DURATION-OF-LOAD AND SIZE EFFECTS ON THE ROLLING SHEAR STRENGTH OF CROSS LAMINATED TIMBER by Yuan Li B.A.Sc., Tongji University, China, 2008 M.A.Sc., Tongji University, China, 2010 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Forestry) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) March 2015 © Yuan Li, 2015 ii Abstract In the beginning of the twenty-first century, the largest mountain pine beetle (MPB) outbreak ever recorded struck western Canada. A huge volume of MPB-attacked lodgepole pine is expected to hit the BC forest industry in the next decade. Technologies to convert MPB-attacked lumber into engineering wood products are urgently required. Cross laminated timber (CLT) is a technology that can produce massive timber members as an engineered wood product for timber structures. In this study, the duration-of-load and size effects on the rolling shear strength of CLT manufactured from MPB-afflicted lumber were evaluated. The study of the duration-of-load effect on the strength properties of wood products is typically challenging; and, additional complexity exists with the duration-of-load effect on the rolling shear strength of CLT, given the necessary consideration of crosswise layups of wood boards, existing gaps and glue bonding between layers. In this research, short-term ramp loading tests and long-term trapezoidal fatigue loading tests (damage accumulation tests) were used to study the duration-of-load behaviour of the rolling shear strength of CLT. In the ramp loading test, three-layer CLT products showed a relatively lower rolling shear load-carrying capacity. Torque loading tests on CLT tubes were also performed. The finite element method was adopted to simulate the structural behaviour of CLT specimens. Evaluation of the rolling shear strength based on test data was discussed. The size effect on the rolling shear strength was investigated. iii A stress-based damage accumulation theory was used to evaluate the duration-of-load effect on CLT rolling shear. The model was first calibrated against the test data and used to investigate the long-term CLT rolling shear strength. A reliability-based method was then applied to assess the CLT rolling shear performance. Finally, a duration-of-load adjustment factor for CLT rolling shear strength was established. The duration-of-load adjustment factor for the three-layer and five-layer CLT is different from that for lumber. The results suggest that the rolling shear duration-of-load strength adjustment factor for CLT is more severe than the general duration-of-load adjustment factor for lumber, and this difference should be considered in the introduction of CLT into the building codes for engineered wood design. iv Preface The objectives of this thesis are to investigate and evaluate the duration-of-load and size effects on the rolling shear strength of mountain pine beetle (MPB) lumber based cross laminated timber (CLT). Under the guidance of Dr. Frank Lam and Dr. Ricardo O. Foschi, this dissertation is an original, unpublished work by the author, Yuan Li. In Chapter 3, Yuan Li was responsible for designing the research program and for the investigation on the model theories. Dr. Frank Lam offered advice and guidance during the design of the research. In Chapter 4, the manufacture of the CLT plate replicates and the transverse vibration tests on the laminated lumber were performed by Dr. Yue Chen. The short-term ramp and long-term trapezoidal fatigue loading tests in Chapter 4 were designed by Yuan Li and Dr. Frank Lam; and, the preparation for the specimens and these experiments were conducted by Yuan Li. The torque tests in Chapter 4 were designed and conducted by Dr. Frank Lam, Yuan Li and Dr. Minghao Li. Drs. Frank Lam and Minghao Li offered advice during the design of the research. The development of the beam finite element model was performed by Yuan Li; and, the torque finite element model was initially developed by Dr. Minghao Li and Yuan Li. In Chapter 5, the rolling shear strength evaluation from the beam finite element model was performed by Yuan Li. The size effect investigation work was performed by Yuan Li. Drs. v Ricardo O. Foschi and Frank Lam offered advice and guidance during the design of the research and the interpretation of the results. In Chapter 6, the development of the damage accumulation model in the duration-of-load investigation was conducted by Yuan Li. The reliability analysis and assessment were performed by Yuan Li. Drs. Ricardo O. Foschi and Frank Lam offered advice and guidance during the design of the research and the interpretation of the results. vi Table of Contents Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iv Table of Contents ......................................................................................................................... vi List of Tables ............................................................................................................................... xii List of Figures ............................................................................................................................ xvii List of Abbreviations ............................................................................................................. xxviii Acknowledgements .................................................................................................................. xxix Dedication ...................................................................................................................................xxx Chapter 1: Introduction and Research Objectives .......................................................................... 1 1.1 Background ................................................................................................................. 1 1.2 Previous Research on Duration-of-Load Effect .......................................................... 6 1.3 Objectives and Methods .............................................................................................. 8 1.4 Thesis Organization .................................................................................................. 10 Chapter 2: Literature Review ........................................................................................................ 12 2.1 Summary ................................................................................................................... 12 2.2 Introduction ............................................................................................................... 12 2.3 Cross Laminated Timber........................................................................................... 12 2.4 Duration of Load ....................................................................................................... 17 2.5 Rheological Constitutive Relationship ..................................................................... 20 2.6 Damage Accumulation Model .................................................................................. 24 2.7 Rolling Shear Behaviour ........................................................................................... 29 vii 2.8 Weakest Link Theory and Size Effect ...................................................................... 30 2.9 Conclusion ................................................................................................................ 34 Chapter 3: Modeling of Duration-of-Load Behaviour of CLT ..................................................... 35 3.1 Summary ................................................................................................................... 35 3.2 Introduction ............................................................................................................... 35 3.3 Finite Element Modeling .......................................................................................... 36 3.4 Rheological Constitutive Relationship Modeling ..................................................... 46 3.4.1 The Modified Five-Parameter Rheological Model under Ramp Loading ............ 48 3.4.2 The Modified Five-Parameter Rheological Model under Trapezoidal Fatigue Loading ............................................................................................................................. 50 3.4.3 The Modified Five-Parameter Rheological Model under Ramp and Constant Loading Protocol ............................................................................................................... 55 3.5 Damage Accumulation Model .................................................................................. 57 3.5.1 Damage under Ramp Loading .............................................................................. 58 3.5.2 Damage under Trapezoidal Fatigue Loading ........................................................ 59 3.5.2.1 Segment One: Uploading in the First Cycle ................................................. 59 3.5.2.2 Segment Two: Constant Loading in the First Cycle ..................................... 60 3.5.2.3 Segment Three: Unloading in the First Cycle............................................... 61 3.5.2.4 Segment Four: Damage Accumulated in the Second Cycle ......................... 62 3.5.2.5 Determination of the Number of Cycles to Failure ...................................... 62 3.5.3 Damage under Ramp and Constant Loading Protocol .......................................... 63 3.6 Conclusion ................................................................................................................ 64 Chapter 4: Testing of Duration-of-Load Behaviour of CLT ........................................................ 66 viii 4.1 Summary ................................................................................................................... 66 4.2 Introduction ............................................................................................................... 66 4.3 Material and Methods ............................................................................................... 67 4.3.1 CLT Specimen ...................................................................................................... 67 4.3.2 Ramp Loading Test ............................................................................................... 72 4.3.3 Torque Loading Test ............................................................................................. 73 4.3.4 Trapezoidal Fatigue Loading Test ........................................................................ 76 4.4 Experimental Results ................................................................................................ 78 4.4.1 Ramp Loading Test Results .................................................................................. 78 4.4.2 Torque Loading Test Results ................................................................................ 91 4.4.3 Analysis and Discussion on the Rolling Shear Failure in Torque Loading Tests 93 4.4.4 Trapezoidal Fatigue Loading Test Results ......................................................... 102 4.5 Conclusion .............................................................................................................. 110 Chapter 5: Size Effect on the Rolling Shear Strength of CLT .................................................... 112 5.1 Summary ................................................................................................................. 112 5.2 Finite Element Model ............................................................................................. 112 5.2.1 Model Calibration ............................................................................................... 113 5.2.2 Calibration Results .............................................................................................. 113 5.2.2.1 Ramp Loading Results ................................................................................ 113 5.2.2.2 Trapezoidal Fatigue Loading Results ......................................................... 115 5.3 Size Effect on the Rolling Shear Strength .............................................................. 124 5.3.1 Rolling Shear Strength Evaluation in Ramp Loading Tests ............................... 124 5.3.2 Size Effect on the Rolling Shear Strength .......................................................... 125 ix 5.3.3 Examination of the Possibility of Longitudinal Shear Failure in Torque Loading Tests Based on Size Effect Model .................................................................................. 142 5.4 Conclusion .............................................................................................................. 145 Chapter 6: Calibration and Verification of CLT Damage Accumulation Model ....................... 146 6.1 Summary ................................................................................................................. 146 6.2 Introduction ............................................................................................................. 147 6.3 Calibration of Damage Accumulation Model ......................................................... 147 6.3.1 Model Calibration ............................................................................................... 147 6.3.2 Calibration Results .............................................................................................. 149 6.4 Verification of Damage Accumulation Model ....................................................... 157 6.5 Duration-of-Load Factor ......................................................................................... 161 6.5.1 Reliability Analysis of Short-Term Rolling Shear Strength of CLT .................. 161 6.5.2 Reliability Analysis of CLT Rolling Shear Strength under Thirty-Year Snow Load ............................................................................................................................. 169 6.5.3 Reliability Analysis of CLT Rolling Shear Strength under Thirty-Year Dead Load Only ............................................................................................................................. 184 6.5.4 Duration-of-Load Factor Based on the Stress Ratio Evaluation Approach from Model Prediction ............................................................................................................. 192 6.6 Conclusion .............................................................................................................. 201 Chapter 7: Concluding Remarks and Future Work..................................................................... 203 7.1 Concluding Remarks ............................................................................................... 203 7.2 Future Work ............................................................................................................ 205 Bibliography ...............................................................................................................................208 x Appendices ..................................................................................................................................216 Appendix A: Rolling Shear Failure Loads in Five-Layer CLT in Ramp Loading Test ...….216 Appendix B: Rolling Shear Failure Loads in Three-Layer CLT in Ramp Loading Test ......218 Appendix C: Number of Cycles to Rolling Shear Failure in Five-Layer CLT in Trapezoidal Fatigue Loading Test (Long Duration in Plateau Part) …...…...…...…...…...…...…...…..220 Appendix D: Number of Cycles to Rolling Shear Failure in Five-Layer CLT in Trapezoidal Fatigue Loading Test (Short Duration in Plateau Part) ...…...…...…...…...…...…...…….221 Appendix E: Number of Cycles to Rolling Shear Failure in Three-Layer CLT in Trapezoidal Fatigue Loading Test (Long Duration in Plateau Part) …...…...…...…...…...…...…...…..222 Appendix F: Number of Cycles to Rolling Shear Failure in Three-Layer CLT in Trapezoidal Fatigue Loading Test (Short Duration in Plateau Part) …...…...…...…...…...…...…...….223 Appendix G: Peak Failure Torque Loads in Five-Layer CLT in Torque Loading Test….....224 Appendix H: Peak Failure Torque Loads in Three-Layer CLT in Torque Loading Test…...225 Appendix I: Rolling Shear Stress Calculation in Different Beam Theories for Three-Layer and Five-Layer CLT…...…...…...…...…...…...…...…...…...…...…...…...…...…...…...…......226 Appendix J: Further Investigation on the Shear Failure Mode in Torque Loading Tests…..234 Appendix K: Introduction of the Fitting Process and Adjusted Parameter Values in Finite Element Beam Modeling for Short-Term Ramp Loading…...…...…...…...…...…...….....238 Appendix L: Introduction of the Fitting Process and Adjusted Parameter Values in Finite Element Beam Modeling for Long-Term Trapezoidal Loading…...…...…...…...…...…...243 Appendix M: Introduction of the Evaluation of the Rolling Shear Strength by Short-Term Ramp Loading…...…...…...…...…...…...…...…...…...…...…...…...…...…...…...…...…..248 Appendix N: Introduction of the Calculated Ratio 𝑇𝑉…...…...…...…...…...…...…...…...…251 xi Appendix O: Introduction of Snow Load in Halifax and Vancouver…...…...…...…...….....252 Appendix P: Statistics on Random Variables and Parameters for Reliability Analysis under Snow Load…...…...…...…...…...…...…...…...…...…...…...…...…...…...…...…...…...…256 Appendix Q: Reliability Analysis Results under Thirty-Year Snow Load Cases from Three Different Locations: Quebec City, Ottawa and Saskatoon…...…...…...…...…...…...…....257 xii List of Tables Table 3.1 Configurations of the dimension in the finite element models ..................................... 38 Table 3.2 Input orthotropic properties of the wood boards .......................................................... 42 Table 3.3 Input shear stiffness of the glue lines............................................................................ 45 Table 4.1 Grades and configurations of the CLT plates ............................................................... 69 Table 4.2 Summary of the vibration MOE of wood boards ......................................................... 70 Table 4.3 Notation of CLT specimens .......................................................................................... 71 Table 4.4 Number of the CLT specimens ..................................................................................... 71 Table 4.5 Information for the torque tube specimens ................................................................... 75 Table 4.6 Number of CLT specimens ........................................................................................... 78 Table 4.7 Summary of the ramp test results ................................................................................. 80 Table 4.8 Summary of the torque test results ............................................................................... 92 Table 4.9 Results of the trapezoidal fatigue test for CLT panels ................................................ 105 Table 5.1 Summary of the cross layer maximum rolling shear stress results from the finite element model ............................................................................................................................. 125 Table 5.2 Size effect model calibration results ........................................................................... 137 Table 5.3 Average material constants for the size effect model ................................................. 139 Table 5.4 Material constants for the longitudinal shear strength of spruce-pine-fir in size effect model........................................................................................................................................... 143 Table 5.5 Predicted longitudinal shear load-carrying capacity in CLT tube specimens ............ 145 Table 6.1 Summary of the calibration results in the five-layer CLT .......................................... 149 Table 6.2 Summary of the calibration results in the three-layer CLT ........................................ 156 xiii Table 6.3 Results between the reliability index and the performance factor in the five-layer CLT..................................................................................................................................................... 164 Table 6.4 Results between the reliability index and the performance factor in the three-layer CLT..................................................................................................................................................... 166 Table 6.5 Curve one and curve two results between the reliability index and the performance factor in the five-layer CLT ........................................................................................................ 172 Table 6.6 Curve one and curve two results between the reliability index and the performance factor in the three-layer CLT ...................................................................................................... 175 Table 6.7 Reliability results for the strength adjustment factors in the five-layer CLT ............. 177 Table 6.8 Reliability results for the strength adjustment factors in the three-layer CLT ........... 178 Table 6.9 Reliability results for the strength adjustment factors in the five-layer CLT for different locations ...................................................................................................................................... 182 Table 6.10 Reliability results for the strength adjustment factors in the three-layer CLT for different locations ....................................................................................................................... 183 Table 6.11 Results between the reliability index and the performance factor in the five-layer CLT..................................................................................................................................................... 185 Table 6.12 Results between the reliability index and the performance factor in the three-layer CLT ............................................................................................................................................. 186 Table 6.13 Curve one and curve two results between the reliability index and the performance factor in the five-layer CLT ........................................................................................................ 189 Table 6.14 Curve one and curve two results between the reliability index and the performance factor in the three-layer CLT ...................................................................................................... 190 Table 6.15 Reliability results for the strength adjustment factor in the dead load case ............. 192 xiv Table 6.16 Summary of the factor calculation in the five-layer CLT ......................................... 194 Table 6.17 Summary of the factor calculation in the three-layer CLT ....................................... 195 Table 6.18 Summary of the reliability results for the strength adjustment factors in CLT ........ 196 Table 6.19 Summary of the factor calculation in CLT ............................................................... 196 Table A.1 Rolling shear failure loads of specimens from No. 1 to No. 34……………………..216 Table A.2 Rolling shear failure loads of specimens from No. 35 to No. 55……………………217 Table B.1 Rolling shear failure loads of specimens from No. 1 to No. 38……………………..218 Table B.2 Rolling shear failure loads of specimens from No. 39 to No. 59……………………219 Table C.1 Number of cycles to rolling shear failure……………………………………………220 Table D.1 Number of cycles to rolling shear failure…………………………………………...221 Table E.1 Number of cycles to rolling shear failure……………………………………………222 Table F.1 Number of cycles to rolling shear failure……………………………………………223 Table G.1 Peak failure torque loads…………………………………………………………….224 Table H.1 Peak failure torque loads…………………………………………………………….225 Table J.1 Results for the Monte Carlo simulation……………………………………………...237 Table K.1 Adjusted parameters in finite element beam modeling for short-term ramp loading.241 Table K. 2 Evaluated maximum rolling shear stresses with or without scaled factors in finite element models..………………………………………………………………………………..242 Table L.1 Adjusted parameters in finite element SPF3 beam model for long-term trapezoidal loading…………………………………………………………………………………………..246 Table L.2 Adjusted parameters in finite element SPF5 beam model for long-term trapezoidal loading…………………………………………………………………………………………..247 xv Table M.1 Rolling shear failure load fitting data and rolling shear strength for three-layer CLT……………………………………………………………………………………………..249 Table M.2 Rolling shear failure load fitting data and rolling shear strength for five-layer CLT……………………………………………………………………………………………..250 Table N.1 Summary of the calculated Tv values for the three-layer and five-layer CLT groups…………………………………………………………………………………………...251 Table P.1 Summary of the random variables and parameters………………………………….256 Table Q.1 Results between the reliability index and the performance factor in the five-layer CLT in curve one (without DOL effect)……………………………………………………………...257 Table Q.2 Curve one and curve two results between the reliability index and the performance factor in the five-layer CLT)……………………………………………………………............258 Table Q.3 Results between the reliability index and the performance factor in the five-layer CLT in curve one (without DOL effect)……………………………………………………………259 Table Q.4 Curve one and curve two results between the reliability index and the performance factor in the five-layer CLT…………………………………………………………………….260 Table Q.5 Results between the reliability index and the performance factor in the five-layer CLT in curve one (without DOL effect)……………………………………………………………...261 Table Q.6 Curve one and curve two results between the reliability index and the performance factor in the five-layer CLT…………………………………………………………………….262 Table Q.7 Results between the reliability index and the performance factor in the three-layer CLT in curve one (without DOL effect).……………………………………………………….263 Table Q.8 Curve one and curve two results between the reliability index and the performance factor in the three-layer CLT…………………………………………………………………...264 xvi Table Q.9 Results between the reliability index and the performance factor in the three-layer CLT in curve one (without DOL effect).……………………………………………………….265 Table Q.10 Curve one and curve two results between the reliability index and the performance factor in the three-layer CLT.…………………………………………………………………..266 Table Q.11 Results between the reliability index and the performance factor in the three-layer CLT in curve one (without DOL effect).……………………………………………………….267 Table Q.12 Curve one and curve two results between the reliability index and the performance factor in the three-layer CLT.…………………………………………………………………..268 xvii List of Figures Figure 1.1 Layering of CLT ............................................................................................................ 2 Figure 1.2 Rolling shear behaviour in CLT .................................................................................... 3 Figure 1.3 MPB-killed lumber ........................................................................................................ 4 Figure 1.4 MPB-killed lumber based CLT product ........................................................................ 4 Figure 2.1 Illustration of one three-element model ...................................................................... 22 Figure 2.2 Illustration of one four-element model ........................................................................ 22 Figure 3.1 Orthotropic material modeling in polar coordinate system ......................................... 39 Figure 3.2 Orthotropic material modeling in Cartesian coordinate system .................................. 41 Figure 3.3 Finite element modeling of the five-layer CLT beam test specimen .......................... 37 Figure 3.4 Finite element modeling of the three-layer CLT beam test specimen......................... 37 Figure 3.5 Gap modeling in the cross layer .................................................................................. 42 Figure 3.6 Multi-linear shear stress-strain relationship ................................................................ 44 Figure 3.7 Schaaf’s study for the torsional shear tests.................................................................. 45 Figure 3.8 The modified five-parameter rheological model ......................................................... 46 Figure 3.9 The strain of the modified five-parameter rheological model ..................................... 47 Figure 3.10 The strain rate of the modified five-parameter rheological model ............................ 47 Figure 3.11 Ramp loading protocol .............................................................................................. 48 Figure 3.12 Strain rate under the ramp loading protocol .............................................................. 49 Figure 3.13 Trapezoidal fatigue loading protocol......................................................................... 50 Figure 3.14 Strain under the trapezoidal loading protocol ........................................................... 55 Figure 3.15 Ramp and constant loading protocol ......................................................................... 55 xviii Figure 4.1 Manufacturing of three-layer and five-layer CLT panels ............................................ 68 Figure 4.2 Two categories of CLT plates ..................................................................................... 68 Figure 4.3 Setup of the MOE vibration test .................................................................................. 69 Figure 4.4 Pair sampling method .................................................................................................. 71 Figure 4.5 Five-layer CLT beam specimen .................................................................................. 72 Figure 4.6 Three-layer CLT beam specimen ................................................................................ 72 Figure 4.7 Ramp loading test setup............................................................................................... 73 Figure 4.8 CLT torque tube specimens ......................................................................................... 74 Figure 4.9 Torque test setup (with Y direction as the direction of wood fibre in the middle cross layer) ............................................................................................................................................. 76 Figure 4.10 Load control in the trapezoidal fatigue test ............................................................... 77 Figure 4.11 Trapezoidal fatigue test of CLT specimens ............................................................... 77 Figure 4.12 Failure modes in the ramp loading test...................................................................... 79 Figure 4.13 Shear stress evaluated in the cross section of the CLT beam (in N/mm2 under 12.51kN centre-point load) ............................................................................................................ 82 Figure 4.14 Shear stress evaluated in the cross section of the CLT beam (in N/mm2 under 19.39kN centre-point load) ............................................................................................................ 83 Figure 4.15 Rolling shear stress distribution in three-layer CLT specimen under 12.51kN load (stress in Pa) .................................................................................................................................. 85 Figure 4.16 Rolling shear stress distribution in five-layer CLT specimen under 12.51kN load (stress in Pa) .................................................................................................................................. 85 Figure 4.17 Rolling shear stress distribution in five-layer CLT specimen under 19.39 kN load (stress in Pa) .................................................................................................................................. 86 xix Figure 4.18 Cumulative distributions of the rolling shear failure loads ....................................... 88 Figure 4.19 Load-displacement curves of the experimental results in the SPF5-0.4 group (the applied load level of trapezoidal fatigue tests was shown as the horizontal dash line) ................ 89 Figure 4.20 Load-displacement curves of the experimental results in the SPF3-0.4 group (the applied load level of trapezoidal fatigue tests was shown as the horizontal dash line) ................ 90 Figure 4.21 Glue failure in the torque test .................................................................................... 91 Figure 4.22 Rolling shear failure in the torque test (with Y direction as the direction of wood fibre in the middle cross layer) ..................................................................................................... 92 Figure 4.23 Finite element modeling of three-layer CLT tube test specimens (orthotropic wood properties with Y direction as the direction of wood fibre in the middle cross layer) .................. 94 Figure 4.24 Shear stress (τxz, rolling shear in orthotropic material modeling) distribution in the cross layer of three-layer tube specimen (Maximum τxz = 3.86 MPa) ....................................... 96 Figure 4.25 Shear stress (τyz, longitudinal shear in orthotropic material modeling) distribution in the cross layer of three-layer tube specimen (Maximum τyz = 7.04 MPa) ................................. 97 Figure 4.26 Shear stress (τxy, longitudinal shear in orthotropic material modeling) distribution in the cross layer of three-layer tube specimen (Maximum τxy = 5.57 MPa) ................................. 