.61, as the rei erence stress in calculating S/Fs. ? Choose th . . . . e elastic neutral axis as a second axis of symmetnc as ,n ;:?1 9Ure 6.6 1c. ?~~ - f mme the values of Aw, g and {3 (equations 3.3 to 3.5) for a sec ion With a two-tip crack and Aw' A, and {3 (equations 3.8 to 3.10) for a section With a th ree-tip crack from the equivalent doubly symmetn?c se ction that is created b Y m1. rroring the lower portion about th e e/ astic neutral axis. ? Ca/cu/at . d 6 7 for two-tip e the correction factors with equations 6.3 an ? and thre t? e- IP cracked /-beams respectively. ? Ca/cu/at E3 8ca e the S/Fs with equations 3.1 and 3.6. llse eq . . II d /-beam or plate 9ir" llations 6. 3 and 6. 7 are for doubly symmetnc ro e ~er t ? se . ' he rec . for singly symmetnc ommended procedure of calculating S/Fs %~ - . t~ is an . d d to determine i approximation. FEA calculations are nee e ccu~ th ? acy of s olution exist for is as llch an approximation. Unfortunately, no s e. Fo r rnor ded e accurate results FEA is recommen ? I 115 6.S.4 C omposite Plate G,.r der with Singly Symmetric Section For th ? is case, SIF Proced s should be calculated using a combination of the ures d escrib e d ? in section 6.5.1 and 6.5.3. 6.6 40 s -Ft s?i mple-span Composite Beam IPs !>. qre c I a cutated i Problem or a three-tip cracked I-beam as an example. Given . has a is a simpl e-span composite girder bridge with 2 lanes of traffic. It deck w? I'A"<, 7 x idth of 33'- 4" and a span length of 40 ft. The girders consi?s t of a 8 cent 4 section w,.t h a cover plate welded to the bottom flange along the 2S-ft er Port1?0 Conn n of th e span. The 7-in. thick concrete slab is composi.t ely ected to th . . F'igure e girders with shear studs. The girder elevation is shown ,n 6. 62 and th . d at the e composite section in Figure 6.63. A three-tip crack is faun Welded W.. ,, end of the cover plate, 7'-6" from the end bearing. 8 apeG eometry Properties 1hevv t?ff ~ '27 x 84 section has a web depth d = 26.71 in ., web thickness 0 ~h -46 i?n ., flang . . k ss t - o 64 in . er e width b = 9 96 ,?n and flange thtc ne ' - ? efoi ' . . , e, the ? ? t ? , ::: Web depth between the upper and tower junction pain s is d - 1, :::: 26 07 . h web and flange are .. ., ~ ? m. and the cross-sectional areas oft e '1; t"" 11 9 t n and Y are c . 9 in .2 and A = t b = 26 71 in .2. parame ers,., lllated as: 13 _ ' f ' . . 62 The section t, .e s 2 A, IAw = 1.06 and y d/b, = 2= ? ? 116 Wneitah r the lo igure 3.4, meaning that the SIF can be calculated wer bound of F equation 6. 7 Crack Lengths length The flange er ack length 1. s assumed to be 2a, ? 5 in . and the web crack they ? uch cracks are typical of those found in the 1-95 bridge over a w -- 3 in S ellow Mill P 0nd r.n Connecticut. ? o .502 and I., ? '!d:,;i. ? ~26.07 ? 0.11s p Th ereforeI A, -- -2a, =- 5 roperties f b, 9.96 o Com pos1. te Section bea own in F igure 6.63, section properties are computed for the ste e As sh I nd 3 2 lllcom positel Y connected to the concrete deck, wi.t h n = 8 a n - 4? lo 2825 A d Ad 0 0 0 24.8 23719 300 23419 1312 26543 73.5 17.85 1312 98.3 . 4 2 - 9028 1n. 13 1 de ::: -1-3=1:2. /NA 26543 - 13.35 < -O d 98.3 - 13 .35 in. l Op Of ateet - 13?35 - 13.35 - O in. - 1 3?35 + 13.35 = 26.70 in. s Qot ? 01 steet ::: -9028 338 in 3 ? 26 .7 A d 117 Stee/ Section 24.8 0 0 0 2825 Con c. 84x7/24 24.5 17.85 437.3 7806 100 7906 t 49.3 437.3 10731 4 /NA = 10731 - 8.87 X 437.3 = 6852 in. d ?op s Of a tee, ::: 1 3. 3 5 - 8. 8 7 = 4.48 in. d = 13.35 + 8.87 = 22.22 in . Bot. of steel ?op Of &lee, ::: 6852 = ~ 308 in. 3 ~ = 1529 in. 3 = SBot. of steel 22 _22 loaa sand IVI 0 ments the mo . . . Ill rnent diagram is shown in Figure 6.64 from which the applied 0rne nts at th D e Weld end of the cover plate are determined as: ead/oad rnornent? Dead ? MoL, = 102 .27 kip - ft load live rnornent? ? M 0L = 18. 18 kip -ft 1 2 0 ad + ? Str impact: M - . LL ? , - 295.45 ktp - ft 8 Sses. in Steel Beam ' he st re a/cu lated for "'1 sses at the top and bottom of the steel beam are c cl, ' M 0L2 , and M LL . I ? 'he dead tee/ beam alone: rh load moment M DL is 5 supported by the erefi - 1 Ore, the calculated for the )p fl stresses at the top and bottom of the beam are an9e: cl, ::: - MOL d/2 ~ 102.27><12><13.35 = - 5.80 ksi d bott 1 2825 om flange : 118 ODL, :: 5.80 ksi The dead Io ad moment MDL is supported by the composite section with 3n:::: 24 Th a ? e stresses at the top and bottom of the beam are calculated for the top flange: a M DL :: - 2 -s- --- D=L-. ?- = 18.18><12 = -0.14ksi and b Top or Slee/ 1529 ottom flange: o M OL, ::: S DLa = 18.18 x12 = 0.71 ksi Bot. or 11ee/ 308 The live load + impact load moment M is supported by the LL? I compo s,? t e section with n = 8. The stresses at the top and bottom of beam are for th e top flange: 0 LL., "' o.o ks; and b 0 tt 0 m flange: OLL~, == LL+t = 295.45x12 = 10_49 ksi 8 aot 338 Th ' or Slee/ . d b ttom flanges of the steel erefore, the stresses acting on the top an ? beam are: a l'op ::: - (5.80 + 0.14 + 0) = - 5.94 ksi 0 801to111 ::: (5.80 + o. 71 + 10.49) = 17.0 ksi A s shown in Figure 6.65, such linear IY d . t ?buted stresses are ,s " deeo d. omponents of: mposec1 into axial tension and pure ben ,ng c 01 o.,. + Orop - 0so11Dm = 11.47 ksi 0 1 1 ::: _op Bottom = 5.53 ksi and 0 b == - 2 2 ?hen, the corresponding reference stresses are: 0 d 26.07 - 11 20 ksi 1 :::: o1 i 11 47" - - . t ::: 5.53 ksi and ob = o~ d = ? 26.71 119 SIFs for Web and Flange Crack Tips The SIFs for the web and flange crack tips are obtained by superposition : K, = <~tat+ ~b ab) fria, With parameter f3 = 1.06,A, = 0.502 and Aw= 0.112, the correction factors are determined from equation 6. 7: t , = 1 o1 96 t w, ? ' w,b = 0.8838, ,,. t = 1.2401 and f f. b = 1.2363. Therefore, the SIFs are given by: Kw= (1.0196lC5.53 +0.9938,c 11.47) ?3.1416lC3 = 52.30 ksi{in: K, = (1 .2408 ,c 5.53 + 1.2363 )( 11.4 7) ?3.1416 x 2.5 = 58.97 ksi fin: 6. 7 Conclusions SIF equations for the two-tip and three-tip cracked I-beams under tension or bending were obtained. For a two-tip cracked I-beam, the best fitting equation is a 15-term polynomial equation of web crack length, eccentricity, anc flanges-to-web area ratio. For a three-tip cracked I-beam, the best fitting equation is a 26-term polynomial ratio equation of web crack length, flange crack length and flanges-to-web area ratio. The maximum positive and negatiw fitting errors are summarized in Table 6. 7. For the interpolation values of the empirical equations between the sample points, no solutions are available for direct comparison, however the equation appears reasonable based on 120 engineering judgement. The obtained SIF equations can be applied to I-beams used in civil engineering from WB to W40 with a wide range of crack geometries. Howe, to use these empirical equations to calculate SIFs for ~elded girders, the relationship between the skeleton ratio y and cross-sectional area ratio 13 should fall into the fuzzy band of Figure 3.4 as discussed in Chapter 3. If the applied loading is simple uniform tension or pure bending, the abovi obtained empirical equations can be applied directly. If the applied loading ca1 be decomposed into the components of uniform tension and pure bending, thE equations can be used based on the principle of superposition. Otherwise, the obtained equations cannot be used directly. In this case other numerical methods, e.g., weight function methods, should be used. The SIF equations presented in this chapter provide the necessary reference solutions. 121 Chapter 7: Summary and Conclusions Based on the results of this study, the following conclusions are dratJ Parameters ? The SIF or correction factor for a two-tip crack in the web of an I-beam ar function of eccentricity e = el(d/2), web crack length Aw= a,)(d/2 - e), anc flanges-to-web cross-sectional area ratio J3 = 2A,IAw. The SIF for a three-ti crack in the web and flange of an I-beam are a function of web crack lengt 1 Aw= awkii, flange crack length>.,= a,l(b/2), and flanges-to-web cross- sectional area ratio /3. ? The depth-width ratio y = ~lb, of a cracked I-beam is not a significant parameter in the calculation of SIFs. Its effect becomes noticeable only for very long cracks. For all I-beams listed in the LRFD Manual, the flanges-to- web area ratio and the depth-width ratio are linearly related, within a bandwidth. Therefore, the I-beams can be characterized in terms of parameter 13 alone. Two-tip Cracked I-beam ? The calculated (FEA) StFs have smooth and flat relationships with parameters Aw, e, and /3. The 15-term polynomial equation 6.3 (a cubic equation of the three parameters) best fits the calculated Sf Fs for the upper or lower crack tips under tension or bending. The prediction errors are withir ?1 % for most of the data. 122 ? The web crack is constrained by the two flanges. As a result, the SIF for the web crack is smaller than that for a single plate. The flange constraint increases as the .crack tip approaches the junction point of web and fla nge. Increasing the flanges-to-web area ratio also increases the constraining effect. Three-tip Cracked I-beam ? The calculated S/Fs have smooth and flat relationships with parameters Aw, A, and fl The 26-term polynomial-ratio equation 6. 7 (a fourth power equation of the three parameters) best fits the calculated SIFs for the web and flange crack tips under tension or bending. The prediction errors are again within ?2% for most data. ? The SIFs are affected by the interaction forces between the web and flange. These forces depend on the compatibility of the web and flange displacements along the junction line. If the displacements along the centerline of the flange are larger than those of the corresponding points along the web edge, the web tends to close the flange crack while the flange tends to open the web crack. Conversely, if the displacements along the centerline of the flange are smaller than those of the corresponding points along the web edge, the web tends to open the flange crack while the flange tends to close the web crack. ? The interaction forces are negligible when the web and flange cracks - in separated plates - have equal CMODs. Such a pair of crack lengths is 123 called non-interacting, and in this case the SIFs of I-beams can be calculated from those of corresponding single plates. Non-interacting crack lengths help in understanding the behavior of three-tip cracked I-beams and assist in developing a rapid, approximate method of calculating SIFs. Applications ? The SIF equations 6.3 and 6. 7 can be applied to I-beams of sizes WB to W40 and welded girders with 0.83 s ~ :s: 2.05. However, these equations were developed for rolled W-shapes, they should be applied with caution to plate girders because the SIFs may be affected by the depth-width ratio. Section 6.4 outlines approximation procedures for calculating SIFs for composite and singly symmetric sections. 2-D Simplified Modeling ? The 3-D problem of a cracked I-beam can be simplified to a 2-D problem by joining the web and flanges only in the direction of the junction lines. For a two-tip crack, 2-D analysis is accurate to 1% for crack lengths Aw s 0.5 and 5% for Aw s 0.9; for a three-tip crack, 2-D analysis is accurate to 1% except in I-beams with long web and short flange cracks. The agreement between 2-D and 3-D analyses strongly suggests that the interaction between the cracked web and flange is controlled mostly by the compatibility of displacements in the direction of the junction line. SIF Calculation from FEA ? The J-integral method yields more accurate SJFs than the displacement- 124 based methods as long as the integral contours are chosen away from the crack tip. ? In the inner region around the crack tip, increasing parameter n, which defines the number of elements around a concentric square, improves the accuracy of SIFs more than increasing parameter m, which defines the number of concentric squares. When n = 8, the mesh is fine enough to give converging results. ? The three displacement-based methods yield nearly equal results. For a coarse mesh of n = 2 or 4, the quarter-point displacement method (equation 2. 12) is more accurate than the nonlinear and linear extrapolation methods (equation 2.7 and 2.10). 125 Appendix A: Benchmark Studies A.1 Objective The objectives of the work described in this appendix are to: ? Choose a suitable mesh pattern - element density and aspect ratio _ in the inner region around the crack tip. ? Examine methods of extracting SIFs from FEA results, including coo (equations 2.7, 2.10, and 2.12) and J-integral methods. A.2 Benchmark Problems The following three geometries/loadings, for which accurate SIF solutions exist, were analyzed: ? Center-cracked finite-width plate under 100-MPa tension (Tada 1973). The plate shown in Figure A.1 a was 2 W =2 00 mm wide, 2h =6 00 mm long, and t = 10 mm thick. ? Edge-cracked finite-width plate under ?100-MPa bending (Brown 1966, Tada 1973). The plate was W= 100 mm wide, 2h = 300 mm long, and t = 10 mm thick Figure A.1b. ? Edge-cracked finite-width plate under 100-MPa tension (Tada 1973). The plate dimensions are shown in Figure A.1 b. For each geometry/loading, the parameters were varied as follows: 126 ? Crack lengths: a/W= 0.1, 0.5 and 0.9 ? Mesh patterns: up to 10 combinations of number of elements around a concentric square m = 2, 3, and 4; and number of concentric squares n = 2, 4, Band 16 (Figure 2.7). ? Methods of calculating SIFs: nonlinear extrapolation (equation 2. 