98 Figure 4.27 Shear stress (τxz, rolling shear in orthotropic material modeling) distribution in the cross layer of five-layer tube specimen (Maximum τxz = 4.83 MPa) ......................................... 99 Figure 4.28 Shear stress (τyz, longitudinal shear in orthotropic material modeling) distribution in the cross layer of five-layer tube specimen (Maximum τyz = 10.30 MPa) ............................... 100 Figure 4.29 Shear stress (τxy, longitudinal shear in orthotropic material modeling) distribution in the cross layer of five-layer tube specimen (Maximum τxy = 6.80 MPa) ............................... 100 xx Figure 4.30 Rolling shear cracks in the trapezoidal fatigue loading test (black marker line in the circle is next to the crack) ........................................................................................................... 103 Figure 4.31 Relationship between the number of cycles to failure (in the logarithm to base 10) and the stress ratio in the SPF5-0.4 group .................................................................................. 108 Figure 4.32 Cumulative distribution of the number of cycles to failure (in the logarithm to base 10) in the SPF5-0.4 group (removed data points of uploading failure specimens but the rank was calculated based on the entire set) .............................................................................................. 108 Figure 4.33 Relationship between the number of cycles to failure (in the logarithm to base 10) and the stress ratio in the SPF3-0.4 group .................................................................................. 109 Figure 4.34 Cumulative distribution of the number of cycles to failure (in the logarithm to base 10) in the SPF3-0.4 group (removed data points of uploading failure specimens but the rank was calculated based on the entire set) .............................................................................................. 109 Figure 5.1 Load-displacement curves of the experimental and simulated results in the SPF5-0.4 group (the horizontal dash line represents the target applied load level) ................................... 114 Figure 5.2 Load-displacement curves of the experimental and simulated results in the SPF3-0.4 group (the horizontal dash line represents the target applied load level) ................................... 115 Figure 5.3 Trapezoidal displacement history from the experimental and simulated results in the SPF5-0.4 (first four cycles) ......................................................................................................... 117 Figure 5.4 Trapezoidal displacement history from the experimental and simulated results in SPF3-0.4 (first five cycles) ......................................................................................................... 117 Figure 5.5 Maximum and minimum trapezoidal displacement curves in SPF5-0.4 ................... 118 Figure 5.6 Minimum trapezoidal displacement experimental and simulated curves in SPF5-0.4 (first three cycles) ....................................................................................................................... 119 xxi Figure 5.7 Minimum trapezoidal displacement experimental and simulated results in SPF5-0.4..................................................................................................................................................... 119 Figure 5.8 Maximum trapezoidal displacement experimental and simulated curves in SPF5-0.4 (first three cycles) ....................................................................................................................... 120 Figure 5.9 Maximum trapezoidal displacement experimental and simulated results in SPF5-0.4..................................................................................................................................................... 120 Figure 5.10 Maximum and minimum trapezoidal displacement curves in SPF3-0.4 ................. 121 Figure 5.11 Minimum trapezoidal displacement experimental and simulated curves in SPF3-0.4 (first six cycles) ........................................................................................................................... 122 Figure 5.12 Minimum trapezoidal displacement experimental and simulated results in SPF3-0.4..................................................................................................................................................... 122 Figure 5.13 Maximum trapezoidal displacement experimental and simulated curves in SPF3-0.4 (first six cycles) ........................................................................................................................... 123 Figure 5.14 Maximum trapezoidal displacement experimental and simulated results in SPF3-0.4..................................................................................................................................................... 123 Figure 5.15 Size effect model calibrated by torque SPF5-0.4 test data ...................................... 129 Figure 5.16 Size effect model prediction for torque SPF3-0.4 test data (calibrated by torque SPF5-0.4 data)............................................................................................................................. 129 Figure 5.17 Size effect model prediction for bending SPF5-0.4 test data (calibrated by torque SPF5-0.4 data)............................................................................................................................. 130 Figure 5.18 Size effect model prediction for bending SPF3-0.4 test data (calibrated by torque SPF5-0.4 data)............................................................................................................................. 130 Figure 5.19 Size effect model calibrated by torque SPF3-0.4 test data ...................................... 131 xxii Figure 5.20 Size effect model prediction for torque SPF5-0.4 test data (calibrated by torque SPF3-0.4 data)............................................................................................................................. 131 Figure 5.21 Size effect model prediction for bending SPF5-0.4 test data (calibrated by torque SPF3-0.4 data)............................................................................................................................. 132 Figure 5.22 Size effect model prediction for bending SPF3-0.4 test data (calibrated by torque SPF3-0.4 data)............................................................................................................................. 132 Figure 5.23 Size effect model calibrated by bending SPF5-0.4 test data ................................... 133 Figure 5.24 Size effect model prediction for torque SPF5-0.4 test data (calibrated by bending SPF5-0.4 data)............................................................................................................................. 133 Figure 5.25 Size effect model prediction for torque SPF3-0.4 test data (calibrated by bending SPF5-0.4 data)............................................................................................................................. 134 Figure 5.26 Size effect model prediction for bending SPF3-0.4 test data (calibrated by bending SPF5-0.4 data)............................................................................................................................. 134 Figure 5.27 Size effect model calibrated by bending SPF3-0.4 test data ................................... 135 Figure 5.28 Size effect model prediction for torque SPF5-0.4 test data (calibrated by bending SPF3-0.4 data)............................................................................................................................. 135 Figure 5.29 Size effect model prediction for torque SPF3-0.4 test data (calibrated by bending SPF3-0.4 data)............................................................................................................................. 136 Figure 5.30 Size effect model prediction for bending SPF5-0.4 test data (calibrated by bending SPF3-0.4 data)............................................................................................................................. 136 Figure 5.31 Size effect model prediction for torque SPF5-0.4 test data (average material constants) .................................................................................................................................... 140 xxiii Figure 5.32 Size effect model prediction for torque SPF3-0.4 test data (average material constants) .................................................................................................................................... 140 Figure 5.33 Size effect model prediction for bending SPF5-0.4 test data (average material constants) .................................................................................................................................... 141 Figure 5.34 Size effect model prediction for bending SPF3-0.4 test data (average material constants) .................................................................................................................................... 141 Figure 5.35 Predicted longitudinal shear load-carrying capacity distribution in SPF3 tube specimen ..................................................................................................................................... 144 Figure 5.36 Predicted longitudinal shear load-carrying capacity distribution in SPF5 tube specimen ..................................................................................................................................... 144 Figure 6.1 Relationships between the number of cycles to failure (in the logarithm to base 10) and the stress ratio from test and model in five-layer CLT ........................................................ 150 Figure 6.2 Cumulative distributions of the experimental and simulated number of cycles to failure (in the logarithm to base 10) in five-layer CLT .............................................................. 151 Figure 6.3 Relationships between the number of cycles to failure (not in the logarithm scale) and the stress ratio from test and model in five-layer CLT ............................................................... 152 Figure 6.4 Cumulative distributions of the experimental and simulated number of cycles to failure (not in the logarithm scale) in five-layer CLT ................................................................. 153 Figure 6.5 Relationships between the number of cycles to failure (in the logarithm to base 10) and the stress ratio from test and model in three-layer CLT....................................................... 156 Figure 6.6 Cumulative distributions of the experimental and simulated number of cycles to failure (in the logarithm to base 10) in three-layer CLT ............................................................. 157 xxiv Figure 6.7 Relationships between the number of cycles to failure (in the logarithm to base 10) and the stress ratio from test and model in five-layer CLT ........................................................ 158 Figure 6.8 Cumulative distributions of the experimental and simulated number of cycles to failure (in the logarithm to base 10) in five-layer CLT .............................................................. 159 Figure 6.9 Relationships between the number of cycles to failure (in the logarithm to base 10) and the stress ratio from test and model in three-layer CLT....................................................... 160 Figure 6.10 Cumulative distributions of the experimental and simulated number of cycles to failure (in the logarithm to base 10) in three-layer CLT ............................................................. 160 Figure 6.11 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Five-layer/Halifax) ......................................................................... 165 Figure 6.12 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Five-layer/Vancouver) .................................................................... 165 Figure 6.13 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Three-layer/Halifax) ....................................................................... 167 Figure 6.14 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Three-layer/Vancouver) ................................................................. 167 Figure 6.15 Average curves between the reliability index and the performance factor without considering the duration-of-load effect ....................................................................................... 168 Figure 6.16 The basic factor determination procedure (curve one-without DOL effect; curve two-with DOL effect) ......................................................................................................................... 171 Figure 6.17 Curves between the reliability index and the performance factor for “Five-layer/Halifax” case (curve one-without DOL effect; curve two-with DOL effect) .................... 173 xxv Figure 6.18 Curves between the reliability index and the performance factor for “Five-layer/Vancouver” case (curve one-without DOL effect; curve two-with DOL effect) .............. 173 Figure 6.19 Curves between the reliability index and the performance factor for “Three-layer/Halifax” case (curve one-without DOL effect; curve two-with DOL effect) .................... 176 Figure 6.20 Curves between the reliability index and the performance factor for “Three-layer/Vancouver” case (curve one-without DOL effect; curve two-with DOL effect) .............. 176 Figure 6.21 Curves between the reliability index and the performance factor for “Five-layer/Quebec City” case (curve one-without DOL effect; curve two-with DOL effect) ............ 179 Figure 6.22 Curves between the reliability index and the performance factor for “Five-layer/Ottawa” case (curve one-without DOL effect; curve two-with DOL effect) .................... 179 Figure 6.23 Curves between the reliability index and the performance factor for “Five-layer/Saskatoon” case (curve one-without DOL effect; curve two-with DOL effect) ............... 180 Figure 6.24 Curves between the reliability index and the performance factor for “Three-layer/Quebec City” case (curve one-without DOL effect; curve two-with DOL effect) ............ 180 Figure 6.25 Curves between the reliability index and the performance factor for “Three-layer/Ottawa” case (curve one-without DOL effect; curve two-with DOL effect) .................... 181 Figure 6.26 Curves between the reliability index and the performance factor for “Three-layer/Saskatoon” case (curve one-without DOL effect; curve two-with DOL effect) ............... 181 Figure 6.27 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Five-layer/Dead Load Only) .......................................................... 186 Figure 6.28 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Three-layer/Dead Load Only) ........................................................ 187 xxvi Figure 6.29 Average curves between the reliability index and the performance factor without considering the duration-of-load effect ....................................................................................... 188 Figure 6.30 Curves between the reliability index and the performance factor for “Five-layer/Dead Load Only” case (curve one-without DOL effect; curve two-with DOL effect) ..... 190 Figure 6.31 Curves between the reliability index and the performance factor for “Three-layer/Dead Load Only” case (curve one-without DOL effect; curve two-with DOL effect) ..... 191 Figure 6.32 Time to failure prediction (minutes in the logarithm to base 10) under the ramp and constant loading protocol in the SPF5-0.4 group........................................................................ 194 Figure 6.33 Time to failure prediction (minutes in the logarithm to base 10) under the ramp and constant loading protocol in the SPF3-0.4 group........................................................................ 195 Figure 6.34 Time to failure prediction (minutes in the logarithm to base 10) under the ramp and constant loading protocol ............................................................................................................ 199 Figure I.1 Shear stress calculation for three-layer CLT…………………………….…………..226 Figure I.2 Shear stress calculation for five-layer CLT…………………………….………...…229 Figure K.1 Stress-strain relationship……………………………………………………………242 Figure Q.1 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Five-layer/ Quebec City, curve one)……………………………...258 Figure Q.2 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Five-layer/ Ottawa, curve one)……………………………………259 Figure Q.3 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Five-layer/ Saskatoon, curve one)………………………………...261 Figure Q.4 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Three-layer/ Quebec City, curve one)…………………………….263 xxvii Figure Q.5 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Three-layer/ Ottawa, curve one)…………………………………..265 Figure Q.6 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Three-layer/ Saskatoon, curve one)……………………………….267 xxviii List of Abbreviations BC: British Columbia CDF: cumulative distribution function CLT: cross laminated timber COV: coefficients of variation DISP: displacement DOL: duration-of-load EDRM: exponential damage rate model FORM: first order reliability method FPL: Forest Products Laboratory Glulam: glued laminated timber LVL: laminated veneer lumber MOE: modulus of elasticity MPB: mountain pine beetle No.: number PSL: parallel strand lumber SPF: spruce-pine-fir STDV: standard deviation UBC: University of British Columbia USDA: U.S. Department of Agriculture xxix Acknowledgements I would like to offer my enduring gratitude to my research supervisor, Dr. Frank Lam, for his academic guidance and support throughout this work. Great appreciation is also given to my supervisory committee member, Dr. Ricardo O. Foschi, who has inspired me to continue my work in this field and provided valuable support, encouragement and help in timber engineering research. I also offer my special thanks to the supervisory committee members, Dr. Stavros Avramidis and Dr. Terje Haukaas, for their advice and guidance in the research. Special thanks go to the Timber Engineering and Applied Mechanics group at UBC for their support. I am particularly grateful to Dr. Minghao Li, Mr. George Lee and Mr. Chao Zhang for their help and time. I am thankful to the NSERC (Natural Sciences and Engineering Research Council of Canada) strategic research Network for Engineered Wood-based Building Systems for supporting this research. Finally I would like to take this opportunity to express my deepest gratitude to my parents for their unconditional love throughout my life. I also offer my thanks to all my friends for their understanding and support. xxx Dedication To my parents 1 Chapter 1: Introduction and Research Objectives 1.1 Background Among the major building materials, wood is the only commonly used renewable material for structural engineering. Wood, which is natural and warm, is often considered an aesthetic building material. In general, wood is stronger under loads of short-term duration and is weaker if the loads are sustained. This phenomenon is called duration of load; and, the primary relationship between the stress ratio, also known as the load ratio (i.e., the value of the applied stress divided by the short-term strength of the timber member), and the time to failure is commonly referred to as the duration-of-load effect. In fact, the duration-of-load effect is not introduced by deterioration, such as biological rot; rather, it is an inherent characteristic of wood. Timber researchers and engineers have devoted great efforts to studying the duration-of-load phenomenon in wood. As a result, different methods based on the theories of damage accumulation, mechano-sorptive behaviour and viscoelastic fracture mechanics have been established to predict the duration-of-load effect in wood. Cross laminated timber (CLT) is a wood composite product suitable for floor, roof and wall applications. The layering of CLT is similar to that of plywood; however, there is a notable difference in that the CLT layers are composed of wood laminates or boards instead of thin 2 veneers. The CLT panel usually includes three to eleven layers, as shown in Figure 1.1. As a new engineered wood product, little research has been done on the duration-of-load effect in CLT products. However, this research area is of great interest in the field of timber engineering, due to the necessity of understanding the structural performance of CLT systems, as well as other issues related to the introduction of this new building product into the building codes for engineered wood design. Figure 1.1 Layering of CLT Specifically, the rolling shear behaviour needs to be investigated, because it plays an important role in CLT structural performance. Rolling shear stress is defined as the shear stress leading to shear strains in a plane perpendicular to the grain (Fellmoser and Blass, 2004). Under out-of-plane bending loads, the CLT panel capacity can sometimes be governed by the rolling shear failure in the cross layers, as shown in Figure 1.2. The orthogonal arrangement of timber laminae in CLT increases the possibility of the rolling shear failure in CLT systems. Research has shown that, under constant loading, the creep deformation of CLT considering the rolling shear behaviour is 140% of the creep deformation of glued laminated timber (glulam) products in the same test conditions (Jöbstl and Schickhofer, 2007). CLT engineering 3 applications in floor or roof components can be used as an example: In North American building codes, the dead load on CLT panels is considered as constant long-term loading, and snow loads on roofs are considered as standard term sustained loads. Although all wood-based products are susceptible to duration of load, research on the CLT duration-of-load property is limited compared with other mechanical properties. Therefore, more research work is needed to quantify the duration-of-load effect and reduce the possibility of CLT rupture under long-term and sustained loading throughout its intended service life. Figure 1.2 Rolling shear behaviour in CLT The largest outbreak of the mountain pine beetle (MPB) ever recorded struck western Canada, including British Columbia (BC), in the beginning of the twenty-first century. Due to the limitations of the annual allowable cut capacity, a huge volume of MPB-attacked lodgepole pine is expected to hit the BC forest industry in the next decade. Technologies to convert the MPB-attacked lumber, as shown in Figure 1.3, into engineering wood products are urgently required to absorb this threat to the profitability of BC forestry (Wang, 2010). 4 Figure 1.3 MPB-killed lumber CLT is one of the available technologies that can produce massive timber members as an engineered wood product for higher timber structures, in both residential and commercial building sectors. In Canada, commercial CLT products made from MPB lumber are currently available, as shown in Figure 1.4. However, little research has been performed on the duration-of-load effect of this product. Figure 1.4 MPB-killed lumber based CLT product One of the key mechanical properties of an engineered wood product is its structural response under long-term loading; therefore, a comprehensive program for understanding the mechanical properties of CLT is being carried out in the Department of Wood Science at the University of 5 British Columbia, Canada. The objectives of this program are to investigate the fundamental mechanical properties of CLT panels and to gain basic knowledge of the duration of load of CLT under long-term loading. As part of the program, this thesis is focused on CLT duration-of-load research with consideration of the rolling shear behaviour under uncontrolled laboratory climatic conditions. To limit the influence of environment variables on the duration-of-load tests, low cycle trapezoidal fatigue loading protocol was adopted to accelerate the damage accumulation process, and the duration of this damage accumulation fatigue test is relatively short. The size effect, or the stress volume effect, in wood products is another well-known phenomenon, where the brittle strength properties in wood are stress volume dependent. For instance, lower longitudinal shear strength is observed when the material stressed volume increases. Considering the rolling shear failure in wood is brittle, the theory of brittle fracture, developed by Weibull (1939), may be applied in the investigation of the conditions under rolling shear failure development. The first study applying Weibull’s brittle fracture theory to wood was done by Bohannan (1966), who used this theory to predict the strength of wood beams with different sizes. This brittle fracture theory has also been applied in the study of longitudinal shear strength of Douglas-fir (Foschi and Barrett, 1976) and other mechanical properties of wood and wood-based products (Lam et al., 1997; Norlin and Lam, 1999; Craig and Lam 2000). Thus far, no research has been performed to investigate the size effect on the rolling shear strength. Therefore, an examination of the size effect on the rolling shear is also presented in this thesis. 6 In this study, duration-of-load tests were first carried out on CLT beam specimens. The finite element method was then adopted to simulate the structural behaviour of the CLT beam specimens. Torque loading tests on CLT tubes were performed; and, the size effect on the rolling shear strength was evaluated. Moreover, a damage accumulation model was developed to describe the creep-rupture rolling shear behaviour of MPB CLT products, with different cross-sectional layups. After the model was calibrated and verified against the test results, the CLT rolling shear long-term behaviour was investigated. A reliability assessment of the CLT rolling shear long-term performance was carried out. 1.2 Previous Research on Duration-of-Load Effect A brief introduction of previous research on duration of load is presented in this section, to establish a background for the studies of this thesis. The brief introduction on the duration of load is useful for a better understanding of the main objective of this thesis and the methods adopted for the CLT research, which will be introduced in Section 1.3. A more detailed literature review on the duration-of-load effect on the CLT rolling shear behaviour will also be presented in Chapter 2 for an understanding of the state of the art. The scientific treatment of the duration-of-load effect can be traced back more than 250 years to a study by Buffon (1740). During the 1940s and 1950s, almost all load duration experiments were carried out using small, clear and straight-grained wood specimens. In the 1970s, Madsen pioneered research work on the duration of load for dimensional lumber rather than small clear specimens. 7 One cooperative research program between the U.S. Department of Agriculture (USDA) Forest Service, Forest Products Laboratory (FPL) and Forintek Canada Corp. included detailed creep-rupture and duration-of-load information for dimensional lumber and composite panel products (Foschi and Barrett, 1982; Gerhards et al., 1987; Laufenberg et al., 1999). In addition, Madsen and Barrett (1976), Foschi and Barrett (1982) and Wang et al. (2012) devoted significant efforts to the research of duration-of-load tests, using reliability-based design methods for reliability assessments of the long-term performance of in-grade lumber and wood composite products. Rheological behaviour, or creep deformation, occurs in wood material given the appropriate conditions. Several detailed mathematical descriptions and empirical models have been proposed for the investigation of this rheological time-dependent behaviour; and, the relative merits of these models depend on how easily their constants can be determined and how well the predictions fit the test results (Dinwoodie, 2000). Damage accumulation models, which are specific theories established to predict the duration-of-load effect in wood, are focused on characterization of the process leading to failure in the wood material with one sophisticated mathematical description. These models characterize the duration-of-load effect, creep effect or both at the same time (Wang, 2010), in order to predict the structural behaviour under long-term loading. There are several different damage accumulation model theories. The Foschi and Yao model, or the Canadian model (Yao, 1987), considers the damage accumulation rate as a function of stress 8 history and the already accumulated damage state. The Gerhards model, or the U.S. model (Gerhards et al., 1987), uses the exponential damage rate to describe the duration-of-load effect on lumber. The Nielsen model, or the damaged viscoelastic material theory (Nielsen, 1986, 2000, 2005), attempts to consider the dependence between the phenomena of creep and creep-rupture at the same time, by employing viscoelastic fracture mechanics to study the process of slow crack growth through timber. More recently, the Wang and Lam model (Wang, 2010) combined the current strain rate, the already accumulated damage state and a five-parameter rheological creep model (Pierce et al. 1985) into one mathematical description. This strain-based damage accumulation model takes into account the deformation history from creep deflection to a final rupture event. As this model is a probabilistic model, it can be incorporated into a time-based reliability study of the structural behaviour of wood composites. 1.3 Objectives and Methods The objectives of this thesis are to investigate and evaluate the duration-of-load and size effects on the rolling shear strength of MPB lumber based CLT. By carrying out duration-of-load tests on CLT beam specimens under ramp loading and trapezoidal fatigue loading protocols, the in-service stiffness and the possibility of the rolling shear failure of such wood-based products can be understood throughout their intended service life. 9 A finite element model was developed to investigate the structural performance of CLT test specimens. By carrying out torque loading tests on CLT tubes, the rolling shear strength under different material volumes was estimated. In order to model the duration-of-load effect on the rolling shear strength of CLT, a stress-based damage accumulation model (i.e., the Foschi and Yao model) was adopted to study the CLT creep-rupture behaviour. Therefore, the effects of the passage of time under different loading histories on the rolling shear strength can be studied as a further step in modeling. MPB CLT products were manufactured, and test samples were designed and processed in the Timber Engineering and Applied Mechanics Laboratory at the University of British Columbia. A series of tests were then carried out to investigate the duration-of-load behaviour of rolling shear strength in CLT. The tests included ramp loading tests and two different types of trapezoidal fatigue loading tests (damage accumulation tests) on CLT products with different cross-sectional layups. Finite element modeling, based on ANSYS and Fortran platforms with customized subroutines, was adopted in simulating the structural behaviour of beam specimens. The finite element modeling results and the rolling shear strength based on the measured test data were examined. CLT tubes were prepared for torque loading tests to investigate the size effect on the rolling shear strength in wood. Finally, a stress-based damage accumulation model was calibrated against one set of the trapezoidal fatigue test data and was verified using another set of the trapezoidal loading results. This model allows one to gain a deeper insight into the reliability assessment of the long-term rolling shear behaviour of CLT products. 10 1.4 Thesis Organization Chapter 2 introduces CLT products and their rolling shear behaviour, and describes previous attempts to understand the duration-of-load behaviour and the size effect in wood and wood-based products. Chapter 3 discusses the modeling approaches for the finite element model theory, rheological model theory and damage accumulation model theory. The damage accumulation process is evaluated mathematically under different loading protocols. Chapter 4 describes the experiments performed to investigate the rolling shear strength of CLT, with beam specimens under ramp loading and trapezoidal fatigue loading and with torque tube specimens under torque loading. The results of these experiments are used to calibrate and verify a stress-based damage accumulation model and a size effect model. Chapter 5 describes a finite element model to simulate a CLT beam specimen’s structural response, and introduces a size effect model that includes Weibull’s theory of brittle fracture to investigate rolling shear strength. Based on the recorded test data, the CLT rolling shear strength is also evaluated in this chapter. 11 In Chapter 6, the developed stress-based damage accumulation model is calibrated using selected experimental results and then verified against the other experimental results. The comparisons between test measurements and predictions show that this damage accumulation model can be used to evaluate the long-term performance of CLT. A reliability assessment of CLT products is then carried out; and, finally, the duration-of-load adjustment factor for the CLT rolling shear strength is discussed. Chapter 7 includes concluding remarks and proposes future research work. 12 Chapter 2: Literature Review 2.1 Summary A literature review on the topics of duration-of-load and size effects on the rolling shear strength of cross laminated timber (CLT) is presented here. This work is helpful in establishing a unified understanding of the research subject. 2.2 Introduction The study of the duration-of-load and size effects on the strength properties of wood products is typically challenging. In the case of CLT, additional complexity exists with the necessary consideration of crosswise layups of wood boards, existing gaps, non-uniform stress distributions in the cross layers, glue bonding between layers and strength variability of timber materials. The following sections introduce and discuss the research efforts performed in related fields. 2.3 Cross Laminated Timber The most widely used multi-layer timber plate products are fabricated of CLT, which is manufactured using low-grade to medium-grade lumber boards and then connected with either metal connections or adhesive in the cross layers (one layer is orthogonal to adjacent layers). CLT is an innovative timber engineering product that was first invented and developed in 13 Germany and Austria in the early 1990s, and it has since been popular in commercial and residential applications in European countries (FPInnovations, 2011). CLT panels usually have three to eleven layers of wood boards; the number of layers in CLT is typically an odd number, and these layers are symmetrical around the middle layer. CLT products have been included in European timber design for more than ten years; they have demonstrated good structural performances, as well as a potential market opportunity for wood industries. The use of CLT panels in North America is fairly recent and has been receiving a lot of interest from the wood industries. In CLT, each layer of boards is placed crosswise to adjacent layers for increased structural rigidity and stability, typically with dimensional lumber as the main input material (FPInnovations, 2011). It is possible to use lower-grade lumber boards for the interior layers and higher-grade boards for the exterior layers. Understanding the structural behaviour and design methods for CLT is essential. Much effort has been devoted into the research on CLT panels. Because the classic composite theory does not consider shear deformation in laminated plates, it cannot give precise results to plates with a low span-to-depth ratio. In order to include shear effect, Blass and Görlacher (2003) proposed a theory of mechanically jointed beams with the concept of a reduction parameter. However, for continuous beams loaded by concentrated loads or for CLT elements with more than five layers, the theory of mechanically jointed beams is not accurate enough. For this case, Kreuzinger (1999) proposed a more precise calculation method called the shear analogy method (German: Schubanalogieverfahren), which has been accepted in the German code for timber structures (DIN1052, 2004-08). 14 Three beam theories (i.e., the layered beam theory, the Gamma method and the shear analogy method), suitable for investigating beam behaviour of CLT products, are introduced in detail as follows. Bodig and Jayne (1982) presented the well known layered beam theory, which requires calculating the transformed cross section in the first step. The derivation of this beam theory does not consider shear deflections and assumes that cross sections remain plane before and after deformation. For a CLT beam, all the laminae need to be converted to an equivalent homogeneous material based on the ratios between the 𝐸 values of individual lamina along the beam axis. The cross layers need to be transformed to a much smaller area because the perpendicular to grain direction 𝐸90 value is much smaller than the parallel to grain direction 𝐸0. Using this method, under the transverse shear load 𝑉, the rolling shear stress 𝜏 in the cross layers of the bending specimen is calculated by: 𝜏 =𝑉𝑄′𝐼′𝑤′ ( 2.1 ) where 𝐼′ is the moment of inertia of the transformed cross section; 𝑄′ is the area moment depending on the distance from the cross layer of interest to the top of the beam. 𝑤′ is the transformed width of the cross layer. In Eurocode 5 (Eurocode 5, 2004), the Gamma method (also known as the mechanically jointed beam theory) can be adopted to calculate wood composite beams with mechanical joints or glued layers. For a glued CLT beam, it can be assumed that the effective bending stiffness comes from 15 the longitudinal layers only and the cross layers are treated as imaginary fasteners to connected the longitudinal layers. Therefore, connection efficiency 𝛾𝑖 factors need to be calculated for the longitudinal layers. For cross layers, the connection efficiency factor 𝛾 equals to zero. Using this method, under the transverse shear load 𝑉, the rolling shear stress 𝜏𝑖 in the cross layer 𝑖 of a CLT beam can be calculated by: 𝜏𝑖 =𝑉∑ 𝛾𝑗𝐸𝑗𝐴𝑗𝑎𝑗𝑛𝑗=𝑖+1(𝐸𝐼𝑒𝑓𝑓) ∙ 𝑤 ( 2.2 ) where 𝑤 is the beam width; 𝛾𝑗 is the connection efficiency factor of longitudinal layer 𝑗; 𝐸𝑗 is the modulus of elasticity of layer 𝑗 along the beam axis; 𝐴𝑗 is the cross section area of layer 𝑗; 𝑎𝑗 is the distance from centroid of layer 𝑗 to the neutral axis of the beam; and, (𝐸𝐼𝑒𝑓𝑓) is the effective stiffness of the beam, calculated by: 𝐸𝐼𝑒𝑓𝑓 =∑(𝐸𝑖𝐼𝑖𝑛𝑖=1+ 𝛾𝑖𝐸𝑖𝐴𝑖𝑎𝑖2) ( 2.3 ) The 𝛾𝑖 factors for the longitudinal layers depend on the slip properties of the cross layers connecting the longitudinal layers. They are calculated by the rolling shear modulus of the cross layers 𝐺𝑅, 𝐸0 value of the longitudinal layer, beam width and span, layer thickness, etc. Detailed calculations of 𝛾 can be found in the literature (Eurocode 5, 2004, Appendix B). 16 Kreuzinger (1999) proposed the shear analogy method, which takes into account the shear deformations of a beam. In this method, a layered beam is separated into two imaginary beams: Beam A and Beam B. Beam A considers flexural stiffness of individual layers about their own neutral axes while their shear rigidity is assumed to be infinite. Beam B considers bending stiffness and shear stiffness of individual layers about their “Steiner” points. These two beams are coupled so that they have equal deflections under the loads. Using this method, under the transverse shear load 𝑉, the rolling shear stress in cross layer 𝑖 is calculated by adding up the shear stresses 𝜏𝐴,𝑖 in Beam A and 𝜏𝐵,𝑖 in Beam B. { 𝜏𝐴,𝑖 =1.5𝑉𝐴ℎ𝑖 ∙ 𝑤∙𝐸𝑖𝐼𝑖𝐵𝐴 𝜏𝐵,𝑖 =𝑉𝐵𝐵𝐵𝑤∙ ∑ 𝐸𝑗𝐴𝑗𝑎𝑗𝑛𝑗=𝑖+1 ( 2.4 ) where 𝐵𝐴 and 𝐵𝐵, are the bending stiffness of Beam A and Beam B, calculated by: { 𝐵𝐴 =∑𝐸𝑖𝐼𝑖𝑛𝑖=1 𝐵𝐵 =∑𝐸𝑖𝐴𝑖𝑎𝑖2𝑛𝑖=1 ( 2.5 ) 𝑉𝐴 and 𝑉𝐵 are the fractions of the shear load 𝑉 on Beam A and Beam B, proportional to the bending stiffness 𝐵𝐴 and 𝐵𝐵; 𝐸𝑖 is the modulus of elasticity of layer 𝑖 along the beam axis; 𝐴𝑖 is the cross section area of layer 𝑖; 𝑎𝑖 is the distance from centroid of layer 𝑖 to the neutral axis of the beam; and, 𝑤 is the beam width. 17 Bejtka and Lam (2008) discussed the calculation methods for CLT elements used for floors loaded in the out-of-plane direction and for walls loaded in the in-plane direction. In out-of-plane loading cases, the theory of mechanically jointed beams and the shear analogy method, both of which require the rolling shear modulus of cross layers, were compared. Axial and shear load cases were considered for CLT panels in the in-plane direction loading cases. Chen (2011) investigated the different ways of making CLT and the structural performance suitable for North America. The main focus of this study was the investigation of the structural performance of box-based CLT systems used in floor applications. Third-point bending tests were conducted on CLT specimens in the Timber Engineering and Applied Mechanics Laboratory at the University of British Columbia. Comprehensive three-dimensional finite element models, which can be used to analyze the mechanical and vibration behaviour of the plate and box type structures, were developed. The numerical analysis agreed well with experimental data, in terms of vertical deflection and bending stiffness. Vibration, which is critical to floor serviceability, was also studied. As pioneering research of CLT materials in North America, this work has contributed to the understanding of the structural performance of floor systems using CLT panels for the commercial and residential applications. 2.4 Duration of Load The first observation of a duration-of-load effect on record was made by the French naval architect Georges Louis Le Clerc, Compte de Buffon in 1740, having discovered structural oak beams’ strength reduced under sustained bending load. From the 1940s to 1950s, almost all load 18 duration experiments were carried out using small, clear and straight-grained wood specimens. Duration-of-load factors were proposed to adjust the design stresses for lumber and wood engineering products for a range of loading conditions. These duration-of-load factors, based on Wood’s (Wood, 1951) work on small defect-free specimens, were applied in U.S. and Canadian allowable stress design codes. During that time, based on the test data from small defect-free specimens, an empirical hyperbolic curve, the Madison curve, was proposed to describe the relationship between the bending stress ratio and the logarithm of the time over which the constant load was applied. The start of the present era of duration-of-load research occurred around 1970, when Madsen pioneered the duration-of-load research work on dimensional timber, rather than on small clear specimens. There were test results indicating that the creep-rupture response of dimensional lumber differed from that of small clear test pieces. Madsen, who carried out the first duration-of-load research on dimensional lumber in Canada, performed the bending test on No. 2 grade nominal 2 by 6 in. hem-fir lumber with different loading procedures (Madsen, 1971, 1973). It was concluded that the time to failure was dependent on the stress level applied and that the duration-of-load effect was greater in high strength timber. Moreover, the results suggested the Madison curve based on small clear wood test was conservative. More research efforts were then devoted to dimensional lumber tests. Madsen and Barrett (1976) and Foschi and Barrett (1982) concluded that the relationship between the time to failure and the applied stress level in in-grade lumber (i.e., the duration-of-load effect in full-sized lumber) was different from that in small clear specimens. 19 There is one ASTM Standard Specification (ASTM D6815) that provides a procedure for the testing and evaluation of duration-of-load and creep effects of wood and wood-based products. The procedure is intended to demonstrate the duration-of-load and creep effects of engineering wood products used in dry service conditions compared to those of solid sawn lumber. This specification is a pass-fail test and has no procedure for developing duration-of-load factors for the tested products. In addition, there is no attempt to evaluate the long-term creep-rupture response of wood products from a three-month test at a relatively low load level. Wang et al. (2012) performed a duration-of-load and creep testing program on a thick strand-based composite product, using wood from forests attacked by the MPB (mountain pine beetle). The constant loading of experimental beams lasted for one year. The long-term deflection was monitored and recorded at a preset frequency. Time to failure data was also obtained for all the broken specimens. The ramp load test was carried out at different rates of loading to investigate its influence on short-term strength. Finally, fatigue tests were also conducted with a triangular cyclic load history, and the strain history was obtained to elucidate the low-cycle fatigue behaviour of the composite. The results from the duration-of-load and creep testing program on MPB strand-based wood composite products were presented. The results provided an experimental database for the calibration and verification of a creep-rupture model proposed by the authors. In order to obtain a more accurate prediction of the duration-of-load effect in wood, research work has started to focus on reliability-based design methods for reliability assessments of the 20 long-term performance of in-grade lumber. The theory for the development of the damage accumulation model is one of the key tools used to investigate the creep-rupture behaviour and to perform a reliability study of the long-term response of timber products. 2.5 Rheological Constitutive Relationship Rheological behaviour, or creep, occurs in most structural materials, given the appropriate environmental conditions (Wang, 2010). The rheological behaviour in the wood (i.e., the deformation increase), which is dependent on the time under constant stresses, is an important material property. The creep deformation is most pronounced in bending members, but it does occur in other types of structural members as well (Madsen, 1992). There are three distinctive types of creep behaviour: time-dependent (viscoelastic) creep, mechano-sorptive (moisture-change) creep, and pseudo-creep and recovery. Timber is described as being neither truly elastic in its behaviour nor truly viscous, but rather a combination of both states. When the research into creep is performed, timber material is usually considered as a viscoelastic material. Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. When subjected to a constant stress loading, viscoelastic material experiences a time-dependent increase in strain. This phenomenon is also known as viscoelastic (i.e., time-dependent) creep. There is a lot of research on modeling the rheological behaviour of timber material. 21 Schniewind (1968) described the classic model that is successfully applied for the simulation of the long-term creep behaviour in the bending of timber. The model is called power-law creep function: 𝜀(𝑡) = 𝜀0 +𝑎𝑡𝑚 ( 2.6 ) where 𝜀(𝑡) is the time-dependent strain; 𝜀0 is the initial elastic strain; 𝑡 is the time step; and, 𝑎 and 𝑚 are material-specific parameters that are determined experimentally. More models with detailed physical interpretation have also been proposed. Some models simulated the creep behaviour in timber by means of the combinations of springs and dashpots. For example, the Kelvin model is a Hookean spring in parallel with a Newtonian dashpot; and, the Maxwell model is a Hookean spring in series with a Newtonian dashpot. A three-element model is a Hookean spring in series with one Kelvin model, as shown in Figure 2.1. The Burger model, which is a four-element model, is one Kelvin model in series with one Maxwell model, as shown in Figure 2.2. The mathematical description of a three-element model is: 𝑌 = 𝛽1 + 𝛽2(1 − 𝑒−𝑡𝛽3) ( 2.7 ) where 𝑌 is the time-dependent deformation; 𝑡 is the time step; and, 𝛽1, 𝛽2 and 𝛽3 are random model parameters. The mathematical description of a four-element model is: 𝑌 = 𝛽1 + 𝛽2(1 − 𝑒−𝑡𝛽3) + 𝛽4𝑡 ( 2.8 ) where 𝛽4 is another random model parameter. 22 Figure 2.1 Illustration of one three-element model Figure 2.2 Illustration of one four-element model Pierce and Dinwoodie (1977, 1979) performed the three-point sustained loading test on five commercially available types of chipboard. The viscoelastic behaviour of the specimens under bending could be represented by three- or four-element spring and dashpot models. The models were calibrated against the overall test data. Results showed that the three-element model did not fit the low side of measured deflection very well. The four-element model provided a consistently better overall fit; however, the four-23 element model always overestimated the end deflection. The high deformation prediction from the four-element model was because the creep rate was assumed to be constant, but in reality the rate decreases. The results showed that 𝛽4, the extra random parameter in the four-element model, may be a function of time. If this is true, neither the three- nor four-element models are adequate to simulate the creep behaviour. Pierce et al. (1985) proposed a modified five-parameter model, with an additional power term introducing the fifth random parameter based on the four-element model, to reduce the contribution of the viscous component over time. The mathematical description of a five-element model is: 𝑌 = 𝛽1 + 𝛽2(1 − 𝑒−𝑡𝛽3) + 𝛽4𝑡𝛽5 ( 2.9 ) where 𝑌 is the time-dependent deformation; 𝑡 is the time step; and, 𝛽1, 𝛽2, 𝛽3, 𝛽4 and 𝛽5 are random model parameters. Results have shown that the five-parameter model was superior to the four-element model for long-term predictions of creep deflection, particularly at lower stress levels. Laufenberg et al. (1999) summarized one cooperative research program between the Forest Products Laboratory, USDA Forest Service and Forintek Canada Corporation. It provided detailed information on the creep-rupture and creep test for plywood, oriented strand board (OSB) and waferboard products. The creep-rupture test used three ramp loading rates and three constant loading levels in only one environmental condition. The creep test used two low-constant load levels under three environmental conditions for a six-month period. One 24 exponential model was then adopted to simulate the creep behaviour, and the Gerhards model was adopted to simulate the damage accumulation process. It was concluded the Gerhards model fitted the test data well, but the model’s interpretation could not include all the specimens that survived or failed on loading. Although the model was trying to describe and predict the specimens’ behaviour from creep to rupture, the creep and rupture behaviours were investigated separately during the mathematical modeling process (Wang, 2010). Jöbstl and Schickhofer (2007) performed long-term four-point bending tests on CLT slab elements. Parallel tests were carried out on glulam timber slab elements with the same testing configurations to enable direct comparison of the test results of CLT specimens. Two different stress levels and two different climatic conditions were applied in the test. After one year of the loading test, the creep deformation of CLT with consideration of the rolling shear behaviour was 140% of the creep deformation of glulam timer products in the same test condition, which showed CLT was more likely to behave like plywood than glulam timber. Based on the test data, the methods for the derivation of creep deformation factors were proposed. 2.6 Damage Accumulation Model Damage accumulation models, using the sophisticated mathematical descriptions, are focused on characterizing the process from the initial state to the failure of the wood material. These models explained the experimental data using the concept of a damage parameter (Yao, 1987), and they were proposed based on the understanding of test phenomena at the macroscopic level, because the cumulative damage at the microscopic level showed incomplete information. 25 Some types of the most important models are summarized in the following paragraphs. Barrett and Foschi (1978) proposed Barrett and Foschi model I, in which the damage rate equaled the product of a damage-dependent factor and a stress-dependent factor. Barrett and Foschi model II was also presented, in which the damage rate equaled the sum of a stress-dependent term and a damage-dependent term (Barrett and Foschi, 1978). The mathematical description of Barrett and Foschi model I is: {𝑑𝛼 / 𝑑𝑡 = 𝑎(𝜏 − 𝜏0)𝑏𝛼𝑐 𝑖𝑓 𝜏 > 𝜏0𝑑𝛼 / 𝑑𝑡 = 0 𝑖𝑓 𝜏 ≤ 𝜏0 ( 2.10 ) where 𝛼 is the damage state variable (𝛼 = 0 in an undamaged state and 𝛼 = 1 in a failure state); 𝑡 is the time; 𝜏 is the stress ratio, defined as the applied stress history 𝜎(𝑡) divided by the short-term strength 𝜎𝑠 (i.e., 𝜏 = 𝜎(𝑡) /𝜎𝑠); 𝜏0 is the stress threshold below which damage will not accumulate; and, 𝑎, 𝑏 and 𝑐 are random model parameters. The mathematical description of Barrett and Foschi model II is: {𝑑𝛼 / 𝑑𝑡 = 𝑎(𝜏 − 𝜏0)𝑏 + 𝜆𝛼 𝑖𝑓 𝜏 > 𝜏0𝑑𝛼 / 𝑑𝑡 = 0 𝑖𝑓 𝜏 ≤ 𝜏0 ( 2.11 ) where 𝜆 is an additional random model parameter. For these two models, three types of loading test were performed on small clear specimens of Douglas-fir in bending: a step function load test, a ramp load test and a combination of a ramp load test followed by a constant loading process. The random model parameters were obtained 26 by calibrating the model with the test data for the constant and ramp types of loading. It was concluded that Barrett and Foschi model II was more flexible, and its prediction showed a better fit with the data from the constant load bending test on small clear specimens. Foschi and Yao (1986) proposed the Foschi and Yao model, in which the damage rate was a function of the stress loading history and the previous accumulated damage state. Compared to Barrett and Foschi model II, the Foschi and Yao model was reasonable: in Barrett and Foschi model II, if the accumulated damage reached a significant level, the slightest damage would continue to grow even under the absence of stress; therefore, material failure could occur under no stress (Wang, 2010). The Foschi and Yao model was based on the expanded series of the damage accumulation rate, and the model included a first-order damage-dependent term. The mathematical description of the expanded series of the damage accumulation rate is: 𝑑𝛼 / 𝑑𝑡 = 𝐹0(𝜎) + 𝐹1(𝜎)𝛼 + 𝐹2(𝜎)𝛼2 + 𝐹3(𝜎)𝛼3 +… ( 2.12 ) The mathematical description of the Foschi and Yao model is: {𝑑𝛼 / 𝑑𝑡 = 𝑎(𝜎(𝑡) − 𝜏0𝜎𝑠)𝑏 + 𝑐(𝜎(𝑡) − 𝜏0𝜎𝑠)𝑛𝛼 𝑖𝑓 𝜎(𝑡) > 𝜏0𝜎𝑠𝑑 𝛼/𝑑𝑡 = 0 𝑖𝑓 𝜎(𝑡) ≤ 𝜏0𝜎𝑠 ( 2.13 ) 𝛼𝑒{− ∫𝑐(𝜎(𝑡)−𝜏0𝜎𝑠)𝑛𝑑𝑡} |0𝑇= ∫ {𝑎(𝜎(𝑡) − 𝜏0𝜎𝑠)𝑏𝑒{− ∫ 𝑐(𝜎(𝑡)−𝜏0𝜎𝑠)𝑛𝑑𝑡}}𝑑𝑡𝑇0 ( 2.14 ) where 𝛼 is the damage state variable (𝛼 = 0 in an undamaged state and 𝛼 = 1 in a failure state); 𝑡 and 𝑇 is the time; 𝜎(𝑡) is the applied stress history; 𝜎𝑠 is the short-term strength; 𝜏0 is a ratio of the short-term strength 𝜎𝑠; thus, the product 𝜏0𝜎s is a threshold stress below which there will be no accumulation of damage; and, 𝑎, 𝑏, 𝑐 and 𝑛 are random model parameters. 27 For the Foschi and Yao model, the formulation was derived considering three types of loading protocol: a ramp load test, a constant load test and an arbitrary load history test. The model was calibrated against the data from two constant stress level loading tests on dimensional Western hemlock lumber conducted by Foschi and Barrett starting in 1977 (Foschi and Barrett, 1982). The result showed that its prediction fit very well with the data from the test on dimensional lumber. Gerhards et al. (1987) proposed an exponential damage rate model (EDRM) to interpret the bending test data on lumber. The model defined that the damage accumulation rate was only a function of applied stress. An iterative reweighted nonlinear least squares procedure was adopted to calibrate the model with ramp and constant load experimental data. The model generated a reasonable fit to the ramp and constant load test data. However, the residual strengths of the surviving specimens under constant load were lower than the prediction from the model. The mathematical description of the Gerhards model is: 𝑑𝛼 / 𝑑𝑡 = 𝑒𝑥𝑝 (−𝑎 + 𝑏𝜏) = 𝑒𝑥𝑝(−𝑎 + 𝑏𝜎(𝑡)/𝜎𝑠) ( 2.15 ) where 𝛼 is the damage state variable (𝛼 = 0 in an undamaged state and 𝛼 = 1 in a failure state); 𝑡 is the time; 𝜏 is the stress ratio, defined as the applied stress history 𝜎(𝑡) divided by the short-term strength 𝜎𝑠 (i.e., 𝜏 = 𝜎(𝑡) /𝜎𝑠); and, 𝑎 and 𝑏 are random model parameters. Yao (1987) observed that the damage accumulation rate was only a function of applied stress in the Gerhards model. Yao calibrated the Gerhards model with the Western hemlock bending data. The results showed that the Gerhards model was too stiff to represent the data trend, because the 28 model neglected the effect from the already accumulated damage in the process of damage accumulation. It was concluded that the Gerhards model could not fit the long-term experimental results, because it lacked a damage-dependent factor in its expression. Brandt and Fridley (2003) performed duration-of-load tests on wood-plastic composites. The test data was fitted into the Gerhards model using the least square regression method. The duration-of-load effect in wood-plastic composites was more predominant than that of solid wood. However, the load-duration behaviour trend from the selected wood-plastic composite formulations was determined to be similar to that of solid sawn lumber, although a rotation of the curves from the prediction of Gerhards model was shown. Wang et al. (2012) developed a creep-rupture model, which included the current strain rate, the already accumulated damage state and a five-parameter rheological creep model into the mathematical description. An experimental database on the time-dependent mechanical properties of strand-based wood composite products was generated for model calibration and verification. The paper presented the first known attempt to evaluate the duration-of-load and creep effects and fatigue behaviour of thick strand-based wood composites made from MPB-killed lumber. The mathematical description of the model is: {𝑑𝛼 / 𝑑𝑡 = 𝐴𝜀̇ + 𝐵 𝜀̇𝛼 𝑖𝑓 𝜀̇ > 0𝑑 𝛼/𝑑𝑡 = 0 𝑖𝑓 𝜀̇ ≤ 0 ( 2.16 ) 𝜀(𝑡) =𝜎𝑎𝜎?̅?[𝛽1 + 𝛽2(1 − 𝑒−𝑡𝛽3) + 𝛽4𝑡𝛽5] ( 2.17 ) where 𝛼 is the damage state variable (𝛼 = 0 in an undamaged state and 𝛼 = 1 in a failure state); 𝑡 is the time; 𝜀 is the time-dependent strain; 𝜀̇ is the strain rate; 𝜎𝑎 is the applied external 29 load; 𝜎?̅? is the average short-term strength of the member; and, 𝐴, 𝐵, 𝛽1, 𝛽2, 𝛽3, 𝛽4 and 𝛽5 are random model parameters. The distinctive advantages of this creep-rupture model are its ability to predict the damage produced by an arbitrary random load sequence and its convenient usage in reliability-based design formats. The model included the short-term strength of material and the load history, and it was able to predict the deflection history and the time to failure. The model can also provide the description of deflection from the initial creep to a final rupture event. 2.7 Rolling Shear Behaviour Rolling shear stress is defined as shear stress leading to shear strains in a plane perpendicular to the grain direction (Fellmoser and Blass, 2004). There has been research performed to measure the rolling shear modulus of CLT, by conducting three-point bending tests with variable span using downscaled sandwich specimens (Zhou et al., 2012). The results showed that the cross layer of in and between growth ring orientations could have a higher rolling shear modulus than that of flat sawn or quarter sawn. Blass and Görlacher (2000) performed the tests on CLT timber elements to determine rolling shear stiffness and rolling shear strength of European spruce timber. The evaluated values were applied in the calculation model for mechanically jointed beams with semi-rigid joints, to design the CLT timber plates. Fellmoser and Blass (2004) studied the influence of rolling shear modulus on strength and stiffness of CLT timber elements. Since the rolling shear modulus is low, shear 30 deformation increases significantly depending on the thickness of the rolling shear layer. The results also show that shear deformation increases significantly when the span to depth ratios are decreasing, due to the low rolling shear modulus in the cross layers. Aicher and Dill-Langer (2000) argued that their research demonstrated that the rolling shear modulus is not an intrinsic material property, but an apparent smeared shear stiffness of structural elements, depending on several elastic coefficients, geometry and size parameters. Jöbstl and Schickhofer (2007) performed long-term four-point bending tests on CLT slab elements. After one-year loading tests, the creep deformation of CLT with consideration of the rolling shear behaviour was 140% of that of a glulam timber product under the same test conditions. The results showed the rolling shear behaviour played an important role in the CLT structural performance. 2.8 Weakest Link Theory and Size Effect Under the same load conditions, small members of brittle material, such as timber, tend to show higher strength than larger members. This can be explained by the conventional brittle fracture theory developed from the weakest link concept. If one structural member with a number (𝑛) of small elements is considered, the member will fail if any one small element fails. If 𝜏 is the strength of each small element and 𝐹0(𝜏) is the probability distribution of the strength 𝜏, the survival probability of each element 𝑝𝑆,𝑖 is: 31 𝑝𝑆,𝑖 = 1 − 𝐹0(𝜏) ( 2.18 ) The probability of the survival of the entire unit is: 𝑝𝑆 =∏𝑝𝑆,𝑖𝑛𝑖=1= [1 − 𝐹0(𝜏)]𝑛 ( 2.19 ) For a large 𝑛, a series expansion yields: 𝑝𝑆 = 𝑒−𝑛∙𝐹0(𝜏) ( 2.20 ) Therefore, the failure probability of the member is: 𝑝𝑓 = 1 − 𝑒−𝑛∙𝐹0(𝜏) ( 2.21 ) According to Weibull’s theory (Weibull, 1939) and assuming the lower tail of 𝐹0(𝜏) governs the strength of the member, the cumulative distribution function for the strength of each small element can be represented by the following approximation: 𝐹0(𝜏) = (𝜏 − 𝜏𝑚𝑖𝑛𝑚)𝑘 ( 2.22 ) where 𝑚, 𝑘, and 𝜏𝑚𝑖𝑛 are the scale, shape and location parameters for the distribution. These parameters are assumed to be material constants. ∴ 𝑝𝑓 = 1 − 𝑒−𝑛∙(𝜏−𝜏𝑚𝑖𝑛𝑚 )𝑘 ( 2.23 ) 32 When 𝜏𝑚𝑖𝑛 = 0 and 𝑛 goes to infinity, the failure probability of the member in a two-parameter Weibull case is: 𝑝𝑓 = 1 − 𝑒−∫ (𝜏𝑚)𝑘𝑑𝑉𝑉 ( 2.24 ) If two different volumes of material (𝑉1 and 𝑉2), each with different stress distributions (𝜏1 and 𝜏2), at an identical probability of failure are considered, these expressions yield: 𝑝𝑓1 = 1 − 𝑒−∫ (𝜏1𝑚)𝑘𝑑𝑉𝑉1 ( 2.25 ) 𝑝𝑓2 = 1 − 𝑒−∫ (𝜏2𝑚)𝑘𝑑𝑉𝑉2 ( 2.26 ) Then, if: 𝑝𝑓1 = 𝑝𝑓2 ( 2.27 ) ∴ ∫(𝜏1)𝑘𝑑𝑉𝑉1= ∫(𝜏2)𝑘𝑑𝑉𝑉2 ( 2.28 ) and Equation ( 2.28 ) must be satisfied at any probability level. Bohannan (1966) studied the Weibull brittle fracture theory to clear wood specimens. These geometrically similar clear wood beams showed both the depth effect and length effect with equal factor values. However, no evidence of a width effect on the strength was detected. 33 Barrett (1974) performed uniform tension perpendicular to grain tests on Douglas-fir to investigate the relationship between the load capacity and the specimen material volume. It was concluded that the tensile strength was affected by the material volume and stress distribution in the specimen. Furthermore, the two-parameter Weibull function was found in good agreement with experimental results. Finally, based on the test data, it was suggested that the magnitude of size effect was dependent on the quality of the investigated material. Foschi and Barrett (1976) applied Weibull’s theory of brittle fracture to the determination of the strength of Douglas-fir wood in longitudinal shear. The theory explained the difference in shear strength between beams and the standard American Society for Testing and Materials (ASTM) shear block, as well as the dependence of the shear strength upon beam size. Finally, the authors derived allowable shear stresses for beams under different loading conditions. Lam et al. (1997) conducted experimental tests on Canadian softwood select structural lumber to evaluate the longitudinal shear strength in wood material. Based on the American Society for Testing and Material (ASTM) shear block test results, the Weibull weakest link theory coupled with finite element analyses was used to predict the shear failure loads. The study showed that the Weibull shape parameter is species dependent. Norlin and Lam (1999) studied the width effect for rolling shear failures of laminated veneer panels under cyclic tests. The width effect was evaluated theoretically, and the modeling results were compared with experimental results. The size adjustment factor was also proposed. The study recommended that, to quantify the width effect for laminated veneer products, a larger test 34 database, which includes specimen types with greater difference of volume and difference of stress levels, are necessary. Craig and Lam (2000) performed shear tests on shear block specimens and bending tests on laminated veneer lumber (LVL) and parallel strand lumber (PSL). The specimens in five-point bending tests had rectangular cross sections, and the specimens in centre-point bending tests were remanufactured into I shapes. One distinct size effect was found in the test results; the shear strength in the shear block tests is generally higher than that in the full-sized bending tests. 2.9 Conclusion To evaluate the duration-of-load and size effects on the CLT rolling shear behaviour, thorough literature reviews in different areas were performed. This knowledge was used to investigate the structural behaviour of the test specimens (as introduced in Chapter 4) and to evaluate the CLT rolling shear failure in the tests. In previous research attempts to model duration-of-load behaviour, the damage accumulation model became a well-established tool to model the duration-of-load effect in lumber. Different rheological models have been found to be efficient and accurate enough in each suitable application. There have been limited investigations of the size effect on the rolling shear strength; therefore, this topic requires further research efforts. 35 Chapter 3: Modeling of Duration-of-Load Behaviour of CLT 3.1 Summary To estimate CLT (cross laminated timber) structural behaviour, the finite element theory was adopted for the simulation of the beam specimens. With regard to the rheological deformation in wood material, the modified five-parameter rheological model was selected and investigated under different loading protocols. In terms of duration-of-load evaluation, a stress-based damage accumulation model was chosen, and the damage accumulation process was evaluated mathematically under different loading protocols. 