7), linear extrapolation (equation 2.10), quarter-point displacement (equation 2.12), and J-integral (equation 2.26). All three geometries were modeled with eight-node, plane stress elements - ABAQUS designation CPSBR - throughout the inner and outer regions of the plates (Figure 2.6). Elements sharing a node with the crack tip were generated with ABAQUS' 11SINGULAR command (Figure 2.4). SlFs calculated with FEA were compared with those from existing solutions in terms of the ratio K FEA / K exist. A.3 Results and Discussions The results are summarized in Tables A 1, A.2 and A.3 for the center- cracked plate under tension, edge-cracked plate under bending, and edge- cracked plate under tension respectively. 127 A.3.1 Mesh Pattern in Inner Region Effect of Parameter m Presented in the upper halves of Tables A.1 and A.2 are the effect of parameter m on the SIFs for the center-cracked plate under tension and edge- cracked plate under bending respectively. This is done for one crack length, a/W = 0.5, and each of the four values of parameter n = 2, 4, 8 and 16. Clearly parameter m has practically no effect on the SIFs for any of then values and four calculation methods. Effect of Parameter n The lower halves of Tables A 1, A.2 and A.3 show the effect of parameter non the SIFs for all three geometries/loadings. This is done for a constant parameter m =3 and three crack lengths a/W =0 .1, 0.5 and 0.9. The following is concluded from the results: ? For the displacement-based methods, increasing parameter n from 2 to 4 and a improves the accuracy of the SIF; but increasing it further to 16 yields no additional gain in accuracy. ? For the J-integraJ method, parameter n has practically no effect on the accuracy of calculating SIFs. 128 A.3.2 Accuracy J-integral Method SIFs calculated from J values are more accurate than those calculated All J-based S/Fs agree with the existing solutions within ?0.5%, even for the coarsest mesh, m x n = 2 x 2. While in theory the J-integra/ is path-independent, in FEA it is only approximately path-independent. Figure A.2 shows the J-integral values as a function of distance from the crack tip to the intersection point of concentric contours with the crack extension line. The first contour - zero distance - is the crack tip. J-integra/ values were determined for a center-cracked plate under tension, with crack length a = 50 mm and half plate width W = 100 mm. Parameter n was varied from n = 2 to 16 while parameter m was kept constant at m = 3. As can be seen, the accuracy increases with mesh refinement. When the mesh is coarse (m x n = 3 x 2 and 3 x 4), J-integral values should be calculated along the outermost contour. When the mesh is fine (m x n = 3 x 8 and 3 x 16), J-integral values rapidly converge to the exact solution. COD Methods Among the COD methods whose results are given in Tab/es A 1, A.2 and A.3, the quarter-point displacement method (equation 2.12) is the most accurate followed closely by linear extrapolation (equation 2.10). Nonlinear extrapolation (equation 2.7) is the least accurate, but only so for the smaller 129 values of parameter n = 2 and 4. For finer meshes, the differences between equations 2.7, 2.10 and 2.12 are smaller. Figure A.3 compares COD profiles calculated with equations 2.4 and 2.11 for a center-cracked plate under tension, with a/W= 0.5 and mesh pattern m x n = 3 x 8. The COD profiles are nearly equal, the maximum difference being about? 0.2%. Therefore, the corresponding SIFs {equations 2. 7 and 2.10) are also nearly equal. Similar trends can be found for other crack lengths. App~rent SIF As stated in section 2.5.1, equations 2.7, 2.10 and 2.12 give equal SIFs only when the condition 2v8 = Ve {equation 2.13) is satisfied. This strict condition requires the size of the crack-tip element to be zero, which cannot be met in FEA. According to the FEA calculations, (2v8 - vc) > 0.0 (Table A.4) for the central and edge cracks with a/W= 0.1 and the central crack with a/W= 0.5. In these cases, the first derivatives of the apparent SIF equations 2.5 and 2.9 with respect to r, which are the slopes of the curves, become negative. So equations 2.5 and 2.9 approach the crack tip from below as shown in Figures A.4 and A.5. The figures also show that the convex curve equation 2.5 (negative curvature) and straight line equation 2.9 intersect at both the quarter point Band corner point C; at the quarter point, equations 2.5, 2.9 and 2.12 have equal SIF values. Therefore, equation 2.12 always predicts the smallest SIF value, and equation 2.5 predicts the largest. The result of equation 2.10 falls inbetween. 130 For longer crack lengths of a/W= 0.5 and 0.9, the FEA results yield 2v _ 8 Ve < 0 (Table A.4) and, therefore, the first derivatives of equations 2.5 and 2.9 with respect to r become positive. As shown in Figure A.6, the concave equation 2.5 (positive curvature) and straight line equation 2.9 approach the crack tip from above. In this case, equation 2.5 produces the smallest SIF value and equation 2.12 produces the largest. Again, the result of equation 2.10 falls inbetween. A.3.3 Additional Cases Having determined the effects of mesh pattern and method of calculating K from FEA results, the focus of the benchmark studies shifted to further ascertaining accuracy for the following cases: ? Center-cracked plate under tension or bending; with crack lengths 81W = 0.1 to 0.9 in steps of 0.1 for tension, and a!W= 0.1 to 0.9 in steps of 0.2 for bending (Figure A.1a). ? Edge-cracked plate under tension or bending; with crack lengths a/W = 0.1 to 0.9 in steps of 0.1 (Figure A 1b ). ? Eccentrically cracked plate under tension or bending; with crack lengths al(W- e) = 0.1 to 0.9 in steps of 0.2; and eccentricities e/W= 0.1 to 0.9 in steps of 0.2 (Figure A. 7). In all subsequent FEAs, the mesh pattern was m x n = 3 x 8 and S/Fs 131 Were determined from J-integral values. FEA SIFs for the center-cracked plate with 0.1 5 a/1111 5 0.9 agree very Wei/ with those reported by Tada (1973) for tension and Chen and Albrecht (1992) for bending as shown in Figure A.8. Maximum absolute differences are 0.2% at a/W= 0.1 in the former and 0.5% at a!W= 0.7 in the latter. FEA clearly yields accurate SIF values. lsida's (1956) and Benthem and Koiter's (1972) solutions for bending are very accurate only for 0. 1 5 aJW 5 0.3, but the solutions become more inaccurate as the crack length increases. In the second evaluation of accuracy, FEA SIFs for the edge-cracked plate are seen in Figure A.9 to agree very well with those from Tada (1973) for tension and Brown (1966) for bending, with maximum differences of 0. 7% at a!W= 0.1 and 0.2% at a/W= 0.3 respectively. The FEA solution for bending agrees very well with Tada's (1973) solution for a short crack, a/1111= 0.1, and long cracks, 0.6 5 a/W 5 0.9; but the differences are larger for intermediate length cracks, 0.2 5 a/W 5 0.5, with a maximum of 2.2% at a/1111= 0.3. Finally, Figures A.1 0 and A 11 compare the FEA SIFs for an eccentrically cracked plate with those reported by lsida (1965) for tension and Chen and Albrecht (1992) for bending respectively. Again, the agreement is very good for combinations of crack lengths, 0.1 5 a/(W- e) 5 0. 7, and eccentricities, 0.1 ~ e/W 5 0. 7; within these ranges, maximum differences are? 0.5%. Larger differences are found for the largest crack length, al(W- e) = 0.9, and/or eccentricity, e/W = 0.9, at which differences are? 3%. 132 In summary, FEA with m x n = 3 x 8 mesh pattern at the crack tip yields very accurate J-based SIFs. This was shown consistently for all combinations of three geometries and two loading conditions. FEA SIFs were compared with those from existing solutions in tenns of the ratio K FEA I K exist. Existing solutions were used as the yardstick simply because they preceded the FEA solutions of this study, not because existing solutions are necessarily more accurate. Since no exact analytical solutions exist for any of the three geometries, it is not known whether FEA or existing solutions are more accurate. A.4 Conclusions Based on the results of the benchmark studies, the following conclusions can be drawn: ? The J-integra/ method yields the most accurate SIF values, even for a coarse mesh. ? Increasing parameter n improves the accuracy of SIFs more than increasing parameter m. ? When n = 8, the mesh is fine enough the results converge. ? The displacement-based methods produce close results. For coarse meshes of n = 2 and 4, the quarter-point displacement method (equation 2.12) is the most accurate. 133 Appendix B: Tables and Figures 134 Table 1.1. Methods of determining stress intensity factors. Category 1 Category 2 Category 3 Handbooks Superposition Collocation (Mapping} Stress Concentration Integral Transform/continuous Dislocation Stress Distribution Body Force Method Green's Function Edge Function Method Weight Function Method of Lines Compounding Finite Elements Method Boundary Element Method Alternating Technique 135 Table 3.1. W-shapes used in calculations of SIFs. W-Shape d1 tw bf t, V Shape (mm) (mm) (mm) (mm) 0.83 W40X 1498 ' b, d 949.2 16.0 300.0 21.1 3.16 1 0.83 W24X628 588.0 10.9 178.8 15.0 3.29 2 1.00 W30X1088'b 738.4 13.8 266.1 19.3 2.77 1 1.00 W40X1678 954.2 16.5 300.0 26.0 3.18 2 1.11 W40X 1928 " b. C 949.2 18.0 449.8 21.1 2.11 1 1.11 W36X17oa,c 890.8 17.3 305.6 27.9 2.91 2 1.25 W21X62a,b 517.5 10.2 209.3 15.6 2.47 1 1.25 w21x114a 669.5 14.5 255.8 23.6 2.62 2 1.37 W40X 1998 ? b, d 955.2 16.5 400.1 27.1 2.39 1 1.37 W24X1038 598.2 14.0 228.6 24.9 2.62 2 1.69 W30X191a. b 749.2 18.0 382.0 30.1 1.96 1 1.69 W36X280a 887.7 22.5 421.5 39.9 2.11 2 1.91 W21X101 8 ' b 522.2 12.7 312.2 20.3 1.67 1 1.91 W21X1478 531.1 18.3 317.8 29.2 1.67 2 1.53 W33X201b 826.26 18.2 399.9 29.2 2.07 2.05 W18X97b,d 450.09 13.6 283.1 22.1 1.59 a W-shapes used in validation of J3 (chapter 3) b W-shapes for which SIFs were calculated (chapter 5) c W-shapes deleted from subsequent analyses (chapters 3 and 5) d W-shapes used in comparison of 2-D and 3-D modelings(chapter 2) 136 Table 4.1. Non-interacting crack lengths for T-section. Non-interacting crack lengths T-flange T-web T-web (tension) (tension) (bending) 2a,IW a, (mm) awlW aw (mm) aw/W aw (mm) 0.0 0.0 0.0 0.0 0.0 0.0 0.05 5.0 0.0171 3.42 0.0172 3.44 0.10 10.0 0.0342 6,84 0.0346 6.92 0.15 15.0 0.0513 10.26 0.0524 10.47 0.20 20.0 0.0684 13.68 0.0704 14.07 0.25 25.0 0.0855 17.10 0.0888 17.75 0.30 30.08 0.1028 20_55? 0.1077 21.53 0.35 35.0 0.1201 24.01 0.1271 25.42 0.40 40.0 0.1377 27.54 0.1473 29.45 0.45 45.0 0.1556 31 .12 0.1681 33.61 0.50 5o.o??b 0.1740 34 .798 ? b 0.1898 37.95 0.55 55.0 0.1929 38.58 0.2124 42.47 0.60 60.0 0.2126 42.51 0.2361 47.21 0.65 65.0 0.2332 46.63 0.2610 52.20 0.70 70.oa 0.2549 50,98a 0.2875 57.49 0.75 75.0 0.2783 55.66 0.3159 63.17 0,80 80.0 0.3039 60.78 0.3468 69.36 0.85 85.0 0.3328 66.56 0.3816 76.32 0,90 90.0 0.3673 73.45 0.4227 84.53 a Crack pairs for comparing SIFs for T-section and single plates. b Crack pairs for comparing CODs and displacements along junction line. 137 Table 4.2. Non-interacting crack lengths for three-tip cracked I-beams . Web Flange Depth- Web Non-interacting flange crack length, 2a1/br depth width width crack ratio length Tension Bending w-shapes di b, d/b, aw/d1 From From From From From From (mm) (mm) CMOD K 1 (FEA) Kw (FEA) CMOD K 1(FEA) Kw (FEA) W40X149 949.2 300.0 3.16 0.05 0.42 . . . . . . 0.41 . . . ... 0.10 0.67 0.60 0.65 0.66 0.60 0.62 W30 X108 738.4 266.1 2.77 0.05 0.37 . . . . . . 0.37 . .. ... 0.10 0.63 0.57 0.60 0.61 0.55 0.57 W21 X 62 517.5 209.3 2.47 0.05 0.34 . . . . . . 0.34 . .. ... 0.10 0.58 0.52 0.56 0.57 0.50 0.52 ...... W40 X199 955.2 400.1 2.39 0.05 0.33 . . . . . . 0.32 . . . ... (..) co 0.10 0.57 0.51 0.54 0.56 0.50 0.51 W33 X201 826.3 399.9 2.07 0.05 0.29 . . . . . . 0.29 . . . ... 0.10 0.52 0.50 0.50 0.51 0.46 0.46 0.15 0.68 . . . . . . 0.66 . . . ... W30 X191 749.2 382.0 1.96 0.05 0.28 . . . . . . 0.27 . . . ... 0.10 0.50 0.47 0.47 0.49 0.45 0.45 0.15 0.67 . . . . . . 0.65 . . . ... W21 X101 522.2 312.2 1.67 0.05 0.24 . . . . . . 0.24 . . . ... 0.10 0.45 0.41 0.42 0.43 0.40 0.40 0.15 0.61 . . . . . . 0.59 . . . ... W18 X 97 450.0 283.1 1.59 0.05 0.23 . . . . . . 0.22 . . . ... 0.10 0.43 0.40 0.40 0.42 0.39 0.38 0.15 0.59 . . . . . . 0.57 . . . ... 0.20 0.69 Table 5.1. Correction factors fA for two-tip cracked I-beam under tension; upper tip. 13 = e;= Aw= 2~/(d 1 - 2e) 2A,/Aw 2e/d 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 1.0019 1.0076 1.0169 1.0307 1.0475 1.0678 1.0866 1.0979 1.0562 0.1 1.0015 1.0062 1.0139 1.0243 1.0375 1.0531 1.0705 1.0875 1.0995 0.3 1.0011 1.0043 1.0098 1.0170 1.0257 1.0357 1.0465 1.0571 1.0653 0.83 0.5 1.0007 1.0031 1.0067 1.0114 1.0172 1.0235 1.0302 1.0358 1.0386 0.7 1.0002 1.0010 1.0024 1.0040 1.0058 1.0078 1.0085 1.0082 1.0052 0.9 0.9995 0.9983 0.9963 0.9936 0.9893 0.9884 0.9783 0.9706 0.9621 0.0 1.0015 1.0060 1.0138 1.0244 1.0371 1.0512 1.0624 1.0595 1.0048 0.1 1.0012 1.0052 1.0111 1.0194 1.0297 1.0413 1.0534 1.0637 1.0671 ...... 0.3 1.0009 1 .0035 1.0077 1.0132 1.0199 1.0275 1.0352 1.0421 1.0463 w 1.00 co 0.5 1.0005 1.0022 1.0049 1.0086 1.0124 1.0166 1.0209 1.0237 1.0235 0.7 1.0001 1.0004 1.0009 1.0015 1.0020 1.0022 1.0014 0.9992 0.9939 0.9 0.9994 0.9979 0.9954 0.9919 0.9870 0.9812 0.9743 0.9658 0.9565 0.0 1.0014 1.0055 1.0122 1.0223 1.0326 1.0464 1.0523 1.0546 0.9895 0.1 1.0012 1.0047 1.0103 1.0178 1.0271 1.0376 1.0483 1.0572 1.0595 0.3 1.0008 1.0033 1.0072 1.0124 1.0184 1.0253 1.0322 1.0383 1.0417 1.11 0.5 1.0006 1.0021 1.0049 1.0080 1.0117 1.0158 1.0195 1.0221 1.0220 0.7 1.0002 1.0006 1.0013 1.0021 1.0030 1.0035 1.0033 1.0013 0.9967 0.9 0.9995 0.9983 0.9965 0.9941 0.9903 0.9855 0.9798 0.9724 0.9642 0.0 1.0011 1.0044 1.0099 1.0170 1.0254 1.0319 1.0352 1.0180 0.9507 0.1 1.0009 1.0038 1.0083 1.0143 1.0213 1.0289 1.0356 1.0393 1.0341 0.3 1.0006 1.0025 1.0055 1.0096 1.0141 1.0189 1.0236 1.0268 1.0268 1.25 0.5 1.0004 1.0014 1.0030 1.0051 1.0074 1.0096 1.0113 1.0109 1.0082 0.7 0.9999 0.9997 0.9994 0.9989 0.9979 0.9964 0.9939 0.9895 0.9822 0.9 0.9992 0.9973 0.9943 0.9903 0.9842 0.9774 0.9694 0.9600 0.9500 Table 5.1. Correction factors fA for two-tip cracked I-beam under tension; upper tip. P= E= >..,, =2 aw/(dl -2e) 2A,/Aw 2e/d 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 1.0009 1.0038 1.0083 1.0142 1.0205 1.0255 1.0246 1.0069 0.9334 0.1 1.0008 1.0031 1.0069 1.0116 1.0172 1.0229 1.0271 1.0276 1.0191 1.37 0.3 1.0005 1.0021 1.0044 1.0075 1.0111 1.0148 1.0180 1.0196 1.0177 0.5 1.0002 1.0009 1.0022 1.0034 1.0050 1.0063 1.0068 1.0057 1.0014 0.7 0.9999 0.9994 0.9987 0.9977 0.9962 0.9943 0.9909 0.9858 0.9778 0.9 0.9992 0.9972 0.9939 0.9894 0.9832 0.9760 0.9677 0.9577 0.9472 0.0 1.0007 1.0031 1.0064 1.0106 1.0147 1.0165 1.0115 0.9882 0.9095 0.1 1.0006 1.0023 1.0051 1.0089 1.0128 1.0164 1.0179 1.0151 1.0025 _.. 1.53 0.3 1.0003 1.0015 1.0034 1.0056 1.0080 1.0104 1.0119 1.0118 1.0081 .t:,,. 0 0.5 1.0002 1.0005 1.0012 1.0019 1.0025 1.0027 1.0021 0.9997 0.9941 0.7 0.9998 0.9990 0.9980 0.9965 0.9944 0.9912 0.9874 0.9815 0.9729 0.9 0.9992 0.9971 0.9934 0.9889 0.9825 0.9749 0.9664 0.9561 0.9455 0.0 1.0005 1.0024 1.0048 1.0075 1.0097 1.0081 0.9997 0.9692 0.8856 0.1 1.0004 1.0018 1.0041 1.0067 1.0092 1.0110 1.0104 1.0049 0.9890 0.3 1.0002 1.0011 1.0025 1.0040 1.0056 1.0067 1.0069 1.0054 1.0001 1.69 0.5 1.0000 1.0001 1.0003 1.0004 1.0001 0.9997 0.9978 0.9944 0.9879 0.7 0.9997 0.9986 0.9972 0.9952 0.9924 0.9886 0.9840 0.9771 0.9681 0.9 0.9991 0.9969 0.9929 0.9880 0.9813 0.9734 0.9644 0.9539 0.9428 0.0 1.0004 1.0017 1.0033 1.0051 1.0051 1.0034 0.9905 0.9632 0.8741 0.1 1.0003 1.0013 1.0028 1.0045 1 .0061 1.0065 1.0042 0.9968 0.9794 0.3 1.0002 1.0007 1.0016 1.0024 1.0035 1.0039 1.0032 1.0004 0.9942 1.91 0.5 0.9999 0.9999 0.9998 0.9996 0.9990 0.9979 0.9956 0.9916 0.9842 0.7 0.9996 0.9987 0.9971 0.9949 0.9921 0.9880 0.9827 0.9755 0.9660 0.9 0.9992 0.9968 0.9933 0.9884 0.9816 0.9738 0.9651 0.9545 0.9438 Table 5.1. Correction factors fA for two-tip cracked I-beam under tension; upper tip. 13 = e: = Aw= 2aw/(d 1 - 2e) 2A,/Aw 2e/d 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 1.0003 1.0011 1.0021 1.0031 1.0022 0.9975 0.9840 0.9507 0.8629 0.1 1.0002 1.0009 1.0020 1.0029 1.0034 1.0024 0.9984 0.9889 0.9686 2.05 0.3 1.0000 1.0004 1.0008 1.0012 1.0014 1.0014 0.9996 0.9959 0.9886 0.5 0.9999 0.9996 0.9991 0.9984 0.9971 0.9953 0.9918 0.9869 0.9788 0.7 0.9997 0.9982 0.9964 0.9937 0.9903 0.9859 0.9801 0.9726 0.9626 0.9 0.9990 0.9967 0.9928 0.9877 0.9807 0.9725 0.9634 0.9526 0.9418 ~ ~ ~ Table 5.2. Correction factors f 8 for two-tip cracked I-beam under tension; lower tip. 13 = e= A.,= 2a,../(d 1 - 2e) 2A /Aw 2e/d 0.1 0 .2 1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 1.0019 1.0076 1.0169 1.0307 1.0475 1.0678 1.0866 1.0979 1.0562 0.1 1.0015 1.0066 1.0149 1.0267 1.0419 1.0598 1.0773 1.0857 1.0476 0.3 1.0012 1.0084 1.0111 1.0202 1.0313 1.0438 1.0540 1.0524 1.0024 0.83 0 .5 1.0008 1.0032 1.0071 1.0124 1.0183 1.0233 1.0232 1.0087 0.9427 0.7 1.0002 1.0008 1.0016 1.0020 1.0007 0.9958 0.9833 0.9531 0.8752 0.9 0.9994 0.9976 0.9942 0.9883 0.9780 0.9623 0.9377 0.8960 0.8149 0.0 1.0015 1.0060 1.0138 1.0244 1.0371 1.0512 1.0624 1.0595 1.0048 0.1 1.0012 1.0052 1.0117 1.0211 1.0324 1.0446 1.0544 1.0515 0.9987 0.3 1.0009 1.0038 1.0085 1.0150 1.0229 1.0304 1.0336 1.0225 0.9605 --" .J:l,.. 1.00 0.5 1.0005 1.0023 1.0049 1.0081 1.0109 1.0117 1.0061 0.9833 0.9096 N 0.7 1.0001 1.0001 0 .9998 0.9985 0.9951 0.9873 0.9708 0.9355 0.8531 0.9 0.9993 0.9970 0 .9930 0.9860 0.9747 0.9573 0.9309 0.8869 0.8031 0.0 1.0014 1.0055 1.0122 1.0223 1.0326 1.0464 1.0523 1.0546 0.9895 0.1 1.0012 1.0047 1.0108 1.0194 1.0297 1.0409 1.0496 1.0472 0.9998 0.3 1.0008 1.0038 1.0079 1.0141 1.0215 1.0285 1.0321 1.0231 0.9680 1.11 0.5 1.0006 1.0022 1.0049 1.0080 1.0112 1.0126 1.0088 0.9899 0.9240 0.7 1.0000 1.0004 1.0005 1.0001 0.9979 0.9924 0.9793 0.9492 0.8740 0.9 0.9995 0.9979 0.9947 0.9896 0.9803 0.9659 0.9433 0.9042 0.8264 0.0 1.0011 1.0044 1.0099 1.0170 1.0254 1.0319 1.0352 1.0180 0.9507 0.1 1.0010 1.0038 1.0087 1.0150 1.0221 1.0289 1.0307 1.0169 0.9502 0.3 1.0006 1.0026 1.0058 1.0101 1.0141 1.0165 1.0128 0.9924 0.9189 1.25 0.5 1.0004 1.0013 1.0025 1.0036 1.0034 0.9999 0.9880 0.9580 0.8759 0.7 0.9999 0.9992 0.9979 0.9948 0.9889 0.9778 0.9570 0.9191 0.8294 0.9 0.9991 0.9966 0.9918 0.9835 0.9704 0.9512 0.9222 0.8754 0.7880 Table 5.2. Correction factors f 8 for two-tip cracked I-beam under tension; lower tip. 13 = e:= A?= 2aw/(dl - 2e) 2At/Aw 2e/d 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 1.0009 1.0038 1.0083 1.0142 1.0205 1.0255 1.0246 1.0069 0.9334 0.1 1.0008 1.0031 1.0071 1.0120 1.0173 1.0212 1.0196 1.0015 0.9291 1.37 0.3 1.0005 1.0021 1.0044 1.0075 1.0099 1.0101 1.0035 0.9791 0.9002 0.5 1.0002 1.0007 1.0012 1.0014 1.0001 0.9946 0.9806 0.9474 0.8611 0.7 0.9996 0.9990 0.9970 0.9933 0.9865 0.9740 0.9514 0.9088 0.8175 0.9 0.9991 0.9962 0.9911 0.9824 0.9687 0.9485 0.9180 0.8690 0.7783 0.0 1.0007 1.0031 1.0064 1.0106 1.0147 1.0165 1.0115 0.9882 0.9095 0.1 1.0006 1.0023 1.0054 1.0089 1.0121 1.0132 1.0076 0.9844 0.9071 0.3 1.0003 ~ 1.0015 1.0030 1.0047 1.0055 1.0030 0.9931 0.9647 0.8830 ~ 1.53 u) 0.5 1.0000 1.0002 1.0002 0.9991 0.9962 0.9888 0.9724 0.9363 0.8491 0.7 0.9995 0.9985 0.9961 0.9916 0.9836 0.9700 0.9458 0.9022 0.8123 0.9 0.9990 0.9963 0.9907 0.9818 0.9679 0.9474 0.9170 0.8683 0.7787 0.0 1.0005 1 .0024 1.0048 1.0075 1.0097 1.0081 0.9997 0.9692 0.8856 0.1 1.0004 1.0018 1.0041 1.0062 1.0078 1.0063 0.9975 0.9699 0.8877 0.3 1.0002 1.0010 1.0021 1.0024 1.0016 0.9971 0.9842 0.9518 0.8664 1.69 0.5 1.0000 0.9998 0.9991 0.9973 0.9927 0.9834 0.9643 0.9253 0.8356 0.7 0.9997 0.9981 0.9951 0.9897 0.9806 0.9654 0.9396 0.8942 0.8026 0.9 0.9990 0.9960 0.9899 0.9807 0.9662 0.9448 0.9135 0.8636 0.7725 0.0 1.0004 1.0017 1.0033 1.0051 1.0051 1.0034 0.9905 0.9632 0.8741 0.1 1.0003 1.0013 1.0028 1.0042 1.0042 1.0014 0.9908 0.9616 0.8793 0.3 1.0001 1.0006 1.0010 1.0011 0.9992 0.9937 0.9796 0.9465 0.8613 1.91 0.5 0.9999 0.9995 0.9986 0.9963 0.9914 0.9818 0.9624 0.9234 0.8341 0.7 0.9996 0.9981 0.9951 0.9897 0.9807 0.9654 0.9399 0.8949 0.8045 0.9 0.9990 0.9961 0.9903 0.9813 0.9671 0.9460 0.9153 0.8662 0.7762 Table 5.2 . Correction factors f 8 for two-tip cracked I-beam under tension; lower tip. P= e: = A.,= 2aw/(d 1 - 2e) 2A /Aw 2e/d 0.1 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 1.0003 1.0011 1.0021 1.0031 1.0022 0.9975 0.9840 0.9507 0.8629 0.1 1.0002 1.0009 1.0017 1.0021 1.0008 0.9959 0.9825 0.9499 0.8642 2.05 0.3 1.0000 1.0002 1.0000 0.9990 0.9960 0.9888 0.9727 0.9365 0.8491 0.5 0.9999 0.9992 0.9975 0.9944 0.9882 0.9767 0.9553 0.9138 0.8246 0.7 0.9997 0.9977 0.9942 0.9879 0.9778 0.9615 0.9347 0.8888 0.7985 0.9 0.9990 0.9959 0.9899 0.9806 0.9657 0.9443 0.9130 0.8632 0.7729 __._ ~ ~ Table 5.3. Correction factors fA for two-tip cracked I-beam under bending; upper tip. 13 = e= >-.,, = 2aw/(d 1 - 2e) 2A /Aw 2e/d 0.1 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 -0.0500 -0.1000 -0.1501 -0.2004 -0.2512 -0.3029 -0.3559 -0.4096 -0.4553 0.1 0.0551 0.0106 -0.0336 -0.0777 -0.1217 -0.1659 -0.2107 -0.2570 -0.3062 0.3 0.2653 0.2313 0.1980 0.1651 0.1328 0.1007 0.0688 0.0363 0.0020 0.83 0.5 0.4754 0.4515 0.4283 0.4058 0.3836 0.3618 0.3399 0.3174 0.2930 0.7 0.6850 0.6707 0.6567 0.6427 0.6290 0.6151 0.6007 0.5851 0.5672 0.9 0.8947 0.8884 0.8816 0.8740 0.8653 0.8556 0.8452 0.8332 0.8205 0.0 -0.0500 -0.1000 -0.1500 -0.2003 -0.2509 -0.3021 -0.3541 -0.4056 -0.4467 0.1 0.0551 0.0105 -0.0339 -0.0781 -0.1224 -0.1668 -0.2119 -0.2583 -0.3071 0.3 -"' 0.2652 0.2310 0.1973 0.1640 0.1310 0.0982 0.0652 0.0315 -0.0041 ..i::-. 1.00 0.5 0.4752 0.4510 0.4274 0.4041 0.3812 0.3582 0.3351 0.3111 0.2849 01 0.7 0.6850 0.6701 0.6555 0.6410 0.6262 0.6113 0.5956 0.5786 0.5592 0.9 0.8945 0.8879 0.8807 0.8725 0.8631 0.8529 0.8416 0.8288 0.8153 0.0 -0.0500 -0.1000 -0.1501 -0.2003 -0.2508 -0.3021 -0.3542 -0.4063 -0.4500 0.1 0.0551 0.0105 -0.0340 -0.0783 -0.1226 -0.1672 -0.2124 -0.2589 -0.3077 0.3 0.2652 0.2310 0.1971 0.1637 0.1305 0.0975 0.0643 0.0304 -0.0053 1.11 0.5 0.4752 0.4511 0.4273 0.4040 0.3808 0.3578 0.3345 0.3104 0.2844 0.7 0.6850 0.6704 0.6558 0.6415 0.6270 0.6122 0.5968 0.5802 0.5613 0.9 0.8947 0.8884 0.8818 0.8746 0.8661 0.8567 0.8466 0.8348 0.8222 0.0 -0.0500 -0.1000 -0.1500 -0.2002 -0.2506 -0.3014 -0.3523 -0.4016 -0.4379 0.1 0.0551 0.0104 -0.0342 -0.0786 -0.1231 -0.1679 -0.2132 -0.2596 -0.3079 0.3 0.2651 0.2307 0.1966 0.1628 0.1292 0.0955 0.0615 0.0266 -0.0102 1.25 0.5 0.4751 0.4506 0.4264 0.4024 0.3785 0.3545 0.3301 0.3045 0.2769 0.7 0.6848 0.6698 0.6544 0.6391 0.6233 0.6071 0.5901 0.5716 0.5510 0.9 0.8945 0.8875 0.8797 0.8710 0.8607 0.8492 0.8373 0.8236 0.8094 Table 5.3. Correction factors fA for two-tip cracked I-beam under bending; upper tip. (3 = ?= >.._ =2 aw/(d 1 - 2e) 2A 0.1 1/Aw 2e/d 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 -0 .0500 -0 .1000 -0.1500 -0.2001 -0.2504 -0 .3010 -0 .3515 -0 .3999 -0.4342 0.1 0 .0551 0.0103 -0 .0343 -0.0789 -0.1235 -0.1684 -0.2138 -0.2602 -0.3083 1.37 0.3 0 .2651 0.2306 0.1963 0.1622 0.1283 0 .0942 0.0597 0.0244 -0.0131 0.5 0.4750 0.4504 0.4260 0.4017 0.3774 0.3529 0.3278 0.3017 0.2733 0.7 0.6849 0.6695 0.6540 0.6383 0.6222 0.6055 0.5880 0.5690 0.5476 0.9 0.8943 0.8871 0.8793 0.8703 0.8597 0.8482 0.8356 0.8215 0.8068 0.0 -0.0500 -0.1000 -0.1500 -0.2001 -0.2503 -0.3006 -0.3506 -0.3980 -0.4305 0.1 0.0550 0.0102 -0.0345 -0.0792 -0.1239 -0.1689 -0.2145 -0.2609 -0.3087 0.3 0.2650 0.2304 .....l,, 0.1959 0.1616 0.1273 0.0928 0.0578 0.0219 -0.0161 ~ 1.53 0.5 0.4749 0.4502 0.4254 0.4008 0.3760 0.3510 0.3253 0.2984 0.2694 0) 0.7 0.6846 0.6691 0.6533 0.6373 0.6207 0.6035 0 .5854 0.5658 0 .5440 0.9 0.8942 0.8871 0.8788 0.8697 0.8589 0.8471 0.8343 0 .8200 0.8053 0.0 -0.0500 -0.1000 -0.1500 -0.2000 -0.2501 -0.3003 -0.3498 -0.3962 -0.4268 0.1 0.0550 0.0102 -0.0346 -0.0794 -0.1242 -0.1694 -0.2150 -0.2615 -0.3090 0.3 0.2650 0.2302 0.1956 0.1611 0.1265 0.0916 0.0563 0.0198 -0.0187 1.69 0.5 0.4748 0.4499 0.4250 0.4000 0.3749 0.3493 0.3231 0.2957 0.2661 0.7 0 .6845 0.6688 0.6530 0.6363 0.6193 0.6016 0.5829 0.5627 0.5405 0.9 0.8940 0.8867 0.8783 0.8688 0.8577 0.8455 0.8324 0.8179 0.8026 0.0 -0.'0500 -0.1000 -0.1500 -0.2000 -0.2500 -0.3001 -0.3496 -0.3959 -0.4266 0.1 0.0550 0.0101 -0.0347 -0 .0796 -0.1245 -0.1698 -0.2155 -0.2620 -0.3095 0.3 0.2649 0.2301 0.1954 0.1607 0.1258 0.0907 0.0550 0.0184 -0.0204 1.91 0.5 0.4748 0.4498 0.4247 0.3996 0.3743 0.3485 0.3219 0.2942 0.2643 0.7 0.6845 0.6689 0.6527 0.6361 0.6190 0.6011 0.5821 0.5617 0.5394 0.9 0.8941 0.8869 0.8786 0.8693 0.8580 0.8459 0.8331 0.8185 0.8037 Table 5.3. Correction factors fA for two-tip cracked I-beam under bending ; upper tip. 13= e= Aw= 2aw/(d 1 - 2e) 2A 1/Aw 2e/d 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 09 1 0.0 -0.0500 -0 .0999 -0 .1499 -0.1999 -0.2500 -0 .2997 -0.3488 -0.3942 -0.4234 0.1 0.0550 0.0101 -0.0348 -0.0797 -0.1248 -0.1701 -0.2159 -0.2623 -0.3096 2.05 0.3 0.2649 0.2300 0.1951 0.1602 0.1252 0.0899 0.0539 0.0168 -0.0224 0.5 0.4746 0.4495 0.4242 0.3989 0.3731 0.3470 0.3200 0.2918 0.2616 0.7 0.6843 0.6684 0.6520 0.6351 0.6176 0.5994 0.5801 0.5593 0.5366 0.9 0.8938 0.8865 0.8780 0.8683 0.8568 0.8445 0.8314 0.8166 0.8016 _.. ~ ---1 Table 5.4. Correction factors f 8 for two-tip cracked I-beam under bending; lower tip. ~= E= A.