3.2 Introduction To better understand the content in this chapter, one brief introduction is presented. To address the complexity existing in CLT, a finite element model was developed with an integrated user-defined FORTRAN subroutine which provided mathematical modeling process for the CLT beam specimens. The modified five-parameter rheological model was chosen in the rheological behaviour simulation process, and this selected model was efficient enough to describe the rheological behaviour of CLT. The response from this rheological model was mathematically investigated under different loading protocols. These loading protocols include ramp loading, trapezoidal 36 fatigue loading and long-term constant loading cases. A stress-based damage accumulation model was also evaluated under these different loading protocols, and the damage from the model was mathematically described for each loading case. 3.3 Finite Element Modeling Additional complexity exists in the investigation of the duration-of-load effect on the rolling shear strength of CLT, given the crosswise layups of wood boards, existing gaps, non-uniform stress distributions in cross layers, glue bonding between layers and strength variability of timber materials. Therefore, the finite element method was adopted to model CLT short-span beam specimens in tests, as shown in Figure 3.1 and Figure 3.2. This modeling work was performed on ANSYS v14.0 (SAS, 2011) and Intel Fortran 11.7 platforms. The detailed structural responses from specimens in short-term ramp tests and long-term trapezoidal fatigue tests, as introduced in Chapter 4, were evaluated. 37 Figure 3.1 Finite element modeling of the five-layer CLT beam test specimen Figure 3.2 Finite element modeling of the three-layer CLT beam test specimen 38 In Table 3.1, dimensions of the finite element model are the same as those in the experimental three-layer and five-layer CLT beam specimens, and these test specimens will be introduced in Chapter 4. Table 3.1 Configurations of the dimension in the finite element models CLT Beam model dimension L×W×H (mm) Lamination thickness in the model (mm) Lamination board width (mm) Five-layer 916.2×50.8×140 34/19/34/19/34 140 Three-layer 688.2×50.8×112 34/34/34 140 Rolling shear behaviour is dependent on the annual ring orientation of wood boards, and a polar coordinate system is suitable for the orthotropic material modeling of cross laminae (Aicher and Dill-Langer, 2000; Blass and Fellmoser, 2004). As shown in Figure 3.3, based on the ring patterns in the radial-tangential plane, the wood material property can be simulated in the polar coordinate system. The stress components for small elements, such as the rolling shear stress 𝜏𝑅𝑇 in the radial-tangential plane and the normal stresses 𝜎𝑅 and 𝜎𝑇 in both radial and tangential directions, can then be investigated. 39 Figure 3.3 Orthotropic material modeling in polar coordinate system However, if such details in the modeling work are considered, it is necessary to follow the rings for each cross laminae in CLT, meaning mapping the ring orientation of each wood board in the cross layer by recording the pith location, grain angles and ring patterns. These observed ring orientations for each board in each beam specimen should then be included in the finite element modeling. A polar orthotropic modeling method is an ideal theory, but in practice using this method may be challenging. For example, for five-layer CLT, each beam specimen includes about 14 pieces of wood boards in the cross layers. There are 60 pieces of specimens in the test group. This means that, to consider the growth rings in the finite element modeling process, simulating around 840 sets of the recorded ring orientations is required. Therefore, mapping of the annual ring orientation of each cross laminae in each specimen would be needed to form a database which would be very time consuming and impractical for modeling consideration. Even though the polar orthotropic modeling method seems to be suited to represent the annual ring orientation for estimating rolling shear stresses, modeling the wood material in Cartesian Ring Pattern Small Element in PolarCoordinate SystemStress Components inThis Small ElementRTRTRTRTTTRR40 coordinate system is also a practical alternative. The difference between a polar and a Cartesian coordinate system will decrease if the radius of curvature of the rings within the piece is larger, as it would be if the wood pieces were cut from larger diameter trees. Keeping track of an additional random factor reflecting ring curvatures adds another level of complexity that would make the problem more difficult to tackle. As shown in Figure 3.4, the adopted assumption is about defining parallel or perpendicular to the grain directions as the X, Y and Z directions in Cartesian coordinate system (i.e., the orthotropic volume modeling method). In Cartesian coordinate system, the rolling shear stress will be 𝜏𝑋𝑌 as given in Figure 3.4. The material properties for this orthotropic volume modeling method are presented in most of the literature (Bodig and Jayne, 1982; FPL, 2010). Also, another reason for adopting the orthotropic volume modeling method is that, in the ANSYS platform the polar orthotropic modeling is more appropriate when the volume geometry is symmetric in polar coordinate system; therefore, SOLID185 elements (based on the orthotropic volume modeling in Cartesian coordinate system) were used to model CLT wood boards with rectangular section in the ANSYS platform. Future research is recommended on the polar orthotropic modeling method to investigate the relationship between rolling shear behaviour and the annual ring orientation in CLT beams. 41 Figure 3.4 Orthotropic material modeling in Cartesian coordinate system As shown in Figure 3.5, small gaps, with an average value of one millimeter, were also simulated in the cross layers of finite element models. Modeling this gap in the cross layer is a conservative assumption, because in the test specimen a very small amount of adhesive might be found in some gaps between the adjacent non-edge glued boards. This small amount of glue could have been squeezed into some of the gaps during the pressing manufacture process. The squeezed glue may influence the continuity between the adjacent boards, and it may influence the apparent rolling shear modulus of the cross laminations. However, there is no research showing any correlation between the small amount of squeezed glue and the rolling shear modulus of non-edge glued laminations. Also, by influencing the shear stress distribution, the gap plays an important role in the rolling shear behaviour of non-edge glued cross layers, because the rolling shear stress is zero near the gap edge. Therefore, a conservative modeling approach was taken by ignoring the small amount of glue between the adjacent boards. The potential influence on CLT structural behaviour from the squeezed glue is recommended to be investigated in the future. Ring Pattern Small Element in CartesianCoordinate SystemStress Components inThis Small ElementxyxYXYXxyxyxyxyyYX42 Figure 3.5 Gap modeling in the cross layer Table 3.2 gives the input elastic orthotropic properties of wood material; in the table, the 𝐸𝐿 value was based on the measurement of boards’ modulus of elasticity (MOE) by transverse vibration tests, as introduced in Chapter 4. The other elastic properties were calculated based on the given ratios in literature (Eurocode 5, 2004). Poisson’s ratios for Spruce-Pine-Fir (SPF) were obtained from the Wood Handbook (FPL, 2010). Table 3.2 Input orthotropic properties of the wood boards Species Grade Elastic properties (GPa) Poisson’s ratios EL ET & ER GLR & GLT GRT νLR νLT νRT SPF No. 2 / better 11.43 0.381 0.714 0.071 0.316 0.347 0.469 stud 10.66 0.355 0.666 0.067 43 The wood mechanical property was assumed to be nonlinear orthotropic. In three orthotropic dimensions, a multi-linear stress-strain relationship was simulated for the timber’s tension, compression and shear behaviour, by defining yield stress and tangent modulus (Patton-Mallory et al., 1997). Existing research has shown the nonlinear stress-strain relationships in timber compressive and tensional behaviour, except for tension perpendicular to the grain stress, which is not considered in the CLT model. The shear behaviour with a brittle failure mode cannot be simply defined as nonlinear, nor can it sustain a high stress level with increasing strain. Ideally, the shear stress should drop to zero once it reaches the maximum shear strength. However, this introduces numerical singularities in the finite element solution. Hence, one modeling method is needed to simulate the global softening effect of local fracture zones, and this model may represent the rolling shear behaviour in the cross layers without actually modeling crack formation and growth in timber. To model the nonlinear shear strength and stress redistribution process once the shear stress reaches the maximum value, a trilinear model, as shown in Figure 3.6, was developed to soften the shear stiffness to zero in the end (Patton-Mallory et al., 1997). This justification is not exactly the same as nonlinear material property; however, it simulated the stress redistribution phenomenon reasonably well and limited the magnitude of shear stress in the numerical solution. The further modeling details are given in Appendix K and Section 5.2.2.1. 44 Figure 3.6 Multi-linear shear stress-strain relationship For modeling the glue line strength between different layers, the glue line shear stiffness was obtained from one test database (Schaaf, 2010). In Schaaf’s study, torsional shear tests were conducted to study the shear strength and stiffness properties of glue lines in three-layered wood composites glued by polyurethane adhesive, as shown in Figure 3.7. Table 3.3 shows the input shear line stiffness values for the finite element models. COMBIN14 linear x-y-z spring pairs were used to model the glue line bonding stiffness. The x- and y-springs considered the glue line shear stiffness, and the z-springs were assigned with high stiffness value representing rigid vertical bonding between wood layers. The boundary condition for the beam model was the same as the support condition in the test setup, i.e., simply supported beam with point load in the middle of span. The test span in the model was 840 mm for five-layer CLT and 612 mm for three-layer CLT, and it was the same as that in the test setup introduced in Chapter 4. 45 Figure 3.7 Schaaf’s study for the torsional shear tests Table 3.3 Input shear stiffness of the glue lines Species Clamping pressure Stiffness (N/mm3) Spruce-Pine-Fir 0.4 MPa 20.6 By coding customized Fortran subroutines of the constitutive relationship based on the modified five-parameter rheological model, user programmable features were selected in the ANSYS v14.0 platform to define the rheological constitutive relationship of wood (SAS, 2011). Therefore, this finite element model could not only mathematically model the CLT specimen with consideration of the rolling shear behaviour, but could also simulate the specimen’s rheological behaviour. Structural responses, including displacements, stresses and strains under different loading protocols, can be investigated based on the finite element theory. The further modeling details are given in Appendix L and Section 5.2.2.2. 46 3.4 Rheological Constitutive Relationship Modeling The modified five-parameter rheological model (Wang, 2010), shown in Figure 3.8, includes an additional power term introducing the fifth random parameter based on the four-element model. This model was adopted for the rheological simulation process. Figure 3.8 The modified five-parameter rheological model In Figure 3.8, 𝐻(𝑡) is Heaviside unit step function: 𝐻(𝑡) = {1, 𝑡 ≥ 00, 𝑡 < 0 ( 3.1 ) 𝑘0 and 𝑘1 are spring constants, and 𝜂0 and 𝜂1 are dashpot damping coefficients. 𝜎𝑎 is the applied external load. 𝜏1 is defined as the retardation time: 𝜏1 =𝜂1𝑘1 ( 3.2 ) and: 𝛽1 =𝜎?̅?𝑘0, 𝛽2 =𝜎?̅?𝑘1, 𝛽3 =1𝜏1=𝑘1𝜂1, 𝛽4 =𝜎?̅?𝜂0, 0 < 𝛽5 < 1 ( 3.3 ) 47 where 𝜎?̅? is one constant characteristic strength value. Then in the modified five-parameter rheological model, the strain 𝜀(𝑡) and strain rate 𝜀̇(𝑡) at time step 𝑡, shown in Figure 3.9 and Figure 3.10, can be expressed as (Wang, 2010): 𝜀(𝑡) = 𝜎𝑎 [1𝑘0+(1 − 𝑒−𝑡/𝜏1)𝑘1+𝑡𝛽5𝜂0] =𝜎𝑎𝜎?̅?[𝛽1 + 𝛽2(1 − 𝑒−𝑡𝛽3) + 𝛽4𝑡𝛽5] ( 3.4 ) 𝜀̇(𝑡) =𝜎𝑎𝜎?̅?∙ (𝛽2𝛽3 ∙ 𝑒−𝑡𝛽3 + 𝛽4𝛽5𝑡𝛽5−1) ( 3.5 ) Figure 3.9 The strain of the modified five-parameter rheological model Figure 3.10 The strain rate of the modified five-parameter rheological model 48 3.4.1 The Modified Five-Parameter Rheological Model under Ramp Loading This section introduces the mathematical derivation for the deformation of the five-parameter rheological model under the ramp loading protocol, shown in Figure 3.11, where 𝐾𝑎 is the ramp loading rate. Figure 3.11 Ramp loading protocol Starting from the model compliance, and based on Boltzmann’s superposition principle (Wang, 2010): 𝐽(𝑡) =𝜀(𝑡)𝜎𝑎=1𝑘0+(1 − 𝑒−𝑡/𝜏1)𝑘1+𝑡𝛽5𝜂0 ( 3.6 ) ∴ 𝜀(𝑡) = ∫ 𝐽(𝑡 − 𝑠)𝑑𝜎𝑑𝑠𝑑𝑠𝑡0= ∫[ 1𝑘0+(1 − 𝑒−(𝑡−𝑠)𝜏1 )𝑘1+(𝑡 − 𝑠)𝛽5𝜂0] ∙ 𝐾𝑎𝑑𝑠𝑡0 = 𝐾𝑎 [(1𝑘0+1𝑘1) 𝑡 −𝜏1𝑘1(1 − 𝑒−𝑡𝜏1) +𝑡𝛽5+1𝜂0(𝛽5 + 1)] =𝐾𝑎𝜎?̅?[(𝜎?̅?𝑘0+𝜎?̅?𝑘1) 𝑡 −𝜏1𝜎?̅?𝑘1(1 − 𝑒−𝑡𝜏1) +𝜎?̅?𝑡𝛽5+1𝜂0(𝛽5 + 1)] 49 =𝐾𝑎𝜎?̅?[(𝛽1 + 𝛽2)𝑡 −𝛽2𝛽3(1 − 𝑒−𝑡𝛽3) +𝛽4𝑡𝛽5+1(𝛽5 + 1)] ( 3.7 ) where: 𝛽1 =𝜎?̅?𝑘0, 𝛽2 =𝜎?̅?𝑘1, 𝛽3 =1𝜏1=𝑘1𝜂1, 𝛽4 =𝜎?̅?𝜂0, 0 < 𝛽5 < 1. Then the strain rate under ramp loading shown in Figure 3.12 is: 𝜀̇(𝑡) = 𝐾𝑎 ∙ 𝐽(𝑡) = 𝐾𝑎 [1𝑘0+(1 − 𝑒−𝑡/𝜏1)𝑘1+𝑡𝛽5𝜂0]=𝐾𝑎𝜎?̅?∙ [𝛽1 + 𝛽2(1 − 𝑒−𝑡𝛽3) + 𝛽4𝑡𝛽5] ( 3.8 ) Figure 3.12 Strain rate under the ramp loading protocol 50 3.4.2 The Modified Five-Parameter Rheological Model under Trapezoidal Fatigue Loading Figure 3.13 Trapezoidal fatigue loading protocol In Figure 3.13, the stress history of the trapezoidal fatigue loading protocol is: 𝜎(𝑡) = {𝐾𝑎(𝑡 − 𝑡𝑠𝐼), 𝑡𝑠𝐼 ≤ 𝑡 ≤ 𝑡𝑠𝐼 + 𝑡𝑚𝜎𝑚𝑎𝑥, 𝑡𝑠𝐼 + 𝑡𝑚 ≤ 𝑡 ≤ 𝑡𝑒𝐼 − 𝑡𝑚−𝐾𝑎(𝑡 − 𝑡𝑒𝐼), 𝑡𝑒𝐼 − 𝑡𝑚 ≤ 𝑡 ≤ 𝑡𝑒𝐼 ( 3.9 ) where, 𝑡𝑠𝐼 and 𝑡𝑒𝐼 are the starting and ending time of the I-th cycle respectively; 𝐾𝑎 is the loading rate; 𝑡𝑚 is the duration of uploading segment, which is equal to the duration of unloading segment; and, 𝜎𝑚𝑎𝑥 is the constant applied load stress. Starting from the model compliance: 𝐽(𝑡) =𝜀(𝑡)𝜎𝑎=1𝑘0+(1 − 𝑒−𝑡/𝜏1)𝑘1+𝑡𝛽5𝜂0 When time step is in the first case: 𝑡𝑠𝐼 ≤ 𝑡 ≤ 𝑡𝑠𝐼 + 𝑡𝑚 51 the strain 𝜀(𝑡) at time step 𝑡 is: 𝜀(𝑡) = 𝜀(𝑡𝑠𝐼) + ∫ 𝐽(𝑡 − 𝑠)𝑑𝜎𝑑𝑠𝑑𝑠𝑡𝑡𝑠𝐼 = 𝜀(𝑡𝑠𝐼) + ∫ [1𝑘0+(1 − 𝑒−(𝑡−𝑠)/𝜏1)𝑘1+(𝑡 − 𝑠)𝛽5𝜂0] ∙ 𝐾𝑎𝑑𝑠𝑡𝑡𝑠𝐼 = 𝜀(𝑡𝑠𝐼) + 𝐾𝑎 [(1𝑘0+1𝑘1) (𝑡 − 𝑡𝑠𝐼) −𝜏1𝑘1(1 − 𝑒−(𝑡−𝑡𝑠𝐼)/𝜏1) +(𝑡 − 𝑡𝑠𝐼)𝛽5+1𝜂0(𝛽5 + 1)] = 𝜀(𝑡𝑠𝐼) +𝐾𝑎𝜎?̅?[(𝜎?̅?𝑘0+𝜎?̅?𝑘1) (𝑡 − 𝑡𝑠𝐼) −𝜏1𝜎?̅?𝑘1(1 − 𝑒−(𝑡−𝑡𝑠𝐼)𝜏1 ) +𝜎?̅?(𝑡 − 𝑡𝑠𝐼)𝛽5+1𝜂0(𝛽5 + 1)] = 𝜀(𝑡𝑠𝐼) +𝐾𝑎𝜎?̅?[(𝛽1 + 𝛽2)(𝑡 − 𝑡𝑠𝐼) −𝛽2𝛽3(1 − 𝑒−(𝑡−𝑡𝑠𝐼)𝛽3) +𝛽4(𝑡 − 𝑡𝑠𝐼)𝛽5+1(𝛽5 + 1)] ( 3.10 ) Hence, the strain value at 𝑡 = 𝑡𝑠𝐼 + 𝑡𝑚 yields: 𝜀(𝑡𝑠𝐼 + 𝑡𝑚) = 𝜀(𝑡𝑠𝐼) +𝑡𝑚𝐾𝑎𝜎?̅?[(𝜎?̅?𝑘0+𝜎?̅?𝑘1) −𝜏1𝜎?̅?𝑡𝑚𝑘1(1 − 𝑒−𝑡𝑚𝜏1 ) +𝜎?̅?(𝑡𝑚)𝛽5𝜂0(𝛽5 + 1)] = 𝜀(𝑡𝑠𝐼) +𝜎𝑚𝑎𝑥𝜎?̅?[(𝛽1 + 𝛽2) −𝛽2𝑡𝑚𝛽3(1 − 𝑒−𝑡𝑚𝛽3) +𝛽4(𝑡𝑚)𝛽5(𝛽5 + 1)] ( 3.11 ) The above results show that, mathematically, the strain increase in the uploading segment in each cycle is one constant value in the trapezoidal loading protocol. Considering the second case in the stress history of the trapezoidal protocol: 𝑡𝑠𝐼 + 𝑡𝑚 ≤ 𝑡 ≤ 𝑡𝑒𝐼 − 𝑡𝑚 the strain, if under the constant load, is given by (Wang, 2010): 52 𝜀(𝑡)𝐶𝑜𝑛𝑠 = 𝜎𝑚𝑎𝑥 [1𝑘0+(1 − 𝑒−𝑡/𝜏1)𝑘1+𝑡𝛽5𝜂0] =𝜎𝑚𝑎𝑥𝜎?̅?[𝛽1 + 𝛽2(1 − 𝑒−𝑡𝛽3) + 𝛽4𝑡𝛽5] ( 3.12 ) where 𝜎𝑚𝑎𝑥 = 𝐾𝑎𝑡𝑚. Therefore, in the second case, the strain 𝜀(𝑡) at time step 𝑡 should be the superposition of the first strain value at t = 𝑡𝑠𝐼 + 𝑡𝑚 and the second strain value caused by the constant load 𝜎(𝑡) =𝜎𝑚𝑎𝑥𝐻(𝑡 − (𝑡𝑠𝐼 + 𝑡𝑚)). The first strain value is calculated from Equation ( 3.11 ). The second strain value is obtained using Equation ( 3.12 ) minus the effect of the instantaneous increase of the stress from zero to 𝜎𝑚𝑎𝑥 at t = 𝑡𝑠𝐼 + 𝑡𝑚 (Wang, 2010). Therefore, the strain 𝜀(𝑡) at time step 𝑡 is: 𝜀(𝑡) = 𝜀(𝑡𝑠𝐼 + 𝑡𝑚) + 𝜀(𝑡 − (𝑡𝑠𝐼 + 𝑡𝑚))𝐶𝑜𝑛𝑠−𝜎𝑚𝑎𝑥𝜎?̅?∙ 𝛽1 = 𝜀(𝑡𝑠𝐼) +𝜎𝑚𝑎𝑥𝜎?̅?[(𝛽1 + 𝛽2) −𝛽2𝑡𝑚𝛽3(1 − 𝑒−𝑡𝑚𝛽3) +𝛽4(𝑡𝑚)𝛽5(𝛽5 + 1)]+𝜎𝑚𝑎𝑥𝜎?̅?[𝛽2(1 − 𝑒−(𝑡−(𝑡𝑠𝐼+𝑡𝑚))𝛽3) + 𝛽4(𝑡 − (𝑡𝑠𝐼 + 𝑡𝑚))𝛽5] ( 3.13 ) Hence, the strain value at 𝑡 = 𝑡𝑒𝐼 − 𝑡𝑚 is: 𝜀(𝑡𝑒𝐼 − 𝑡𝑚) = 𝜀(𝑡𝑠𝐼) +𝜎𝑚𝑎𝑥𝜎?̅?[(𝛽1 + 𝛽2) −𝛽2𝑡𝑚𝛽3(1 − 𝑒−𝑡𝑚𝛽3) +𝛽4(𝑡𝑚)𝛽5(𝛽5 + 1)+ 𝛽2(1 − 𝑒−(𝑡𝑒𝐼−𝑡𝑠𝐼−2𝑡𝑚)𝛽3) + 𝛽4(𝑡𝑒𝐼 − 𝑡𝑠𝐼 − 2𝑡𝑚)𝛽5] ( 3.14 ) Considering the third case in the stress history of the trapezoidal protocol: 53 𝑡𝑒𝐼 − 𝑡𝑚 ≤ 𝑡 ≤ 𝑡𝑒𝐼 the strain 𝜀(𝑡) at time step 𝑡 is: 𝜀(𝑡) = 𝜀(𝑡𝑠𝐼) + ∫ 𝐽(𝑡 − 𝑠)𝑑𝜎𝑑𝑠𝑑𝑠𝑡𝑠𝐼+𝑡𝑚𝑡𝑠𝐼+ 𝜀(𝑡𝑒𝐼 − 𝑡𝑠𝐼 − 2𝑡𝑚)𝐶𝑜𝑛𝑠 −𝜎𝑚𝑎𝑥𝜎?̅?∙ 𝛽1+∫ 𝐽(𝑡 − 𝑠)𝑑𝜎𝑑𝑠𝑑𝑠𝑡𝑡𝑒𝐼−𝑡𝑚 = 𝜀(𝑡𝑠𝐼) + 𝐾𝑎 [(1𝑘0+1𝑘1) 𝑡𝑚 −𝜏1𝑘1[𝑒−(𝑡−𝑠)/𝜏1]𝑡𝑠𝐼𝑡𝑠𝐼+𝑡𝑚−[(𝑡 − 𝑠)𝛽5+1]𝑡𝑠𝐼𝑡𝑠𝐼+𝑡𝑚𝜂0(𝛽5 + 1)]+𝜎𝑚𝑎𝑥𝜎?̅?[𝛽2(1 − 𝑒−(𝑡𝑒𝐼−𝑡𝑠𝐼−2𝑡𝑚)𝛽3) + 𝛽4(𝑡𝑒𝐼 − 𝑡𝑠𝐼 − 2𝑡𝑚)𝛽5]− 𝐾𝑎 [(1𝑘0+1𝑘1) (𝑡−𝑡𝑒𝐼 + 𝑡𝑚) −𝜏1𝑘1[𝑒−(𝑡−𝑠)/𝜏1]𝑡𝑒𝐼−𝑡𝑚𝑡−[(𝑡 − 𝑠)𝛽5+1]𝑡𝑒𝐼−𝑡𝑚𝑡𝜂0(𝛽5 + 1)] ( 3.15 ) Then, the strain value at 𝑡 = 𝑡𝑒𝐼 is: 𝜀(𝑡𝑒𝐼) = 𝜀(𝑡𝑠𝐼)+ 𝐾𝑎 [−𝜏1𝑘1(𝑒−(𝑡𝑒𝐼−𝑡𝑠𝐼−𝑡𝑚)𝜏1 − 𝑒−(𝑡𝑒𝐼−𝑡𝑠𝐼)𝜏1 )−1𝜂0(𝛽5 + 1)((𝑡𝑒𝐼 − 𝑡𝑠𝐼 − 𝑡𝑚)𝛽5+1 − (𝑡𝑒𝐼 − 𝑡𝑠𝐼)𝛽5+1)]− 𝐾𝑎 [−𝜏1𝑘1(1 − 𝑒−𝑡𝑚𝜏1 ) −1𝜂0(𝛽5 + 1)(−(𝑡𝑚)𝛽5+1)]+𝜎𝑚𝑎𝑥𝜎?̅?[𝛽2(1 − 𝑒−(𝑡𝑒𝐼−𝑡𝑠𝐼−2𝑡𝑚)𝛽3) + 𝛽4(𝑡𝑒𝐼 − 𝑡𝑠𝐼 − 2𝑡𝑚)𝛽5] ( 3.16 ) 54 ∴ 𝜀(𝑡𝑒𝐼) = 𝜀(𝑡𝑠𝐼)+ 𝐾𝑎 [−𝜏1𝑘1(𝑒−(𝑡𝑒𝐼−𝑡𝑠𝐼−𝑡𝑚)𝛽3 − 𝑒−(𝑡𝑒𝐼−𝑡𝑠𝐼)𝛽3 + 𝑒−𝑡𝑚𝛽3 − 1)−1𝜂0(𝛽5 + 1)((𝑡𝑒𝐼 − 𝑡𝑠𝐼 − 𝑡𝑚)𝛽5+1 − (𝑡𝑒𝐼 − 𝑡𝑠𝐼)𝛽5+1+(𝑡𝑚)𝛽5+1)]+𝜎𝑚𝑎𝑥𝜎?̅?[𝛽2(1 − 𝑒−(𝑡𝑒𝐼−𝑡𝑠𝐼−2𝑡𝑚)𝛽3) + 𝛽4(𝑡𝑒𝐼 − 𝑡𝑠𝐼 − 2𝑡𝑚)𝛽5] = 𝜀(𝑡𝑠𝐼) +𝜎𝑚𝑎𝑥𝜎?̅?[−𝛽2𝛽3𝑡𝑚(𝑒−(𝑡𝑒𝐼−𝑡𝑠𝐼−𝑡𝑚)𝛽3 − 𝑒−(𝑡𝑒𝐼−𝑡𝑠𝐼)𝛽3 + 𝑒−𝑡𝑚𝛽3 − 1)−𝛽4𝑡𝑚(𝛽5 + 1)((𝑡𝑒𝐼 − 𝑡𝑠𝐼 − 𝑡𝑚)𝛽5+1 − (𝑡𝑒𝐼 − 𝑡𝑠𝐼)𝛽5+1+(𝑡𝑚)𝛽5+1)]+𝜎𝑚𝑎𝑥𝜎?̅?[𝛽2(1 − 𝑒−(𝑡𝑒𝐼−𝑡𝑠𝐼−2𝑡𝑚)𝛽3) + 𝛽4(𝑡𝑒𝐼 − 𝑡𝑠𝐼 − 2𝑡𝑚)𝛽5] ( 3.17 ) Therefore, the strain increase in the I-th cycle is: ∆𝜀𝐼 = 𝜀(𝑡𝑒𝐼) − 𝜀(𝑡𝑠𝐼)=𝜎𝑚𝑎𝑥𝜎?̅?[−𝛽2𝛽3𝑡𝑚(𝑒−(𝑡𝑒𝐼−𝑡𝑠𝐼−𝑡𝑚)𝛽3 − 𝑒−(𝑡𝑒𝐼−𝑡𝑠𝐼)𝛽3 + 𝑒−𝑡𝑚𝛽3 − 1)−𝛽4𝑡𝑚(𝛽5 + 1)((𝑡𝑒𝐼 − 𝑡𝑠𝐼 − 𝑡𝑚)𝛽5+1 − (𝑡𝑒𝐼 − 𝑡𝑠𝐼)𝛽5+1+(𝑡𝑚)𝛽5+1)]+𝜎𝑚𝑎𝑥𝜎?̅?[𝛽2(1 − 𝑒−(𝑡𝑒𝐼−𝑡𝑠𝐼−2𝑡𝑚)𝛽3) + 𝛽4(𝑡𝑒𝐼 − 𝑡𝑠𝐼 − 2𝑡𝑚)𝛽5] ( 3.18 ) The above results show that the strain increase in each cycle is one constant value in the trapezoidal loading protocol, as illustrated in Figure 3.14. 55 Figure 3.14 Strain under the trapezoidal loading protocol 3.4.3 The Modified Five-Parameter Rheological Model under Ramp and Constant Loading Protocol Figure 3.15 Ramp and constant loading protocol As shown in Figure 3.15, the ramp and constant loading protocol is the combination of the ramp load and the constant load. This loading protocol can simulate the dead load history. The stress history of the ramp and constant loading protocol is: 56 𝜎(𝑡) = {𝐾𝑎𝑡, 0 ≤ 𝑡 ≤ 𝑡𝑚𝜎𝑚𝑎𝑥, 𝑡 > 𝑡𝑚 ( 3.19 ) where, 𝐾𝑎 is the loading rate, and 𝑡𝑚 is the duration of uploading segment. 𝜎𝑚𝑎𝑥 = 𝐾𝑎𝑡𝑚 is the constant applied load stress. Starting from the model compliance: 𝐽(𝑡) =𝜀(𝑡)𝜎𝑎=1𝑘0+(1 − 𝑒−𝑡/𝜏1)𝑘1+𝑡𝛽5𝜂0 When: 0 ≤ 𝑡 ≤ 𝑡𝑚 the strain 𝜀(𝑡) at time step 𝑡 is: 𝜀(𝑡) = ∫ 𝐽(𝑡 − 𝑠)𝑑𝜎𝑑𝑠𝑑𝑠𝑡0 =𝐾𝑎𝜎?̅?[(𝛽1 + 𝛽2)𝑡 −𝛽2𝛽3(1 − 𝑒−𝑡𝛽3) +𝛽4𝑡𝛽5+1(𝛽5 + 1)] ( 3.20 ) Hence, the strain value at 𝑡 = 𝑡𝑚 yields: 𝜀(𝑡𝑚) =𝜎𝑚𝑎𝑥𝜎?̅?[(𝛽1 + 𝛽2) −𝛽2𝑡𝑚𝛽3(1 − 𝑒−𝑡𝑚𝛽3) +𝛽4(𝑡𝑚)𝛽5(𝛽5 + 1)] ( 3.21 ) Considering the second case in the stress history of the ramp and constant loading protocol, when: 𝑡 > 𝑡𝑚 the strain under constant load is given by Equation ( 3.12 ): 57 𝜀(𝑡)𝐶𝑜𝑛𝑠 = 𝜎𝑚𝑎𝑥 [1𝑘0+(1 − 𝑒−𝑡/𝜏1)𝑘1+𝑡𝛽5𝜂0] =𝜎𝑚𝑎𝑥𝜎?̅?[𝛽1 + 𝛽2(1 − 𝑒−𝑡𝛽3) + 𝛽4𝑡𝛽5] where 𝜎𝑚𝑎𝑥 = 𝐾𝑎𝑡𝑚. Then, the strain 𝜀(𝑡) at time step 𝑡 is: 𝜀(𝑡) = 𝜀(𝑡𝑚) + 𝜀(𝑡 − 𝑡𝑚)𝐶𝑜𝑛𝑠 −𝜎𝑚𝑎𝑥𝜎?̅?∙ 𝛽1 = 𝜎𝑚𝑎𝑥𝜎?̅?[(𝛽1 + 𝛽2) −𝛽2𝑡𝑚𝛽3(1 − 𝑒−𝑡𝑚𝛽3) +𝛽4(𝑡𝑚)𝛽5(𝛽5 + 1)]+𝜎𝑚𝑎𝑥𝜎?̅?[𝛽2(1 − 𝑒−(𝑡−𝑡𝑚)𝛽3) + 𝛽4(𝑡 − 𝑡𝑚)𝛽5] ( 3.22 ) 3.5 Damage Accumulation Model The stress-based damage accumulation model, i.e., the Foschi and Yao model in which the damage rate is a function of stress history and the previous accumulated damage state, is: {𝑑𝛼𝑑𝑡= 𝑎(𝜎(𝑡) − 𝜏0𝜎s)𝑏 + 𝑐(𝜎(𝑡) − 𝜏0𝜎s)𝑛𝛼, 𝜎(𝑡) > 𝜏0𝜎s𝑑𝛼𝑑𝑡= 0, 𝜎(𝑡) ≤ 𝜏0𝜎s ( 3.23 ) where 𝛼 is the damage state variable (𝛼 = 0 in an undamaged state and 𝛼 = 1 in a failure state); 𝑡 is the time; 𝜎(𝑡) is the applied shear stress history; 𝜎𝑠 is the short-term rolling shear strength; 𝜏0 is a ratio of the short-term strength 𝜎𝑠; thus, the product 𝜏0𝜎s is a threshold stress below which there will be no accumulation of damage; and, 𝑎, 𝑏, 𝑐 and 𝑛 are random model parameters. 58 3.5.1 Damage under Ramp Loading For the ramp loading case, as shown in Figure 3.11, when 𝜎(𝑡) = 𝐾𝑠𝑡, where 𝐾𝑠 is the constant loading rate, assuming there is damage accumulated only when 𝜎(𝑡) − 𝜏0𝜎s > 0, then one time point 𝑡0 is defined when 𝜎(𝑡0) = 𝜏0𝜎s. Another time point 𝑇 is defined when 𝛼(𝑇) = 1.0 meaning the specimen fails. By integrating the expression of the damage accumulation model from 𝑡 = 0.0 to 𝑡 = 𝑇 in Equation ( 3.23 ), the equation yields as follows: 𝛼𝑒{− ∫𝑐(𝜎(𝑡)−𝜏0𝜎𝑠)𝑛𝑑𝑡} |0𝑇= ∫ {𝑎(𝜎(𝑡) − 𝜏0𝜎𝑠)𝑏𝑒{− ∫ 𝑐(𝜎(𝑡)−𝜏0𝜎𝑠)𝑛𝑑𝑡}}𝑑𝑡𝑇0 𝛼𝑒{−𝑐𝐾𝑠(1+𝑛)[𝐾𝑠𝑡−𝜏0𝜎𝑠](1+𝑛)} |𝑡0𝑇= ∫ 𝑎[𝐾𝑠𝑡 − 𝜏0𝜎𝑠]𝑏𝑒{−𝑐𝐾𝑠(1+𝑛)[𝐾𝑠𝑡−𝜏0𝜎𝑠](1+𝑛)}𝑑𝑡𝑇𝑡0 ( 3.24 ) Foschi and Yao (1986) showed that 𝐾𝑠 is as a rule large compared to the model parameter 𝑐 and consequently, with 𝛼(𝑡0) = 0.0, Equation ( 3.24 ) yields: 1.0 ≅ ∫ 𝑎[𝐾𝑠𝑡 − 𝜏0𝜎𝑠]𝑏𝑑𝑡𝑇𝑡0=𝑎𝐾𝑠(1 + 𝑏)[𝜎𝑠 − 𝜏0𝜎𝑠](1+𝑏) Therefore, 𝑎 is related to the ramp rate, the short-term strength value and the other model parameters: 𝑎 ≅𝐾𝑠(1 + 𝑏)[𝜎𝑠 − 𝜏0𝜎𝑠](1+𝑏) ( 3.25 ) 59 3.5.2 Damage under Trapezoidal Fatigue Loading One question is how much damage will be accumulated under given cycles in the trapezoidal fatigue loading protocol, as shown in Figure 3.13. To answer this question, first, what needs to be investigated is how much damage is accumulated in the first two cycles. For the trapezoidal loading, when 𝜎(𝑡) = 𝐾𝑠𝑡, where 𝐾𝑠 is the uploading and unloading rate, assuming there is damage accumulation only when 𝜎(𝑡) − 𝜏0𝜎s > 0, the investigation of damage accumulation in the first two trapezoidal cycles is divided into four parts, i.e., damage accumulation in 1) the uploading segment in the first cycle, 2) the constant loading segment in the first cycle, 3) the unloading segment in the first cycle, and 4) the second cycle. These four parts will provide mathematical foundation for the damage accumulation research in the stress-based model under the trapezoidal fatigue loading protocol. 3.5.2.1 Segment One: Uploading in the First Cycle One time point 𝑡0 is defined when 𝜎(𝑡0) = 𝜏0𝜎s, and another time point 𝑡1 is defined when 𝜎(𝑡1) = 𝐾s𝑡1 = 𝜎𝑐, where 𝜎𝑐 is the peak stress in any cycle. By integrating the expression of the damage accumulation model from 𝑡 = 0.0 to 𝑡 = 𝑡1, the equation is shown as follows: 𝛼𝑒{−𝑐𝐾𝑠(1+𝑛)[𝐾𝑠𝑡−𝜏0𝜎𝑠](1+𝑛)} |𝑡0𝑡1= ∫ 𝑎[𝐾𝑠𝑡 − 𝜏0𝜎𝑠]𝑏𝑒{−𝑐𝐾𝑠(1+𝑛)[𝐾𝑠𝑡−𝜏0𝜎𝑠](1+𝑛)}𝑑𝑡𝑡1𝑡0 ( 3.26 ) 60 𝐾𝑠 is large compared to the model parameter 𝑐 (Foschi and Yao, 1986), and with 𝛼(𝑡0) = 0.0 Equation ( 3.26 ) yields: 𝛼(𝑡1) ≅ ∫ 𝑎[𝐾𝑠𝑡 − 𝜏0𝜎𝑠]𝑏𝑑𝑡𝑡1𝑡0=𝑎𝐾𝑠(1 + 𝑏)[𝐾𝑠𝑡1 − 𝜏0𝜎𝑠](1+𝑏) with 𝛼(𝑡0) = 0.0 and according to Equation ( 3.25 ): ∴ 𝛼(𝑡1) ≅ [𝜎𝑐 − 𝜏0𝜎𝑠𝜎𝑠 − 𝜏0𝜎𝑠](1+𝑏) ( 3.27 ) 3.5.2.2 Segment Two: Constant Loading in the First Cycle The stress remains one constant level 𝜎(𝑡) = 𝜎𝑐 during the time segment from 𝑡 = 𝑡1 to 𝑡 = 𝑡2, where 𝑡2 is the end time point for the constant loading segment. By integrating the expression of the damage accumulation model from 𝑡 = 𝑡1 to 𝑡 = 𝑡2: 𝛼𝑒{−𝑐[𝜎𝑐−𝜏0𝜎𝑠]𝑛𝑡} |𝑡1𝑡2= ∫ 𝑎[𝜎𝑐 − 𝜏0𝜎𝑠]𝑏𝑒{−𝑐[𝜎𝑐−𝜏0𝜎𝑠]𝑛𝑡}𝑑𝑡𝑡2𝑡1 𝛼(𝑡2) = 𝛼(𝑡1)𝑒{𝑐[𝜎𝑐−𝜏0𝜎𝑠]𝑏∆𝑡} +𝑎𝑐[𝜎𝑐 − 𝜏0𝜎𝑠](𝑏−𝑛) [𝑒{𝑐[𝜎𝑐−𝜏0𝜎𝑠]𝑏∆𝑡} − 1] ( 3.28 ) where ∆𝑡 = 𝑡2 − 𝑡1, then according to Equation ( 3.25 ): ∴ 𝛼(𝑡2) = 𝛼(𝑡1)𝑒{𝑐[𝜎𝑐−𝜏0𝜎𝑠]𝑏∆𝑡} +𝐾𝑠(1 + 𝑏)𝑐 [𝜎𝑐 − 𝜏0𝜎𝑠](𝑏−𝑛)[𝜎𝑠 − 𝜏0𝜎𝑠](1+𝑏)[𝑒{𝑐[𝜎𝑐−𝜏0𝜎𝑠]𝑏∆𝑡} − 1] ( 3.29 ) 61 3.5.2.3 Segment Three: Unloading in the First Cycle One time point 𝑡3 is defined when 𝜎(𝑡3) = 𝜏0𝜎s in the unloading segment, where the unloading rate is constant 𝐾𝑠. Therefore, the unloading history is 𝜎(𝑡) = 𝜎𝑐 − 𝐾𝑠(𝑡 − 𝑡2). Then, the damage accumulation in this segment is: 𝛼𝑒{𝑐𝐾𝑠(1+𝑛)[𝜎𝑐−𝐾𝑠(𝑡−𝑡2)−𝜏0𝜎𝑠](1+𝑛)} |𝑡2𝑡3= ∫ 𝑎[𝜎𝑐 − 𝐾𝑠(𝑡 − 𝑡2) − 𝜏0𝜎𝑠]𝑏𝑒{𝑐𝐾𝑠(1+𝑛)[𝜎𝑐−𝐾𝑠(𝑡−𝑡2)−𝜏0𝜎𝑠](1+𝑛)}𝑑𝑡𝑡3𝑡2 𝐾𝑠 is large compared to the model parameter 𝑐 (Foschi and Yao, 1986), and the damage accumulated in one intact trapezoidal cycle (i.e., 𝛼1) is: 𝛼1 = 𝛼(𝑡3) = 𝛼(𝑡2) + ∫ 𝑎[𝜎𝑐 − 𝐾𝑠(𝑡 − 𝑡2) − 𝜏0𝜎𝑠]𝑏𝑑𝑡𝑡3𝑡2= 𝛼(𝑡2) +𝑎𝐾𝑠(1 + 𝑏)[𝜎𝑐 − 𝜏0𝜎𝑠](1+𝑏) ( 3.30 ) according to Equation ( 3.25 ), Equation ( 3.27 ) and Equation ( 3.29 ), 𝛼1 is expressed as follows: 𝛼1 = [𝜎𝑐 − 𝜏0𝜎𝑠𝜎𝑠 − 𝜏0𝜎𝑠](1+𝑏)(1 + 𝑒{𝑐[𝜎𝑐−𝜏0𝜎𝑠]𝑛∆𝑡})+𝐾𝑠(1 + 𝑏)𝑐 [𝜎𝑐 − 𝜏0𝜎𝑠](𝑏−𝑛)[𝜎𝑠 − 𝜏0𝜎𝑠](1+𝑏)[𝑒{𝑐[𝜎𝑐−𝜏0𝜎𝑠]𝑛∆𝑡} − 1] ( 3.31 ) 62 3.5.2.4 Segment Four: Damage Accumulated in the Second Cycle As before, based on the same calculation process in Sections 3.5.2.1 to 3.5.2.3, the damage accumulated in the second cycle under cyclic loading (i.e., 𝛼2) is: 𝛼2 = [𝜎𝑐 − 𝜏0𝜎𝑠𝜎𝑠 − 𝜏0𝜎𝑠](1+𝑏)(1 + 𝑒{𝑐[𝜎𝑐−𝜏0𝜎𝑠]𝑛∆𝑡})2+𝐾𝑠(1 + 𝑏)𝑐 [𝜎𝑐 − 𝜏0𝜎𝑠](𝑏−𝑛)[𝜎𝑠 − 𝜏0𝜎𝑠](1+𝑏)[𝑒{2𝑐[𝜎𝑐−𝜏0𝜎𝑠]𝑛∆𝑡} − 1] ( 3.32 ) 3.5.2.5 Determination of the Number of Cycles to Failure Since the trapezoidal fatigue loading test includes the identical loading shape in different cycles, the damage after the I-th cycle, defined as 𝛼𝐼, holds the relationship as: 𝛼𝐼 = 𝛼𝐼−1𝐾0(𝐼) + 𝐾1(𝐼) ( 3.33 ) where 𝐾0(𝐼) and 𝐾1(𝐼) remain constants as 𝐾0 and 𝐾1. Considering 𝛼0 = 0, then: 𝐾0 =𝛼2𝐾1− 1 = 𝑒{𝑐[𝜎𝑐−𝜏0𝜎𝑠]𝑛∆𝑡} ( 3.34 ) 𝐾1 = 𝛼1 ( 3.35 ) If the failure of the specimen occurs after 𝑁 + 1 cycles of loading, with 𝛼1+𝑁 = 1.0 and Equation ( 3.33 ), the expression is shown as follows: 63 𝛼1+𝑁 = 𝛼𝑁𝐾0(1 + 𝑁) + 𝐾1(1 + 𝑁) = 𝛼0𝐾0𝑁 + 𝐾1(𝐾0𝑁−1 + 𝐾0𝑁−2+. . . +𝐾0 + 1) ∴ 1.0 = 𝐾1 (1 − 𝐾0𝑁1 − 𝐾0) Therefore, the number of cycles to failure (i.e., 𝑁𝑓) will be determined by: 𝑁𝑓 =log(𝐾1 + 𝐾0 − 1𝐾1)log(𝐾0)+ 1 ( 3.36 ) The damage accumulation model in Equation ( 3.23 ) is time based. Since 𝑁𝑓 was expressed as a positive integer, the exact time to failure within the cycle of failure was not considered. However, as the 𝑁𝑓 increases, this error becomes trivial. For example, when 𝑁𝑓 = 10, the error is close to 1/10=10%; when 𝑁𝑓 = 100, this error is only less than 1/100=1%. The 𝑁𝑓 calculation at the time basis is further explained and discussed in Section 6.3.2. 3.5.3 Damage under Ramp and Constant Loading Protocol In the ramp and constant loading test, as shown in Figure 3.15: 𝜎(𝑡) = {𝐾𝑎𝑡, 0 ≤ 𝑡 ≤ 𝑡𝑚𝜎𝑚𝑎𝑥, 𝑡 > 𝑡𝑚 where, 𝐾𝑎 is the loading rate, and 𝑡𝑚 is the duration of uploading segment. 𝜎𝑚𝑎𝑥 = 𝐾𝑎𝑡𝑚 is the constant applied load stress. 64 According to the stress-based damage accumulation theory under the ramp loading protocol as introduced before, when 0 ≤ 𝑡 ≤ 𝑡𝑚, the damage accumulated at time step 𝑡 = 𝑡𝑚 is: 𝛼(𝑡𝑚) ≅ [𝜎𝑚𝑎𝑥 − 𝜏0𝜎𝑠𝜎𝑠 − 𝜏0𝜎𝑠](1+𝑏) When 𝑡 > 𝑡𝑚, the time to failure 𝑇𝑓 under the ramp and constant loading protocol is expressed as follows (Foschi, 1989): 𝑇𝑓 = 𝑡𝑚 +1c(𝜎𝑚𝑎𝑥 − 𝜏0𝜎𝑠)n× ln [𝑐 + 𝑎(𝜎𝑚𝑎𝑥 − 𝜏0𝜎𝑠)b−n𝛼(𝑡𝑚)𝑐 + 𝑎(𝜎𝑚𝑎𝑥 − 𝜏0𝜎𝑠)b−n] ( 3.37 ) However, when the applied maximum stress exceeds the short-term capacity (i.e., 𝜎𝑚𝑎𝑥 > 𝜎𝑠), the time to failure 𝑇𝑓 will be: 𝑇𝑓 = 𝜎𝑠/𝐾𝑎 ( 3.38 ) When the applied maximum stress does not exceed the threshold (i.e., 𝜎𝑚𝑎𝑥 < 𝜏0𝜎𝑠), there will be no damage accumulated. 3.6 Conclusion Not enough research has been documented to understand the time-dependent performance of MPB (mountain pine beetle) CLT products. Rolling shear behaviour is certainly playing an important role in the structural performance of CLT panels, especially under out-of-plane 65 bending loads. Therefore, it is necessary to perform duration-of-load tests on CLT, with consideration of the rolling shear behaviour. To understand the structural performance of the CLT beam specimens, a finite element model with customized Fortran subroutines was chosen for the simulation process. This model included the modified five-parameter rheological modeling theory. In order to investigate the duration-of-load effect from a mathematical perspective, a stress-based damage accumulation model was adopted; and, the mathematical description for this model under different loading protocols was derived. All of the above-mentioned model theories have been applied in the analysis and evaluation of the test results from the duration-of-load testing program on MPB CLT, which will be introduced in Chapter 4. 66 Chapter 4: Testing of Duration-of-Load Behaviour of CLT 4.1 Summary This chapter introduces the results from a duration-of-load testing program on MPB (mountain pine beetle) CLT (cross laminated timber) with different cross-sectional layups. These CLT panels were manufactured in collaboration between the Timber Engineering and Applied Mechanics Laboratory at the University of British Columbia (UBC) and a CLT manufacturing factory in the province of British Columbia. CLT beam specimens with small span-to-depth ratios were prepared and divided into different groups using the pair sampling method. Ramp loading tests and trapezoidal fatigue tests (damage accumulation tests) with concentrated loading protocol were performed on the CLT beams. Specifically, in each CLT configuration (i.e., three-layer or five-layer CLT products), two sets of the trapezoidal fatigue test results were obtained to evaluate the relationship between the load ratio and the time to failure; i.e., the number of cycles to rolling shear failure. CLT tube specimens were manufactured for the torque loading tests, in order to investigate the rolling shear strength in timber material. 4.2 Introduction Considering that CLT is one of the available production technologies to absorb a large volume of MPB-attacked lumber, further effort is required to evaluate the duration of load for MPB CLT 67 products. In this chapter, after MPB CLT panels were prepared and CLT beam specimens were sampled in the laboratory, duration-of-load experiments were carried out. The experimental data was collected and used to calibrate and verify the developed damage accumulation model for simulating duration-of-load behaviour, as described in Chapter 6. CLT tube torque tests were also carried out for further investigation of the size effect on the rolling shear strength. The evaluation of the size effect is introduced in Chapter 5. 4.3 Material and Methods 4.3.1 CLT Specimen Two categories of the non-edge glued CLT plates laminated with polyurethane adhesive, i.e., five-layer Spruce-Pine-Fir (SPF5) plates and three-layer Spruce-Pine-Fir (SPF3) plates, are shown in Figure 4.1 and Figure 4.2. Within each category, one rigid mechanical clamping pressure level (i.e., 0.4 MPa) was applied for manufacturing. 38 mm × 140 mm No.2 grade dimension lumber was used for the laminations in the top and bottom layers of three-layer and five-layer CLT, and was also used in the middle core layer of five-layer CLT, whereas 38 mm × 140 mm stud grade lumber was used for the other cross lamination layers in three-layer and five-layer CLT. The wood boards were cut and planed into the cross section of 34 mm × 140 mm and 19 mm ×140 mm for the cross layers of three-layer and five-layer CLT panels, respectively. For convenience, a five-layer Spruce-Pine-Fir plate is simply denoted as a SPF5 plate. Considering the 0.4 MPa pressure applied, this plate is further denoted as SPF5-0.4. Similarly, a three-layer 68 Spruce-Pine-Fir plate pressed under 0.4 MPa is denoted as a SPF3-0.4 plate. Table 4.1 below shows the configurations of these CLT plates including the board grades, the thickness of laminations, the width of laminations and the plate dimensions (Yawalata and Lam, 2011; Chen, 2011). Figure 4.1 Manufacturing of three-layer and five-layer CLT panels Figure 4.2 Two categories of CLT plates 69 Table 4.1 Grades and configurations of the CLT plates CLT category Lamination grade Lamination thickness (mm) Lamination board width (mm) Plate dimension L×W×H (mm) SPF5-0.4 No.2/Stud/No.2/Stud/No.2 34/19/34/19/34 140 3658×1219×140 SPF3-0.4 No.2/Stud/No.2 34/34/34 140 3658×1219×102 As shown in Figure 4.3, the modulus of elasticity (MOE) of the boards was measured by transverse vibration tests following the test standard (ASTM D6874-03, 2009). Table 4.2 gives the vibration test results for different grades (Chen, 2011); the mean values of measured MOE were used for finite element modeling purpose, as given in Table 3.2. Figure 4.3 Setup of the MOE vibration test 70 Table 4.2 Summary of the vibration MOE of wood boards Species Grade MOE Sample size n Mean (GPa) COV SPF No. 2 or better 11.43 16.4% 256 Stud 10.66 18.5% 280 Both five-layer and three-layer CLT plates were investigated; in each type there were three plate replicates selected for sampling, such as panel SPF5-0.4-1, SPF5-0.4-2 and SPF5-0.4-3. As shown in Figure 4.4, forty short-span CLT beam specimens were cut from each panel in such a manner that beams had face layers parallel to the span direction. The pair sampling method shown in Figure 4.4 was adopted, meaning the corresponding specimens for the trapezoidal fatigue loading test in each group were cut from the same CLT panel where the ramp loading test specimens were cut. Therefore, in one plate replicate, twenty specimens out of those forty were selected in a staggered way for the short-term ramp loading test, and this side-by-side sampling method assured random matching formula. Then, the other twenty were prepared for the trapezoidal fatigue loading test. In total, in each CLT configuration, there were sixty specimens for the short-term ramp loading test and sixty specimens for the trapezoidal fatigue test. Matrices about CLT beam specimens are shown in Table 4.3 and Table 4.4. The test span-to-depth ratio is 6.0. The length of the prepared beam specimen was 76.2mm (3 inches) longer than test span considering the width of the support in the test setup. 71 Figure 4.4 Pair sampling method Table 4.3 Notation of CLT specimens Species No. of layers Press pressure (MPa) Notation of the specimen plate replicates Length(mm) Depth(mm) Width(mm) SPF 5 0.4 SPF5-0.4-1 / -2 / -3 916.2 140 50.8 SPF 3 0.4 SPF3-0.4-1 / -2 / -3 688.2 102 50.8 Table 4.4 Number of the CLT specimens Notation of specimen groups Test span(mm) Depth(mm) No. for ramp loading No. for trapezoidal loading SPF5-0.4 840 140 60 60 SPF3-0.4 612 102 60 63 72 4.3.2 Ramp Loading Test The short-term ramp loading tests were displacement controlled until specimen’s failure. The time-dependent load value and deformation were recorded for each specimen. The sixty specimens in each category of CLT, designed in a span-to-depth ratio of 6.0 to encourage the rolling shear behaviour in the cross layers, are shown in Figure 4.5 and Figure 4.6. The rolling shear failure load was recorded when the first rolling shear crack occurred in the cross layers. Figure 4.5 Five-layer CLT beam specimen Figure 4.6 Three-layer CLT beam specimen 73 In the ramp loading test, the loading speed was 2 mm/min (0.08 inch/min) for five-layer CLT specimens and 1.5 mm/min (0.06 inch/min) for three-layer CLT specimens. The total time to failure for one specimen in the ramp loading test was about 5 min, and the centre-point load was applied to the top of the CLT beam, as shown in Figure 4.7. The displacement in the middle of the beam span was also recorded. The loading rate for the short-term ramp tests was a deformation rate, so the load or stress loading rate changed from specimen to specimen. Figure 4.7 Ramp loading test setup 4.3.3 Torque Loading Test To investigate the rolling shear failure, CLT tube specimens were prepared for the torque test, as shown in Figure 4.8. 74 Figure 4.8 CLT torque tube specimens The torque tube specimen included three layers of laminated wood. First, the sampled CLT cubic block was prepared. Then, the middle cross layer was cut and shaped into cylinder volume with an outer diameter of 52 millimeters by Computer Numerical Control Machine. The whole cylinder middle layer came from one piece of cross laminated board, and this middle layer was continuous with no gaps guaranteed by the quality control process. Although some surfaces of the middle layer appeared to be charred as shown in Figure 4.8, the material strength of specimens was assumed to be unaffected. Then, one hole with an inner diameter of 19.0 millimeters was drilled throughout the torque specimen crossing all the three layers of laminated wood. Table 4.5 shows the detailed information for the CLT torque tube specimens. 75 Table 4.5 Information for the torque tube specimens CLT category Lamination grade Number of specimen Lamination thickness (mm) Upper and bottom layers Middle cylinder layer Length Width (mm) Outer diameter (mm) Inner diameter (mm) SPF5-0.4 No.2/Stud /No.2 33 34/19/34 100×100 52 19.0 SPF3-0.4 No.2/Stud /No.2 33 34/34/34 100×100 52 19.0 After one torque tube specimen was fixed on the loading machine platform, the upper layer of the specimen was locked on one steel frame with lever welded, as shown in Figure 4.9. Then, vertical load was applied to the other end of the lever. The jack kept loading vertically at a constant loading rate until the specimens failed. The vertical load on the lever transformed into the torque load on the specimen, leaving the middle cylinder layer in shear status. The loading test adopted the displacement control method, and the speed was about 4.8 mm/min (0.2 inch/min). The total time to failure for one tube specimen was about 4 min. The specimen would fail when the shear stress from the torque loading exceeded the rolling shear strength in the timber material, as discussed in Section 4.4.3. In the test, the loading history and the failure modes for the specimens were recorded. 76 Figure 4.9 Torque test setup (with Y direction as the direction of wood fibre in the middle cross layer) 4.3.4 Trapezoidal Fatigue Loading Test To limit the influence of environment variables on the damage accumulation process, low cycle trapezoidal fatigue loading protocol was adopted since the duration of this damage accumulation fatigue test is relatively short (Norlin, 1997). For the trapezoidal fatigue loading test with the load control method, as shown in Figure 4.10 and Figure 4.11, the loading and recording device setup on CLT beam specimens was the same as that in the short-term ramp loading test. The jack applied the load from zero to a target value, i.e., the load level in the plateau loading part. Then, the jack held the load which equaled this target load value in the plateau loading process. Finally, the jack released the load level to zero. The loading proceeded periodically (Norlin, 1997). With regard to the constant load level in the plateau part, the load value applied was chosen as the 25th percentile of the rolling shear capacity obtained in the short-term ramp loading test from Section 4.3.2. The uploading rates, which are equal to the unloading rates, were 37.5 kN/min for the five-layer CLT specimens and 27.0 kN/min for the three-layer CLT specimens. The short-X Y Z 77 term ramp tests adopted the displacement control method (with a constant deformation rate), so the stress rate was different from one specimen to another, while the loading rate for the trapezoidal test specimens is given in terms of stress. The load was cyclically applied until the first rolling shear crack was observed with careful examination in the cross layer, defined as the rolling shear failure. In the same time, the number of cycles to the rolling shear failure, i.e., the time to failure, was recorded. The displacement in the middle of the beam span was also recorded. Figure 4.10 Load control in the trapezoidal fatigue test Figure 4.11 Trapezoidal fatigue test of CLT specimens 78 Two types of the trapezoidal fatigue loading tests were performed, and these two tests had different load duration in the plateau part when the constant loading was applied. The first one, i.e., the trapezoidal short plateau test, includes the constant loading part with a duration of 0.5tm, where tm is the duration in the uploading segment shown in Figure 3.13 and Figure 4.10. The second type has a longer plateau part, which is equal to 2.0tm. Table 4.6 gives the sample size for these two different CLT trapezoidal fatigue loading tests. Table 4.6 Number of CLT specimens Notation of specimen groups Span(mm) Depth(mm) No. for short plateau test No. for long plateau test SPF5-0.4 840 140 30 30 SPF3-0.4 612 102 30 33 4.4 Experimental Results 4.4.1 Ramp Loading Test Results In the ramp loading tests, results showed that rolling shear failure was the major failure mode. The failure process typically started with rolling shear cracks in the cross layers at an inclined angle, followed by horizontal cracks in the timber material near the glue lines between the different layers. Finally, the capacity-reduced specimens experienced tension parallel to the grain failure in the bottom layers. The failure mode is shown in Figure 4.12. 79 Fifty-five specimens in the SPF5-0.4 group showed rolling shear failure, as did fifty-nine specimens in the three-layer CLT group. The rolling shear failure load was recorded when the first rolling shear crack, as shown in Figure 4.12, was observed. Table 4.7 provides a summary of the rolling shear failure loads in the tests, including the mean value and standard deviation. The Weibull fitting parameters for the cumulative distribution of these rolling shear failure loads are also included. Figure 4.12 Failure modes in the ramp loading test 80 Table 4.7 Summary of the ramp test results CLT type Span depth (mm mm) Sample size n Rolling shear loads (kN) Weibull fitting parameters Mean COV 5th percentile Location Scale Shape SPF5-0.4 840x140 55 19.39 12.6% 14.74 0.0000 20.4230 9.2193 SPF3-0.4 612x102 59 12.51 24.3% 7.96 0.0000 13.5510 5.0045 The results show the capacity of the SPF3-0.4 group was much lower than that in the five-layer CLT beams. This may have been due to the smaller cross section and the cross layer’s middle location in the SPF3-0.4 beams. Considering the relationship between the location of the cross layers and the shear stress distribution pattern, where the maximum shear stress point is located in beam section’s centre, the cross layer in the three-layer CLT takes all the high shear stress right in the middle, which leads to the lower capacity value in the test. This conclusion can be better explained by evaluating the sectional shear stress distribution, as introduced in the next paragraph. Also, the width to thickness ratio of the middle cross lamination board in three-layer CLT (140𝑚𝑚 ÷ 34𝑚𝑚 = 4.12) is smaller than that in five-layer CLT (140𝑚𝑚 ÷ 19𝑚𝑚 =7.37). This lower ratio is another reason for the lower capacity in three-layer CLT beams, because the rolling shear strength is significantly influenced by the width to thickness ratio (Blass and Fellmoser, 2004). To investigate shear stress levels in the cross layer, different beam theories (i.e., the layered beam theory, the gamma beam theory and the shear analogy theory) were used to evaluate the shear stress distribution in the cross section of CLT beams. Figure 4.13 shows the calculated 81 shear stress value in the section of three-layer and five-layer CLT beams, under an identical applied point load, i.e., the mean value of the rolling shear failure load of three-layer CLT (12.51 kN from Table 4.7). Detailed examples of calculations based on the different beam theories were given in Appendix I. The results in Figure 4.13 suggest that the mean rolling shear strength for three-layer CLT, which was evaluated from the different beam theories, ranges from 1.23 MPa to 1.68 MPa. Similarly, Figure 4.14 shows the calculated shear stress values in the cross section of five-layer CLT beams, under the mean value of the rolling shear failure load of five-layer CLT (19.39 kN from Table 4.7). The results in Figure 4.14 suggest that the mean rolling shear strength for five-layer CLT ranges from 1.63 MPa to 1.73 MPa. 82 Figure 4.13 Shear stress evaluated in the cross section of the CLT beam (in 𝑵/𝒎𝒎𝟐 under 12.51kN centre-point load) Layered beam Gamma beam Shear analogy1.681.661.04 1.191.23Layered beam0.641.101.111.20 1.26Gamma beam Shear analogy1.05 1.06ABCThree-layer CLT17341717171.68ABCABCABCABDCEFive-layer CLT17179,59,517ABDCABDCABDCE E0.641.101.111.05 1.061.661.04 1.1983 Figure 4.14 Shear stress evaluated in the cross section of the CLT beam (in 𝑵/𝒎𝒎𝟐 under 19.39kN centre-point load) In Figure 4.13, in the same beam theory, shear stress in the cross layer is lower in five-layer CLT beam (1.05 MPa to 1.11 MPa) than that in three-layer CLT. The cross layer in three-layer CLT is under higher shear stress, and this can explain why the cross layer in three-layer CLT takes all the high shear stress right in the middle, which leads to the lower load-carrying capacity. Based on the finite element model, a linear elastic analysis was performed to evaluate the rolling shear stress distribution with consideration of the orthotropic wood material property. Rather than the analytical beam theories which only show the shear stress distribution in cross sections, the finite element model can give the rolling shear stress distribution in different locations in the Layered beam0.991.711.731.86 1.96Gamma beam Shear analogy1.63 1.65ABDCEFive-layer CLT17179,59,517ABDCABDCABDCE E0.991.711.731.63 1.6584 volume. Figure 4.15 and Figure 4.16 show the results about the rolling shear stress distribution in the cross layers of three-layer and five-layer CLT specimens, when the applied load is the mean rolling shear failure load of three-layer CLT (12.51 kN from Table 4.7). As shown in the figures, the maximum rolling shear stress occurs near the one third of span in the lamination boards. The gaps in the cross layer have a clear influence on the rolling shear stress distribution, considering the rolling shear stress near the edge of the gap is zero. Also, under the same applied load (12.51 kN), the evaluated maximum rolling shear stress in cross layers of five-layer CLT (1.13 MPa) is lower than that in three-layer CLT (1.41 MPa). This result again explains why the rolling shear load-carrying capacity in three-layer CLT in Table 4.7 is lower. The evaluated mean rolling shear strength (1.41 MPa) in three-layer CLT is also in the range of the results from Figure 4.13 (1.23 MPa to 1.68 MPa). Similarly, Figure 4.17 shows the result of the rolling shear stress distribution in cross layers of five-layer CLT, when the applied load is the mean rolling shear failure load of five-layer CLT (19.39 kN from Table 4.7). The maximum rolling shear stress in the cross layers of five-layer CLT is 1.76 MPa in the figure. The evaluated mean rolling shear strength (1.76 MPa) in five-layer CLT is close to the results from Figure 4.14 (1.63 MPa to 1.73 MPa). 85 Figure 4.15 Rolling shear stress distribution in three-layer CLT specimen under 12.51kN load (stress in Pa) Figure 4.16 Rolling shear stress distribution in five-layer CLT specimen under 12.51kN load (stress in Pa) 86 Figure 4.17 Rolling shear stress distribution in five-layer CLT specimen under 19.39 kN load (stress in Pa) It was also observed that the coefficient of variation was relatively higher in the SPF3-0.4 group. The reason might be that one piece of specimen in the SPF3-0.4 group, as shown in Figure 4.18, had a very high rolling shear strength compared to the others; this specimen increased the variability of test data. During the ramp test, it was observed that one set of ten specimens cut from one three-layer CLT plate replicate had a relatively low rolling shear strength. Finally, the coefficient of variation value is supposed to depend on the number of layers in CLT. The more layers in CLT, the more lumber pieces are involved in the cross layer, reducing the variability. All these factors contributed to the larger coefficient of variation value observed. In certain failed specimens, the rolling shear failure cracks seemed to be coupled with tension perpendicular to the grain failure in the cross layers. Involving the component of the tension perpendicular to grain failure mode is more complicated than simply investigating the rolling 87 shear behaviour of the laminated lumber in the cross layers. Therefore, further investigation is recommended for a deeper evaluation of the wood fibre behaviour in various failure modes under different stress combinations. Cumulative distributions of these rolling shear failure loads are shown in Figure 4.18. This figure gives the 25th percentile rolling shear capacity to define the applied load level in the plateau part of the trapezoidal fatigue loading tests. The ramp test results for the recorded displacement and load are shown in Figure 4.19 and Figure 4.20. These figures show that, after the rolling shear failure load was recorded when the first rolling shear crack occurred, the final ultimate load of beam specimens in ramp loading tests was not yet reached. The ultimate load is about 30% more than the mean rolling shear failure load from Table 4.7, and in the end most of the beam specimens had the tension parallel to grain failure in the bottom layer of CLT. The rolling shear failure in one cross laminated board will not result in the failure or collapse of the whole CLT beam member, but investigation on the rolling shear failure behaviour is important. The occurrence of rolling shear cracks will change the stress distribution in CLT cross layers, and the adjacent board will carry more loads and shear stresses, resulting in more rolling shear cracks in laminations. The CLT member is then weakened with the soften stiffness due to these rolling shear cracks. This is an issue of concern specifically for CLT panels under concentrated shear loads, for example, the panel area near the lifting point (the case of concentrated loading), or in tall timber buildings with CLT slabs supported directly above columns (the case of point supports). Because the axial force in the columns is large in the lower stories of tall buildings, the CLT panel supporting area has high shear stresses. Too many rolling 88 shear cracks may reduce the punching shear resistance in this CLT supporting area, which can lead to potential punching shear failure. The tension parallel to grain failure in the bottom layer of CLT beams, which is different from rolling shear failure, was observed under the peak failure load level. The objective of this research is focused on rolling shear properties of CLT; however, future research is recommended on the evaluation of ultimate load-carrying capacities of CLT, considering one laminated board failure in the cross layer does not mean the CLT member is collapsing. Based on these results in Figure 4.19 and Figure 4.20, a finite element model was also developed to simulate the CLT beam specimen’s ramp loading behaviour, as discussed in Chapter 5. Figure 4.18 Cumulative distributions of the rolling shear failure loads 00.10.20.30.40.50.60.70.80.910 5 10 15 20 25 30CDFRolling Shear Failure Load (kN)CDF of Rolling Shear Failure LoadSPF5-0.4SPF3-0.489 Figure 4.19 Load-displacement curves of the experimental results in the SPF5-0.4 group (the applied load level of trapezoidal fatigue tests was shown as the horizontal dash line) 90 Figure 4.20 Load-displacement curves of the experimental results in the SPF3-0.4 group (the applied load level of trapezoidal fatigue tests was shown as the horizontal dash line) 91 4.4.2 Torque Loading Test Results There were two failure modes in the torque tests, shown in Figure 4.21 and Figure 4.22. Most of the tube specimens failed in rolling shear as further discussed in Section 4.4.3. The rolling shear failure showed the typical brittle failure behaviour with 45-degree inclined angle observing from the top or bottom face of the tube specimen. The resistance of the specimens to carry load suddenly dropped to zero, when the failure occurred. The other specimens showed the glue strength failure in the tests. Only those specimens with the rolling shear failure were included in the analysis. In Table 4.8, the mean value, standard deviation and coefficient of variation of the measured torque failure loads are given. Table 4.8 also includes the Weibull fitting parameters for the cumulative distribution of the torque test results. Figure 4.21 Glue failure in the torque test 92 Figure 4.22 Rolling shear failure in the torque test (with Y direction as the direction of wood fibre in the middle cross layer) Table 4.8 Summary of the torque test results CLT type Peak failure torque load (N×mm) Weibull fitting parameters No. of specimen with rolling shear failure Mean STDV COV Location Scale Shape SPF5-0.4 103989 17058 16.40% 0.0000 110378 7.4229 31 SPF3-0.4 84561 14048 16.61% 0.0000 89210 7.8630 25 Y X Z X Z X Y Z Z X 93 4.4.3 Analysis and Discussion on the Rolling Shear Failure in Torque Loading Tests In torque loading tests, most of the CLT tube specimens had brittle shear failure. At failure the resistance to carry load sharply dropped to zero. The cracks were developed at an inclined angle in the middle layer of the specimen. The brittle shear failure mode in the torque tube specimens was not as clear as the rolling shear failure observed in the beam specimens. In order to investigate and verify that the observed shear failure mode is rolling shear failure in the torque tube specimens, further modeling work was performed and summarized as follows with consideration of different shear stress components (longitudinal shear). Considering the anisotropic material property in timber material and the grain influences on the strength, the evaluation of the stress state of the CLT torque tube specimens requires the finite element method for further investigation, since the elementary mechanical torque theory equations (calculated by hand) could not accurately evaluate the structural response of the CLT tube specimen. Therefore, based on the finite element model developed in the UBC Timber Engineering and Applied Mechanics group (Li et al., 2014), one linear elastic finite element analysis was carried out. To discuss and verify that the observed brittle failure mode in the cross layer is rolling shear failure, the finite element model, as shown in Figure 4.23, was developed in ANSYS platform (SAS, 2011) to model the CLT tube specimens, considering the glue line shear stiffness and the orthotropic wood properties (with Y direction in the figure as the direction of wood fibre in the middle cross layer) (Li et al., 2014). Dimensions of the tube finite element models are the same 94 as those in the three-layer and five-layer CLT tube specimens, introduced in Table 4.5. Table 3.2 provides information on the input wood elastic orthotropic properties. Solid volume elements were used to model the wood boards, and linear spring elements were used to model the glue line shear stiffness. The metal part of the test jig was also simulated. The fibre direction of the cross layer in the model was the same as that in the test setup, i.e., the grain direction (Y direction) was perpendicular to the longitudinal direction of the steel lever (X direction in Figure 4.22 and Figure 4.23). Figure 4.23 Finite element modeling of three-layer CLT tube test specimens (orthotropic wood properties with Y direction as the direction of wood fibre in the middle cross layer) Figure 4.24 to Figure 4.26 show the shear stress (𝜏𝑥𝑧 , 𝜏𝑥𝑦, 𝜏𝑦𝑧) distribution in the cross layer of three-layer CLT tube specimens, when the applied torque load level is the mean peak failure load (84561 N×mm from Table 4.8) for three-layer CLT. The shear stress 𝜏𝑥𝑧 (the same as Fibre direction of the middle cross layer (Y direction) Fibre directions of these two side layers (X direction) Lever 95 engineering definitions in Cartesian coordinate system in terms of directions) is the shear stress directed parallel to a given plane (YZ plane) which is perpendicular to the X direction, and this 𝜏𝑥𝑧 is parallel to the Z direction. Therefore, since rolling shear is defined as a shear stress leading to the shear strain in the planes perpendicular to the grain, 𝜏𝑥𝑧 is the rolling shear stress, and 𝜏𝑥𝑦 and 𝜏𝑦𝑧 are the longitudinal shear stresses. These two longitudinal shear stresses (𝜏𝑥𝑦 and 𝜏𝑦𝑧) are also uncoupled according to the defined wood orthotropic material property (Bodig and Jayne, 1982). The maximum value of the shear stress (𝜏𝑥𝑧 , 𝜏𝑥𝑦 𝑜𝑟 𝜏𝑦𝑧) is also given in the figures (in Pa). 96 Figure 4.24 Shear stress (𝝉𝒙𝒛, rolling shear in orthotropic material modeling) distribution in the cross layer of three-layer tube specimen (Maximum 𝝉𝒙𝒛 = 𝟑. 𝟖𝟔 𝑴𝑷𝒂) 0xz:rolling shear stressYXZFiber directionY in the crosslayerSmall cubic elementin the orthotropicmaterial modeling oftube volumeStress Component Introduction in Small ElementFiber directionFibre direction of the middle cross layer (Y direction) 97 Figure 4.25 Shear stress (𝝉𝒚𝒛, longitudinal shear in orthotropic material modeling) distribution in the cross layer of three-layer tube specimen (Maximum 𝝉𝒚𝒛 = 𝟕. 𝟎𝟒 𝑴𝑷𝒂) 0yz:longitudinal shear stressYXZFiber directionY in the crosslayerSmall cubic elementin the orthotropicmaterial modeling oftube volumeStress Component Introduction in Small ElementFiber directionFibre direction of the middle cross layer (Y direction) 98 Figure 4.26 Shear stress (𝝉𝒙𝒚, longitudinal shear in orthotropic material modeling) distribution in the cross layer of three-layer tube specimen (Maximum 𝝉𝒙𝒚 = 𝟓. 𝟓𝟕 𝑴𝑷𝒂) Figure 4.24 shows that the maximum rolling shear stress (3.86 MPa) occurred near the ring-edge of the cross layer volume close to the top or bottom face of the tube specimen, where the initial 0xy:longitudinal shear stressYXZFiber directionY in the crosslayerSmall cubic elementin the orthotropicmaterial modeling oftube volumeStress Component Introduction in Small ElementFiber directionFibre direction of the middle cross layer (Y direction) 99 shear crack typically started in the tests (brittle shear failure mode as given in Figure 4.22). As shown in Figure 4.25, the maximum longitudinal shear stress is 7.04 MPa. Figure 4.27 to Figure 4.29 show the shear stress (𝜏𝑥𝑧 , 𝜏𝑥𝑦, 𝜏𝑦𝑧) distribution in the cross layer of five-layer CLT tube specimens, when the applied load level is the mean value of peak failure load (103989 N×mm from Table 4.8) for five-layer CLT. The maximum value of the shear stress (𝜏𝑥𝑧 , 𝜏𝑥𝑦 𝑜𝑟 𝜏𝑦𝑧) is also given in the figures (in Pa). Figure 4.27 Shear stress (𝝉𝒙𝒛, rolling shear in orthotropic material modeling) distribution in the cross layer of five-layer tube specimen (Maximum 𝝉𝒙𝒛 = 𝟒. 𝟖𝟑 𝑴𝑷𝒂) xz:rolling shear stressSmall cubic elementin the orthotropicmaterial modeling oftube volumeFiber directionYXZFibre direction of the middle cross layer (Y direction) 100 Figure 4.28 Shear stress (𝝉𝒚𝒛, longitudinal shear in orthotropic material modeling) distribution in the cross layer of five-layer tube specimen (Maximum 𝝉𝒚𝒛 = 𝟏𝟎. 𝟑𝟎 𝑴𝑷𝒂) Figure 4.29 Shear stress (𝝉𝒙𝒚, longitudinal shear in orthotropic material modeling) distribution in the cross layer of five-layer tube specimen (Maximum 𝝉𝒙𝒚 = 𝟔. 𝟖𝟎 𝑴𝑷𝒂) yz:longitudinal shear stressSmall cubic elementin the orthotropicmaterial modeling oftube volumeFiber directionYXZxy:longitudinal shear stressSmall cubic elementin the orthotropicmaterial modeling oftube volumeFiber directionYXZFibre direction of the middle cross layer (Y direction) Fibre direction of the middle cross layer (Y direction) 101 Similarly, Figure 4.27 shows that the maximum rolling shear stress (4.83 MPa) occurred near the ring-edge of the cross layer volume. So did the longitudinal shear stress (10.30 MPa) as shown in Figure 4.28. For the three-layer CLT tube specimens, the maximum rolling shear stress in the cross layer is 𝜏𝑥𝑧𝑚𝑎𝑥 = 3.86 𝑀𝑃𝑎, and the maximum longitudinal shear stress (𝜏𝑥𝑦𝑚𝑎𝑥 or 𝜏𝑦𝑧𝑚𝑎𝑥) is 𝜏𝑦𝑧𝑚𝑎𝑥 = 7.04 𝑀𝑃𝑎. The ratio between the maximum rolling shear stress and longitudinal shear stress is RShearThree−layer =τxzmaxτyzmax= 0.55 ≈ 1 1.82⁄ . For five-layer CLT tube specimens, the maximum rolling shear stress in the cross layer is 𝜏𝑥𝑧𝑚𝑎𝑥 = 4.83 𝑀𝑃𝑎, and the maximum longitudinal shear stress (𝜏𝑥𝑦𝑚𝑎𝑥 or 𝜏𝑦𝑧𝑚𝑎𝑥) is 𝜏𝑦𝑧𝑚𝑎𝑥 = 10.30 𝑀𝑃𝑎. The ratio between the maximum rolling shear stress and longitudinal shear stress is RShearFive−layer =τxzmaxτyzmax= 0.47 ≈12.13⁄ . Rolling shear strength and stiffness of wood are much lower than its longitudinal shear strength and stiffness. According to Wood Handbook (FPL, 2010), rolling shear strength typically is between 18% and 28% (about 1/5 to 1/3) of parallel-to-grain shear strength based on limited test data (Aicher and Dill-Langer, 2000; Blass and Görlacher, 2003; Fellmoser and Blass, 2004; Lam et al., 1997; Li et al., 2014). Therefore, the ratio 𝑅𝑆ℎ𝑒𝑎𝑟 = 0.47 ≈12.13⁄ (i.e., the minimum of RShearThree−layer and RShearFive−layer) is much higher than the ratio of rolling shear strength and 102 longitudinal shear strength (18%~28%, which is about 1/5 to 1/3). In the torque loading tests, the rolling shear stress will reach the material strength limit earlier than longitudinal shear. Therefore, the brittle failure mode observed in the cross layer of CLT tube specimens is rolling shear failure. Based on a Monte Carlo simulation procedure, a further investigation on the probability of rolling shear failure in the torque loading tests is also given and included in Appendix J. The results also suggest that the probability of rolling shear failure in the torque tube specimen is at a very high level, when the randomness of the rolling shear strength and longitudinal shear strength in wood material is considered. Meanwhile, the longitudinal shear behaviour in the tube specimens is predicted by applying Weibull’s theory of brittle fracture in Section 5.3.3. Since the predicted longitudinal shear load-carrying capacity is significantly higher than the torque test measurements, the results suggest that the observed shear failures are most unlikely to be longitudinal shear. Therefore, the results from these works provide additional evidence that the observed failure mode in the CLT torque tube specimen is indeed rolling shear failure. 4.4.4 Trapezoidal Fatigue Loading Test Results In the trapezoidal fatigue loading test, rolling shear failure occurred during the loading process. The shape of these rolling shear cracks in the cross layer was typically very similar to that in the ramp test; however, these rolling shear cracks appeared relatively slowly in the cross layer. The number of cycles to rolling shear failure was recorded when the first rolling shear crack was 103 observed with careful examination. Different types of rolling shear cracks occurred during the test, as shown in Figure 4.30. Some cracks were continuous, while others were not. Some cracks occurred in the lower density part of the timber material, following the wood ring pattern in cross laminae. Figure 4.30 Rolling shear cracks in the trapezoidal fatigue loading test (black marker line in the circle is next to the crack) The test results as presented in Table 4.9, show the numbers of cycles to rolling shear failure, i.e., 𝑁𝑓, which failed within that 𝑁𝑓𝑡ℎ cycle of loading in time scale. The number of cycles to failure varied greatly from one specimen to another, as small as 1 or larger than 280. Failure 104 cycles in the short plateau test were relatively larger than those in the long plateau test, given the smaller damage accumulation in the short plateau. The 25th percentile rolling shear failure loads from the ramp tests (from Table 4.7 and Figure 4.18) were applied in the trapezoidal fatigue tests, as shown in Table 4.9. However, less than 25% of the specimens failed in the first uploading process, as given in Table 4.9. This difference was attributed to the use of a significantly higher rate of loading (the uploading rate) in the trapezoidal fatigue test compared to the short-term ramp loading test; i.e., with a higher rate of loading it is expected that the apparent short-term strength would increase compared to the case with a lower loading rate (Madsen, 1992). The uploading rates in the ramp and trapezoidal fatigue loading tests were different. In the ramp loading tests, the displacement control methods were adopted; and, the uploading rates were around 3.9 kN/min for the five-layer CLT specimens and 2.3 kN/min for the three-layer CLT specimens, which were considerably lower than the values of 37.5 kN/min for the five-layer CLT specimens and 27.0 kN/min for the three-layer CLT specimens in the trapezoidal fatigue loading tests. Meanwhile, the short-term ramp tests adopted the displacement control method (with a constant deformation rate), so the stress rate was different from one specimen to another, while the loading rate for the trapezoidal test specimens is given in terms of stress. Short-term strength property in ramp loading is typically recognized to be increasing when the loading rate is higher. Considering the relationship between the loading rate and the strength property increase, therefore, less than 25% of specimens failed in the first cycle of the 105 trapezoidal fatigue tests. The strength property under higher loading rate is expected to be increased based on the ramp and trapezoidal loading test results as shown in Table 4.9, and there is research showing that about 15% shear strength increase was observed when the loading rate was increased ten times higher in the previous performed tests (Madsen, 1992). Therefore, further modeling process considering this relationship between the loading rate and strength increase, based on the damage accumulation model and test results, will be introduced in Section 6.3. Table 4.9 Results of the trapezoidal fatigue test for CLT panels Test type: Short / Long plateau test CLT group No. of specimens with rolling shear failure (No. of specimens failed in the first cycle) Applied load level Number of cycles to the rolling shear failure 25th% rolling shear failure loads(kN) Mean STDEV Maximum cycle Short SPF5-0.4 28 (2) 17.69 66.1 76.5 281 Short SPF3-0.4 30 (0) 10.33 38.5 50.3 212 Long SPF5-0.4 29 (0) 17.69 15.2 18.5 88 Long SPF3-0.4 32 (4) 10.33 12.8 23.2 92 Considering that one specimen in the trapezoidal fatigue loading test could not be broken twice to determine both its short-term strength and its response under fatigue load, one technique, known as the equal rank assumption, was employed to estimate this failed member’s short-term 106 rolling shear strength (Barrett, 1996). The short-term rolling shear strength distribution was first established in the ramp loading tests. If one group was subjected to a ramp uploading to a constant stress level A, clearly all specimens with a strength of less than A would fail upon uploading. Typically, additional specimens fail under cyclic long-term loading; therefore, corresponding to the constant stress level A, the numbers of cycles to failure are known. If one-to-one correspondence between the short- and long-term strengths is assumed, the specimens should fail in the order of increasing short-term strength. With the rank of the long-term data known, its corresponding short-term strength can be established from the matched control group’s short-term strength distribution. The load ratio, also known as the stress ratio applied to one specimen, is defined as the ratio of the applied stress level to the short-term strength value established from the equal rank assumption. The applied stress level was evaluated from the maximum cross-layer rolling shear stress, and it was calculated from the finite element model under a centre-point load, with the model load input value as the trapezoidal constant load level. The established short-term rolling shear strength distribution for specimens in trapezoidal tests, which was assumed to be lognormally distributed, was developed from the same finite element evaluation method (introduced in Chapter 5) with consideration of the influence of higher loading rate, meaning the obtained finite element results on the rolling shear strength were corrected for the trapezoidal test specimens with the expected 15% strength increase due to the higher loading rate for modeling purpose. 