= 2a/(d 1- 2e) 2A 1/Aw 2e/dl 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.0500 0.1000 0.1501 0.2004 0.2512 0.3029 0.3559 0.4096 0.4553 0.1 0.1448 0.1906 0.2365 0.2829 0.3301 0.3782 0.4269 0.4743 0.5076 0.3 0.3353 0.3715 0.4084 0.4462 0.4848 0.5240 0.5625 0.5960 0.6058 0.83 0.5 0.5253 0.5515 0.5786 0.6062 0.6342 0.6615 0.6861 0.7012 0.6840 0.7 0.7148 0.7306 0.7461 0.7613 0.7753 0.7867 0.7922 0.7838 0.7376 0.9 0.9041 0.9077 0.9095 0.9092 0.9049 0.8956 0.8782 0.8446 0.7741 0.0 0.0500 0.1000 0.1500 0.2003 0.2509 0.3021 0.3541 0.4056 0.4467 0.1 0.1451 0.1905 0.2362 0.2823 0.3289 0.3759 0.4230 0.4674 0.4954 ~ 0.3 0.3352 0.3710 0.4075 0.4446 0.4821 0.5194 0.5553 0.5848 0.5886 ..i:.. 1.00 CX> 0.5 0.5252 0.5510 0.5773 0.6040 0.6303 0.6555 0.6768 0.6874 0.6650 0.7 0.7149 0.7299 0.7447 0.7588 0.7712 0.7804 0.7830 0.7710 0.7209 0.9 0.9040 0.9072 0.9086 0.9071 0.9019 0.8911 0.8718 0.8363 0.7632 0.0 0.0500 0.1000 0.1501 0.2003 0.2508 0.3021 0.3542 0.4063 0.4500 0.1 0.1451 0.1905 0.2361 0.2821 0.3286 0.3756 0.4227 0.4678 0.4993 0.3 0.3352 0.3711 0.4074 0.4443 0.4816 0.5190 0.5552 0.5861 0.5946 1.11 0.5 0.5252 0.5510 0.5774 0.6040 0.6306 0.6562 0.6789 0.6922 0.6754 0.7 0.7151 0.7301 0.7453 0.7600 0.7733 0.7842 0.7894 0.7816 0.7376 0.9 0.9042 0.9081 0.9102 0.9103 0.9071 0.8990 0.8832 0.8522 0.7848 0.0 0.0500 0.1000 0.1500 0.2002 0.2506 0.3014 0.3523 0.4016 0.4379 0.1 0.1451 0.1904 0.2359 0.2816 0.3276 0.3736 0.4189 0.4603 0.4832 0.3 0.3351 0.3707 0.4066 0.4429 0.4792 0.5147 0.5478 0.5732 0.5712 1.25 0.5 0.5250 0.5504 0.5761 0.6016 0.6263 0.6490 0.6671 0.6733 0.6451 0.7 0.7146 0.7292 0.7434 0.7561 0.7668 0.7735 0.7729 0.7570 0.7029 0.9 0.9036 0.9066 0.9072 0.9051 0.8980 0.8854 0.8640 0.8257 0.7492 Table 5.4. Correction factors f 8 for two-tip cracked I-beam under bending; lower tip. 13 = e:= Aw= 2a/(d 1 - 2e) 2A /Aw 2e/d 0.1 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 0.0500 0.1000 0.1500 0.2001 0.2504 0.3010 0.3515 0.3999 0.4342 0.1 0.1451 0.1903 0.2357 0.2812 0.3270 0.3726 0.4171 0.4574 0.4780 1.37 0.3 0.3351 0.3706 0.4063 0.4421 0.4778 0.5126 0.5445 0.5683 0.5636 0.5 0.5250 0.5503 0.5755 0.6006 0.6246 0.6464 0.6632 0.6675 0.6366 0.7 0.7148 0.7291 0.7427 0.7551 0.7651 0.7709 0.7690 0.7512 0.6940 0.9 0.9038 0.9064 0.9066 0.9039 0.8966 0.8830 0.8601 0.8198 0.7402 0.0 0.0500 0.1000 0.1500 0.2001 0.2503 0.3006 0.3506 0.3980 0.4305 0.1 0.1450 0.1902 0.2355 0.2808 0.3262 0.3714 0.4151 0.4540 0.4729 ........ 0.3 0.3350 0.3703 0.4058 0.4412 0.4763 0.5102 0.5409 0.5627 0.5566 ~ 1.53 .., = 2a/(d 1 - 2e) 2A 1/Aw 2e/d 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0 0.0500 0.0999 0.1499 0.1999 0.2500 0.2997 0.3488 0.3942 0.4234 0.1 0.1449 0.1899 0.2350 0.2800 0.3247 0.3688 0.4109 0.4472 0.4627 2.05 0.3 0.3348 0.3698 0.4047 0.4393 0.4731 0.5055 0.5337 0.5523 0.5431 0.5 0.5245 0.5492 0.5734 0.5954 0.6183 0.6366 0.6492 0.6488 0.6154 0.7 0.7141 0.7280 0. 7402 0.7509 0.7586 0.7616 0.7566 0.7363 06796 0.9 0.9032 0.9055 0.9052 0.9018 0.8933 0.8788 0.8551 0.8142 0.7350 -" (J'1 0 Table 5.5. Correction factors fw for three-tip cracked I-beam under tension; web crack tip. 13= ~ w = At= 2a1/b1 2A 1/Aw aw/d 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.7019 0.7558 0.8252 0.9092 1.0093 1.1296 1.2782 1.4724 1.7585 0.2 0.7017 0.7260 0.7596 0.8024 0.8568 0.9258 1.0163 1.1426 1.3438 0.3 0.7091 0.7389 0.7532 0.7839 0.8234 0.8750 0.9449 1.0575 1.2151 0.83 0.4 0.7255 0.7387 0.7571 0.7816 0.8138 0.8565 0.9153 1.0024 1.1538 0.5 0.7363 0.7473 0.7627 0.7833 0.8107 0.8472 0.8983 0.9752 1.1127 0.6 0.7461 0.7553 0.7684 0.7859 0.8093 0.8407 0.8850 0.9526 1.0762 0.7 0.7518 0.7640 0.7750 0.7897 0.8093 0.8359 0.8734 0.9483 1.0387 0.1 0.7031 0.7664 0.8475 0.9451 1.0603 1.1984 1.3686 1.5915 1.9207 ........ 0.2 0.6967 0.7249 0.7633 0.8122 0.8741 0.9524 1.0550 1.1982 1.4269 U1 ........ 0.3 0.7055 0.7242 0.7500 0.7843 0.8283 0.8861 0.9638 1.0765 1.2662 1.00 0.4 0.7155 0.7298 0.7500 0.7769 0.8122 0.8590 0.9235 1.0193 1.1864 0.5 0.7244 0.7362 0.7529 0.7752 0.8046 0.8443 0.8995 0.9830 1.1326 0.6 0.7325 0.7423 0.7562 0.7750 0.8000 0.8336 0.8810 0.9536 1.0865 0.7 0.7405 0.7486 0.7601 0.7757 0.7965 0.8246 0.8646 0.9261 1.0408 0.1 0.7263 0.8147 0.9236 1.0494 1.1938 1.3615 1.5631 1.8201 2.1889 0.2 0.7078 0.7479 0.7996 0.8650 0.9455 1.0451 1.1727 1.3460 1.6155 0.3 0.7117 0.7374 0.7726 0.8179 0.8758 0.9497 1.0480 1.1875 1.4168 1.11 0.4 0.7188 0.7382 0.7653 0.8008 0.8468 0.9071 0.9890 1.1088 1.3142 0.5 0.7257 0.7414 0.7635 0.7927 0.8311 0.8820 0.9523 1.0574 1.2436 0.6 0.7321 0.7451 0.7635 0.7879 0.8202 0.8634 0.9239 1.0156 1.1821 0.7 0.7386 0.7492 0.7643 0.7846 0.8113 0.8474 0.8983 0.9763 1.1208 Table 5.5. Correction factors fw for three-tip cracked I-beam under tension; web crack tip. 13 = Aw = ht= 2a,lb1 2A /Aw aw/d1 0.1 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.7035 0.7765 0.8699 0.9814 1.1132 1.2705 1.4653 1.7204 2.1006 0.2 0.6916 0.7232 0.7664 0.8218 0.8912 0.9793 1.0949 1.2561 1.5154 0.3 0.6974 0.7177 0.7464 0.7841 0.8328 0.8963 0.9823 1.1071 1.3180 1.25 0.4 0.7052 0.7206 0.7423 0.7715 0.8097 0.8606 0.9307 1.0349 1.2173 0.5 0.7124 0.7248 0.7425 0.7662 0.7978 0.8402 0.8993 0.9887 1.1497 0.6 0.7188 0.7290 0.7436 0.7634 0.7896 0.8252 0.8754 0.9523 1.0933 0.7 0.7243 0.7328 0.7448 0.7610 0.7828 0.8122 0.8540 0.9186 1.0391 0.1 0.7022 0.7781 0.8754 0.9924 1.1307 1.2962 1.5016 1.7729 2.1802 _,,_ 0.2 0.6886 0.7209 0.7654 0.8227 0.8947 0.9863 1.1068 1.2759 1.5495 CJ1 N 0.3 0.6932 0.7140 0.7431 0.7816 0.8315 0.8969 0.9858 1.1149 1.3347 1.37 0.4 0.7002 0.7156 0.7376 0.7671 0.8059 0.8578 0.9294 1.0363 1.2244 0.5 0.7076 0.7190 0.7366 0.7606 0.7924 0.8352 0.8952 0.9862 1.1506 0.6 0.7122 0.7223 0.7368 0.7565 0.7829 0.8187 0.8693 0.9469 1.0898 0.7 0.7167 0.7249 0.7369 0.7530 0.7746 0.8041 0.8461 0.9110 1.0326 0.1 0.7113 0.8030 0.9185 1.0546 1.2134 1.4138 1.6342 1.9383 2.3924 0.2 0.6899 0.7290 0.7845 0.8489 0.9327 1.0382 1.1797 1.3673 1.6754 0.3 0.6918 0.7163 0.7505 0.7953 0.8530 0.9282 1.0294 1.1758 1.4236 1.53 0.4 0.6970 0.7150 0.7405 0.7745 0.8191 0.8782 0.9595 1.0802 1.2920 0.5 0 .7021 0 .7164 0.7367 0.7641 0.8002 0.8487 0 .9164 1.0189 1.2034 0.6 0.7065 0.7180 0.7347 0.7570 0.7869 0.8272 0.8840 0.9710 1.1311 0.7 0.7095 0.7190 0.7324 0.7506 0.7749 0.8080 0.8549 0.9274 1.0629 Table 5.5. Correction factors rw for three-tip cracked I-beam under tension; web crack tip. '3 = >,. w = Ai= 2ai/b, 2A,/Aw a_/di 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.7130 0.8112 0.9344 1.0879 1.2481 1.4485 1.6951 2.0194 2.5051 0.2 0.6882 0.7297 0.7859 0.8571 0.9458 1.0574 1.2030 1.4059 1.7330 0.3 0.6888 0.7147 0.7507 0.7978 0.8586 0.9375 1.0438 1.1979 1.4590 1.69 0.4 0.6930 0.7119 0.7387 0.7741 0.8207 0.8823 0.9672 1.0934 1.3150 0.5 0.6974 0.7123 0.7334 0.7618 0.7992 0.8497 0.9200 1.0263 1.2183 0.6 0.7011 0.7132 0.7302 0.7534 0.7842 0.8258 0.8845 0.9744 1.1400 0.7 0.7032 0.7129 0.7267 0.7454 0.7705 0.8045 0.8527 0.9273 1.0667 0.1 0.7278 0.8473 0.9937 1.1630 1.3571 1.5854 1.8642 2.2274 2.7675 ~ 0.2 0.6935 0.7440 0.8116 0.8961 1.0001 1.1297 1.2972 1.5285 1.8979 (JI (.,J 0.3 0.6906 0.7218 0.7648 0.8207 0.8918 0.9839 1.1065 1.2832 1.5799 1.91 0.4 0.6930 0.7154 0.7471 0.7887 0.8430 0.9147 1.0125 1.1571 1.4092 0.5 0.6959 0.7134 0.7381 0.7712 0.8147 0.8730 0.9539 1.0756 1.2940 0.6 0.6984 0.7123 0.7322 0.7590 0.7945 0.8425 0.9098 1.0125 1.2011 0.7 0.6992 0.7103 0.7264 0.7479 0.7767 0.8156 0.8708 0.9558 1.1144 0.1 0.7319 0.8599 1.0157 1.1944 1.3994 1.6400 1.9331 2.3155 2.8845 0.2 0.6934 0.7475 0.8195 0.9094 1.0194 1.1559 1.3322 1.5756 1.9642 0.3 0.6890 0.7222 0.7680 0.8270 0.9023 0.9990 1.1280 1.3133 1.6243 2.05 0.4 0.6910 0.7142 0.7477 0.7916 0.8489 0.9240 1.0263 1.1776 1.4415 0.5 0.6928 0.7114 0.7374 0.7721 0.8177 0.8786 0.9629 1.0898 1.3177 0.6 0.6954 0.7095 0.7303 0.7583 0.7955 0.8452 0.9153 1.0221 1.2195 0.7 0.6946 0.7063 0.7232 0.7456 0.7756 0.8160 0.8731 0.9612 1.1253 Table 5.6. Correction factors f' for three-tip cracked I-beam under tension; flange crack tip . 13 = Aw = Ai= 2a1/b1 2A 1/Aw aw/d1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 1.4339 1.2977 1.2487 1.2384 1.2580 1.3108 1.4055 1.5739 1.9403 0.2 1.6683 1.5114 1.4589 1.4544 1.4881 1.5630 1.6971 1.9394 2.4698 0.3 1.8640 1.7077 1.6388 1.6405 1.6881 1.7851 1.9584 2.2910 2.9667 0.83 0.4 2.0483 1.8637 1.8098 1.8305 1.8793 1.9982 2.2092 2.5923 3.4563 0.5 2.2269 2.0300 1.9755 1.9901 2.0645 2.2046 2.4527 2.9065 3.9405 0.6 2.3986 2.1891 2.1339 2.1545 2.2414 2.4016 2.6851 3.2054 4.4053 0.7 2.5571 2.3374 2.2817 2.3148 2.4055 2.5923 2.9000 3.5651 4.8346 0.1 1.3773 1.2544 1.2111 1.2045 1.2268 1.2816 1.3773 1.5472 1.9151 ....,. 0.2 1.5886 1.4445 1.3979 1.3966 1.4312 1.5067 1.6399 1.8775 2.3991 u, 0.3 1.7576 1.6009 1.5533 1.5578 1.6049 1.7000 1.8678 2.1700 2.8330 ~ 1.00 0.4 1.9141 1.7458 1.6983 1.7085 1.7677 1.8819 2.0834 2.4482 3.2724 0.5 2.0634 1.8848 1.8364 1.8527 1.9232 2.0561 2.2897 2.7162 3.6901 0.6 2.2056 2.0166 1.9682 1.9892 2.0709 2.2210 2.4851 2.9697 4 0873 0.7 2.3382 2.1393 2.0905 2.1157 2.2069 2.3728 2.6639 3.2002 4.4504 0.1 1.3647 1.2260 1.1733 1.1597 1.1739 1.2198 1.3028 1.4514 1.7755 0.2 1.5854 1.4255 1.3665 1.3558 1.3810 1.4456 1.5629 1.7743 2.2386 0.3 1.7581 1.5833 1.5235 1.5175 1.5543 1.6378 1.7890 2.0628 2.6118 1.11 0.4 1.9143 1.7278 1.6663 1.6655 1.7137 1.8164 2.0006 2.3364 3.0942 0.5 2.0601 1.8628 1.8011 1.8051 1.8642 1.9850 2.2014 2.5981 3.5061 0.6 2.1983 1.9900 1.9272 1.9360 2.0054 2.1432 2.3899 2.8448 3.8975 0.7 2.3245 2.1075 2.0435 2.0562 2.1346 2.2879 2.5613 3.0688 4.2550 Table 5.6. Correction factors f' for three-tip cracked I-beam under tension; flange crack tip. P= >-. w = "-t = 2.3i/b1 2A 1/Aw aw/d 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 1.3247 1.2133 1.1770 1.1756 1.2012 1.2590 1.3582 1.5323 1.9079 0.2 1.5111 1.3808 1.3408 1.3436 1.3815 1.4581 1.5916 1.8287 2.3483 0.3 1.6570 1.5146 1.4738 1.4822 1.5301 1.6244 1.7891 2.0839 2.7423 1.25 0.4 1.7892 1.6366 1.5954 1.6084 1.6667 1.7782 1.9719 2.3221 3.1135 0.5 1.9133 1.7518 1.7102 1.7276 1.7960 1.9232 2.1447 2.5475 3.4692 0.6 2.0304 1.8600 1.8184 1.8400 1.9176 2.0593 2.3064 2.7581 3.8018 0.7 2.1386 1.9604 1.9177 1.9435 2.0289 2.1836 2.4537 2.9503 4.1050 0.1 1.2925 1.1982 1.1686 1.1707 1.1989 1.2607 1.3644 1.5448 1.9349 ....,. 0.2 1.4473 1.3441 1.3143 1.3231 1.3637 1.4440 1.5808 1.8225 2.3527 (J1 (J1 0.3 1.5670 1.4585 1.4310 1.4458 1.4980 1.5945 1.7607 2.0573 2.7176 0.4 1.37 1.6769 1.5631 1.5374 1.5578 1.6194 1.7323 1.9257 2.2738 3.0600 0.5 1.7768 1.6610 1.6373 1.6630 1.7352 1.8619 2.0808 2.4773 3.3845 0.6 1.8729 1.7531 1. 7311 1.7618 1.8428 1.9827 2.2248 2.6668 3.6864 0.7 1.9604 1.8379 1.8173 1.8521 1.9409 2.0928 2.3562 2.8371 3.9581 0.1 1.2714 1.1778 1.1485 1.1497 1.1767 1.2488 1.3391 1.5151 1.8950 0.2 1.4206 1.3178 1.2924 1.2958 1.3351 1.4101 1.5516 1.7816 2.2959 0.3 1.5321 1.4252 1.3972 1.4110 1.4610 1.5550 1.7164 2.0038 2.6439 1.53 0.4 1.6305 1.5202 1.4944 1.5140 1.5739 1.6825 1.8697 2.2064 2.9666 0.5 1.7217 1.6086 1.5846 1.6094 1.6785 1.8008 2.0119 2.3943 3.2693 0.6 1.8069 1.6909 1.6689 1.6980 1.7755 1.9100 2.1434 2.5676 3.5494 0.7 1.8847 1.7662 1.7454 1.7787 1.8635 2.0090 2.2621 2.7239 3.7997 Table 5.6. Correction factors f' for three-tip cracked I-beam under tension; flange crack tip . 13 = Aw = At= 2ailb1 2A,/Aw aw/dl 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 1.2524 1.1646 1.1378 1.1384 1.1700 1.2331 1.3363 1.5150 1.9009 0.2 1.3923 1.2959 1.2700 1.2795 1.3198 1.3995 1.5338 1.7698 2.2871 0.3 1.4950 1.3948 1.3705 1.3864 1.4373 1.5320 1.6929 1.9794 2.6174 1.69 0.4 1.5844 1.4817 1.4598 1.4811 1.5416 1.6499 1.8354 2.1690 2.9209 0.5 1.6668 1.5621 1.5421 1.5681 1.6375 1.7584 1.9666 2.3427 3.2035 0.6 1.7431 1.6359 1.6181 1.6490 1.7260 1.8585 2.0870 2.5028 3.4626 0.7 1.8126 1.7035 1.6872 1.7219 1.8058 1.9484 2.1950 2.6450 3.6938 0.1 1.2326 1.1402 1.1117 1.1149 1.1419 1.2015 1.2999 1.4714 1.8414 __.. 