107 Based on the equal rank assumption, the relationship between the number of cycles to rolling shear failure and the stress ratio applied in the SPF5-0.4 group is shown in Figure 4.31. The figure shows that, under the same stress ratio, the time to failure was shorter in the long plateau test, since more damage is accumulated in this test category for each loading cycle. The cumulative distribution of the number of cycles to rolling shear failure in the SPF5-0.4 group is shown in Figure 4.32. In Figure 4.31 and Figure 4.32, the number of cycles to failure is shown in the logarithm to base 10. Uploading failure refers to the specimen failed in the uploading segment of the first cycle in the trapezoidal fatigue tests. As shown in Table 4.9, less than 25% of specimens failed during uploading. In Figure 4.32, the value of number of cycles to failure that can be attributed to the specimen with uploading failure was between zero and one. However, the exact value (the number of cycles to failure) is ambiguous for such cases. Therefore, the specimens that failed during the first uploading sequence (𝑁𝑓 < 1) were not included in the cumulative distribution plots in Figure 4.32; furthermore, the rank of cumulative probability of the specimens that failed beyond the initial uploading was calculated based on the whole dataset including the data points not shown. For example, in Table 4.9 for the SPF5-0.4 short plateau test, 2 specimens out of 28 failed in the uploading segment; in Figure 4.32, only 26 data points (referring to the specimens that failed beyond uploading) were shown for the SPF5-0.4 short plateau test, and the first point in this group started from the rank of 3/(28+1)=10.3%. Similarly, the relationship between the number of cycles to rolling shear failure and the stress ratio applied in the three-layer CLT is shown in Figure 4.33; and, the cumulative distribution of 108 the number of failure cycles is shown in Figure 4.34. This information will be adopted in the damage accumulation modeling process introduced in Chapter 6. Figure 4.31 Relationship between the number of cycles to failure (in the logarithm to base 10) and the stress ratio in the SPF5-0.4 group Figure 4.32 Cumulative distribution of the number of cycles to failure (in the logarithm to base 10) in the SPF5-0.4 group (removed data points of uploading failure specimens but the rank was calculated based on the entire set) 109 Figure 4.33 Relationship between the number of cycles to failure (in the logarithm to base 10) and the stress ratio in the SPF3-0.4 group Figure 4.34 Cumulative distribution of the number of cycles to failure (in the logarithm to base 10) in the SPF3-0.4 group (removed data points of uploading failure specimens but the rank was calculated based on the entire set) 110 4.5 Conclusion The results from the duration-of-load testing program on MPB lumber based CLT have been presented. CLT beam specimens with small span-to-depth ratios were prepared. Concentrated loads were then applied in ramp loading and trapezoidal fatigue loading tests. In the ramp loading tests, the basic short-term rolling shear strength distribution for the test specimens was first developed; and, the three-layer CLT product showed a lower rolling shear load-carrying capacity. In the trapezoidal fatigue loading tests, the time to failure data was obtained to understand the development of deflection and the damage accumulation process. This experimental research contributed to the study of the CLT rolling shear duration-of-load behaviour, i.e., the results from the ramp and trapezoidal fatigue tests provided a database for both the calibration and verification of the damage accumulation model, which will be introduced in Chapter 6. CLT tube specimens were also sampled from the full size panels, and torque loading tests were performed. The results from the torque tests contributed to the investigation of the size effect on the rolling shear strength in timber material, as described in Chapter 5. According to certain failed specimens in the ramp loading tests, the rolling shear failure mode seemed to be coupled with tension perpendicular to grain failure in the cross layer. Inclusion of the tension perpendicular to grain failure mode is very complicated; therefore, further 111 investigation is recommended for a deeper evaluation of the failure mode in terms of different stress combinations. 112 Chapter 5: Size Effect on the Rolling Shear Strength of CLT 5.1 Summary Finite element models are first developed to simulate CLT (cross laminated timber) beam specimens. The results of both ramp and trapezoidal fatigue loading tests are interpreted using the finite element model with a customized Fortran subroutine. The evaluation of the rolling shear strength based on the collected test data is then discussed. Finally, from the finite element results, stress integration in timber volume is performed for each small volume element, so that the size effect on the rolling shear strength can be investigated based on the data from the ramp and torque loading tests. By applying Weibull’s theory of brittle fracture to determine the CLT rolling shear strength, the developed size effect model can be used to explain the differences in the rolling shear strength between the CLT beam specimens and the torque tube specimens. This developed size effect model is suitable to predict the rolling shear capacity in CLT members. 5.2 Finite Element Model Considering the complicated failure mechanism in CLT, the elementary elastic beam theory and the other CLT beam design methods, such as the shear analogy method, could not accurately evaluate the structural response of the CLT beam specimen, with consideration of the rolling 113 shear behaviour. Thus, one finite element model based on ANSYS v14.0 and Intel Fortran platforms was developed. 5.2.1 Model Calibration The developed finite element model was calibrated against both the ramp loading and the trapezoidal fatigue loading test data, based on the criteria defined below. 5.2.2 Calibration Results 5.2.2.1 Ramp Loading Results The finite element models were calibrated against the ramp loading test data from the SPF5-0.4 group and the SPF3-0.4 group under the ramp loading history. There are two criteria for this calibration. First, the model should show approximately the same initial stiffness compared with the test data. Then, the model displacement in the middle of the beam span, at the target applied load level in the trapezoidal fatigue loading, i.e., the applied load level in the plateau part, relatively should match the average value of the maximum and the minimum measured displacement values in the tests; the maximum and minimum measured values were collected in the ramp loading tests when the ramp load was in the same target applied load level. Figure 5.1 and Figure 5.2 show the calibration results. The results showed that the load-displacement curve from the finite element model (with triangle black marker) agreed well with the defined criteria. 114 The detailed fitting process (in Figure 5.1 and Figure 5.2) and the corresponding adjusted model parameter values (i.e., elastic properties in Table 3.1 and the multi-linear stress-strain relationship simulating timber mechanical behaviour) are given in Appendix K. Figure 5.1 and Figure 5.2 show that the finite element model, with consideration of the gaps in adjacent boards, can predict the short-term load-displacement curve with a good agreement with test results. The agreement was achieved even though the influence of the possible presence of glue squeezed into the gaps between the edges of the laminae was ignored in the finite element models. Figure 5.1 Load-displacement curves of the experimental and simulated results in the SPF5-0.4 group (the horizontal dash line represents the target applied load level) 115 Figure 5.2 Load-displacement curves of the experimental and simulated results in the SPF3-0.4 group (the horizontal dash line represents the target applied load level) 5.2.2.2 Trapezoidal Fatigue Loading Results Based on the User programmable features in the ANSYS v14.0 platform (SAS, 2011), the finite element model was also integrated with the rheological constitutive relationship behaviour for wood by coding customized Fortran subroutines, i.e., the constitutive relationship based on the modified five-parameter rheological model in Equation ( 3.4 ) and ( 3.5 ). The detailed modeling process is introduced in Appendix L. This finite element model was calibrated against the trapezoidal fatigue loading test data under the trapezoidal fatigue loading protocol. There are three criteria of this calibration. First, in the 116 first cycle, the model should show relatively the same initial stiffness compared with the test data. Then, the model displacement value in the middle span at the controlled load level, i.e., the applied load level in the plateau part, should be equal to the recorded displacement value of the tested specimen under the same load level. Finally, since the finite element model was incorporated with the customized Fortran codes to simulate the rheological behaviour, the model displacement should be increasing in the time history. This increasing trend in the model, related to the rheological behaviour, is required to be well fitted with the measurements in the test. The detailed fitting process (for the results in this section) and the corresponding adjusted model parameter values (i.e., elastic properties in Table 3.1, the multi-linear stress-strain relationship simulating timber mechanical behaviour and the parameters of the rheological model) are given in Appendix L. Comparison between the model result and the test data is presented as follows. For one selected specimen in the SPF5-0.4 group, Figure 5.3 shows the calibration result in the first four cycles, and this result suggests that the finite element model displacement is correlated with the test result. Similarly, Figure 5.4 shows the calibration result for one selected specimen in the SPF3-0.4 group, and the model result agrees well with the test data. 117 Figure 5.3 Trapezoidal displacement history from the experimental and simulated results in the SPF5-0.4 (first four cycles) Figure 5.4 Trapezoidal displacement history from the experimental and simulated results in SPF3-0.4 (first five cycles) Then, specifically, in the SPF5-0.4 group, two test specimens from the trapezoidal long plateau tests, i.e., one specimen with the maximum measured displacement value and the other specimen with the minimum measured displacement value, were selected as shown in Figure 5.5. Figure 5.6 to Figure 5.9 show the model simulations and the test data for these two selected specimens. For instance, Figure 5.6 and Figure 5.7 show the specimen case with the minimum displacement value, and this specimen failed in six cycles in the test. Figure 5.6 shows the model prediction 118 agrees well with the test data in the first three cycles. Figure 5.7 includes the comparison between the test data and the model prediction during the whole loading history within six cycles. To clarify, the model prediction in this figure is flipped over to make the clear comparison, meaning the positive predicted model displacement value was multiplied by minus one before it was adopted into Figure 5.7. Therefore, by integrating the wood material properties into the finite element model, it can predict the specimen’s displacement in time history. The figures introduced above show the realistic time dependent displacement, not only initially, but also over time. These model predictions proved that the finite element method was one feasible method for selection. Figure 5.5 Maximum and minimum trapezoidal displacement curves in SPF5-0.4 119 Figure 5.6 Minimum trapezoidal displacement experimental and simulated curves in SPF5-0.4 (first three cycles) Figure 5.7 Minimum trapezoidal displacement experimental and simulated results in SPF5-0.4 -5-4-3-2-10123450 100 200 300 400 500 600 700DISPLACEMENT (mm)TIME (Sec)DISPLACEMENT HISTORY FROM SPF5-0.4 GROUPTEST SPECIMEN WITH MINIMUM DISP VALUE (FAILURE IN 6 CYCLES)MODEL FOR TEST SPECIMEN WITH MINIMUM DISP VALUE (TIMES MINUS ONE)120 Figure 5.8 Maximum trapezoidal displacement experimental and simulated curves in SPF5-0.4 (first three cycles) Figure 5.9 Maximum trapezoidal displacement experimental and simulated results in SPF5-0.4 Similarly, in the SPF3-0.4 group in the trapezoidal short plateau tests, Figure 5.10 shows the selected two test specimens, one with the maximum measured displacement value and the other one with the minimum measured displacement value. Figure 5.11 to Figure 5.14 show the 121 comparisons between the model simulations and the test data for these two specimens. The prediction, shown in Figure 5.12, fitted well with the test data. In this model prediction, not only the initial stiffness and the target displacement in the first cycle agreed well with the test measurement, but also the predicted rheological displacement increase agreed well with the test data. The prediction in Figure 5.12 also included the nonlinear behaviour of the specimens. Therefore, these results showed the finite element method was able to simulate realistic results based on the material property input. Figure 5.10 Maximum and minimum trapezoidal displacement curves in SPF3-0.4 122 Figure 5.11 Minimum trapezoidal displacement experimental and simulated curves in SPF3-0.4 (first six cycles) Figure 5.12 Minimum trapezoidal displacement experimental and simulated results in SPF3-0.4 123 Figure 5.13 Maximum trapezoidal displacement experimental and simulated curves in SPF3-0.4 (first six cycles) Figure 5.14 Maximum trapezoidal displacement experimental and simulated results in SPF3-0.4 124 5.3 Size Effect on the Rolling Shear Strength 5.3.1 Rolling Shear Strength Evaluation in Ramp Loading Tests In order to establish the basic short-term rolling shear strength distribution by the short-term ramp loading, a linear elastic finite element beam analysis (the similar process as shown from Figure 4.15 to Figure 4.17) was performed. First, to generate rolling shear failure load input values in the finite element model, a non-parametric fitting technique was adopted (Madsen, 1992). This fitting technique generated thirty data points representing the rolling shear failure load distribution according to Figure 4.18. The detailed introduction of this fitting process is explained in Appendix M. For each finite element analysis, one of the thirty generated rolling shear failure loads was used as the input load, and the maximum rolling shear stress can be evaluated in the cross layer under the concentrated load. The shear stress distribution pattern is the same as that as shown in Figure 4.15 and Figure 4.16. For example, when the concentrated load input was 7.24 kN in the three-layer CLT beam model (the first fitted data point given in Table M.1 in Appendix M for three-layer CLT beam specimens), the model showed that the maximum rolling shear stress evaluated in the cross layer was 0.82 MPa, as given in Table M.1 in Appendix M. Table 5.1 gives the summary of the rolling shear strength in the cross layer, including the mean and standard deviation for the three-layer and the five-layer CLT. The Weibull fitting parameters for the cumulative distribution of the rolling shear strength are also given. 125 Table 5.1 Summary of the cross layer maximum rolling shear stress results from the finite element model CLT type Maximum rolling shear stress value in the cross layer (MPa) Weibull fitting parameters Mean COV 25th percentile 5th percentile Location Scale Shape SPF5-0.4 1.76 12.2% 1.61 1.34 0.0000 1.8584 9.2765 SPF3-0.4 1.41 23.3% 1.16 0.90 0.0000 1.5288 5.0032 The results in Table 5.1 show that the mean rolling shear strength value ranges from 1.41 MPa to 1.76 MPa. The rolling shear strength of these specimens is believed to depend on the wood species. It can also be affected by specimen sizes, loading types and loading protocols. Since the input load value came from the CLT beam ramp loading test under the concentrated load protocol, different results might be observed in the other cases, such as the CLT two-dimensional panel under the uniformly distributed loads. Therefore, future research is recommended in this area, to study the influence on the rolling shear strength from different loading protocols. 5.3.2 Size Effect on the Rolling Shear Strength As shown in Figure 4.15, Figure 4.17, Figure 4.24 and Figure 4.27, based on the rolling shear failure load data, the evaluated rolling shear strengths were different. This difference mainly is due to the different material stressed volume between each testing case. Therefore, to consider the size dependence and to explain the strength differences between the CLT ramp loading tests and the torque loading tests, as shown in Table 4.7 and Table 4.8, the weakest link theory has been applied, and the size effect on the rolling shear strength was evaluated. 126 According to the weakest link theory, it is assumed that wood material under shear stress in tests has a non-uniform stress volume. The probability of failure (i.e., 𝑝) of a homogeneous isotropic material of a given volume 𝑉 can be predicted as: 𝑝 = 1 − e−1V0∫ (τ−τminm )kdVV where 𝑚, 𝑘 and 𝜏𝑚𝑖𝑛 are the scale, shape and location parameters for the distribution, and these parameters are assumed to be material constants. The material strength is expressed by 𝜏, and 𝑉0 is a reference volume. A simpler, two-parameter model may be used by assuming 𝜏𝑚𝑖𝑛 = 0. If we consider the reference volume, 𝑉∗, as a unit volume under a uniform rolling shear stress, 𝜏∗, volume 𝑉 will have the same probability of failure if the following equation is true between 𝜏∗ and stress 𝜏: ∫ (τ)kdV𝑉= ∫ (τ∗)kdV𝑉∗= (τ∗)k ( 5.1 ) If the structural component under stress 𝜏 is evaluated as: τ = ρ ∙ θ[x, y, z] ( 5.2 ) where ρ is one characteristic load or stress value of this component. For example, if ρ is one applied characteristic load (in kN) as shown in Figure 4.15 and τ[x, y, z] is the calculated stress value in the location [x, y, z] also shown in Figure 4.15, θ[x, y, z] represents the ratio between τ[x, y, z] and ρ in Cartesian coordinate system. Then, Equation ( 5.1 ) may be written as: ρk ∙ I[k] = (τ∗)k ( 5.3 ) 127 with τ∗ = m[−ln(1 − 𝑝)]1/k ( 5.4 ) where I[k] = ∫ (θ[x, y, z])kdV𝑉, and 𝑝 is the probability of failure. If we assume the rolling shear strength of unit volume under uniform shear is known at different levels of survival probability and if the component of rolling shear stresses 𝜏 is also informed so that I[k] may be integrated from the theory calculation or the finite element method, the rolling shear strength of volume 𝑉 at different levels of survival probability can, therefore, be evaluated as: ρ = τ∗I[k]1/k⁄ ( 5.5 ) It is not easy to evaluate the unit volume strength from an experimental study. However, the material constant and the unit volume strength may be obtained by fitting a theoretical size effect model to the experiment results with the recorded failure loads and given configurations. Based on the weakest link and size effect theories, material constants can first be calibrated by one set of test data, for example, the torque test on a five-layer CLT as shown in Figure 5.15. The calibrated constants could be searched by minimization of the following sum of squares: S =∑(ρiTest − ρiModel)2Ni=1 ( 5.6 ) where 𝑁 is the number of probability levels considered; ρiTest is the test characteristic strength at a given survival probability; ρiModel is the theoretical strength evaluated in the size effect model at the 𝑖𝑡ℎ probability level from Equation ( 5.5 ), where I[k] is integrated from the finite element 128 analysis results, meaning that the stress volume integration is the summation of products of volume and the rolling shear stress value in each tiny small element. The material constants in the size effect modeling process were calibrated by part of the experimental data, and the model was able to predict the CLT rolling shear strength in the other part of the experimental tests. The material constants (𝑚 and 𝑘), corresponding to the minimum of Equation ( 5.6 ), were able to be determined by an iterative procedure. With the calibrated results, predictions from the size effect model were then compared with test results shown in Table 4.7 and Table 4.8. Figure 5.15 shows the calibration results based on the torque SPF5-0.4 test data; and, with calibrated material constants, Figure 5.16 gives the predictions of the size effect model in comparison with test data for the torque SPF3-0.4. The prediction agreed quite well with the torque SPF3-0.4 group when the calibration was based on the torque SPF5-0.4 test data as given in Figure 5.16. The error between the predicted and observed results at either the 5th or 50th percentile level was also given in the figures. These errors were estimated as the absolute difference between predicted and observed results divided by the observed results in percent. Then, with the calibrated material constants from the torque test, it was decided to check if the calibrated model was also applicable to the beam cases. Figure 5.17 to Figure 5.18 give the predictions of the size effect model in comparison with test data for the bending SPF5-0.4 and bending SPF3-0.4 categories. The results show the model predictions fitted well with the test 129 data, with small differences between the predictions and the test data in the bending SPF5-0.4 and bending SPF3-0.4 categories. Figure 5.15 Size effect model calibrated by torque SPF5-0.4 test data Figure 5.16 Size effect model prediction for torque SPF3-0.4 test data (calibrated by torque SPF5-0.4 data) 2.5% error at the 50th% level 6.6% error at the 5th% level 3.6% error at the 50th% level 12.7% error at the 5th% level 130 Figure 5.17 Size effect model prediction for bending SPF5-0.4 test data (calibrated by torque SPF5-0.4 data) Figure 5.18 Size effect model prediction for bending SPF3-0.4 test data (calibrated by torque SPF5-0.4 data) The calibration results based on the torque SPF5-0.4 data seem to provide good predictions. Rather than just fitting the torque SPF5-0.4 data, it was decided to see if the size effect model could be calibrated by other test groups, such as the torque SPF3-0.4 data. Figure 5.19 to Figure 5.22 include the calibration and predictions based on the torque SPF3-0.4 test data. Similarly, Figure 5.23 to Figure 5.26 present the calibration and predictions based on the bending SPF5-0.4 4.9% error at the 50th% level 12.4% error at the 5th% level 0.6% error at the 50th% level 5.0% error at the 5th% level 131 data, and Figure 5.27 to Figure 5.30 present the same for the bending SPF3-0.4 data. Table 5.2 shows the calibrated material constants for different cases. Figure 5.19 Size effect model calibrated by torque SPF3-0.4 test data Figure 5.20 Size effect model prediction for torque SPF5-0.4 test data (calibrated by torque SPF3-0.4 data) 2.9% error at the 50th% level 17.4% error at the 5th% level 0.0% error at the 50th% level 9.5% error at the 5th% level 132 Figure 5.21 Size effect model prediction for bending SPF5-0.4 test data (calibrated by torque SPF3-0.4 data) Figure 5.22 Size effect model prediction for bending SPF3-0.4 test data (calibrated by torque SPF3-0.4 data) When the calibration was based on torque SPF3-0.4 test data, predictions were reasonably good for the torque SPF5-0.4, bending SPF5-0.4 and bending SPF3-0.4 groups, as shown in Figure 5.19 to Figure 5.22. Therefore, from Figure 5.15 to Figure 5.18 and Figure 5.19 to Figure 5.22, it can be concluded that the prediction from the size effect model calibrated from either torque SPF5-0.4 or torque SPF3-0.4 test data gave good agreement for the different groups. The prediction also agreed quite well with the torque SPF5-0.4 group when the calibration was based 8.2% error at the 50th% level 15.9% error at the 5th% level 4.3% error at the 50th% level 0.4% error at the 5th% level 133 on the torque SPF3-0.4 test data, so the size effect model fitted well between the torque test groups as given in Figure 5.16 and Figure 5.20. Figure 5.23 Size effect model calibrated by bending SPF5-0.4 test data Figure 5.24 Size effect model prediction for torque SPF5-0.4 test data (calibrated by bending SPF5-0.4 data) 1.2% error at the 50th% level 1.2% error at the 5th% level 13.0% error at the 50th% level 11.8% error at the 5th% level 134 Figure 5.25 Size effect model prediction for torque SPF3-0.4 test data (calibrated by bending SPF5-0.4 data) Figure 5.26 Size effect model prediction for bending SPF3-0.4 test data (calibrated by bending SPF5-0.4 data) On the other hand, when calibration was based on bending SPF5-0.4 test data, meaning that only minor adjustments had to be made to improve the fitting, as shown in the comparison of the results in Figure 5.21 and Figure 5.23, the predictions were not as good as those from the case of calibration by the other torque tests. With the calibrated material constants by bending SPF5-0.4, the differences from the resulting predictions were relatively small for the bending SPF3-0.4 00.10.20.30.40.50.60.70.80.910 50000 100000 150000 200000CDFTorque Load (N.mm)Torque SPF3SPF3-TESTSPF3-MODEL52.2% error at the 50th% level 28.3% error at the 5th% level 9.5% error at the 50th% level 28.5% error at the 5th% level 135 group as shown in Figure 5.26, but the prediction was not accurate for the torque SPF5-0.4 and torque SPF3-0.4 groups, as shown in Figure 5.24 and Figure 5.25 with a significant difference. Figure 5.27 Size effect model calibrated by bending SPF3-0.4 test data Figure 5.28 Size effect model prediction for torque SPF5-0.4 test data (calibrated by bending SPF3-0.4 data) 2.6% error at the 50th% level 11.4% error at the 5th% level 54.7% error at the 50th% level 14.9% error at the 5th% level 136 Figure 5.29 Size effect model prediction for torque SPF3-0.4 test data (calibrated by bending SPF3-0.4 data) Figure 5.30 Size effect model prediction for bending SPF5-0.4 test data (calibrated by bending SPF3-0.4 data) However, when calibration was based on the bending SPF3-0.4 test data, meaning there was one fairly good fit in the calibration as shown in Figure 5.27 by calibrating the material constants, the differences from the predictions were large for the torque SPF5-0.4, torque SPF3-0.4 and bending SPF5-0.4 groups, as shown in Figure 5.28 to Figure 5.30. The size effect model calibrated from bending SPF3-0.4 test data did not predict good approximations for other torque and bending groups. 00.10.20.30.40.50.60.70.80.910 50000 100000 150000CDFTorque Load (N.mm)Torque SPF3SPF3-TESTSPF3-MODEL37.5% error at the 50th% level 47.4% error at the 5th% level 9.9% error at the 50th% level 17.2% error at the 5th% level 137 Table 5.2 Size effect model calibration results Test type which the calibration is based on Material constants Scale m (𝑁 𝑚2⁄ ) Shape k Torque SPF5-0.4 423510.7 6.600 Torque SPF3-0.4 400851.3 6.500 Bending SPF5-0.4 622337.6 9.000 Bending SPF3-0.4 255326.9 4.375 Therefore, if the calibration was based on the torque test groups (torque SPF5-0.4 and torque SPF3-0.4), the predictions agreed well with different test groups. The size effect model worked well on the torque test groups in these calibrations, as shown in Figure 5.15 to Figure 5.18 and in Figure 5.19 to Figure 5.22. One reason for the good predictions of the torque test based size effect model is that rolling shear failures were observed only in the torque tests as discussed in Section 4.4.3. The predictions based on calibrations using the bending SPF5-0.4 and bending SPF3-0.4 test data however were not very accurate, especially when the calibration was based on the bending SPF3-0.4 test data, as shown in Figure 5.28 to Figure 5.30. The result from the calibration based on the bending SPF5-0.4 was also not good as shown in Figure 5.24 and Figure 5.25. Therefore, the size effect model did not predict very well when the calibration was based on the bending test data. 138 This might be due to the recorded rolling shear failure in the bending tests was influenced by the tension perpendicular to grain behaviour. The combined stress state in the bending case was not considered in this study and was not considered in size effect model thus leading to the poor predicted response. Again for the torque tests, there was rolling shear failure observed only in wood material. When the calibration was based on one torque test group, for example, with calibration from the torque SPF5-0.4 group as shown in Figure 5.15 to Figure 5.18, there was a good agreement for the torque SPF3-0.4 group; there was also fairly good agreement for the other two bending test groups. The differences between the predictions and the test data from these two bending tests in Figure 5.17 and Figure 5.18 probably came from the interaction between the rolling shear and the tension perpendicular to grain failure modes in the bending tests. Therefore, this result suggests that the torque test on CLT tube specimen is a viable option for the investigation of the size effect on rolling shear strength of CLT elements, because the size effect model calibrated by the torque tests provided good predictions for the bending tests. Particularly because the failure mode in the bending specimens is complicated, and the rolling shear failure mode can be combined with tension perpendicular to grain behaviour. Models calibrated by the torque tests provided good predictions for the bending tests. Although not perfect, the predictions are better than the results from the bending calibration with the interaction of the different failure modes. In the future research, it is suggested to investigate the combined failure criterion for the bending specimens regarding the combination of rolling shear 139 and tension perpendicular to grain behaviour; the research on this combined failure criterion and the complicated failure mode is not within the scope and objective of this thesis. Therefore, it is recommended to use the calibrated parameters from the torque tests to evaluate the rolling shear capacity of CLT elements. Calibration using either of the torque groups, the torque SPF5-0.4 group or the torque SPF3-0.4 group, allowed for a reasonable prediction of the capacities of the other torque group as well as the other two bending tests. The calibrated material constants from the torque SPF3-0.4 and torque SPF5-0.4 tests, or the average of both cases, could provide good predictions that fit relatively well with the test measurements. The average material constants are shown in Table 5.3. Based on these average constants, Figure 5.31 to Figure 5.34 show the fitting results for the different test data sets. The results show a good fit for each case. By applying Weibull’s theory of brittle fracture to determine the CLT rolling shear strength, the size effect model, which takes into account different material stressed volumes and stress distributions, can be used to explain the difference of the rolling shear strength between the beam specimens and the torque tube specimens. Table 5.3 Average material constants for the size effect model Material constants Scale m (𝑁 𝑚2⁄ ) Shape k Average 412181.0 6.550 140 Figure 5.31 Size effect model prediction for torque SPF5-0.4 test data (average material constants) Figure 5.32 Size effect model prediction for torque SPF3-0.4 test data (average material constants) 00.10.20.30.40.50.60.70.80.910 50000 100000 150000CDFTorque Load (N.mm)Torque SPF3SPF3-TESTSPF3-MODEL1.3% error at the 50th% level 8.1% error at the 5th% level 0.8% error at the 50th% level 15.0% error at the 5th% level 141 Figure 5.33 Size effect model prediction for bending SPF5-0.4 test data (average material constants) Figure 5.34 Size effect model prediction for bending SPF3-0.4 test data (average material constants) Weibull’s theory can also be applied to CLT engineering design purposes in the future, and the results summarized in Table 5.3 are recommended during rolling shear strength design. For example, with these same material constants (i.e., 𝑚 and 𝑘 in Table 5.3) for the unit volume material, the CLT structural rolling shear capacity distribution may be determined in the case of a two-dimensional panel under a uniformly distributed load. 6.6% error at the 50th% level 14.1% error at the 5th% level 2.4% error at the 50th% level 2.7% error at the 5th% level 142 This application of Weibull’s theory can be summarized as follows: First, the rolling shear strength distribution for the unit volume material can be developed from the material constants and Equation ( 5.4 ); and, then, in order to calculate the rolling shear load-carrying capacity distribution, a finite element analysis can be performed to give the rolling shear stress distribution information in the CLT components, contributing to the integral calculation in Equation ( 5.5 ) giving the load capacities. Therefore, for different cases of interest, Weibull’s theory of brittle fracture is one recommended approach in the CLT rolling shear capacity design process. 5.3.3 Examination of the Possibility of Longitudinal Shear Failure in Torque Loading Tests Based on Size Effect Model Based on the research performed on the size effect of longitudinal shear strength of Canadian softwood structural lumber (Madsen, 1992; Foschi, 1989; Foschi and Barrett, 1976; Lam et al., 1997), and based on the finite element analysis on the longitudinal shear stress distribution in CLT torque specimens (in Figure 4.25 and Figure 4.28), the longitudinal shear behaviour in the tube specimens can be predicted by applying Weibull’s theory of brittle fracture. This application of Weibull’s size effect theory can be summarized as follows: First, the longitudinal shear strength distribution for the unit volume material can be developed from the material constants and Equation ( 5.4 ) (Foschi and Barrett, 1976; Foschi, 1989; Lam et al., 1997), and these material constants were based on the previous test results on the longitudinal shear strength of Spruce-Pine-Fir (SPF) species, as given in Table 5.4 (Lam et al., 1997). In 143 order to calculate the longitudinal shear load-carrying capacity distribution, a finite element analysis was then performed to give the longitudinal shear stress distribution information in CLT torque components, as shown in Figure 4.25 and Figure 4.28, contributing to the integral calculation in Equation ( 5.5 ). The predicted longitudinal shear load-carrying capacity can then be calculated from Equation ( 5.5 ). Table 5.4 Material constants for the longitudinal shear strength of spruce-pine-fir in size effect model Material constants (Lam et al., 1997) Longitudinal shear strength of a unit volume (𝑖𝑛3) Median 𝜏0.5∗ (𝑀𝑃𝑎) Shape k 17.76 6.95 The results are summarized in Figure 5.35, Figure 5.36 and Table 5.5. They show that the mean values of the predicted longitudinal shear load-carrying capacities for CLT torque tube specimens are higher (about three times more) than the recorded test peak failure loads; and, the coefficient of variation (15.29% to 15.52%) is close to that of the test data (16.40% to 16.61%). Therefore, since the predicted longitudinal shear load-carrying capacity is significantly higher than the measurements, the shear failure mode is very unlikely to be longitudinal shear. These results suggest that CLT tube specimens were showing the rolling shear failure in the tests. 