0.2 1.3808 1.2742 1.2435 1.2500 1.2877 1.3637 1.4920 1.7179 2.2128 (.J1 0.3 1.4889 0) 1.3746 1.3440 1.3552 1.4018 1.4922 1.6463 1.9210 2.5327 1.91 0.4 1.5815 1.4579 1.4306 1.4466 1.5020 1.6056 1.7831 2.1027 2.8248 0.5 1.6654 1.5408 1.5109 1.5302 1.5933 1.7089 1.9080 2.2689 3.0955 0.6 1.7422 1.6130 1.5840 1.6068 1.6766 1.8029 2.0216 2.4200 3.3422 0.7 1.8118 1.6784 1.6496 1.6755 1.7514 1.8871 2.1226 2.5535 3.5601 0.1 1.2205 1.1313 1.1044 1.1080 1.1362 1.1968 1.2962 1.4687 1.8411 0.2 1.3639 1.2608 1.2318 1.2397 1.2776 1.3545 1.4831 1.7093 2.2045 0.3 1.4672 1.3565 1.3276 1.3398 1.3875 1.4781 1.6320 1.9056 2.5149 2.05 0.4 1.5551 1.4385 1.4108 1.4268 1.4825 1.5859 1.7627 2.0798 2.7960 0.5 1.6327 1.5124 1.4852 1.5057 1.5688 1.6836 1.8810 2.2381 3.0556 0.6 1.7066 1.5803 1.5536 1.5773 1.6471 1.7722 1.9882 2 3811 3.2956 0.7 1.7683 1.6409 1.6142 1.6416 1.7171 1.8511 2.0831 2.5068 3.4972 Table 5.7. Correction factors fw for three-tip cracked I-beam under bending; web crack tip. ~= Aw = Ai= 2a,/b1 2A /Aw aw/d 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 0.8 0.9 1 0.1 0.5977 0.6504 0.7186 0.8014 0.9004 1.0189 1.1662 1.3587 1.6423 0.2 0.4909 0.5139 0.5458 0.5869 0.6388 0.7054 0.7929 0.9147 1.1095 0.3 0.3913 0.4080 0.4292 0.4592 0.4938 0.5440 0.6068 0.7009 0.8589 0.83 0.4 0.2931 0.3044 0.3205 0.3421 0.3705 0.4084 0.4607 0.5381 0.6732 0.5 0.1903 0.1992 0.2119 0.2291 0.2526 0.2827 0.3257 0.3906 0.5070 0.6 0.0845 0.0916 0.1017 0.1155 0.1339 0.1588 0.1940 0.2478 0.3464 0.7 -0.0253 -0.0187 -0.0107 -0.0020 0.0146 0.0321 0.0623 0.1056 0.1861 0.1 0.5992 0.6611 0.7409 0.8371 0.9512 1.0881 1.2565 1.4776 1.8044 0.2 ....I, 0.4873 0.5138 0.5504 0.5975 0.6571 0.7329 0.8323 0.9712 1.1936 u, 0.3 0.3875 --.J 0.4044 0.4283 0.4600 0.5011 0.5551 0.6280 0.7340 0.9123 1.00 0.4 0.2870 0.2995 0.3173 0.3412 0.3728 0.4148 0.4728 0.5592 0.7102 0.5 0.1845 0.1938 0.2078 0.2267 0.2518 0.2857 0.3331 0.4048 0.5338 0.6 0.0788 0.0865 0.0975 0.1125 0.1326 0.1599 0.1985 0.2577 0.3663 0.7 -0.0290 -0.0230 -0.0144 -0.0027 0.0132 0.0347 0.0654 0.1128 0.2014 0.1 0.6213 0.7081 0.8154 0.9396 1.0826 1.2489 1.4489 1.7042 2.0704 0.2 0.4971 0.5344 0.5846 0.6477 0.7255 0.8221 0.9458 1.1145 1.3770 0.3 0.3929 0.4162 0.4488 0.4910 0.5450 0.6146 0.7071 0.8387 1.0555 1.11 0.4 0.2900 0.3070 0.3310 0.3628 0.4041 0.4586 0.5326 0.6414 0.8283 0.5 0.1858 0.1988 0.2175 0.2424 0.2753 0.3192 0.3800 0.4711 0.6330 0.6 0.0796 0.0899 0.1046 0.1244 0.1506 0.1861 0.2357 0.3114 0.4492 0.7 -0.0286 -0.0206 -0.0092 0.0062 0.0269 0.0549 0.0945 0.1555 0.2688 Table 5.7. Correction factors rw for three-tip cracked I-beam under bending; web crack tip . 13 = >-. w = >,., = 2a1/b1 2A 1/Aw aw/d 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.6000 0.6714 0.7635 0.8737 1.0042 1.1600 1.3534 1.6069 1.9847 0.2 0.4832 0.5132 0.5547 0.6080 0.6753 0.7609 0.8732 1.0306 1.2834 0.3 0.3818 0.4005 0.4271 0.4623 0.5080 0.5680 0.6493 0.7674 0.9675 1.25 0.4 0.2808 0.2943 0.3138 0.3399 0.3746 0.4207 0.4846 0.5797 0.7468 0.5 0.1780 0.1883 0.2034 0.2238 0.2511 0.2880 0.3397 0.4181 0.5594 0.6 0.0731 0.0812 0.0931 0.1092 0.1309 01605 0.2023 0.2665 0.3847 0.7 -0 .0338 -0.0275 -0.0183 -0.0057 0.0113 0.0346 0.0677 0.1191 0.2150 0.1 0.5989 0.6733 0.7695 0.8848 1.0218 1.1863 1.3904 1.6599 2.0649 _,,_ 0.2 0.4809 0.5116 0.5544 0.6098 0.6798 0.7690 0.8864 1.0514 1.3190 u, CX) 0.3 0.3790 0.3980 0.4251 0.4612 0.5084 0.5703 0.6545 0.7774 0.9868 0.4 1.37 0.2777 0.2913 0.3111 0.3378 0.3732 0.4206 0.4864 0.5847 0.7580 0.5 0.1749 0.1853 0.2005 0.2213 0.2491 0.2867 0.3396 0.4200 0.5658 0.6 0.0703 0.0784 0.0903 0.1067 0.1288 0.1588 0.2015 0.2672 0.3885 0.7 -0.0362 -0.0299 -0.0206 -0.0079 0.0094 0.0330 0.0667 0.1191 0.2173 0.1 0.6077 0.6977 0.8117 0.9463 1.1040 1.2913 1.5221 1.8246 2.2763 0.2 0.4824 0.5196 0.5705 0.6355 0.7171 0.8198 0.9543 1.1417 1.4436 0.3 0.3783 0.4009 0.4329 0.4751 0.5298 0.6012 0.6975 0.8375 1.0744 1.53 0.4 0.2760 0.2921 0.3151 0.3460 0.3836 0.4412 0.5162 0.6281 0.8246 0.5 0.1729 0.1850 0.2026 0.2265 0.2584 0.3014 0.3617 0.4530 0.6183 0.6 0.0682 0.0776 0.0914 0.1101 0.1353 0.1696 0.2181 0.2927 0.4302 0.7 -0.0380 -0.0307 -0.0201 -0.0056 0.0141 0.0410 0.0793 0.1388 0.2502 Table 5.7. Correction factors fw for three-tip cracked I-beam under bending; web crack tip. ~= >-. w = Ai= 2a1/b1 2A 1/A.,, a.,,/d 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.6094 0.7059 0.8276 0.9701 1.1384 1.3375 1.5830 1.9056 2.3895 0.2 0.4812 0.5208 0.5750 0.6440 0.7303 0.8395 0.9818 1.1807 1.5015 0.3 0.3762 0.4001 0.4339 0.4784 0.5361 0.6113 0.7129 0.8605 1.1111 1.69 0.4 0.2737 0.2905 0.3147 03472 0.3900 0.4470 0.5257 0.6431 0.8497 0.5 0.1705 0.1832 0.2016 0.2266 0.2599 0.3048 0.3678 0.4634 0.6363 06 0.0661 0.0758 0.0901 0.1097 0.1359 0.1717 0.2223 0.3000 0.4436 0.7 -0.0398 -0.0323 -0.0213 -0.0061 0.0143 0.0423 0.0822 0.1441 0.2602 0.1 0.6239 0.7414 0.8862 1.0538 1.2467 1.4742 1.7512 2.1130 2.6507 .....,. 0.2 0.4863 0.5340 0.5999 0.6821 0.7837 0.9107 1.0748 1.3009 1.6649 C.11 co 0.3 0.3783 0.4071 0.4476 0.5006 0.5685 0.6566 0.7743 0.9440 1.2299 1.91 0.4 0.2743 0.2943 0.3231 0.3615 0.4118 0.4784 0.5697 0.7052 0.9419 0.5 0.1703 0.1853 0.2070 0.2364 0.2754 0.3278 0.4008 0.5113 0.7099 0.6 0.0655 0.0770 0.0938 0.1167 0.1473 0.1889 0.2475 0.3374 0.5027 0.7 -0.0404 -0.0317 -0.0188 -0.0011 0.0228 0.0553 0.1015 0.1731 0.3070 0.1 0.6278 0.7538 0.9078 1.0850 1.2884 1.5278 1.8197 2.2005 2.7675 0.2 0.4863 0.5381 0.6078 0.6951 0.8026 0.9364 1.1094 1.3486 1.7309 0.3 0.3772 0.4080 0.4511 0.5073 0.5790 0.6717 0.7955 0.9740 1.2743 2.05 0.4 0.2729 0.2942 0.3247 0.3652 0.4182 0.4882 0.5842 0.7261 0.9743 0.5 0.1689 0.1847 0.2076 0.2386 0.2795 0.3344 0.4110 0.5265 0.7346 0.6 0.0641 0.0762 0.0939 0.1179 0.1501 0.1936 0.2549 0.3488 0.5218 0.7 -0.0416 -0.0324 -0.0188 0.0002 0.0246 0.0586 0.1069 0.1816 0.3213 Table 5.8. Correction factors f' for three-tip cracked I-beam under bending; flange crack tip. 13= Aw = At= 2a,/b, 2A,/Aw aw/d 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 1.4090 1.2792 1.2334 1.2255 1.2460 1.2993 1.3936 1.5613 1.9254 0.2 1.5914 1.4475 1.4011 1.4002 1.4352 1.5099 1.6419 1.8775 2.3926 0.3 1.7110 1.5620 1.5174 1.5262 1.5699 1.6674 1.8275 2.1214 2.7739 0.83 0.4 1.8013 1.6455 1.6023 1.6135 1.6709 1.7802 1.9712 2.3157 3.0899 0.5 1.8604 1.7019 1.6601 1.6762 1.7406 1.8641 2.0768 2.4631 3.3416 0.6 1.8923 1.7325 1.6928 1.7125 1.7842 1.9156 2.1447 2.5625 3.5227 0.7 1.8985 1.7401 1.7019 1.7173 1.7994 1.9297 2.1763 2.6144 3.6318 0.1 1.3574 1.2389 1.1986 1.1942 1.2175 1.2723 1.3681 1.5370 1.9030 0.2 ~ 1.5205 1.3889 1.3487 1.3502 1.3866 1.4620 1.5929 1.8256 2.3343 0) 0.3 1.6259 0 1.4879 1.4489 1.4569 1.5047 1.5970 1.7575 2,0439 2.6794 1.00 0.4 1.7001 1.5582 1.5211 1.5346 1.5913 1.6981 1.8829 2.2154 2.9639 0.5 1.7490 1.6047 1.5697 1.5873 1.6514 1.7697 1.9741 2.3441 3.1873 0.6 1.7747 1.6300 1.5962 1.6174 1.6870 1.8136 2.0325 2.4308 3.3488 0.7 1.7794 1.6360 1.6029 1.6266 1.6995 1.8315 2.0598 2.4765 3.4459 0.1 1.3421 1.2103 1.1620 1.1502 1.1656 1.2120 1.2949 1.4431 1.7658 0.2 1.5148 1.3697 1,3186 1.3122 1.3399 1.4052 1.5214 1.7290 2.1835 0.3 1.6233 1.4707 1.4215 1.4216 1.4605 1.5434 1.6891 1.9506 2.5292 1.11 0.4 1.6977 1.5412 1.4941 1.4995 1.5483 1,6463 1.8177 2.1274 2.8210 0.5 1.7455 1.5875 1.5423 1.5528 1.6093 1. 7196 1.9120 2.2616 3.0556 0.6 1.7713 1.6122 1.5692 1.5831 1.6458 1.7654 1.9739 2.3543 3.2300 0.7 1.7755 1.6181 1.5764 1.5932 1.6601 1.7860 2.0052 2.4066 3.3418 Table 5.8. Correction factors f' for three-tip cracked I-beam under bending; flange crack tip. 13 = Aw = A,= 2a1/b1 2A1/Aw a_./dl 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 1.3056 1.2001 1.1667 1.1670 1.1939 1.2519 1.3506 1.5240 1.8981 0.2 1.4508 1.3326 1.2985 1.3050 1.3436 1.4206 1.5524 1.7851 2.2941 0.3 1.5418 1.4175 1.3848 1.3964 1.4455 1.5382 1.6968 1.9788 2.6047 1.25 0.4 1.6040 1.4764 1.4453 1.4619 1.5190 1.6247 1.8051 2.1283 2.8559 0.5 1.6442 1.5149 1.4859 1.5059 1.5695 1.6849 1.8828 2.2394 3.0523 0.6 1.6648 1.5350 1.5069 1.5302 1.5988 1.7218 1.9320 2.3136 3.1922 0.7 1.6680 1.5388 1.5125 {5374 1.6090 1.7367 1.9554 2.3532 3.2775 0.1 1.2786 1.1885 1.1610 1.1643 1.1934 1.2552 1.3587 1.5385 1.9264 0.2 ~ 1.3992 1.3037 1.2788 1.2897 1.3315 1.4115 1.5465 1.7840 2.3041 CJ) ~ 0.3 1.4737 1.3767 1.3546 1.3716 1.4237 1.5184 1.6786 1.9630 2.5951 1.37 0.4 1.5246 1.4272 1.4074 1.4295 1.4894 1.5961 1.7768 2.1008 2.8277 0.5 1.5573 1.4597 1.4421 1.4681 1.5343 1.6501 1.8470 2.2012 3.0081 0.6 1.5739 1.4765 1.4606 1.4895 1.5600 1.6827 1.8912 2.2684 3.1367 0.7 1.5761 1.4798 1.4647 1.4956 1.5691 1.6958 1.9118 2.3044 3.2144 0.1 1.2603 1.1710 1.1437 1.1461 1.1739 1.2348 1.3356 1.5110 1.8895 0.2 1.3771 1.2822 1.2575 1.2675 1.3084 1.3860 1.5176 1.7487 2.2549 0.3 1.4465 1.3511 1.3288 1.3450 1.3953 1.4879 1.6442 1.9211 2.5359 1.53 0.4 1.4929 1.3966 1.3775 1.3990 1.4488 1.5612 1.7376 2.0528 2.7610 0.5 1.5222 1.4264 1.4090 1.4344 1.4989 1.6120 1.8042 2.1497 2.9363 0.6 1.5368 1.4413 1.4260 1.4541 1.5230 1.6429 1.8464 2.2148 3.0619 0.7 1.5385 1.4442 1.4298 1.4597 1.5314 1.6558 1.8668 2.2498 3.1398 Table 5.8. Correction factors f' for three-tip cracked I-beam under bending; flange crack tip . 13 = },. w = ~ = 2a1/b1 2A 1/Aw a.ld 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 1.2440 1.1599 1.1353 1.1368 1.1688 1.2314 1.3341 1.5122 1.8969 0.2 1.3534 1.2646 1.2426 1.2543 1.2958 1.3751 1.5079 1.7406 2.2500 0.3 1.4175 1.3277 1.3088 1.3271 1.3780 1.4712 1.6275 1.9043 2.5197 1.69 0.4 1.4595 1.3699 1.3536 1.3768 1.4356 1.5397 1.7152 2.0280 2.7333 0.5 1.4859 1.3967 1.3826 1.4093 1.4740 1.5867 1.7772 2.1197 2.8992 0.6 1.4988 1.4102 1.3979 1.4272 1.4960 1.6152 1.8166 2.1798 3.0176 0.7 1.5002 1.4126 1.4012 1.4320 1.5041 1.6271 1.8356 2.2132 3.0910 0.1 1.2211 1.1334 1.1073 1.1111 1.1393 1.1986 1.2969 1.4679 1.8370 ...... 0.2 1.3372 1.2400 1.2155 1.2248 1.2642 1.3403 1.4677 1.6898 2.1797 0) 0.3 1.4047 1.3055 1.2824 1.2971 1.3456 1.4353 1.5859 1.8526 2.4442 N 1.91 0.4 1.4484 1.3865 1.3266 1.3462 1.4021 1.5030 1.6726 1.9750 2.6564 0.5 1.4751 1.3744 1.3554 1.3783 1.4401 1.5495 1.7342 2.0654 2.8215 0.6 1.4882 1.3878 1.3699 1.3962 1.4619 1.5778 1.7737 2.1268 2.9410 0.7 1.4896 1.3900 1.3735 1.4014 1.4702 1.5904 1.7936 2.1616 3.0173 0.1 1.2120 1.1272 1.1023 1.1070 1.1337 1.1958 1.2949 1.4668 1.8381 0.2 1.3244 1.2312 1.2072 1.2180 1.2574 1.3340 1.4615 1.6850 2.1739 0.3 1.3889 1.2928 1.2715 1.2871 1.3360 1.4258 1.5765 1.8423 2.4334 2.05 0.4 1.4301 1.3329 1.3135 1.3341 1.3902 1.4907 1.6598 1.9609 2.6392 0.5 1.4551 1.3578 1.3404 1.3647 1.4262 1.5352 1.7190 2.0482 2.7997 0.6 1.4670 1.3702 1.3542 1.3812 1.4473 1.5625 1.7570 2.1075 2.9155 0.7 1.4681 1.3721 1.3576 1.3863 1.4549 1.5746 1.7762 2.1413 2.9898 Table 6.1. Fitting coefficients for two-tip cracked I-beam. Tension Bending Coef. Upper Lower Upper Lower Crack Tip Crack Tip Crack Tip Crack Tip ao 1.0 1.0 0.0 0.0 81 -0.07184 -0.03591 1.02395 1.02052 82 0.05916 0.03257 -0.02824 -0.03142 83 0.07266 0.01609 -0.02660 -0.02841 84 0.16801 0.17113 -0.51095 0.48403 85 -0.15810 -0.17469 -0.00309 -0.02169 as -0.09645 0.00540 0.66587 -0.19538 87 0.13248 0.19882 0.02106 0.10116 as 0.11124 0.10355 -0.03243 -0.02670 ag -0.01464 -0.01573 0.00337 0.00206 810 -0.03299 0.00399 0.02660 0.03282 811 0.04288 0.05901 0.00483 0.01704 8 12 -0.14373 -0.13149 -0.14302 -0.20652 813 -0.09648 -0.16125 -0.01281 -0.06069 814 -0.03380 -0.27916 -0.04610 -0.28079 163 Table 6.2. Fitting coefficients for three-tip cracked I-beam. Tension Bending Coef. Web Crack Flange Web Crack Flange Tip Crack Tip Tip Crack Tip a1 0.82991 1.48266 0.82922 1.39283 82 -0.67499 0.29636 -0.58851 0.06970 83 0.21031 0.07549 0.16098 0.04610 84 5.48154 4.44880 3.18865 3.37633 85 -7.24174 2.74806 -1.92470 -2.78526 aa 6.51764 -8.92438 -5.45652 -1 .04763 87 -1 .22688 -9.01819 -0.87320 -7.16218 ae 2.27070 10.67798 1.30689 9.95421 89 -0.04335 -2.79219 0.75006 -3.94745 810 -3.53419 39.40912 1.92329 24.63804 8 11 5.23305 -33.56530 -1.28348 -20.37830 8 12 1.11541 -32.7345 -2.84890 -19.76580 8 13 -2.53266 30.52621 3.09458 18.05231 814 -0.67489 0.31779 -0.67725 0.14847 8 15 0.19608 0.08256 0.17624 0.04106 81a 9.15477 2.56783 7.93375 1.27056 817 -13.48190 -6.28871 -0.68127 -3.72795 8 1e 10.74853 3.32973 1.25324 2.41420 8 19 -2.99836 -6.42741 -2.47185 -5.07106 820 5.46394 8.44963 4.36222 7.53843 8 -2.86079 -2.83403 -2.15147 -3 .33554 21 822 -6.27909 38.11440 -3.59192 23.75073 8 23 10.41571 -35.78230 11.11301 -20.20160 8 24 3.65483 -38.73570 1.49894 -23.70230 8 25 -7.95641 36.97307 -11 .80580 20.59509 164 Table 6.3. Three-tip cracks under tension with fitting error for web crack tip IAI ~ 3%. J3 A,, At fFEM fprec1 Diff. A% 1.69 0.1 0.9 2.5051 2.5391 -0.0340 -3.58 0.83 0.1 0.3 0.8252 0.8590 -0.0338 -3.55 2.05 0.7 0.1 0.6946 0.7273 -0.0328 -3.44 1.53 0.2 0.9 1.6754 1.6436 0.0318 3.34 1.00 0.1 0.9 1.9207 1.8893 0.0313 3.29 1.53 0.1 0.6 1.4138 1.3830 0.0309 3.24 0.83 0.1 0.4 0.9092 0.9389 -0.0297 -3.12 0.83 0.1 0.2 0.7558 0.7852 -0.0294 -3.09 1.37 0.2 0.8 1.2759 1.3046 -0.0287 -3.02 Table 6.4. Three-tip cracks under tension with fitting error for flange crack tip!AI ~ 3%. J3 ~w ~, fFEM fl!!!! Diff. A% 1.37 0.6 0.1 1.872~ 1.9620 -0.0891 -4.76 1.69 0.1 0.9 1.9009 1.8169 0.0840 4.49 1.37 0.5 0.1 1.7768 1.8548 -0.0781 -4.17 1.37 0.1 0.9 1.9349 1.8612 0.0737 3.94 1.53 0.6 0.1 1.8069 1.8802 -0.0733 -3.92 0.83 0.2 0.1 1.6683 1.5957 0.0726 3.88 0.83 0.7 0.1 2.5571 2.4889 0.0682 3.65 1.69 0.6 0.1 1.7431 1.8110 -0.0679 -3.63 2.05 0.1 0.9 1.8411 1.7735 0.0676 3.61 0.83 0.3 0.1 1.8640 1.7973 0.0666 3.56 1.53 0.5 0.1 1.7217 1.7844 -0.0627 -3.35 0.83 0.1 0.8 1.5739 1.6361 -0.0621 -3.32 0.83 0.2 0.8 1.9394 2.0003 -0.0608 -3.25 1.69 0.5 0.1 1.6668 1.7242 -0.0574 -3.07 1.53 0.1 0.9 1.8950 1.8382 0.0568 3.03 1.00 0.2 0.8 1.8775 1.9339 -0.0564 -3.02 165 Table 6.5. Three-tip cracks under bending with fitting error for web crack tip Ill.I ~ 3%. ~ >--w >--t fFEM fpred Diff. Ll. % 1.37 0.2 0.8 1.0514 1.0846 -0.0332 -6.50 1.53 0.1 0.9 2.2763 2.2441 0.0323 6.31 1.00 0.1 0.9 1.8044 1.7733 0.0311 6.08 1.37 0.1 0.8 1.6599 1.6908 -0.0309 -6.04 1.53 0.2 0.9 1.4436 1.4136 0.0300 5.87 1.37 0.3 0.8 0.7774 0.8036 -0.0262 -5.12 0.83 0.2 0.8 0.9147 0.9406 -0.0259 -5.07 1.69 0.1 0.8 1.9056 1.9311 -0.0255 -4.98 1.00 