144 Figure 5.35 Predicted longitudinal shear load-carrying capacity distribution in SPF3 tube specimen Figure 5.36 Predicted longitudinal shear load-carrying capacity distribution in SPF5 tube specimen 145 Table 5.5 Predicted longitudinal shear load-carrying capacity in CLT tube specimens CLT Predicted longitudinal shear load-carrying capacity (N×mm) Peak failure torque load in tests (N×mm) Mean COV Mean COV Torque SPF3-0.4 380753 15.29% 103989 16.40% Torque SPF5-0.4 381546 15.52% 84561 16.61% 5.4 Conclusion The results from the CLT ramp loading and trapezoidal fatigue loading tests were well interpreted by the finite element models coupled with customized Fortran subroutines; and, the rolling shear strength was evaluated according to the test data. The size effect on the rolling shear strength was then investigated, based on the ramp loading and torque loading test measurements. The material constants in the size effect modeling process were calibrated by part of the experimental data, and the model was able to predict the CLT rolling shear strength in the other part of the experimental tests. The size effect model, which takes into account the different material stressed volume and the different stress distribution, explained the difference of the rolling shear strength between the beam specimens and the torque tube specimens; therefore, this model provides a good tool to predict the rolling shear capacity in CLT’s engineering applications. 146 Chapter 6: Calibration and Verification of CLT Damage Accumulation Model 6.1 Summary A stress-based damage accumulation model is chosen to evaluate the duration-of-load effect on the rolling shear strength of CLT (cross laminated timber). This model incorporates the established short-term rolling shear strength and predicts the time to failure under arbitrary loading history. For each configuration of the specimens (i.e., the three- or five-layer CLT products), calibration of the damage accumulation model is based on one set of the trapezoidal fatigue test data, and the prediction of this model is verified with the other set of the trapezoidal test data. The long-term rolling shear behaviour of CLT can be further evaluated from this verified model. As the damage accumulation model is a probabilistic model, it can be incorporated into a time-reliability study. Therefore, a reliability assessment of the CLT products is performed. Based on the experimental and modeling results, the duration-of-load adjustment factor for the CLT rolling shear strength is discussed. The results suggest that the duration-of-load strength adjustment factor for CLT is more severe than the general duration-of-load adjustment factor for lumber; and, this difference should be considered when introducing CLT into the building codes for engineered wood design. 147 6.2 Introduction The damage accumulation model is a well-established tool to predict the duration-of-load effect in lumber. Based on the duration-of-load test results, the output of the selected stress-based damage accumulation model, after calibration and verification, could provide predictions of the long-term performance of CLT products under arbitrary loading protocol. A time-reliability study of CLT was incorporated, considering short-term, snow and dead load only loading cases. The reliability analysis results and the factors reflecting the duration-of-load effect on the rolling shear strength of CLT are compared and discussed. 6.3 Calibration of Damage Accumulation Model 6.3.1 Model Calibration The stress-based damage accumulation model was calibrated against the trapezoidal fatigue test data from both the SPF5-0.4 group and the SPF3-0.4 group, and the calibration program was developed in Fortran codes. This calibration procedure was based on the damage accumulation theory and the algorithm developed by Foschi (1989). The lognormal distributed rolling shear strength 𝜎𝑠 was based on the maximum cross layer rolling shear stresses calculated from the finite element model (introduced in Chapter 5) with consideration of the influence of higher loading rate, which used each individual ramp rolling shear failure load as the load input. The short-term rolling shear strength in Table 5.1 was then corrected with a 15% strength increase due to the higher loading rate for modeling purpose (Madsen, 1992). The applied stress history 148 𝜎(𝑡) was evaluated by the finite element models as well, as shown in Table 5.1. The random parameters (i.e., 𝑏, 𝑐, 𝑛 and 𝜏0 in the damage accumulation model) were assumed to be lognormally distributed. By employing a nonlinear function minimization procedure using the quasi-Newton method, the mean value and standard deviation of the lognormal distribution for each model parameter were estimated. Therefore, a vector, X: 𝑋 = [𝑀𝑒𝑎𝑛𝑏 , 𝑆𝑡𝑑𝑣𝑏 , 𝑀𝑒𝑎𝑛𝑐, 𝑆𝑡𝑑𝑣𝑐 , 𝑀𝑒𝑎𝑛𝑛, 𝑆𝑡𝑑𝑣𝑛 , 𝑀𝑒𝑎𝑛𝜏0 , 𝑆𝑡𝑑𝑣𝜏0]𝑇 ( 6.1 ) with unknowns matching the mean value and standard deviation of these independent model parameters, could be found through the minimization procedure. The procedure programmed in Fortran codes is summarized as follows: 1) First, initial estimates of the lognormal distributions in the X vector were provided. 2) Based on the initial distribution parameters, a random sample size of 𝑁𝑅𝑛𝑑 = 2500 numbers of cycles to failure (𝑁𝑓 values) was generated. 3) The cumulative distribution of the randomly generated 𝑁𝑓 values was obtained and compared to the experimental data by computing a objective function (i.e., 𝜙) as: 𝜙 =∑(1.0 −𝑁𝑓𝑖𝑠𝑁𝑓𝑖𝑎)2𝐿𝑖=1 ( 6.2 ) where 𝐿 is the number of probability levels considered, 𝑁𝑓𝑖𝑠 from Equation ( 3.36 ) is the simulated number of cycles to failure at the 𝑖𝑡ℎ probability level, and 𝑁𝑓𝑖𝑎 is the actual number of cycles to failure obtained from the experimental data at the same probability level. 4) Equation ( 6.2 ) was minimized according to the quasi-Newton method. The initially chosen mean value and standard deviation of the lognormal distribution for each model parameter were 149 modified automatically through an iteration procedure. The convergence criterion for the objective function was set to 𝜀 = 0.001. The increment was defined as ∆𝑋𝑖 = 0.01𝑋𝑖. An optimal solution was considered to be found when changes in 𝑋𝑖 of a magnitude ∆𝑋𝑖, did not reduce the objective function value between two consecutive iterations more than 𝜀𝜙. 6.3.2 Calibration Results The stress-based damage accumulation model was calibrated against the trapezoidal long plateau test data from the SPF5-0.4 group, and the calibrated results for each model parameter are shown in Table 6.1. Then, based on the calibrated model parameters, simulated 𝑁𝑓 values were produced and compared to the experimental data. Figure 6.1 shows the relationships between the number of cycles to failure and the stress ratio of the experimental and the simulated 𝑁𝑓 values (in the logarithm to base 10) based on the model calibration. Table 6.1 Summary of the calibration results in the five-layer CLT Parameter Mean STDV 𝑏 39.857 2.219 𝑐 3.483 × 10−3 2.446 × 10−3 𝑛 6.754 0.117 𝜏0 0.194 0.247 150 Figure 6.1 Relationships between the number of cycles to failure (in the logarithm to base 10) and the stress ratio from test and model in five-layer CLT Figure 6.2 shows the cumulative distributions of the experimental data and the simulated 𝐿𝑜𝑔(𝑁𝑓) values based on the model calibration. The results from Figure 6.1 and Figure 6.2 show that the calibration fits the test data well. 151 Figure 6.2 Cumulative distributions of the experimental and simulated number of cycles to failure (in the logarithm to base 10) in five-layer CLT Figure 6.2 shows the CDFs of 𝐿𝑜𝑔(𝑁𝑓) values. It shows there are small differences between the calibrated model output and the test measurement when log(𝑁𝑓) < 0.7, i.e., 𝑁𝑓 < 5. However, the model fitted well with the test data when log(𝑁𝑓) > 0.7. At the lower tail when 𝑁𝑓 is small, the difference between the model output and the test results has been magnified by the logarithm scale. To clarify and better understand the fitting results, Figure 6.3 and Figure 6.4 show the calibration results with horizontal axis as 𝑁𝑓 rather than in the logarithm scale. These results suggest the calibration agrees with the test data well. Also, the other reason for the small differences between the fitting and test data when log(𝑁𝑓) <0.7 is that, in the lower tail of the distribution (when 𝑁𝑓 < 5) in Figure 6.4, when the calibration 152 was performed at the time basis, there was uncertainty of how to define the specific time to failure point within the 𝑁𝑓𝑡ℎ cycle, because only 𝑁𝑓 value (in Table 4.9) with rolling shear failure was measured in the tests. However, as the 𝑁𝑓 increases, this error becomes trivial because the 𝑁𝑓 value is much larger. For example, when 𝑁𝑓 = 100, this error is only less than 1/100=1%. In general, the fitting was also quite acceptable in the upper tail; therefore, it is a viable option to investigate the duration-of-load behaviour based on the measured number of cycles to failure, which is also in time scale basis (Norlin, 1997). Figure 6.3 Relationships between the number of cycles to failure (not in the logarithm scale) and the stress ratio from test and model in five-layer CLT 153 Figure 6.4 Cumulative distributions of the experimental and simulated number of cycles to failure (not in the logarithm scale) in five-layer CLT In the trapezoidal loading tests, the recorded 𝑁𝑓 value (in Table 4.9) for the test specimens with rolling shear failure represents the time to failure, which failed within that 𝑁𝑓𝑡ℎ cycle in time scale. Similarly, in the model calibration process, the randomly simulated 𝑁𝑓 values were generated based on Equation ( 3.36 ), representing the approximations of the time to failure, because Equation ( 3.36 ) from the damage accumulation model is derived based on the function of time, meaning that the damage accumulation model is developed on the time basis as well. To calibrate on the basis of time scale, the simulated time to failure was based on the recorded number of cycles to failure test data. There was uncertainty in it, because there was no measurement of the time to failure within the last cycle of the test. However, this error was only 154 significant when failure occurred in the smaller number of cycles to failure, for example the first or the second cycle. This error becomes less significant when the recorded number of cycles to failure was large. Therefore, the calibrated damage accumulation model was able to estimate the time to failure of specimens under different long-term loading protocols. Also, as discussed in Section 4.4.4, because of the inaccuracy of the assigned 𝑁𝑓 value for test specimens that failed during the first uploading sequence (𝑁𝑓 < 1), these specimens were not shown in the cumulative distribution plots in Figure 6.2; furthermore, the rank of cumulative probability of the specimens that failed beyond the initial uploading was calculated based on the whole dataset including the data points not shown. To be consistent, the simulated 𝑁𝑓 values (model output under the trapezoidal loading protocol) referring to the specimens failed during uploading (𝑁𝑓 < 1) were also not included in the cumulative distribution plots in Figure 6.2, but the value of the simulated specimens that failed beyond uploading was calculated based on the whole predicted dataset including the uploading failure specimens. For example, from Table 4.9 for the five-layer long plateau test, the first test data point in Figure 6.2 started from the rank of 1/(29+1)=3.3%. As shown in Figure 6.2 for the five-layer long plateau test group, the first model predicted data point in the figure started from the rank of 4%, which is the same as the test results. As discussed in Section 4.4.4, less than 25% of the test specimens failed in the first uploading process when the 25th percentile rolling shear failure loads from the ramp tests were applied in the trapezoidal fatigue tests. This difference was attributed to the use of a significantly higher rate of loading in the fatigue trapezoidal tests compared to the short-term tests; i.e., with a higher 155 rate of loading it is expected that the apparent short-term strength would increase. With the increased short-term strength under the higher loading rate, for the same applied plateau level (25th percentile rolling shear failure loads), the rank of the uploading failure specimen is expected to be lower than 25th percentile, meaning there are fewer specimens failed during the uploading segment, as the test data suggested. Therefore, from the fitting results in Figure 6.2, only 4% of the simulated specimens failed in the first uploading process when the 25th percentile rolling shear failure loads from the ramp tests were applied. The simulation result was also close to the test results with rank of 3.3% from the uploading failure test specimens, as shown in Figure 6.2 and Figure 6.4. These simulation results from the calibrated damage accumulation model further confirmed the influence of rate of loading on the short-term strength. The damage accumulation model was also calibrated against the trapezoidal short plateau test data from the SPF3-0.4 group. Table 6.2 gives the mean and standard deviation values of the calibrated model parameters. Simulated 𝑁𝑓 values were produced and compared to the experimental data, as shown in Figure 6.5 and Figure 6.6. From the calibration results, the mean value of threshold ratio (𝜏0 = 0.194) in SPF5-0.4 is higher than that (𝜏0 = 0.059) in SPF3-0.4. In Figure 6.6, from Table 4.9 for the three-layer short plateau test, the first test data point in Figure 6.6 started from the rank of 1/(30+1)=3.2%. As shown in Figure 6.6 for the three-layer short plateau test group, the first model predicted data point started from the rank of 9%, which is relatively higher than the test data. However, similarly as discussed and shown in Figure 6.4 before, this difference is not so significant considering the general good fitting in Figure 6.6. 156 Table 6.2 Summary of the calibration results in the three-layer CLT Parameter Mean STDV 𝑏 257.249 229.738 𝑐 9.861 × 10−2 1.104 × 10−5 𝑛 14.911 0.045 𝜏0 0.059 0.001 Figure 6.5 Relationships between the number of cycles to failure (in the logarithm to base 10) and the stress ratio from test and model in three-layer CLT 157 Figure 6.6 Cumulative distributions of the experimental and simulated number of cycles to failure (in the logarithm to base 10) in three-layer CLT 6.4 Verification of Damage Accumulation Model The stress-based damage accumulation model was verified against the trapezoidal short plateau test data from the SPF5-0.4 group. Figure 6.7 shows the relationships between the number of cycles to failure and the stress ratio of the experimental and the simulated 𝑁𝑓 values. Figure 6.8 gives the cumulative distribution results based on the model calibration and the model verification. The prediction from the calibrated model agrees well with the test data, as shown in Figure 6.7. Also, from Table 4.9 for the five-layer short plateau test, the first test data point in Figure 6.8 started from the rank of 3/(28+1)=10.3%. As shown in Figure 6.8 for the five-layer short plateau test group, the first model predicted data point in the figure started from the rank of 4%. 158 Figure 6.7 Relationships between the number of cycles to failure (in the logarithm to base 10) and the stress ratio from test and model in five-layer CLT 159 Figure 6.8 Cumulative distributions of the experimental and simulated number of cycles to failure (in the logarithm to base 10) in five-layer CLT Similarly, for the SPF3-0.4 group, the damage accumulation model was verified against the long plateau test data. Figure 6.9 and Figure 6.10 include the relationships between the test data and the model calibration and verification results. As shown in Figure 6.9, the verification results agreed reasonably well with the test measurement. Again in Figure 6.8 and Figure 6.10, because of the inaccuracy of the assigned 𝑁𝑓 values, the cumulative distribution results do not show the specimens with uploading failure in the first cycle. From Table 4.9 for the three-layer long plateau test, the first test data point in Figure 6.10 started from the rank of 5/(32+1)=15.1%. As shown in Figure 6.10 for the three-layer long plateau test group, the first model predicted data point in the figure started from the rank of 9%, which is close to the test results. 160 Figure 6.9 Relationships between the number of cycles to failure (in the logarithm to base 10) and the stress ratio from test and model in three-layer CLT Figure 6.10 Cumulative distributions of the experimental and simulated number of cycles to failure (in the logarithm to base 10) in three-layer CLT 161 6.5 Duration-of-Load Factor 6.5.1 Reliability Analysis of Short-Term Rolling Shear Strength of CLT Section 6.5.1 introduces the reliability analysis on the limit state of the short-term rolling shear strength of CLT products, when the duration-of-load effect is not considered. The objective of this reliability analysis is to evaluate the relationship between the reliability index and the performance factor in the design codes. To clarify, the reliability analysis with consideration of the effect of load duration on the rolling shear strength will be addressed in Section 6.5.2. First, based on the ultimate strength limit state design equation from the design code: 1.25𝐷𝑛 + 1.50𝑄𝑛 = 𝜙𝑅𝑆(0.05)𝑇𝑉 ( 6.3 ) where 𝐷𝑛 is the design dead load which is normally computed using average weights of materials, and 𝑄𝑛 is the design live load which, in the case of snow plus associated rain for example, is taken from the distributions of annual maxima and corresponds to loads with a 1/30 probability of being exceeded (i.e., 30 years return); 𝑇𝑉 is defined as the ratio between the sectional load-carrying capacity calculated from different beam theories (i.e., the layered beam theory, the gamma beam theory and the shear analogy theory) and the shear stress value; and, 𝜙 is the performance factor applied to the characteristic strength (i.e., 𝑅𝑆(0.05)). This characteristic rolling shear strength is chosen to be the parametric 5th percentile rolling shear stress value with consideration of the influence of higher loading rate (consistent with the model calibration process in Section 6.3.1), as obtained from the finite element evaluation results on the rolling 162 shear strength (as shown in Table 5.1) corrected with the expected 15% strength increase due to the higher loading rate for modeling purpose (Madsen, 1992). Therefore, 𝑅𝑆(0.05) is not dependent on the ratio 𝑇𝑉 used. The information of this ratio 𝑇𝑉 is introduced in detail in Appendix N. From Equation ( 6.3 ), the performance factor 𝜙 will affect reliability index 𝛽. For instance, with a given 𝜙, the performance function G for the calculation of the reliability index 𝛽 is: 𝐺 = 𝑅 − (𝐷 + 𝑄) ( 6.4 ) in which, R is the random variable related to the rolling shear load-carrying capacity (based on the observation from the short-term ramp loading tests in Chapter 4) corrected with the expected 15% strength increase due to the higher loading rate for modeling purpose, which is consistent with the term 𝑅𝑆(0.05) in Equation ( 6.3 ). 𝐷 is the random dead load, and 𝑄 is the random live load. Then, the ratio of the design dead load to the design live load is defined as ( here chosen to be 0.25 ): 𝑟 =𝐷𝑛𝑄𝑛 ( 6.5 ) Therefore, the performance function G is: 𝐺 = 𝑅 −𝜙𝑅𝑆(0.05)𝑇𝑉(1.25𝑟 + 1.50)(𝑑𝑟 + 𝑞) ( 6.6 ) where the random variables d and q are: 𝑑 =𝐷𝐷𝑛 𝑞 =𝑄𝑄𝑛 163 With regard to the short-term rolling shear strength design method for the CLT beam under the concentrated load, three beam theories (i.e., the layered beam theory, the gamma beam theory and the shear analogy theory) were adopted in the reliability analysis. Also, two different site snow loads (i.e., snow loads in Halifax and Vancouver) were investigated and introduced in the following reliability analysis process. The snow load information comes from the statistics on the maximum annual snow depth, the snow duration and the ground-to-roof snow conversion factors provided by the National Research Council of Canada (Foschi, 1989). This snow load information for Halifax and Vancouver is introduced in detail in Appendix O. The objective of this reliability analysis, adopting the First Order Reliability Method (FORM), is to evaluate the relationship between the reliability index 𝛽 and the performance factor 𝜙. Table 6.3 gives the results on the relationship between 𝛽 and 𝜙 in five-layer CLT products under the different snow load cases, and these results are different because of the different assumptions (from the different beam theories) about the ratio 𝑇𝑉, which is defined in Equation ( 6.3 ). The average 𝛽 in this table is summarized from the 𝛽 calculated from the different beam theories to get an average estimation over the error from the different assumptions. Curves with the same information in Table 6.3 are displayed in Figure 6.11 and Figure 6.12. Similarly, the reliability analysis results without considering the duration-of-load effect for the three-layer CLT can be found in Table 6.4, Figure 6.13 and Figure 6.14. 164 Table 6.3 Results between the reliability index and the performance factor in the five-layer CLT Five-layer Using the 𝑇𝑉 from Layered beam Using the 𝑇𝑉 from Gamma beam Using the 𝑇𝑉 from Shear analogy Average 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 Halifax 0.6 3.538 0.6 3.423 0.6 3.449 0.6 3.470 0.7 3.230 0.7 3.112 0.7 3.138 0.7 3.160 0.8 2.956 0.8 2.836 0.8 2.863 0.8 2.885 0.9 2.710 0.9 2.587 0.9 2.614 0.9 2.637 1.0 2.485 1.0 2.360 1.0 2.388 1.0 2.411 1.1 2.278 1.1 2.151 1.1 2.179 1.1 2.203 1.2 2.086 1.2 1.956 1.2 1.985 1.2 2.009 1.3 1.906 1.3 1.774 1.3 1.803 1.3 1.828 Vancouver 0.6 3.760 0.6 3.629 0.6 3.658 0.6 3.682 0.7 3.403 0.7 3.262 0.7 3.293 0.7 3.319 0.8 3.071 0.8 2.920 0.8 2.954 0.8 2.982 0.9 2.760 0.9 2.600 0.9 2.636 0.9 2.665 1.0 2.466 1.0 2.298 1.0 2.335 1.0 2.366 1.1 2.187 1.1 2.012 1.1 2.051 1.1 2.083 1.2 1.922 1.2 1.740 1.2 1.780 1.2 1.814 1.3 1.668 1.3 1.480 1.3 1.521 1.3 1.556 165 Figure 6.11 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Five-layer/Halifax) Figure 6.12 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Five-layer/Vancouver) 166 Table 6.4 Results between the reliability index and the performance factor in the three-layer CLT Three-layer Using the 𝑇𝑉 from Layered beam Using the 𝑇𝑉 from Gamma beam Using the 𝑇𝑉 from Shear analogy Average 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 Halifax 0.6 3.474 0.6 3.044 0.6 3.474 0.6 3.331 0.7 3.267 0.7 2.815 0.7 3.267 0.7 3.116 0.8 3.080 0.8 2.607 0.8 3.080 0.8 2.922 0.9 2.907 0.9 2.415 0.9 2.907 0.9 2.743 1.0 2.748 1.0 2.237 1.0 2.748 1.0 2.578 1.1 2.598 1.1 2.070 1.1 2.598 1.1 2.422 1.2 2.457 1.2 1.912 1.2 2.457 1.2 2.275 1.3 2.323 1.3 1.763 1.3 2.323 1.3 2.136 Vancouver 0.6 3.340 0.6 2.883 0.6 3.340 0.6 3.188 0.7 3.122 0.7 2.636 0.7 3.122 0.7 2.960 0.8 2.922 0.8 2.406 0.8 2.922 0.8 2.750 0.9 2.736 0.9 2.191 0.9 2.736 0.9 2.554 1.0 2.562 1.0 1.986 1.0 2.562 1.0 2.370 1.1 2.396 1.1 1.790 1.1 2.396 1.1 2.194 1.2 2.238 1.2 1.601 1.2 2.238 1.2 2.026 1.3 2.085 1.3 1.419 1.3 2.085 1.3 1.863 167 Figure 6.13 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Three-layer/Halifax) Figure 6.14 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Three-layer/Vancouver) From the results in Table 6.3 and Table 6.4, using different beam theories (i.e., the layered beam theory, the gamma beam theory and the shear analogy theory), the calculated reliability index 𝛽 is different. This difference comes from the different interpretations of the defined term 𝑇𝑉 in 168 Equation ( 6.6 ), and this 𝑇𝑉 is different when different beam theories are adopted. Also, Equation ( 6.6 ), i.e., the performance function, depends only on the random variables d and q in the reliability calculation. Therefore, to clarify, the reliability analysis process is independent of the chosen beam theory for defining the term 𝑇𝑉 in Equation ( 6.6 ). From Figure 6.15 which shows the average curves between the reliability indices and the performance factors from the previous results, in different snow load cases when the performance factor is less than 0.8, the probability of rolling shear failure in the three-layer CLT is higher than that in five-layer products. This is consistent with the short-term ramp loading test results, where the three-layer CLT products showed the lower rolling shear load-carrying capacity. Also, this reliability analysis on the short-term rolling shear strength will provide necessary information for the following investigation on duration of load. Figure 6.15 Average curves between the reliability index and the performance factor without considering the duration-of-load effect 169 6.5.2 Reliability Analysis of CLT Rolling Shear Strength under Thirty-Year Snow Load Section 6.5.2 will introduce the reliability analysis on the limit state of CLT products under a thirty-year snow load, with consideration of load duration effect on the rolling shear strength. The objective of this reliability analysis is to evaluate the relationship between the reliability index and the performance factor, when duration-of-load effect is included. A Monte Carlo simulation procedure, incorporating the verified damage accumulation model in Sections 6.3 and 6.4, was used to determine the probability of the rolling shear failure of a single bending CLT beam specimen under load for a prescribed service life (Foschi, 1989). Then, based on the previous results from the short-term rolling shear strength reliability analysis (without considering the duration-of-load effect) as shown in Table 6.3 and Table 6.4, the duration-of-load adjustment factor for the rolling shear strength can be obtained with one margin of safety. The Monte Carlo simulation was used to determine the probability of rolling shear failure for a service life ranging from one year to thirty years, considering the duration-of-load effect on the rolling shear strength. Based on the verified model in Table 6.1 and Table 6.2 (from Sections 6.3 and 6.4), a sample size of NR=1000 replications was chosen. Then, these simulated samples were tested under the thirty-year snow loading history as introduced in Appendix O (Foschi, 1989). Consistent with the reliability analysis procedure in Section 6.5.1, two different site snow loads (i.e., snow loads in Halifax and Vancouver) were considered. This snow load information for Halifax and Vancouver is introduced in detail in Appendix O. Dead load was also included in the service life. Then, the performance function G is: 170 𝐺 = 1 − 𝛼 ( 6.7 ) where 𝛼 is the damage parameter from the damage accumulation model. If 𝐺>0, the sample will survive. If 𝐺 <0, the sample will fail. After performing the Monte Carlo simulation, the relationship between the reliability index and the performance factor can be evaluated. The duration-of-load strength adjustment factor 𝐾𝐷 can then be derived. The basic determination procedure for this factor is shown in Figure 6.16. In this figure, two cases are displayed for the relationship between the reliability index 𝛽 and the performance factor 𝜙. The first case in the figure is known as curve one, when the duration-of-load effect is not considered and only the short-term rolling shear strength is analyzed. This information comes from the previous reliability analysis on the short-term rolling shear strength of CLT, i.e., the results of the average reliability index 𝛽 from Table 6.3 and Table 6.4. The second curve, i.e., curve two in Figure 6.16, includes the performed Monte Carlo simulation results, and the duration-of-load effect has been taken into account. Based on the Monte Carlo simulation results (points in curve two), curve two is calculated from the exponential regression fitting method. 171 Figure 6.16 The basic factor determination procedure (curve one-without DOL effect; curve two-with DOL effect) As shown in Figure 6.16, at the same reliability index level 𝛽 (the target reliability), the performance factor for curve one is defined as 𝜙𝐼, and 𝜙𝐼𝐼 is the factor calculated from curve two. Then the strength adjustment factor 𝐾𝐷 for the rolling shear strength is defined as: 𝐾𝐷 =𝜙𝐼𝐼𝜙𝐼 ( 6.8 ) For example, Table 6.5 below shows the relationship between the reliability index 𝛽 and the performance factor 𝜙 in the five-layer CLT products, for both curve one and curve two. These results are also included in Figure 6.17 and Figure 6.18 for different snow load cases. In Table 6.5, it shows that the 𝜙𝐼 factor is around 0.9, at a reliability target index 𝛽 = 2.80. For the short-term bending strength of lumber in the Canadian design code, the performance factor is 𝜙 = 0.9. Therefore, the obtained 𝜙𝐼 in Table 6.5 for CLT is close to the 𝜙 in the code for lumber. 172 Table 6.5 Curve one and curve two results between the reliability index and the performance factor in the five-layer CLT Five-layer Curve one (without DOL effect) Curve two (with DOL effect) 𝜙 𝛽 𝜙 𝛽 Halifax 0.6 3.470 0.35 3.090 0.7 3.160 0.4 2.652 0.8 2.885 0.5 2.257 0.9 2.637 1.0 2.411 1.1 2.203 1.2 2.009 1.3 1.828 Vancouver 0.6 3.682 0.5 2.948 0.7 3.319 0.55 2.687 0.8 2.982 0.6 2.428 0.9 2.665 0.65 2.125 1.0 2.366 1.1 2.083 1.2 1.814 1.3 1.556 173 Figure 6.17 Curves between the reliability index and the performance factor for “Five-layer/Halifax” case (curve one-without DOL effect; curve two-with DOL effect) Figure 6.18 Curves between the reliability index and the performance factor for “Five-layer/Vancouver” case (curve one-without DOL effect; curve two-with DOL effect) 174 Also, the relationship between the reliability index 𝛽 and the performance factor 𝜙 in three-layer CLT products is shown in Table 6.6, and curves with the same information in this table are shown in Figure 6.19 and Figure 6.20. 175 Table 6.6 Curve one and curve two results between the reliability index and the performance factor in the three-layer CLT Three-layer Curve one (without DOL effect) Curve two (with DOL effect) 𝜙 𝛽 𝜙 𝛽 Halifax 0.6 3.331 0.4 2.748 0.7 3.116 0.5 2.409 0.8 2.922 0.6 2.457 0.9 2.743 0.7 1.799 1.0 2.578 1.1 2.422 1.2 2.275 1.3 2.136 Vancouver 0.6 3.188 0.5 2.536 0.7 2.960 0.6 2.418 0.8 2.750 0.7 2.165 0.9 2.554 0.8 1.817 1.0 2.370 1.1 2.194 1.2 2.026 1.3 1.863 176 Figure 6.19 Curves between the reliability index and the performance factor for “Three-layer/Halifax” case (curve one-without DOL effect; curve two-with DOL effect) Figure 6.20 Curves between the reliability index and the performance factor for “Three-layer/Vancouver” case (curve one-without DOL effect; curve two-with DOL effect) Then, from Equation ( 6.8 ), the derived duration-of-load rolling shear strength adjustment factor 𝐾𝐷 is shown in Table 6.7. Take the five-layer CLT under the thirty-year Halifax snow load 177 combined with the dead load case as an example, 𝐾𝐷 is 0.466 when the reliability index 𝛽 =2.80. On the other hand, for the three-layer CLT, Table 6.8 shows that 𝐾𝐷 is around 0.462 under the same circumstances. Table 6.7 Reliability results for the strength adjustment factors in the five-layer CLT Five-layer Reliability results β ϕI ϕII KD Halifax 3.0 0.758 0.354 0.467 2.8 0.834 0.388 0.466 2.5 0.961 0.444 0.463 2.0 1.205 0.554 0.460 Vancouver 3.0 0.794 0.496 0.625 2.8 0.855 0.528 0.617 2.5 0.953 0.580 0.609 2.0 1.131 0.683 0.604 178 Table 6.8 Reliability results for the strength adjustment factors in the three-layer CLT Three-layer Reliability results β ϕI ϕII KD Halifax 3.0 0.760 0.346 0.456 2.8 0.868 0.402 0.462 2.5 1.050 0.492 0.469 2.0 1.398 0.670 0.480 Vancouver 3.0 0.682 0.378 0.553 2.8 0.776 0.440 0.566 2.5 0.929 0.542 0.583 2.0 1.216 0.743 0.611 Based on Equation ( 6.8 ) and the same reliability analysis process in Sections 6.5.1 and 6.5.2, the reliability results for the rolling shear strength adjustment factors are summarized in Table 6.9 and Table 6.10 for another three different locations: Quebec City, Ottawa and Saskatoon (The snow load information for these cities is introduced in detail in Appendix O). Curves with the same information in these tables are shown from Figure 6.21 to Figure 6.26 (The calculated values from Figure 6.21 to Figure 6.26 are given in Appendix Q). 179 Figure 6.21 Curves between the reliability index and the performance factor for “Five-layer/Quebec City” case (curve one-without DOL effect; curve two-with DOL effect) Figure 6.22 Curves between the reliability index and the performance factor for “Five-layer/Ottawa” case (curve one-without DOL effect; curve two-with DOL effect) 180 Figure 6.23 Curves between the reliability index and the performance factor for “Five-layer/Saskatoon” case (curve one-without DOL effect; curve two-with DOL effect) Figure 6.24 Curves between the reliability index and the performance factor for “Three-layer/Quebec City” case (curve one-without DOL effect; curve two-with DOL effect) 181 Figure 6.25 Curves between the reliability index and the performance factor for “Three-layer/Ottawa” case (curve one-without DOL effect; curve two-with DOL effect) Figure 6.26 Curves between the reliability index and the performance factor for “Three-layer/Saskatoon” case (curve one-without DOL effect; curve two-with DOL effect) 182 Table 6.9 Reliability results for the strength adjustment factors in the five-layer CLT for different locations Five-layer Reliability results β ϕI ϕII KD Quebec City 3.0 0.838 0.357 0.426 2.8 0.916 0.392 0.428 2.5 1.041 0.450 0.432 2.0 1.291 0.563 0.436 Ottawa 3.0 0.768 0.334 0.436 2.8 0.844 0.366 0.434 2.5 0.971 0.419 0.431 2.0 1.216 0.522 0.430 Saskatoon 3.0 0.780 0.361 0.463 2.8 0.857 0.394 0.460 2.5 0.984 0.449 0.457 2.0 1.229 0.557 0.453 183 Table 6.10 Reliability results for the strength adjustment factors in the three-layer CLT for different locations Three-layer Reliability results β ϕI ϕII KD Quebec City 3.0 0.810 0.347 0.429 2.8 0.925 0.391 0.422 2.5 1.115 0.462 0.414 2.0 1.469 0.603 0.410 Ottawa 3.0 0.766 0.353 0.461 2.8 0.875 0.392 0.448 2.5 1.058 0.456 0.431 2.0 1.407 0.581 0.413 Saskatoon 3.0 0.774 0.362 0.468 2.8 0.884 0.411 0.464 2.5 1.068 0.490 0.459 2.0 1.419 0.646 0.455 Take the five-layer CLT under the thirty-year snow load combined with the dead load case as an example, 𝐾𝐷 is around 0.428 to 0.617 shown in Table 6.7 and Table 6.9 when the reliability index 𝛽 = 2.80. The factor difference comes from the different snow load in each location, and the average factor from the five cities is 0.481. On the other hand, for the three-layer CLT, Table 6.8 and Table 6.10 show that 𝐾𝐷 is around 0.422 to 0.566 under the same circumstances; the average factor from the five cities is 0.472. In the Canadian design code, for lumber, the factor 𝐾𝐷 is 0.8. Therefore, the results suggest that the duration-of-load strength adjustment factor for 184 rolling shear strength in CLT products seems to be very different from that in lumber. Specifically, the rolling shear duration-of-load strength adjustment factor for CLT was found to be more severe compared to the general duration-of-load adjustment factor for lumber. 