0.1 0.6 1.0881 1.0631 0.0250 4.89 1.53 0.3 0.9 1.0744 1.0495 0.0249 4.88 1.91 0.2 0.9 1.6649 1.6405 0.0244 4.77 1.00 0.1 0.7 1.2565 1.2324 0.0241 4.71 1.37 0.2 0.7 0.8864 0.9094 -0.0230 -4.50 1.37 0.4 0.8 0.5847 0.6064 -0.0217 -4.24 1.37 0.1 0.9 2.0649 2.0866 -0.0217 -4.24 1.53 0.1 0.7 1.5221 1.5011 0.0210 4.11 1.00 0.3 0.9 0.9123 0.8914 0.0209 4.10 1.69 0.1 0.9 2.3895 2.4104 -0.0209 -4.10 0.83 0.1 0.2 0.6504 0.6710 -0.0207 -4.04 1.00 0.4 0.9 0.7102 0.6895 0.0207 4.04 1.69 0.2 0.8 1.1807 1.2004 -0.0197 -3.86 1.37 0.1 0.7 1.3904 1.4100 -0.0197 -3.85 1.53 0.1 0.6 1.2913 1.2718 0.0195 3.82 1.37 0.5 0.8 0.4200 0.4385 -0.0185 -3.61 1.00 0.1 0.8 1.4776 1.4592 0.0183 3.59 1.00 0.1 0.5 0.9512 0.9333 0.0179 3.51 1.91 0.3 0.9 1.2299 1.2121 0.0178 3.48 1.53 0.4 0.9 0.8246 0.8068 0.0178 3.48 1.37 0.3 0.7 0.6545 0.6721 -0.0176 -3.43 1.53 0.1 0.8 1.8246 1.8076 0.0171 3.33 1.00 0.5 0.9 0.5338 0.5168 0.0170 3.32 1.25 0.2 0.8 1.0306 1.0472 -0.0166 -3.25 1.69 0.3 0.8 0.8605 0.8771 -0.0166 -3.25 0.83 0.1 0.3 0.7186 0.7352 -0.0166 -3.24 1.91 0.1 0.7 1.7512 1.7348 0.0165 3.22 2.05 0.5 0.9 0.7346 0.7501 -0.0154 -3.02 0.83 0.3 0.8 0.7009 0.7162 -0.0154 -3.00 166 Table 6.6. Three-tip cracks under bending with fitting error for flange crack tip l~I ~ 3%. p >-.w At fFEM fpred Diff. ~% 1.69 0.1 0.9 1.8969 1.8273 0.0695 4.14 0.83 0.2 0.1 1.5914 1.5253 0.0661 3.94 1.37 0.1 0.9 1.9264 1.8615 0.0649 3.86 1.37 0.6 0.1 1.5739 1.6353 -0.0614 -3.66 0.83 0.3 0.1 1.7110 1.6513 0.0597 3.56 1.37 0.5 0.1 1.5573 1.6128 -0.0555 -3.31 Table 6.7. Fitting errors for two-tip and three-tip cracked I-beams. Maximum fitting error Crack Loading Crack tip ~% Positive Negative 2-tip Tension Upper 1.45 -2.98 2-tip Tension Lower 1.29 -1.93 2-tip Bending Upper 2.25 -2.73 2-tip Bending Lower 1.95 -2.82 3-tip Tension Flange 4.49 -4.76 3-tip Tension Web 3.34 -3.58 3-tip Bending Flange 4.14 -3.66 3-tip Bending Web 6.31 -6.50 167 Table A 1. Comparison of SIFs for center-cracked plate under tension. Mesh Crack SIF Ratio, KFEA/KTada Pattern Length Nonlinear Linear Quarter-point J m xn a/W Extrapolation Extrapolation Displacement Integral Eq. 2.7 Eq. 2.10 Eq. 2.12 Effect of Parameter m 2x2 0.5 1.0468 1.0116 0.9941 0.9974 3x2 0.5 1.0472 1.0118 0.9941 0.9974 2x4 0.5 1.0192 1.0104 1.0061 1.0000 3x4 0.5 1.0193 1.0106 1.0062 1.0001 4x4 0.5 1.0192 1.0105 1.0062 1.0001 2x8 0.5 1.0122 1.0099 1.0088 1.0002 3x8 0.5 1.0122 1.0010 1.0089 1.0004 4x8 0.5 1.0121 1.0099 1.0088 1.0003 3 X 16 0.5 1.0110 1.0104 1.0101 1.0004 4 X 16 0.5 1.0109 1.0104 1.0101 1.0004 Effect of Parameter n 3x2 0.1 1.0475 1.0125 0.9950 0.9974 3x4 0.1 1.0188 1.0103 1.0060 0.9997 3x8 0.1 1.0125 1.0103 1.0092 1.0000 3 X 16 0.1 1.0107 1.0101 1.0098 1.0000 3x2 0.5 1.0472 1.0118 0.9941 0.9974 3x4 0.5 1.0193 1.0106 1.0062 1.0001 3x8 0.5 1.0122 1.0100 1.0089 1.0004 3 X 16 0.5 1.0110 1.0104 1.0101 1.0004 3x2 0.9 1.0143 1.0101 1.0079 0.9982 3x4 0.9 1.0112 1.0103 1.0099 1.0008 3x8 0.9 1.0114 1.0111 1.0110 1.0010 3 X 16 0.9 1.0113 1.0111 1.0111 1.0010 168 Table A.2. Comparison of SIFs for edge-cracked plate under bending. Mesh Crack SIF Ratio, i 1 1 2 2 Figure 2.4. Degenerated quarter-point element with1/Jf singularity. 183 1 st el. 2nd el. 3rd el. nth el. - -- - - .. ? ~ ? 0 ? 0 ? C ? -- ~ - j_L I 4 ~ ~ 2 (X) 2 L ~ 4 . , -. .. -- ... 32 L - -- - n 2L = I -- I Figure 2.5. Element sizes in inner region generated by ABAQUS' * SINGULAR command. I. Centerline of center-cracked flange; or i edge of edge-cracked web; or centerline of eccentric crack in web I o < a/W < 0.5 a/W = 0.5 0.5 < a/W < 1 ~ CX) CJ'I Crack tip (Typ.) ct ,T ~ Symm. a w w w Figure 2.6. Mesh scheme for different crack lengths. 21 (Typ.) ~ 2m?4 1 2mz 4 2m? 4 N N N II II II E E E I (Typ.) n?2 n=4 n?8 Selected for I-beam analysis 2m"'6 2m?6 2m?6 2m? 6 ~ I') I') I') I') (X) ? II ? II CJ) E E E E ""'2 n=4 n?B n ? 16 2m?8 2m?8 2m::8 ... ... ... H ? ? E E E n ? 4 n?B n ? 16 Figure 2.7. Mesh patterns in inner region around crack tip. Quarter-point Element y, V ...... - C...D., Crack Surfaces '- L I ' / ~ / ~ L a Figure 2.8. Nodes used for calculating SIFs. 1.10 -------------------, ~ -- -- -1.05 to- ~ u: -- -- -- -- -- -- -- -- -- C-l) t- -- ---a;:-?? -a-?=???? en? ? eeeeftftmeeeeeeeeeeeeeeeeeeee eeee Cl -----?????????1 w -N 1.00 - -~ ? - ? - ? - ? - ? - ? - ? - ? - ? - ? - ? - ? - ? - ? - ? ? ....... Cl) ...J Cl) ~ ~ 0.95 - - Equation 2.17 to- 0:: 0 ------ Equation 2.18 z - . - Equation 2.19 0.90 L,__ __ __._ __. ...L...----'----___.----:-- 0 0.1 0.2 0.3 0.4 0.5 CRACK TIP ELEMENT LENGTH, Ua Figure 2.9. SIFs calculated from displacement-based methods. Q) O') C 0 ro 0 ii= 0) ~-? ci> ~?? ? -?? -?? -?? -?? -?? -?? .... -?? -?? -?? -?, "'C . . 0 ? E ' .. ' .. E ' .. ' . Q)."c' ". . .. ? ...!. . ? ' . C":' 0 ' .. "in ~-? -?? -?? -?? ~-- . -?? -?? -?? -?? -?? - C: .. -?? -?? Q) E "'C I ~-1- X j \ot- (.) \N 1ii 0 E 0 Q) r0) .l:. (.) . . .0 en Q) 0 I ' . ~ . 3: ' .. .. ? ' ?. ' ' .. ? , .. -?? -?? -?? -?? -?? -?? -?? -?? -?? -?? -.. -? .;\ Q) 0) C ro ti= 189 1.08 1 Upper lip 0 1.06 ct) .~...... I ? W40 X 149 E= 0.0 0 C\I r 0 W40 X 199 ~ 1.04 ----??-???? E= 0.3 .. 0 ? W18x 97 - - - - E= 0.7 1.02 ~ Lower Tip Cl 1.06 L C') ~........ I ? W40 X 149 e= 0.0 Cl C'\I 1.04 f- 0 W40 x 199 ~ ----------- e= 0.3 - 0- I ? W18 X 97 - - - ~ 1.02 ...... .c..o 0::: ... -u.. en 1.00 0.98 L__...J.__....J._ _. .___...J.__....J._ _, L,__.J-_--lo,_ _~ --=--- 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 CRACK LENGTH, Aw Figure 2.12. Comparison of 2-0 and 3-0 model results for SIFs of two-tip cracked I-beam under tension; lower tip. 1.08 ,------------------------..... Lower Tip 0 1.06 L M ~..._ 0 I ? W40 x 149 e= 0.0 C\I 1.04 ~ I 0 W40 x 199 ??????----- e= 0.3 ~ 0 I ? W18 x 97 - - - ~ 1.02 ~ CD a:: N L e- L n 1.00 0.98 L...._...J..._......1,_ _ J.__...J..._ _,.1._ _L -_ ....._ __: --~~-=-- 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 CRACK LENGTH, Aw Figure 2.13. Comparison of 2-D and 3-D model results for SIFs of two-tip cracked I-beam under bending; lower tip. 1.08 ,------------------------- Q 1.06 ~ Web Crack ('I) ~........ Q N 1.04 ~ ? W40 X 149 - Aw= 0.1 0- 0 W40 X 199 1.02 ?-----?? Aw= 0.5 ~ ? W18 X 97 ...... c.o 0:: (,.) -u. en 1.00 0.98 L._..J...._.....1._ _L __...J__.....J.._ _JL---'--_._ _. ._----:-- 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FLANGE CRACK LENGTH, "A. t Figure 2.14. Comparison of 2-D and 3-D model results for SIFs of three-tip cracked I-beam under tension; web crack tip. 1.08 I Flange Crack 0 1.06 . (W) ~....... ? W40x 149 0 N 1.04 A.w = 0.1 ~ ~II - II .. 0 W40x 199 II 0- G Aw= 0.5 .... ..? .? ? .. ???-----~ ? W18x97 ~ 1.02 ~-.. ~ a:: .. ??t;.1 .?. ..... co ?-~~ .... .. ~ e- u. n 1.00 0.980.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FLANGE CRACK LENGTH, A t Figure 2.15. Comparison of 2-D and 3-D model results for SIFs of three-tip cracked I-beam under tension; flange crack tip. 1.08 r--- ---------------------- Q 1.06 ~ Web Crack C') .~....... Q W40 X 149 N 1.04 ? ~ Aw=0.1 - 0 W40 X 199 -0 -------- Aw= 0.5 ~ 1.02 ? W18 X 97 ~ co 0::: CJ1 L-L en 1.00 0.980.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FLANGE CRACK LENGTH, A t Figure 2.16. Comparison of 2-D and 3-D model results for SIFs of three-tip cracked I-beam under bending; web crack tip. 1.08 r--------------------- Flange Crack 0 1.06 L (") ~..._ 0 ? W40 X 149 C'\I 1.04 Aw= 0.1 ~ - 0 W40 X 199 0- -------- Aw= 0.5 ~ 1.02 G:?. ? W18 X 97 _., 0:: ??=? ?.:;;:::, (0 0) u.. bi:-. - ~ en .. --- ?-:?~ ??--? ~1.00 ..- -..?.?? ?- 0.98 L---L----L--..L----L--L--....L-----L--......__. ....... __ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FLANGE CRACK LENGTH, A t Figure 2.17. Comparison of 2-D and 3-D model results for SIFs of three-tip cracked I-beam under bending; flange crack tip. bt 1 ? .. bt 1 ? .. E ~ ,r, '., % ,1/, t E tw A z l'w z ..... dj d dj d co --J e aw af \ a f X X Figure 3.1. Dimensions of two-tip web crack. Figure 3.2. Dimensions of symmetric three-tip crack. 1.05 ,--------------- ---------..... --N ~ 1.00 I Iii! ? ssss ncsn.11111111111111111.lllllllltl? ..- -".'-yWce???- I -~ '+- ...... co OJ ~= 0.83 1.00 1.11 1.25 1.37 1.69 1.91 ?-(;-? -~?? ????? ??~?- ?EJ-? ?+? ??O?? I . 0.95 - - - 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 CRACK LENGTH, 2a w/d j Figure 3.3. Effect of parameter~ on correction factor for tvlo-tip center-cracked I-beams under tension. 4.0 .-c-- I ..... ?- ..... "'C ll '?- 3.0 o-- ? ~ 2.0 I ~ \- CD 0- I - ..... CD ..... ..... ~ I I 1.0 \- AISC Shape 1 Shape 2 a.. w \ 0 -+- ?-??? 0 0.00.5 1.0 1.5 2.0 2.5 AREA RATIO, p = 2A Flange /A Web Figure 3.4. Selected W-shapes for validation of p parameter. 1.05 ,------------------------- --N I ..?. ?? -..-.-??.?.?.?.? ? .00 eatL a 1 . II?????? ?????? -~ 1 ~ 0 $$$$ ?- ~ ?ttltlllllllll!C;i'i,i;----?????? I -T'9 '+- N 0 0 ~ = 0.83 1.37 1.92 -~- --o?? -e- I . . 0.95 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 CRACK LENGTH, 2a w/d j Figure 3.5. Effect of parameter~ on correction factor for tv?>-tip center-cracked I-beams under bending. 1.05 ,-------------------- Web crack tip Flange crack tip / --- / ; , '> -N -,:':..-+::--- 1 . 00 ?????????????? .. . ~ ... .::........ ? ????????????????? ::::??? I---! '+- ?????-??-??----?LJ ????????v N 0 -" ~= 0.83 1.91 ?v? -e- 0.95 ________. ....._ _ __._ _. ..__ _ _..._ ____. ....._ _____ ___. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FLANGE CRACK LENGTH, 2a lb t Figure 3.6. Effect of parameter ~ on correction factor for three-tip cracked I-beams under tension; Aw= 0.1. 1.05 ,---------------------------. Webaack tip ....... Flange crack tip l..:J-???? -.......... ????? .. . N ??? --~......... 1 I 00 ... ... ... ????. . ????????????????? ...... '+- I\.) 0 I\.) P= 0.83 1.91 -~- ?-8-? 0.95 L---'------L--.,__-...&.._---,1_ _. .__ _ __._ _______ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FLANGE CRACK LENGTH, 2a lb t Figure 3.7. Effect of parameter p on correction factor for three-tip cracked I-beams under tension; Aw= 0.5. 1.05 1 -------------------- Web crack tip ....... Flange crack tip v ' --N ~ 1.00 "' ~............. . G. ?????????????????B ????????????????? t:..1?????????????????????????????????? =????????-?-?-?-??-?-?-?-?r. N 0 w ~= 0.83 1.91 -~- ?-El-? 0.95 L-.--L----'---'---.......L--"----'--------........ ------- 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FLANGE CRACK LENGTH, 2a lb t Figure 3.8. Effect of parameter ~ on correction factor for three-tip cracked I-beams under bending; Aw= 0.1. 1.05 ,--------------------- Web crack tip Flange crack tip __,;;,,,---___- e,-----_____: --- -N ....!-..-.i....t ...-. 1 .0 0 . ... . B. -,r- ?????????????????Q ................. ~ -------??? ??????-??-??-- - 't- I\.) 0 ~ ~= 0.83 1.91 0.95 ______ _._ _. ...._ ___-~_-__?-8_-? ___________ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FLANGE CRACK LENGTH, 2a lb t Figure 3.9. Effect of parameter ~ on oorredion fader for three-tip cracked I-beams under bending; Aw= 0.5. 0 0 N llllllllllllllllllllllilllllllli 0 LO ~ ~ZM 'vZM Q) ~ C) 0 C 0 rn ~ ~ Cl:'. LZM cc <( w LL 0?M ?? 0 LO 9?M O'vM 0 0 LO 0 LO 0 LO 0 ('I') N N 0 0 205 Junction Point z X Figure 4.1. Center-cracked infinite plate joined with edge-cracked semi-infinite plate. 206 Y, V y, V a a t t t t t t t t t t t t t s I\) l> {= ! f ?o z 0... ... 2ac .. X, U 4 l l l l l l l l l l l l l l (a) (b) Figure 4.2. (a) Infinite plate with central crack under tension; (b) Semi-infinite plate with edge crack under tension. 12 ,,.-.... w .t5 10 ................ . ,,.-.... Q) ...r..o... 8 ...>..... .. .,_: z 6 w rE dge crack (Eq. 4.28) N ~ 0 w ;,,,,-0) 4 (_) ::i ---?--l Central crack Q_ 2 C-l) (Eq. 4.27) 0 0 0 2 4 6 8 10 DISTANCE, y/a e Figure 4.3. Displacements along junction line of center-cracked infinite plate and edge-cracked semi-infinite plate under tension. -0 - . -0 C: 0 Q) C) ?en C: C: m s ii= I- I I- Q) -g . ::, g> ,Ll?- Q) -c 3~ I _Ll I- 1... -c Q) C: -c m c: t I I I I I I II I Il Q) ::, C>,Ll C: Q) - Jg 3. I 1-a 1- -0 -c: Q) Bm C) ID ? - C: C: QtJ) _0_ m ii= J-I 2 tJ) C: I- a. Q) Q)- :t= a; ~-g 0 ::, ~& -c C: G> m C: ii= ?--o, ~ -m - 209 100 ........... E ..E__.. . 80 T-web: ? ;: . ??? ro ? - ??????? under bending ? ? ? . J: - - under tension .? ? / I- C) 60 ?. ?? ? / z .,? / w ? _J ? .?? / I'\.) ~ ?. ? / ~ 0 () 40 .??/ <( ? 0:: .. ??/ :?, () ,.~? cc w 20 .,> ? s _,; ?" T-flange: under tension I _/ I- ? ~ 0 0 20 40 60 80 100 T-FLANGE CRACK LENGTH, a f (mm) Figure 4.5. Relationship between non-interacting T-flange and T-web crack lengths. 80 I .,;;: I T-section ~ ~ ....-... 60 ~ - - Single plate ~ ~ / ~ .E am.. 40 .~._. . I_\.J, _., ~ '(!!" ~ l T-flange crack L-a.: T-web crack Central crack en 20 Edge crack a---------_.__ ________ __. 0 20 40 60 80 CRACK LENGTH, aw and at (mm) Figure 4.6. Comparison of SIFs for non-interacting T-section and single plates under tension. 0.5 --------------------, Central crack 1 T-flange crack-, _ 0.4 - 27"1 T-web crack ..-... [ 0.3 Edge crack ..__... 0 T-section N 0 ...... 0 0.2 - - Single plates Cl N 1./ aw= 34.79 .I ~ 0.1 I Crack ti~ \{ v Cra~k tip I_ __ _;___ 0 I 1 -100 -80 -60 -40 -20 0 DISTANCE, x (mm) Figure 4.7. Comparison of CODs for non-interacting T-section and single plates under tension. 1.5 r--------------------- ........... E I T-section ...:_::_i,., I - - Single plates ._>- 1.0 z UJ ~ uUJ "-> ...II. w :5 a.. ac= 50.0 mm e-n 0 I l L~~~ ...._ .._ ........_1~ a8 = 34.79 mm 0.0 L,__ _. .J...... __- 1..._ __ __,,1_ __. ....._ ______~ 0 50 100 150 200 250 300 DISTANCE, y (mm) Figure 4.8. Comparison of displacements along junction line of non-interacting T-section and single plates under tension. 4,--------------------- T-section 3 - - --Single plates ,,, -- -- -- ,,,,,....., E -- -- -- ..:..1._., _,,,,. _,,,,. -- -- 0 2 _,,,,. _,,,,. -- ae= 100.0 0 CJ _,,,,. N --" ~ ~ _,,, ac= 50.0 _,,, 1 / Edge crack Central crack / ------- 0 ~--..1,__ __ _.__~~~~-~--~ -100 -80 -60 -40 -20 0 DISTANCE, x (mm) Figure 4.9. Comparison of CODs for T-section and single plates under tension. 4,----------------------- ....-... - - E - - - - - - - - - ----1- - - ::1. ~ - Edge crack ....._,, 3 ... > T-section ._:- z - - Single plates w ~ 2 .... w a C = 50.0 mm 0 I\.) ....i.. ::j 01 a. a = 100.0 mm _ - -1 Cf) 1 t e - - - - - 0 -- - - - _ _ _ _ _ - - - - L_ Central crack o-------------------- 0 50 100 150 200 250 300 DISTANCE, y (mm) Figure 4.10. Comparison of displacements along junction line of T-section and single plates under tension. 4,-------------------- ~ I-beam Single plate --9? -,f- ~ 3 It- ~ e- u: n C 2 .. ?? w ?? ?? N I\J ::i ...Ji. 0) <( ~ 1 0:: 0z i.o 0.2 0.4 0.6 0.8 1.0 FLANGE CRACK LENGTH, A f Figure 4.11. Non-interacting flange crack length determined by equating K t; W33 x 201 , Aw= 0.1 , under tension . 4------------------ ~ I-beam Single plate --~-- ~ -E 3 ... 3: :x::: u- : ?? en 2 .. ?? ?? -? .... e Cl w ... e ? . -??? N r.????? N ,, ....I,, -....J r:::..????????~ '\. , -..J ~ 1 ... ?""' ,--, = ........ =. ........ .. / ~ ................ ~ .............. 'IJ.... 'IJ 0:: 0z I I I I i.o 0.2 0.4 0.6 0.8 1.0 FLANGE CRACK LENGTH, At Figure 4.12. No-interacting flange crack length determined by equating Kw; W33 x 201, "A.w= 0.1, under tension. 1.0 ,-------------------- '+- c<. . :I: 0.8 I- C) z w ???-? 0.6 _J ~ ??????"?"?"? ~;?I?ii????????????-?????????????? ,~? ??..?...... ~.? ???????????..~,,, -???????????????-?> ~ (.) N ~ 0.4 -" CX) (.) w C) z 0.2 ::i From CMOD From Kt From Kw u.. ~ -?- --~-- 0.~.5 2.0 2.5 3.0 3.5 DEPTH-WIDTH RATIO, y= d j/b f Figure 4.13. Comparison of non-interacting flange crack lengths determined from CMOD and SIFs; 'A.w = 0.1, under tension. 1.0 ,--- ----------------- ,.._ c< :c- 0.8 I- C) z w 0.6 ...J "" = ...~.. .~ ? ?????? ???.?.?.? ~????????????????)< ???????????????-..> ~ (.) JQ.=???????????=-??"" ????????? .. ????????? N ~ 0.4 .....t,, co (.) w C) z 0.2 ~ From CMOD Fr-o~m -Kt From Kw u.. --e- --~-- 0.0 L------'-------'--- - ---:-~---~ 1.5 2.0 2.5 3.0 3.5 DEPTH-WIDTH RATIO, y= d j/b f Figure 4.14. Comparison of non-interacting flange crack lengths determined from CMOD and SIFs; "Aw= 0.1, under bending. a: 0 1.1 I- (.) <( LL z 1 0 I- (.) 0.9 UJ a: a: 0 (.) 0.8 0.7 N N 0 2 1.5 ~,9?"-<:._ -->< 0.1 @ 0.9 I ~~ ~ ~ ----------- ~ ~ 0 .3 a: I ~ ~~ ><_ 0.5 a: 0 o 0.8 --1 ~ ..________ ~ ~ ~__,._ 0.7 0.9 I N 0.7 N N 1.5 0.6 ~/;,?-'~ J;,~ '!Jo 0 .4 1 '/.. \..E.~G1\-\, }.~ , Ii C \f-JE.\3 c\'.\~ Figure 5.3. Correction factor for two-tip cracked I-beam under tension; eccentric crack, lower tip. 3.0 r---------------- <( 't-.. 2.5 r ? e= 0.0 a:: 0 \ X e= 0.1 ------- Single plate I- ~ 2.0 r ? e= 0.5 I-beam u..: z 0 \ e= 0.9 ? . 0 ? ? ? ? J\:) - 1.5 J\:) I- (.,J u w a:: ~ 1.0 u o.1>.o 0.2 0.4 0.6 0.8 1.0 CRACK LENGTH, 'Aw Figure 5.4. Comparison of I-beam and single plate under tension; upper tip, ~ = 0.83. 3.0 r---------------- a) '+-.. 2.5 ? E = 0.0 a:: ..:.?,t?. 0 X = 0.1 ------- Single plate -.... .. ...' E ~ -. l) 2.0 ?- ?. if ? e,= 0.5 I-beam ? ? .. ? ?? ? ?? z 0 e,= 0.9 .??. ~: ??" 0 I'\.) - 1.5 .. ~::???? ?? .?? ?? I'\.) ~ -? ~ l) ????????? ? w ~-- .. ? ?? ... a:: ___ .i.-1?????! ~ ?????????????? ?= . ..? ???? ./?!? a:: 0 1.0 l) 0?1l.o 0.2 0.4 0.6 0.8 1.0 CRACK LENGTH, Aw Figure 5.5. Comparison of I-beam and single plate under tension; lower tip, J3 = 0.83 . .... 1.2 r-------------------, '+-- - 1.1 0:: 0.,_ (.) 1.0 ~ ---~-----???o.. - ---...... ..... ,. z ~ E 0 "'?}... ?? .,_ ?? 0.9 ?? ? N ?? N -~ ? ? ?. u, (.) .... _ ~ w ? ? ? 0:: E, = ? 0.0 0.1 0.3 0.5 0.7 0.9 ? ? 0:: ~ 0 0.8 Upper tip ~ ~ ~ -& ~ ~ (.) Lower tip ????- --~-- ?v- --B-- ??- --o-- 0.70.0 0.2 0.4 0.6 0.8 1.0 CRACK LENGTH, Aw Figure 5.6. Correction factors for two-tip cracked I-beam under tension; J3 = 0.83. 1.2 I ?= -0-.0- 0.1 0.3 0.5 0.7 0.9 Upper tip I- -,.f-1.1 ~ -B- ~ ~ '+-- Lower tip 0:: ???-- --~-- -?- ?-Er- ?*? --o-- 0 I- (..) 1.0 K2': ? j (!!!' ,,,,,,~ :?:?:;- :-?:??! -?:?1 ::i.-l ;.ill. ? ?? ? ? - ~?? I-::e. e..: Etf .. :::: E ~ z ????????-e.... )'!<..??--- ...... ...?,,.,,,, -.. -----6._ "?? --- =- . .. .. ?. . . .. ..._ 0 ??????"El.. ?? ? ,,,_ ?? ? -~ N I-- 0.9 -- .. .. N 0) (..) ?--- .. .. w "''G) .. .. .. .. .. ..~ .. ....... L_J .... 0:: 0:: 0.8 .. .. . * 0 .. .. ~ (..) 0.70.0 0.2 0.4 0.6 0.8 1.0 CRACK LENGTH, 'Aw Figure 5.7. Correction factors for two-tip cracked I-beam under tension; ~ = 1.37. 1.2 I e= 0.0 0.1 0.3 0.5 0.7 0.9 Upper tip -e- ~ -a- --e-'+- - 1.1 ~ ~ -+- ~ Lower tip --?-- --~-- -?- -e- ?*? --o-- 0 I- (_) (t 1.0 z 0- N I- 0.9 N -.J (_) w ~ ~ 0 0.8 (_) 0.70.0 0.2 0.4 0.6 0.8 1.0 CRACK LENGTH, Aw Figure 5.8. Correction factors for two-tip cracked I-beam under tension; J3 = 2.05. 1.2 .----------------- Upper tip <( '+- 1.1 I .. ~ 0 t5 1.0 Lt z 0 N -.,_ 0.9 N CX> (.) w I ~ 0:: 0:: 0 0.8 t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (.) )( 0 0 ? ? ? ~ 0.70 .0 0.2 0.4 0.6 0.8 1.0 ECCENTRICITY, E Figure 5.9. Correction factors for two-tip cracked I-beam under tension; upper tip, f3 = 0.83. 1.2 ,------------------ Lower tip co '+- 1.1 - .:???????????? :a ... a:: ????????????? ?!? .?.?.=...=..??.??. I..?.?.?..?.?.. ...... , 0 ????????? ?..?.?.. ....?.?..?.?. .?.?..?.. :.......? .?.?.-.?.?.?. . ~ !..1 ...1..1??? ?????? I I- .......................................... :??-???? ........................... !...I..?. :?:?:1??1?:?? (..) 1.0 ::::.?.?.?.- -????????????????-????????-?-?-?-?-?-1 ??.?.?? ????.?.??.?..1.1.1 1111?1 ?? ?-?- iill!!????i:E ?-?-?- ?--- ???!!~!!?: ???--?--.-_~ . !.!.!.! !::!???--? !?!-!-!-!-!-!-::~ ?--. -?---. ??,: z --?? tL.l.._ T???? - m?. ?- ????? 0- 0.9 ?-?-?? .. ??-?- ??-N N I- ?? c.o --~ ?----(..) 7"'- .. w Aw ?-?-?-?-.. ... et: .. --.:-\- et: 0.8 0.1 0 0. 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (..) ????? --~-- ...?..? --~-- --+?- ??B-- ....... ??*?? --+?- \ 0.7 0.0 0.2 0.4 0.6 0.8 1.0 ECCENTRICITY, E Figure 5.10. Correction factors for two-tip cracked I-beam under tension; lower tip, ~ = 0.83. 1.2 r----------------- Upper tip <( '+- 1. 1 I a::- 0.... .. (.) 1.0 Lt z 0 N - (.,.) ...... 0.9 0 (.) w r Aw a:: a:: 0 0.8 t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (.) )( 0 0 ? ? ? ~ 0.70.0 0.2 0.4 0.6 0.8 1.0 ECCENTRICITY, E Figure 5.11. Correction factors for two-tip cracked I-beam under tension; upper tip, ~ = 2.05. 1.2 I 0.1 0.2 0.3 0.4 0.5 0.6 0. 7 0.8 0.9 cc '+- 1.1 r ??*?? ??0-- ??????- --0-- ??+-- --B?? ??+?? ??*?? ??+?- - 0:: Aw 0 \- 0 1.0 ???????????~:-:-:-:??????-??-???-??-???-t??????????????????:?:?:?:?:????1--?-???????t?1?????t??:?.:?:??:?:?:?:i:?:? ?????1??:?:?:??:?:?:??:??:?::??:?:t. ~ ??????????-????? . ----- ????????? ??????--?------ z -------~ ----- ?-----------------... ---------?-??????-- ----------...........?.-.-.. ?. .................. .......... :;;;.i.L_ -- ???-?-- - ... 0 7rv'?????------?------~ ......... . ???---------- 0.9 ???-??? N (.,.) \- ???-?????"'\.L(_....,,. ------- ~? -??????--- 0 ............. _* ' w 0:: -4.----------------?--+----??-????--?--?--+.--.-.._ __________ + ????--??--???-??-?-r- 0:: 0.8 0 Lower tip 0 0.70.0 0.2 0.4 0.6 0.8 1.0 ECCENTRICITY, E Figure 5.12. Correction factors for two-tip cracked I-beam under tension; lower tip, ~ = 2.05. 1.2 ,----------------- < 1.1 Upper tip ..... L ~ C( g 0 1.0 r !l( , ~ I if ? ? l !: z 0 Nw - 0.9 \ ....... N I-0w ~ 0::: 0::: 0.8 t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0 )( 0 ~( ? I ? ? ~ 0.70.5 1.0 1.5 2.0 2.5 AREA RATIO, ~ Figure 5.13. Correction factors for two-tip cracked I-beam under tension; upper tip, ? = 0.5. 1.2 I 0.1 0.2 0.3 0.4 0.5 0.6 0. 7 0.8 0.9 CX) '+- 1.1 ~ --~-- --?-- ...?..? ??*?? ????? ??+?? ??+?- ??B?? --0-- - 0:: Aw 0 t; 1.0 ~ lti1i~~~??????!!!!,~~~.-::::::?!!!!!~!::::::!!f:::::? tf ~------????.-;J __ _ D ... ?? ?G-- ---+--------?-----? ?r.:l. z G... c.:.r ?--. ?????e---?-???B?-??-EJ N o 0.9 ????.t:: ~ ~1 --?--... ~ (,.) (,.) b --~~ --?o---" ' w ----~~- -????o??-----??-f9.?-???o Cl'.. Cl'.. 0.8 i 0 Lower tip (._) 0.7 l 0.5 1.0 1.5 2.0 2.5 AREA RATIO, {3 Figure 5.14. Correction factors for two-tip cracked I-beam under tension; lower tip, e = 0.5. <- er 1 E 0 ~ o.s 2 ECCENTRIC/TY 0 ~ C 0 0 ~ 0.9 a: 0 o -0.S 0.7 N o.s w ~ 0.3 2 -1~ 1.5 ~.l?-1-.. 'IQ 1 ? I) 0.2 0.4 0.6 0 'l'JE.'3 Cf\~C\( LE.~G1\-\, ~YI Figure 5.15. Correction factor for lwo-fip cracked I-beam under bending; upper lip. C-D ECCENTRICITY C ,e_r 0 1 0.9 0 <{ 0.7 LL z 0.5 0.5 0.,_ 0.3 0 0.1 w aa: 0 0.0 : 0 0 -0.5 N v) U'I 2 1 1.5 "1A?"~ ~ ...,.,a 0 4 0.6 :..q ? /1 0.2 . CY... \..'c.~G1\'\, 1'-.. 0 \f'-1\:.ec~i>-l Figure 5.16. Correction factor for two-tip cracked I-beam under bending; lower tip. 2.0 Upper tip 1.5 ~ a:- 0.,_ 1.Q I (3 ??????-~--==?---:?: -------?3----????-?1????????e????????-E)????????"U & 0 0 0 ?O"\ (.) ~ 0.5 z 0-.,_ 0.0 N <,.) 0) hl -o.5 ? e= 0.0 0:: X e= 0.1 ?----- Single p\ate 0:: o -1 .0 D e,= 0.5 \-beam (.) o e= 0.9 -1.50.0 0.2 0.4 0.6 0.8 1.0 CRACK LENGTH, "Aw Figure 5.17. Comparison of \-beam and single web plate under bending; upper tip, ~ = 0.83. 2.0 Lower tip 1.5 ? ? .. ??? ? co ?"? '+- - .... ????? ?? ?? .. .. a:: 1.0 0 \- (.) ~ 0.5 z 0 0.0 N w - -...a \- (.) w -0.5 ? e= 0.0 0:: a:. x e= 0_1 ?????? Single plate 8 -1.0 D e= 0 .5 -- \-beam 0 e= 0.9 -1.50.0 0.2 0.4 0.6 0.8 1.0 CRACK LENGTH1 Aw Figure 5.18. Comparison of \-beam and single web plate under bending; lower tip, 13 = 0.83. 1.5 ------------------, 1.0 \to-.. a=: 0 0.5 ~ (._) i:E z 0.0 0 N v,) - CX) ~ (._) w -0.5 I a: e.= 0.0 0.1 0.3 0.5 0.7 0.9 a: -@o- 0 -1.0 \- Upper tip -?-- ~ ~ -B- ~ (._) lower tip --?-- --~-- -?- --a-- ?-)\(-- ??e?- \ -1.1) .0 0.2 0.4 0.6 0.8 1.0 CRACK LENGTH, Aw Figure 5.19. Correction factors for two-tip cracked I-beam under bending; 13 = 0.83. 1.5 r------------------- '+-.. 1.0 0:: 0'G 0.5 i:E ~ 0.0 - 1') ~ v,) (0 l) w -0.5 I ci:: e.= 0.0 0.1 0.3 0.5 0.7 0.9 ci:: 0 Upper tip --e- ~ l) -1.0 \- ~ ~ -B- ~ Lower t\p --?-- --~-- -~- -e- ?*? ??O?- \ -1.5 0.0 0.2 0.4 0.6 0.8 1.0 CRACK LENGTH, Aw Figure 5.20. Correction factors for two-tip cracked I-beam under bending; 13 = 1.37. 1.5 ------------------, '+- 1.0 0:::- 0 \u- 0.5 ~ z 0.0 0- N I- ~ 0 u w -0.5 a:: I a:: e= 0.0 0.1 0.3 0.5 0.7 0.9 0 -1.0 \- Upper tip --e-u ~ ~ -&- ~ ~ Lowerttp -??-- --~-- -?- ?-Er- -+- --e?- \ -1.5 0.0 0.2 0.4 0.6 0.8 1.0 CRACK LENGTH, Aw Figure 5.21. Correction factors for two-tip cracked I-beam under bending; ~ = 2.05. 1.5 r-----------------7 <( ~.. 1.0 0:: 0 I- 0.5 (.) if z 0.0 0 I- N - ~ (.) ~ w -o.s r a:: Aw a:: 0 (.) -1.0 r 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ~ 0 0 ? ~ \ ? ? -1.5 0.0 0.2 0.4 0.6 0.8 1.0 ECCENTRICITY, E Figure 5.22. Correction factors for two-tip cracked I-beam under bending; upper tip, ~ =0 .83 . 1.5 -----------------, ~. . 1.0 ~ 0:: 0 1- ~:::::::!~::11??????????=?=:1:=:=111?????11111111111111111111111111??????????????1!!!~ l) 0.5 ?????????????? ??????????????:-----?? ???-?????:?:?:?:?::?:?:?:?:?.?: :: ?????????:::====?-?? ::::?:?::?:?:?:?:?:?=?i.?.? ?; ?? ?????????? ii:a if t????????????????? ?:?:?:=? ?? ?????????????:?:?:=?:?:?:? ? ?????? :::--????? . z o. o r::::::. ??????????? _ I 0- I- N .;:.. t) N w -0.5 ~ ~ ~ ~ 0 l) -~.O ... 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 --~-- --~-- ...?... --0-- ??+?? --8-- --?-- ??*?? -?+?- -1?1l.o 0.2 0.4 0.6 0.8 1.0 ECCENTRICITY, E Figure 5.23. Correction factors for two-tip cracked I-beam under bending; lower tip, ~ = 0.83 . 1.5 r------------------- <( '+- 1.0 a::- 0 r- t) 0.5 if z 0 0.0 r- N t) ~ (.,) w a:: -o.5 r Aw a:: 0 t) -1.0 r 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ~ 0 0 \ ? ? ? ~ -1.5 0.0 0.2 0.4 0.6 0.8 1.0 ECCENTRICITY, E Figure 5.24. Correction factors for two-tip cracked I-beam under bending; upper tip, (3 = 2.05. 1.5 ----------------7 c:o '+- 1.0 .... 0::- 11 11 0 lo- 0.5 --?????? ???????? 111111111111?11111 111111111111111111 ??? ???????? ? ~?1????1?1???1?????? ???????? ?1??????????????????????????????1?1?1:1:1:1:1:1:1:1: ????::ss?~' l ? ????? ??????????????' .?.?....... ..., ,. ?? ' - ?????????????????? ???????I???I ~~-- i:E fE??????????????? ? ????:::::::.?????:. ::==?????? .. , 0 ::==????????????? z 0 . ????????? I 0- N 1- ~ (._) ~ w -0.5 ... a:: - ~ a:: 0 (._) -1.0 ... 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 --~-- --~-- ---??-- --~-- --+-- ??B?? --+-- ??*?? --+-- -1?1l.o 0.2 0.4 0.6 0.8 1.0 ECCENTR\C\TY, E Figure 5.25. Correction factors for two-tip cracked I-beam under bending; lower tip, ~ = 2.05. 1.5 ------------------, <( 1.0 Upper tip '+- a::- 0 I- 0.5 ~ t) l~. ?- ? __ :,;;;;;;;;;;;;;,:_ :---:;...: ---~~-:_:=::.a1 if z 0.0 L-----------------~ 0 N I-~ - CJl t) -0.5 w Aw a::: a::: 0 -1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <..) ~ A ~ ? ? ~ 9 -1.50.5 . 1.0 1.5 2.0 2.5 AREA RATIO, ~ Figure 5.26. Correction factors for two-tip cracked I-beam under bending; upper tip, e = 0.5. 1.5 -----------------, a:l 1.0 r Lower tip '+- er:-: 0 o.5 l 11111111)11111111111,,,,,,,,,,111111111111111,,,,, I- l) ~ z 0.0 0 N - ..t:,.. I- O') l) w -o.5 r Aw a:: a:: -1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 t) ???-- --~-- ????--- ??*?? ....... --+-- --+-- --B-- ??-@>-?- -1.5 I 0.5 1.0 1.5 2.0 2.5 AREA RATIO, ~ Figure 5.27. Correction factors for two-tip cracked I-beam under bending; lower tip, e= 0.5. 7 -- 6 .e_r 0 CJ 5 WEB CRACK LENGTH, Aw <( u.. z 4 0 -i ? 0.7 \- 0.6 (.) 3 w 0.5 a: cc 2 0.4 0 0.3 (.) 1 0.2 0.1 N ~ 0 ---- -....1 2 1 1.5 ..q~~..q 0 4 0.6 ~,q?-10 0.2 . c'I--\..'c.~G,'f\. l--' ? 13 0 r'-"~G\:. c?-~ Figure 5.28. Correction factor for three-tip cracked I-beam under tension; flange crack tip. e- ~ r 6 0 ~ 5 CJ <( LL z 4 0 ~ 3 WEB CRACK LENGTH, >-. w CJ w a: a: 2 0.1 0 (.) 1 0.2 0.4 0 0.7 0 N ~ 0.2 0) ~ 0.4 Lqi\'G~ 0.6 C~..qCJr 0.8 1.2 1.4 l. ?21\tG-,.lt 1 0.8 ? ,i ""\:.""""''a.~ I Figure 5.29. Correction factor for three-tip cracked I-beam under tension; web crack tip. 7.0 r----------------- - 6.0 0.1 0.2 0.3 0.4 0.5 0.6 0. 7 Tada 't-.. ??-@>-?? ??*?? --~-- ??*?? -?+?- ???-- ??+?? ~ ~ 5.0 Aw b . ... ?.. ? ? ? (t 4.0 ? ?? ? ? ....,L ... ..M.? z .?. ? ? ? ?? ,,,, ? ? ? ?- ?? '\ ,L/' ?? J ? ~ Q 3.0 ? b ?. ?-?-. ....-.. . ?.- :t.. .. ...... .... ???????? ........ 1.... ..... .., ?... . ... ..,.,~----? .. .. ;(_ """)" N ..s=,... l ??? .. . ......... ... ....... .... ..... ... .. . .. . .. _..._,, (0 ~ 2.0 ???-?-???? .? ... .. i ??....?..?. ..? .. . .. ....... ..................... ??..?..?.?..?.. .... -......... ............. . ... ?..?.?.?......?..? ..... .. .. .. . ..... .. ................ ....... ..????? ~ ... . . . ... .. . ... . 0::. ~c, ?????? ??????~ .. =~?? ??????: :??;.'?~ ??? .?.??? ?..?.: .~ I-:? ..?.?..?..?.?..?:.~.l.~..:.? ..?.?..?..?.?..?..?.~.?. ..? ..?.?..?..?..?..?._ .i~- ?.-??- .-... .... .. ___ ..~......., ...... ,(e)o........................... .. ............ -?, .. .. 8 1.0 0.00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FLANGE CRACK LENGTH, A f Figure 5.30. Correction factor for flange crack of three-tip cracked I-beam under tension ; (}= 0.83. 7.0 r----------------_.,, - 6.0 0.1 0.2 0.3 0.4 0.5 0.6 0. 7 Tada ~ - --~-- --~-- --~-- ??*?? ??+?? ????? ??+?? -B ~ 5.0 lw I- (.) Lt 4.0 z Q 3.0 i- l'v (J1 (.) 0 ~ 2.0 et:. 0u 1.0 O.?o.o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FLANGE CRACK LENGTH, "- f Figure 5.31. Correction factor for flange crack of three-tip cracked I-beam under tension; ~ = 2.05. 6 - 5 e-r 0 I- 4~ / ~W EB CRACK LENGTH, J,..w (.) <( LL z 3 0 21 ~ ~/ ~ / / ~0.7 ~ (.) w a: 17 0.3 a: 0.2 0 (.) 0.1 0 N u, -1 -"' 2 ~ ~ - 1.5 ------- 1 4~? 0.6 4 ~4 }) 1 0.4 ~\\ ~, 0. 13 0.2 \..~~G ' 0 cf\P...C"' r~~G~ Figure 5.32. Correction factor for three-tip cracked \-beam under bending; flange crack tip. 6 -~ 5 er r0- 4 0 ct: LL 3 WEB CRACK LENGTH, >-. w z 0 ~ 2 0.1 t) w cc cc 1 0.2 0 0.3 t) 0 0.4 0.7 N (J'\ -1 N 0 0.2 ~ 0.4 ~tvG/2 0.6 C. 0 0 0 -3 0 -4 -5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 AREA RAT\O, \3 Figure 6.6. Variation in prediction error with area ratio~ two-tip cracked I-beam under tension, upper crack tip. 5 4 - 3 - ,,-.... 2 - cf!. 9 .._.. 1 - I., r "" ~ <1 ~ 0 0 .. ~ wt - - .... ~ (JI co ~ - ? 3 - 8 _A - l 5 ? I I I I 0 0.2 0.4 0.6 0.8 1 ECCENTRICITY, E Figure 6.7. Variation in prediction error with eccentricity; two-tip cracked I-beam under tension, upper crack tip. 5 4 3 2 ....--. 0 ~0 1 0 '- - 0) - '9 0 ~ 3 - -4 - a I 5 I I ? ? I I ? 0.6 0.8 '\ '\.2 '\ .4 '\ .6 '\.8 2 2.2 2.4 AREA RATIO, ~ Figure 6.12. Variation in prediction error with area ratio; two-tip cracked I-beam under tension, lower crack tip. 5 4 ... 3 - 2 ,,,...... - ,.. ,.. I- - ~1 .... ~ i,:: 0 f ~ l ... ......... - ~l .. ~ la, '- '- ..... '- ~ 0 w 0 ) .... N ~ O') c.n 3 ~ -4 ~ \ I I I I I 5 0 0.2 0.4 0.6 0.8 1 ECCEN"TR\CrTY, E F,gure 6.13. Variation in prediction error with eccentricity~ two-tip cracked \-beam under tension, \ower crack tip. 5 4 3 2 ~ ~0... ..... . 1 -2 O> -3 -4 -5 0 0.2 0.4 0.6 0.8 1 CRACK LENGTH, lw Figure 6.14. Variation in prediction error with web crack length~ two-tip cracked I-beam under tension, lower crack tip. 1.2 C-D a: 0r- ~ 1.1 u.. z ;, 0 5 w er. er. 0 N (.) / /6.a CJ) -...i 0 9 0.6 0.4 ,,...J ~ - _l~_ o.6~A~""----o/o 0 ~ ,, .2 ~~G 0<:c,~ WEB CRACK LENGTH, A. - - -...i ~ 3 ... 0 .4 - 5 \ I ' ' ? ? l 0 0.2 0.4 0.6 0.8 1 ECCENTR\C\TY, E Figure 6.19. Variation in prediction error with eccentricity; two-tip cracked I-beam under bending, upper crack tip. 5 4 3 2 ? 8 ..-. 0 ? ~._, 1 w'- 0 --l - ~ "-> 3 0 A 5 0 0.2 0.4 0.6 0.8 1 CRACK LENGTH, lw Figure 6.20. Variation in prediction error with web crack length; two-tip cracked I-beam under bending, upper crack tip. 1.4 <- a: g 0.85 - ~u. 5 0.3 lo- N w --.l w aa:: -0.25 - 0 0 ~~ -0.8 0 .6--,__ - --;:;.--,_____ ~ ~6 0 .4 ~ '?, Wta CAA 0.2 - -- / 0.2 :<,.~c,\ Cf( LENG ~ 0 CJ~. o~----- ~-6 _, /L_ o. 4 ~ ,'tJ WE:a CRA o2~ 0.2 ~-<..~c,' Cl( LENG; 0 0 <.:,Ve,<.::: H,A 1? Figure 6.22. Comparison of predicted and calculated correction factors for two-tip cracked I-beam under bending; W18 x 9 7, upper tip. 5 4 3 2 ...-.. ~0.. ._. . 1 - @ ~ ~ ' ~ !!I ~ ~ " .. ~ ! 0.:::_ !e... . 1 ,. t Ii <:l 0 ~e - I - l :.... 0 0 ? '- .I0 .i.i. N wL.---l 2. - 0 8 O> - 0 0 0 3 ~ 0 -4 ~ I I 5 I I I I I ? 0.6 0.8 '\ '\ .2 '\ .4 '\ .6 '\ .8 2 2.2 2.4 AREA RATIO, ~ Figure 6.24. Variation in prediction error with web crack length; two-tip cracked \-beam under bending, \ower crack tip. 5 4 - 3 ... 2 ... 0 0 .~ I r? ~ I ~ ? ._.. . 1 - a: 0 a: 0 / 0.8 (.) -0 .8 ---------- 0.6 - - ---;r 0.4 0.6 "4 _ ~ 6 ,.'..',l, "-' v. 0.2 ---- 2 ~~ o o !-..j:,~ LENGT 0.1 0.1 <;,.~G 1-f, ,\ t ?-~f<<-~~~ - 0.4 X':""' E c 1'14cK O. 4 '0::.--3:: --, ----~. ,,: i;? ""'B' // 0.3 ~~ lENGr 0.2 n . 0.2 f,4- H, A, O. 1 0. 1 ~~ 0 w 0 N -2 -:\ i 0 \.) 0 0 0 ~ 0 0 0 -5 0 0 .'\ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 WEB CRACK LENGTH, "-w Figure 6.50. Var,ation in prediction error with web crack length~ three-tip cracked I-beam under bending, web crack tip . s---------------0- ------ 4 0 0 0 0 3 0 8 @ ij 0 8 0 0 @ 2 ~ ~ ,,,.,......, @ 0 8 cF i ? ....._-_,.,. 1 \ 1:1 <\ 0\ ,:s I iii 8 ~ I @! es 9 I iiil c5 ~ -1 l;;;J '- v) '- 0 v) W -2 ~ ~ '~ I 0 ij 0 8 ij 0 0 0 -34 8' 0 } 0 ? \ ' ' C) - .., ? ~<$> Figure 6.53. Comparison of predicted and calculated correction factors for three-tip cracked I-beam under bending; W18 x 9 7, web crack tip. ~ 5 4k 0 0 0 I 3. 2t 00 0 0 0 O 0 0 00 ...--... mo 00 ,___ n 0 8 0 ~0... .....,, ,. '\ -- Ul -2~ 0 i 0 ~ r 8 9 0 0 0 -3 0 0 -4 . 0 O:\ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 WEB CRACK LENGTH, ')._w F\gure 6.56. \Jartation in prediction error with web crack length; three-tip cracked I-beam under bending, flange crack tip. 5 4 r- 0 8 0 I 3 ? 2 0 ,,,-.... ~ '\ 0.. _. . f\gure 6.59. Compar\son of pred\cted and ca\cu\ated correction factors for three-tip cracked \-beam under bending: W'\8 x 9 7, f\ange crack tip. Concrete (.,) ~ N ta) tb) tc) td) figure 6.60. ta) Composite ro\\ed beam?, tb) composite p\ate girder with doub\y symmetric section~ tc) noncomposite sing\y symmetric section; td) composite p\ate girder with sing\y symmetric section. r - - - - J I I I I ',\ (.,) ~ (.,) \ la) lb) le) r\gure S.S'\. la) Noncompos\te s\ng\y symmetr\c sect\on; lb) stress d\str,but,on; and (c) equ,va\ent doub\y symmetrtc sect\on. t End Bear\ng Sym. about t Span I I \ 10@ 12" = 10'-0" \ 8@ 15" = 10'-0" \ Shear Connector Spacing i-- -\ 7 W27x84 <_.,.,). .i,,.. Cover Plate S"x. ?" '~ ?" 7'-6" '\2'-6" 20'-0" Figure 6.62. Composite, simp\e supported, ro\\ed beam. ?? II 11? \ ? ? 4. ..; ? ?? . ? I' I .? .. . - -~ II N? -c, -CD ~ 0 e0. 0 J 0.. . .? -0 ?";c' ! ' 'ii c-.o.. .. ?:::, i z 315 - sym. about t span I 2a::: 4.67' t End Bearing Half Parabola 441 441 43,7:.. _..o-__- 1-_--o . 6 64 sending moment diagrarns. F1gure . . -- -5.94 +5.53 -11.47 t.,....).. --.1 + +'\1.0 +5.5'3 +'\ '\ .47 f\gure 6.65. Oecoml)os\t\on o1 \\near\~ d\str\buted \oad\ng \nto ax\a\ tens\on and pure bend\ng. Note: All dimensions are in mm. 0 LO ~ II ~_,,. .c ct) a \ a \ \ \ \ I a \ - Equation 2.11 ,-..... E 40 ..'_i_, a 0 (.) (.,l N C> Crack: a/W = 0.5, W = 100 mm Load\ng: tension, a = 100 MPa Mesh: _m _= 3, n = 8, L = 781 ?m ________O\s p:. v8 = 5(.) _.... _'\ .'\_ ?m_, "_c =_ '\0_1.9_ ?m_ __ __ (.) 0.2 0.4 0.6 0.8 1 D\S1'ANCE, r/L f\gure A.3. COO pro1\\es with\n crack tip e\ement for center-cracked p\ate under tension, a/W = 0.5. I . 1.02 ------------------. ~ r Equation 2.12 Equation 2. 9 1.01 I ?????--?='=l ro -0 8t .? re JC ~ -~.... ,Jc w ~ : LE ::a~o~ 2.5 N ~ 1.00 I I Crack: a/W =0 .5, W =1 00 mm I Loading: tension, cr = 100 MPa Mesh: m = 3, n = 8, L = 781 ?m I 0.99 .__ __________D_isp_. :_ v_8 =_ 5_1_.1_ _?_,m_ , Ve_= 1_01.9. . _?_m_ _- J 0.0 0.2 0.4 0.6 0.8 1.0 DISTANCE, r/L Figure A.4. Normalized apparent SIF within crack tip element for center-cracked plate under tension; a/W = 0.5. 1.02 r Equation 2.12 r Equation 2.9 - . - C B' 1.01 - I ~ I 0 L.. I CD LE quation 2.5 ~.._, I ,tc I ~ (.,) N I N 1.00 I I Crack: a/W = 0.1, W = 100 mm I Loading: in-plane bending, cr = 100 MPa Mesh: m = 3, n = 8, L = 156 ?m I Disp.: v8 = 0.90 ?m, vc= 1.81 ?m I t, I 0.99 11 I 0.0 0.2 0.4 0.6 0.8 1.0 DISTANCE, r/L Figure A.5. Normalized apparent SIF within crack tip element for edge-cracked plate under bending; a/W = 0.1 . 1.02 C Equation 2.1 Equation 2.5 B I Equation 2. 9 ffl "'C I ~ ~ I -... I ~ (,J N 1.00 I (,J I I Crack: a/W = 0.9, W = 100 mm I Loading: in-plane bending, cr = 100 MPa Mesh: m = 3, n = 8, L = 156 ?m I 0.99 ~-_.. ......... ...,_ _D_isp.. .:. .v_8_ = _32_.3_2 ?_m_, V_e=_ 6_ 5_.1_6. .?_m_ __ _. 0.0 0.2 0.4 0.6 0.8 1.0 DISTANCE, r/L Figure A.6. Normalized apparent SIF within crack tip element for edge-cracked plate under bending; a/W = 0.9. 0 0 C") II .c. ..e. ..-~w----e- -- 2W = 200 - Note: All dimensions in mm Figure A. 7. Plate with eccentric crack. 324 1.05 II II II II II ?? FEA/Tada for tension / ? ?? ? - ?? ??Cl) ? ?x ?? ~-... ..~ ----------~--?- --?- ... --~ i1l ----B 1 .... - .... -?-~-~~ ---?-? ... ... --??9-??????;. ........ t!J , v) ~ ? 1111 , ... ~ N U1 ...... .. .. -?-&-- .... FEN\sida for bending ?-........ ---e--- -41,~ .. FEA/Chen for bending .. ----~--- FEA/Benthern for bending ???-...... __ 0.95 - 't.~ 0 0.2 0.4 0.6 0.8 1 CRACK LENGTH, a/W Figure A.8. Comparison of FEA and existing solutions for center-cracked plate under tension or bending. 1.os ,------------------ ??O ?~?-???-"=' r-.????? ~--- +J ~ -- (/) .. .? ? --s .. -~ -. .. . ??-?'-~.::'??? ~ ?---....... 1 ,;o.<, "' ? , ...." .........~~ .---?-?v ?. -e-........ ?-.-.::._,- ??-??---" ,~------- -~- -- i <( w u. w ~ N 0) - FEA/Tada for tension ---0--- FEA/Tada for bending ---~?-- FEA/Brown for bending 0.95 .___ __. ..__ ___. .__ ___. i..-_______ __ 0 0.2 0.4 0.6 0.8 1 CRACK LENGTH, a/W Figure A.9. Comparison of FEA and existing solutions for edge-cracked plate under tension or bending. 1.05 .-----------------------, e/W=0.1 0 Crack tip A - . - . - e/W = 0.3 -?? .... -- e/W = 0.5 ? Crack tip B - - - e/W = 0.7 -??????--? e/W = 0.9 ~ -0 (J) ~..._ :(?!e?c? ????I ??????-???????01 ~ ????????????????-0- ? - 8 ??? 'v - ???-?-?-?'". ,, - ~-?---???_"-_ .... ... - #~"-:? .. .. .. : -- ? :- s? <( ~' ~---? _..__ ??:y(w . .~.-..- .? ? &.: ! ::?.-.- --.--1 I --- ? -----0 .. w ~ -. .. .-.-. .?- N . .?. - -..J ..... .. ?v ??~ 0.95 L-------1.------'----__.__ ___ _.__ _~ 0 0.2 0.4 0.6 0.8 1 CRACK LENGTH, a/(W-e) Figure A.1 O. Comparison of FEA and lsida's solution {1965) for plate with eccentric crack under tension. 1.05 -------------------~ e/W = 0.1 o Crack tip A e/W = 0.3 e/W = 0.5 ? Crack tip B e/W = 0.7 C e/W = 0.9 (l) ..c. --2<- 1 ( ll.l <.,J ~ N co 0.95 .___ __ .__ ___________ __.__ __ ___. 0 0.2 0.4 0.6 0.8 1 CRACK LENGTH, a/(W-e) Figure A.11. Comparison of FEA and Chen and Albrecht's (1994) solution for plate with eccentric crack under bending. - - REFERENCE ABAQUS/Standard User's Manual, (1993). Version 5.3, Hibbitt, Karlsson & Sorensen, Inc., Rhode Island. Aliabadi, M. H., and Rooke, D. P. (1991). Numerical Fracture Mechanics. Kluwer Academic Publishers. Banks-Sills, L., and Sherman, D. (1992). 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