6.5.3 Reliability Analysis of CLT Rolling Shear Strength under Thirty-Year Dead Load Only One question raised is what the duration-of-load adjustment factor is for the CLT rolling shear strength under a thirty-year dead load only case. To answer this question, the rolling shear strength adjustment factor 𝐾𝐷 with consideration of the duration-of-load effect for this constant dead loading case was evaluated. The performed procedure for this evaluation was similar to that introduced in Sections 6.5.1 and 6.5.2, except the dead load only case was characterized not only by changing the load factor 1.25 in Equation ( 6.6 ) in Section 6.5.1 and in Equation (O.1) in Appendix O to be 1.40 (considering the different load combination factor from the design code for the dead load only loading case) (CSA, 2009), but also by letting 𝑟 tend to infinity ( here chosen to be 1000) (Foschi, 1989). In the reliability analysis on the short-term rolling shear strength without considering the duration-of-load effect, the performance function G is: 𝐺 = 𝑅 −𝜙𝑅𝑆(0.05)𝑇𝑉(1.40𝑟 + 1.50)(𝑑𝑟 + 𝑞) where 𝑟 = 1000. 185 Consistent with the reliability analysis procedure in Section 6.5.1, three beam theories (i.e., the layered beam theory, the gamma beam theory and the shear analogy theory) were adopted in the reliability analysis on the short-term rolling shear strength without considering the duration-of-load effect. These reliability analysis results are summarized in Table 6.11, Table 6.12, Figure 6.27 and Figure 6.28. Table 6.11 Results between the reliability index and the performance factor in the five-layer CLT CLT Using the 𝑇𝑉 from Layered beam Using the 𝑇𝑉 from Gamma beam Using the 𝑇𝑉 from Shear analogy Average 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 Five-layer 0.6 3.943 0.6 3.813 0.6 3.842 0.6 3.866 0.7 3.589 0.7 3.448 0.7 3.479 0.7 3.505 0.8 3.255 0.8 3.101 0.8 3.136 0.8 3.164 0.9 2.935 0.9 2.767 0.9 2.804 0.9 2.835 1.0 2.622 1.0 2.438 1.0 2.479 1.0 2.513 1.1 2.312 1.1 2.109 1.1 2.154 1.1 2.192 1.2 2.001 1.2 1.776 1.2 1.826 1.2 1.868 1.3 1.685 1.3 1.435 1.3 1.492 1.3 1.537 186 Figure 6.27 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Five-layer/Dead Load Only) Table 6.12 Results between the reliability index and the performance factor in the three-layer CLT CLT Using the 𝑇𝑉 from Layered beam Using the 𝑇𝑉 from Gamma beam Using the 𝑇𝑉 from Shear analogy Average 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 Three-layer 0.6 3.287 0.6 2.819 0.6 3.287 0.6 3.131 0.7 3.064 0.7 2.562 0.7 3.064 0.7 2.897 0.8 2.858 0.8 2.323 0.8 2.858 0.8 2.680 0.9 2.667 0.9 2.095 0.9 2.667 0.9 2.476 1.0 2.485 1.0 1.876 1.0 2.485 1.0 2.282 1.1 2.312 1.1 1.662 1.1 2.312 1.1 2.095 1.2 2.145 1.2 1.453 1.2 2.145 1.2 1.914 1.3 1.983 1.3 1.245 1.3 1.983 1.3 1.737 187 Figure 6.28 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Three-layer/Dead Load Only) Figure 6.29 shows the average curves from the previous results in Table 6.11 and Table 6.12. When the performance factor is less than 1.1, the probability of rolling shear failure in three-layer CLT products is higher than that in the five-layer CLT. This is consistent with the previous test results that the three-layer CLT showed the lower rolling shear load-carrying capacity in the short-term ramp loading tests. 188 Figure 6.29 Average curves between the reliability index and the performance factor without considering the duration-of-load effect Similar to the reliability analysis procedure in Section 6.5.2, the Monte Carlo simulation was used to determine the probability of rolling shear failure for a service life ranging from one year to thirty years on the limit state of CLT products under a thirty-year dead load only case, with consideration of load duration effect on the rolling shear strength. Based on the verified model in Table 6.1 and Table 6.2 (from Sections 6.3 and 6.4), a sample size of NR=1000 replications was chosen. Then, these simulated samples were tested under the thirty-year dead load only loading history as introduced in Appendix O (Foschi, 1989). The simulation of thirty-year dead load history is briefly introduced in the end of Appendix O. Then, the performance function G is: 𝐺 = 1 − 𝛼 where 𝛼 is the damage parameter from the damage accumulation model tested under the dead load only history. If 𝐺>0, the sample will survive. If 𝐺 <0, the sample will fail. 189 After performing the Monte Carlo simulation, the following Monte Carlo simulation results considering the duration-of-load effect are shown in Table 6.13, Table 6.14, Figure 6.30 and Figure 6.31. Table 6.13 Curve one and curve two results between the reliability index and the performance factor in the five-layer CLT CLT Curve one (without DOL effect) Curve two (with DOL effect) 𝜙 𝛽 𝜙 𝛽 Five-layer 0.6 3.866 0.35 2.457 0.7 3.505 0.375 2.197 0.8 3.164 0.4 1.607 0.9 2.835 0.425 1.359 1.0 2.513 1.1 2.192 1.2 1.868 1.3 1.537 190 Figure 6.30 Curves between the reliability index and the performance factor for “Five-layer/Dead Load Only” case (curve one-without DOL effect; curve two-with DOL effect) Table 6.14 Curve one and curve two results between the reliability index and the performance factor in the three-layer CLT CLT Curve one (without DOL effect) Curve two (with DOL effect) 𝜙 𝛽 𝜙 𝛽 Three-layer 0.6 3.131 0.35 2.652 0.7 2.897 0.4 2.457 0.8 2.680 0.5 2.257 0.9 2.476 0.55 1.751 1.0 2.282 1.1 2.095 1.2 1.914 1.3 1.737 191 Figure 6.31 Curves between the reliability index and the performance factor for “Three-layer/Dead Load Only” case (curve one-without DOL effect; curve two-with DOL effect) Based on Equation ( 6.8 ), the rolling shear strength adjustment factor 𝐾𝐷 for the thirty-year dead load application case is summarized in Table 6.15. From this table, 𝐾𝐷 is from 0.371 to 0.445 at a target reliability level 𝛽 = 2.80 for five -layer and three -layer CLT products. This result suggests that there is an approximate 55% to 63% strength reduction from the short-term rolling shear strength when duration-of-load effect is considered in the thirty-year dead load only case. 192 Table 6.15 Reliability results for the strength adjustment factor in the dead load case CLT products Reliability results β ϕI ϕII KD Five-layer 3.0 0.850 0.330 0.388 2.8 0.911 0.338 0.371 2.5 1.004 0.352 0.350 2.0 1.159 0.378 0.326 Three-layer 3.0 0.656 0.294 0.448 2.8 0.745 0.331 0.445 2.5 0.888 0.393 0.443 2.0 1.153 0.515 0.447 6.5.4 Duration-of-Load Factor Based on the Stress Ratio Evaluation Approach from Model Prediction Sections 6.5.1 to 6.5.3 introduced the evaluation of the duration-of-load effect on CLT rolling shear strength based on the reliability analysis methods. The duration-of-load strength adjustment factor can also be evaluated by another approach, i.e., the stress ratio evaluation approach introduced as follows. In Sections 6.3 and 6.4, the damage accumulation model was calibrated and verified with the test data from the short-term ramp and long-term trapezoidal fatigue loading tests, as shown in 193 Figure 6.7 and Figure 6.9. This verified model could be used to predict the duration-of-load behaviour under arbitrary loading history. By modeling duration-of-load behaviour under the long-term constant loading protocol shown in Figure 3.15 (Foschi, 1989), which can simulate a dead load (with ramp loading rate in Figure 3.15 as 37.5 kN/min for five-layer CLT and 27.0 kN/min for three-layer CLT), the rolling shear duration-of-load effect of CLT could also be quantified in terms of the relationship between time to failure and stress ratio. Based on the verified model in Table 6.1 and Table 6.2 (from Sections 6.3 and 6.4), a sample size of NR=500 replications was chosen. Then, these simulated samples were tested under the ramp and constant loading protocol shown in Figure 3.15 (Foschi, 1989). Based on this stress ratio evaluation approach, Figure 6.32 shows the predicted relationship between the time to failure and the stress ratio applied in the verified SPF5-0.4 model. Based on the prediction, Table 6.16 gives the stress ratio corresponding to different load duration cases. Assuming that the factor from the strength in a short-term case (i.e., ten-minute duration) is equal to one, factors can be derived for each load duration case, as shown in Table 6.16 (for example, in standard-term case, 0.3942/0.7967 ≈ 0.49). Similarly, the predicted results for three-layer CLT specimens are shown in Figure 6.33 and Table 6.17. 194 Figure 6.32 Time to failure prediction (minutes in the logarithm to base 10) under the ramp and constant loading protocol in the SPF5-0.4 group Table 6.16 Summary of the factor calculation in the five-layer CLT Loading Duration Log(Time) (in Min) Stress ratio Factor Short term 10 minutes 1.0000 0.7967 1.00 Standard term 3 months 5.1126 0.3942 0.49 Long term 10 years 6.7207 0.3145 0.39 30 years 7.1978 0.2978 0.37 50 years 7.4196 0.2916 0.37 195 Figure 6.33 Time to failure prediction (minutes in the logarithm to base 10) under the ramp and constant loading protocol in the SPF3-0.4 group Table 6.17 Summary of the factor calculation in the three-layer CLT Loading Duration Log(Time) (in Min) Stress ratio Factor Short term 10 minutes 1.0000 0.7050 1.00 Standard term 3 months 5.1126 0.4298 0.61 Long term 10 years 6.7207 0.3575 0.51 30 years 7.1978 0.3387 0.48 50 years 7.4196 0.3315 0.47 Based on the factor calculation from the different verified models shown in Table 6.16 and Table 6.17, the ratio of the thirty-year long-term strength to the ten-minute short-term strength is 37% for five-layer CLT and is 48% for three-layer CLT, which agrees with the results from the 196 reliability analysis in Table 6.15 (between 37% and 45% when β = 2.8). Table 6.18 summarizes the results from Table 6.7, Table 6.8, Table 6.9, Table 6.10 and Table 6.15; and, Table 6.19 includes information from Table 6.16 and Table 6.17. Table 6.18 Summary of the reliability results for the strength adjustment factors in CLT Load case Factor KD when β = 2.8 in the reliability analysis Five-layer Three-layer Thirty-year snow load in Quebec City 0.428 0.422 Thirty-year snow load in Ottawa 0.434 0.448 Thirty-year snow load in Saskatoon 0.460 0.464 Thirty-year snow load in Halifax 0.466 0.462 Thirty-year snow load in Vancouver 0.617 0.566 Thirty-year dead load only 0.371 0.445 Table 6.19 Summary of the factor calculation in CLT Loading Duration Factor between the calculated stress ratio Five-layer Three-layer Short term 10 minutes 1.00 1.00 Standard term 3 months 0.49 0.61 Long term 30 years 0.37 0.48 Based on previous reliability research of the duration-of-load effect on lumber (Foschi, 1989), the strength adjustment factor (𝐾𝐷) for lumber in a thirty-year snow load case is around 0.7 to 197 0.8, and in a thirty-year dead load only case is around 0.5. From Table 6.18, five-layer CLT shows smaller values, e.g., 𝐾𝐷 in a thirty-year snow load case is around 0.428 to 0.617 (the average factor from the five cities is 0.481) and 𝐾𝐷 is 0.371 which is less than 0.5 in a thirty-year dead load only case. The decreasing trend in lumber 𝐾𝐷 from snow to dead load cases in the reliability analysis (i.e., 0.7~0.8 to 0.5) was consistent with that of five-layer CLT (0.428~0.617 to 0.371). However, the specific values and the decreasing magnitudes of five-layer CLT were different from those of lumber. The five-layer CLT specimens also exhibited similar results in another approach (0.49 to 0.37), as presented in Table 6.19, through the evaluation of the factors between the different load duration cases. Three-layer CLT specimens exhibited the similar results as five-layer CLT; i.e., factor 𝐾𝐷 in the thirty-year snow load case was around 0.422 to 0.566 (the average factor from the five cities is 0.472) and 𝐾𝐷 in the thirty-year dead load only case (𝐾𝐷 = 0.445) was also low, which is close to but relatively lower than 𝐾𝐷 in snow load case as shown in Table 6.18 (around 0.422 to 0.566 with an average of 0.472). These results are also different from those of lumber. In the stress ratio investigation approach for three-layer CLT, as shown in Figure 6.33 and Table 6.19, the factors between the different load duration cases had similar results as those from the reliability analysis in Table 6.18. For example, the factor for standard term loading, which included the snow load case, was 0.61 (in Table 6.19) which was larger than but close to the range of 0.422 to 0.566 (in Table 6.18); and, the factor for the long-term constant loading case (0.48 in Table 6.19) was also lower than 0.5 (it is 0.445 in Table 6.18). 198 In summary, the five-layer and three-layer CLT results in Table 6.18 and Table 6.19 were different from those of lumber, but the five-layer and three-layer products showed similar results in both the reliability analysis and the stress ratio evaluation approach. This stress ratio evaluation approach presented very close output to that of the reliability theory based method. Figure 6.34 shows the relationship of the model prediction between the three-layer and five-layer CLT specimens under ramp and constant loading protocol in Figure 3.15 (long-term constant loading case). It is suggested that, under the same stress ratio, the three-layer CLT product will fail sooner than five-layer CLT when the duration is between ten minutes and two days (i.e., the short-term duration when 1.0 < 𝐿𝑂𝐺(𝑇𝑖𝑚𝑒) < 3.5 in Figure 6.34). This may be due to the single middle layer in three-layer CLT takes almost all the high shear stress (as shown in the stress evaluation results in Figure 4.13), which is consistent with the short-term ramp loading test results. Also, in the long-term duration, five-layer CLT and three-layer CLT show similar long-term behaviour with small difference, as suggested in Figure 6.34. This small difference in long-term duration-of-load behaviour might be influenced by the different width to thickness ratios of the cross laminated boards in three and five-layer CLT products, considering that the rolling shear strength is influenced by the width to thickness ratio (Blass and Fellmoser, 2004). Compared to three-layer CLT products, the five-layer CLT also includes more pieces of laminated boards in the cross layer. This difference might increase the probability of the rolling shear failure in the long-term behaviour of five-layer CLT considering more pieces of boards under rolling shear stress, as suggested in Figure 6.34. 199 Figure 6.34 Time to failure prediction (minutes in the logarithm to base 10) under the ramp and constant loading protocol Figure 6.34 also shows that the curves of the SPF5-0.4 group and SPF3-0.4 group do not have the same shape as lumber, based on previous research of the duration-of-load effect on lumber (Foschi, 1989). This result suggests that the duration-of-load behaviour of rolling shear strength in CLT is different from that of lumber. Moreover, this duration-of-load test was performed under concentrated loading on the CLT short-span beam, which may be different from other cases, such as the uniformly distributed loading pattern on a two-dimensional CLT panel. Future research is recommended on the load protocol influence on the duration-of-load effect on the rolling shear strength of CLT. Based on the reliability-based approach (as shown in Table 6.18), the load duration adjustment factor from short-term test duration (ten minutes for test specimen failure) to 30-year return 200 period snow loads is approximately between 0.4 and 0.6 (the average factor from the five cities is 0.48) for the five-layer CLT, and is also approximately between 0.4 and 0.6 (the average from the five cities is 0.47) for the three-layer CLT. Considering the stress ratio investigation approach (based on the rolling shear failure result at different load levels as shown in Table 6.19), the load duration adjustment factor from short-term test duration (ten minutes) to standard term (three months) is approximately 0.5 for the five-layer CLT, and is 0.6 for the three-layer CLT. The evaluated adjustment factor from the stress ratio investigation approach is approximately in the range of the calculated factors from the reliability-based approach. For code implementation, further adjustments are needed to convert the values from short-term test duration (ten minutes) to the short-term seven-day duration (seven days); typically, an adjustment factor of 1.25 1.15⁄ = 1.087 for such load-case conversion can be assumed. In this study, the rolling shear failure was defined when the first rolling shear crack was observed. After the first crack occurred, the specimen could still carry a further but limited load. However, under long-term sustained loads, the rolling shear cracks, although small, may reduce the long-term panel stiffness of CLT. The reduced stiffness issue should raise concerns about the overall structural performance in CLT systems, such as the system influence from floor panels with the soften stiffness. Therefore, the choice of first observable rolling shear crack as the failure criteria is conservative; future research is needed for the analysis of the rolling shear behaviour in terms of different failure criteria and its impact on the CLT system performance. 201 6.6 Conclusion In Chapter 6, the stress-based damage accumulation theory was adopted to evaluate the duration-of-load effect on the rolling shear strength of MPB (mountain pine beetle) CLT. This model was calibrated and verified based on the test results; the model predictions fitted the measured data well. As the developed duration-of-load model is a probabilistic model, a time-reliability study of the CLT products was performed. The reliability results provided further information about the duration-of-load effect on the rolling shear behaviour of CLT. The verified model was also able to predict the CLT duration-of-load behaviour under arbitrary loading history, such as long-term dead load history. Therefore, the predictions of the time to failure from this model and this investigation process elucidated the duration-of-load effect and provided guidance for the evaluation of the CLT rolling shear duration-of-load effect. The duration-of-load adjustment factors on the rolling shear strength of CLT were discussed, and it is suggested that this adjustment factor for CLT is smaller than the general duration-of-load adjustment factor for lumber. Therefore, when CLT is introduced into building codes for engineered wood design, the duration-of-load adjustment factor on the rolling shear strength should be considered wherever appropriate. The results also suggest that the duration-of-load adjustment factor for the three-layer CLT is close to that for the five-layer CLT. 202 This study considered the duration-of-load effect only for CLT beam specimens under concentrated load cases; therefore, different loading patterns may influence CLT duration-of-load behaviour. Since the rolling shear failure was defined at the time point when the first rolling shear crack was observed, the duration-of-load behaviour for CLT might be different based on different failure definitions. Therefore, future research to understand the rolling shear failure mechanism and its impact on the structural performance of CLT systems is suggested. In summary, this research investigated the rolling shear behaviour of CLT under long-term loading and provided the fundamental knowledge of a method to establish the duration-of-load adjustment factor for the rolling shear strength of CLT. The basic understanding of the duration-of-load effect on rolling shear strength of CLT developed in this research can lead to rational design of the rolling shear strength of CLT at target performance level. 203 Chapter 7: Concluding Remarks and Future Work 7.1 Concluding Remarks In the beginning of the twenty-first century, the largest outbreak of the mountain pine beetle (MPB) ever recorded struck western Canada. Technologies that are capable of converting MPB-attacked lumber into engineering wood products are urgently required. Manufacturing of cross laminated timber (CLT) using MPB wood as value added products provides one available option. In Canada, commercial production of MPB CLT products has recently been established. However, no research has been undertaken on the duration-of-load and size effects for this product. The objectives of this work are to investigate and evaluate the duration-of-load and size effects on the rolling shear strength of these CLT products. Through collaboration between the Timber Engineering and Applied Mechanics Laboratory at the University of British Columbia and a CLT manufacturing factory in British Columbia, MPB lumber based CLT panels were manufactured with different cross-sectional layups, and CLT beam specimens were cut from the panels and divided into different groups by the pair sampling method. Ramp loading tests and trapezoidal fatigue loading tests with centre-point loads were performed. From the ramp loading tests, the three-layer CLT product showed a lower rolling shear load-carrying capacity, due to the only middle layer taking all the high shear stresses in the beam section. 204 CLT tube specimens were also designed for torque tests, in order to investigate the rolling shear strength of the timber materials. For each configuration (i.e., three- or five-layer CLT products), two sets of the trapezoidal fatigue loading test results were used to evaluate the relationship between the load ratio and the rolling shear time to failure. To estimate CLT beam structural behaviour, the finite element theory was adopted during the simulation process. The modified five-parameter rheological model under different loading protocols was investigated with regard to the rheological deformation of the CLT products. Finally, the duration-of-load effect was evaluated with the stress-based damage accumulation theory under various loading histories from a mathematical perspective. The results from both ramp and trapezoidal fatigue loading tests were well interpreted by a finite element model that was coupled with customized Fortran subroutine codes. The evaluation of the rolling shear strength based on the measured test data was discussed. The size effect model for the rolling shear strength was also investigated, and this model could explain the strength differences between the ramp bending and the torque loading test results. This model is, therefore, suitable for the prediction of the rolling shear capacity in CLT engineering applications. The developed damage accumulation model evaluated the duration-of-load effect on the CLT rolling shear strength. This model incorporated the established short-term rolling shear strength of the material, and it could predict the time to failure under arbitrary loading history. For each 205 CLT configuration, the calibration of the damage accumulation model was based on one set of the trapezoidal fatigue test data; and, the prediction of the model was verified using the test results from the other set of the trapezoidal fatigue tests. The long-term behaviour of CLT was then evaluated. As the verified damage accumulation model was a probabilistic model, it was incorporated into a time-reliability study. By performing a reliability assessment of the CLT products, the duration-of-load adjustment factor for the rolling shear strength of CLT was analyzed and discussed. The duration-of-load adjustment factor for the three-layer CLT and five-layer CLT is different from that for lumber. The results also suggest that the duration-of-load adjustment factor for the rolling shear strength of CLT is smaller than the general duration-of-load adjustment factor for lumber. This rolling shear strength adjustment factor for CLT should be considered when CLT is introduced into the building codes for engineered wood design wherever appropriate. 7.2 Future Work In this research, the testing program on the duration-of-load and size effects on the rolling shear strength of MPB CLT consisted of the short-term ramp loading tests on CLT beam specimens, the long-term trapezoidal fatigue loading tests on CLT beam specimens and the torque loading tests on CLT tube specimens. In the future research, more duration-of-load tests should be carried out under different loading histories, such as long-term creep tests with a constant loading protocol. 206 A larger and more comprehensive test database is recommended, in order to contribute to further reliability assessments and analyses. In the studies of this thesis, CLT beam members were tested under concentrated loads; however, different loading patterns, such as uniformly distributed loading on two-dimensional CLT panels, may influence the duration-of-load behaviour of CLT. This area needs more investigation in the future. In the finite element modeling of CLT beam specimens, the possible presence of glue, which might be squeezed into the gaps between adjacent non-edge glued boards, may influence the shear stress distribution. This squeezed glue could be taken into consideration in the future modeling process. Considering the dependence between rolling shear behaviour and the annual ring orientation, the polar orthotropic finite element modeling method is suggested in simulating the material properties of the CLT cross layers. Since the rolling shear failure was defined at the time point when the first rolling shear crack was observed, more research into the rolling shear failure mechanism and the impact of the rolling shear failure on the structural performance of CLT is suggested. Because one laminated board with rolling shear failure in the cross layer does not mean the CLT member is collapsing, future research is also recommended on the evaluation of ultimate load-carrying capacities of CLT elements. 207 In certain failed CLT beam specimens in the ramp loading tests, the rolling shear failure cracks seemed to be coupled with tension perpendicular to grain failure in the cross layer. Therefore, further investigation is recommended for the failure mode and wood fibre behaviour under combined stresses. 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(2012) Measurement of rolling shear modulus of Cross Laminated Timber: Exploratory study using downscaled specimens under variable span bending tests. 2012 International Conference on Biobase Material Science and Engineering (BMSE), Oct. 2012, Changsha, China 216 Appendices Appendix A Rolling Shear Failure Loads in Five-Layer CLT in Ramp Loading Test Table A.1 Rolling shear failure loads of specimens from No. 1 to No. 34 No. Notation of specimen Rolling shear failure loads (kN) No. Notation of specimen Rolling shear failure loads (kN) 1 SPF5-0.4-1-01 22.19 18 SPF5-0.4-1-18 14.78 2 SPF5-0.4-1-02 21.67 19 SPF5-0.4-1-19 16.57 3 SPF5-0.4-1-03 20.95 20 SPF5-0.4-1-20 15.38 4 SPF5-0.4-1-04 17.94 21 SPF5-0.4-2-01 13.39 5 SPF5-0.4-1-05 20.70 22 SPF5-0.4-2-02 21.74 6 SPF5-0.4-1-06 19.69 23 SPF5-0.4-2-03 20.38 7 SPF5-0.4-1-07 18.49 24 SPF5-0.4-2-04 20.77 8 SPF5-0.4-1-08 19.74 25 SPF5-0.4-2-05 17.75 9 SPF5-0.4-1-09 22.20 26 SPF5-0.4-2-06 20.85 10 SPF5-0.4-1-10 20.90 27 SPF5-0.4-2-07 20.79 11 SPF5-0.4-1-11 19.93 28 SPF5-0.4-2-08 21.49 12 SPF5-0.4-1-12 19.62 29 SPF5-0.4-2-09 17.23 13 SPF5-0.4-1-13 19.79 30 SPF5-0.4-2-10 19.82 14 SPF5-0.4-1-14 17.09 31 SPF5-0.4-2-11 22.55 15 SPF5-0.4-1-15 16.23 32 SPF5-0.4-2-12 19.87 16 SPF5-0.4-1-16 24.21 33 SPF5-0.4-2-13 21.40 17 SPF5-0.4-1-17 19.90 34 SPF5-0.4-2-14 16.65 217 Table A.2 Rolling shear failure loads of specimens from No. 35 to No. 55 No. Notation of specimen Rolling shear failure loads (kN) No. Notation of specimen Rolling shear failure loads (kN) 35 SPF5-0.4-2-15 18.73 46 SPF5-0.4-3-09 21.89 36 SPF5-0.4-2-16 18.54 47 SPF5-0.4-3-10 16.62 37 SPF5-0.4-2-17 21.89 48 SPF5-0.4-3-12 14.69 38 SPF5-0.4-2-18 17.64 49 SPF5-0.4-3-13 19.63 39 SPF5-0.4-2-19 19.99 50 SPF5-0.4-3-14 20.19 40 SPF5-0.4-2-20 20.90 51 SPF5-0.4-3-15 21.28 41 SPF5-0.4-3-01 22.99 52 SPF5-0.4-3-17 19.25 42 SPF5-0.4-3-02 19.45 53 SPF5-0.4-3-18 21.43 43 SPF5-0.4-3-05 20.62 54 SPF5-0.4-3-19 13.79 44 SPF5-0.4-3-06 15.40 55 SPF5-0.4-3-20 17.53 45 SPF5-0.4-3-08 21.19 218 Appendix B Rolling Shear Failure Loads in Three-Layer CLT in Ramp Loading Test Table B.1 Rolling shear failure loads of specimens from No. 1 to No. 38 No. Notation of specimen Rolling shear failure loads (kN) No. Notation of specimen Rolling shear failure loads (kN) 1 SPF3-0.4-1-01 11.55 20 SPF3-0.4-1-20 8.44 2 SPF3-0.4-1-02 8.86 21 SPF3-0.4-2-01 7.26 3 SPF3-0.4-1-03 10.39 22 SPF3-0.4-2-02 14.78 4 SPF3-0.4-1-04 11.47 23 SPF3-0.4-2-03 9.74 5 SPF3-0.4-1-05 12.57 24 SPF3-0.4-2-04 14.82 6 SPF3-0.4-1-06 11.06 25 SPF3-0.4-2-05 16.88 7 SPF3-0.4-1-07 11.05 26 SPF3-0.4-2-06 15.04 8 SPF3-0.4-1-08 10.27 27 SPF3-0.4-2-07 11.90 9 SPF3-0.4-1-09 10.97 28 SPF3-0.4-2-08 12.57 10 SPF3-0.4-1-10 13.42 29 SPF3-0.4-2-09 18.32 11 SPF3-0.4-1-11 8.98 30 SPF3-0.4-2-10 11.33 12 SPF3-0.4-1-12 10.80 31 SPF3-0.4-2-11 13.96 13 SPF3-0.4-1-13 9.19 32 SPF3-0.4-2-12 15.95 14 SPF3-0.4-1-14 7.98 33 SPF3-0.4-2-13 17.56 15 SPF3-0.4-1-15 9.68 34 SPF3-0.4-2-14 14.21 16 SPF3-0.4-1-16 8.52 35 SPF3-0.4-2-15 16.01 17 SPF3-0.4-1-17 11.55 36 SPF3-0.4-2-16 15.85 18 SPF3-0.4-1-18 12.28 37 SPF3-0.4-2-17 15.79 19 SPF3-0.4-1-19 7.84 38 SPF3-0.4-2-18 21.22 219 Table B.2 Rolling shear failure loads of specimens from No. 39 to No. 59 No. Notation of specimen Rolling shear failure loads (kN) No. Notation of specimen Rolling shear failure loads (kN) 39 SPF3-0.4-2-19 14.70 50 SPF3-0.4-3-10 14.05 40 SPF3-0.4-2-20 17.35 51 SPF3-0.4-3-11 15.29 41 SPF3-0.4-3-01 13.25 52 SPF3-0.4-3-12 13.34 42 SPF3-0.4-3-02 13.70 53 SPF3-0.4-3-14 14.24 43 SPF3-0.4-3-03 13.42 54 SPF3-0.4-3-15 10.12 44 SPF3-0.4-3-04 11.56 55 SPF3-0.4-3-16 11.20 45 SPF3-0.4-3-05 7.02 56 SPF3-0.4-3-17 12.81 46 SPF3-0.4-3-06 15.63 57 SPF3-0.4-3-18 10.88 47 SPF3-0.4-3-07 11.28 58 SPF3-0.4-3-19 8.90 48 SPF3-0.4-3-08 16.43 59 SPF3-0.4-3-20 9.85 49 SPF3-0.4-3-09 12.92 220 Appendix C Number of Cycles to Rolling Shear Failure in Five-Layer CLT in Trapezoidal Fatigue Loading Test (Long Duration in Plateau Part) Table C.1 Number of cycles to rolling shear failure No. Notation of specimen Number of cycles to rolling shear failure No. Notation of specimen Number of cycles to rolling shear failure 1 SPF5-0.4-1-21 2 16 SPF5-0.4-2-26 6 2 SPF5-0.4-1-22 13 17 SPF5-0.4-2-27 31 3 SPF5-0.4-1-23 7 18 SPF5-0.4-2-28 6 4 SPF5-0.4-1-24 4 19 SPF5-0.4-2-29 16 5 SPF5-0.4-1-25 5 20 SPF5-0.4-2-30 26 6 SPF5-0.4-1-26 23 21 SPF5-0.4-3-21 3 7 SPF5-0.4-1-27 50 22 SPF5-0.4-3-22 88 8 SPF5-0.4-1-28 6 23 SPF5-0.4-3-23 28 9 SPF5-0.4-1-29 4 24 SPF5-0.4-3-24 9 10 SPF5-0.4-1-30 10 25 SPF5-0.4-3-25 8 11 SPF5-0.4-2-21 41 26 SPF5-0.4-3-27 5 12 SPF5-0.4-2-22 4 27 SPF5-0.4-3-28 13 13 SPF5-0.4-2-23 3 28 SPF5-0.4-3-29 15 14 SPF5-0.4-2-24 3 29 SPF5-0.4-3-30 5 15 SPF5-0.4-2-25 8 221 Appendix D Number of Cycles to Rolling Shear Failure in Five-Layer CLT in Trapezoidal Fatigue Loading Test (Short Duration in Plateau Part) Table D.1 Number of cycles to rolling shear failure No. Notation of specimen Number of cycles to rolling shear failure No. Notation of specimen Number of cycles to rolling shear failure 1 SPF5-0.4-1-31 173 16 SPF5-0.4-2-37 132 2 SPF5-0.4-1-32 26 17 SPF5-0.4-2-38 85 3 SPF5-0.4-1-33 55 18 SPF5-0.4-2-39 18 4 SPF5-0.4-1-34 201 19 SPF5-0.4-2-40 74 5 SPF5-0.4-1-35 140 20 SPF5-0.4-3-31 2 6 SPF5-0.4-1-36 16 21 SPF5-0.4-3-32 1 7 SPF5-0.4-1-37 5 22 SPF5-0.4-3-33 5 8 SPF5-0.4-1-38 22 23 SPF5-0.4-3-34 1 9 SPF5-0.4-1-39 11 24 SPF5-0.4-3-35 21 10 SPF5-0.4-1-40 15 25 SPF5-0.4-3-37 67 11 SPF5-0.4-2-31 38 26 SPF5-0.4-3-38 230 12 SPF5-0.4-2-32 281 27 SPF5-0.4-3-39 104 13 SPF5-0.4-2-33 12 28 SPF5-0.4-3-40 65 14 SPF5-0.4-2-34 8 15 SPF5-0.4-2-35 43 222 Appendix E Number of Cycles to Rolling Shear Failure in Three-Layer CLT in Trapezoidal Fatigue Loading Test (Long Duration in Plateau Part) Table E.1 Number of cycles to rolling shear failure No. Notation of specimen Number of cycles to rolling shear failure No. Notation of specimen Number of cycles to rolling shear failure 1 SPF3-0.4-1-21 2 17 SPF3-0.4-2-26 8 2 SPF3-0.4-1-22 2 18 SPF3-0.4-2-27 5 3 SPF3-0.4-1-23 3 19 SPF3-0.4-2-28 9 4 SPF3-0.4-1-24 2 20 SPF3-0.4-2-29 3 5 SPF3-0.4-1-25 1 21 SPF3-0.4-2-31 15 6 SPF3-0.4-1-26 1 22 SPF3-0.4-3-21 1 7 SPF3-0.4-1-27 3 23 SPF3-0.4-3-22 89 8 SPF3-0.4-1-28 4 24 SPF3-0.4-3-23 20 9 SPF3-0.4-1-29 3 25 SPF3-0.4-3-24 13 10 SPF3-0.4-1-30 2 26 SPF3-0.4-3-25 92 11 SPF3-0.4-1-31 2 27 SPF3-0.4-3-26 4 12 SPF3-0.4-2-21 31 28 SPF3-0.4-3-27 7 13 SPF3-0.4-2-22 4 29 SPF3-0.4-3-28 7 14 SPF3-0.4-2-23 3 30 SPF3-0.4-3-29 12 15 SPF3-0.4-2-24 2 31 SPF3-0.4-3-30 2 16 SPF3-0.4-2-25 57 32 SPF3-0.4-3-31 1 223 Appendix F Number of Cycles to Rolling Shear Failure in Three-Layer CLT in Trapezoidal Fatigue Loading Test (Short Duration in Plateau Part) Table F.1 Number of cycles to rolling shear failure No. Notation of specimen Number of cycles to rolling shear failure No. Notation of specimen Number of cycles to rolling shear failure 1 SPF3-0.4-1-32 51 16 SPF3-0.4-2-37 81 2 SPF3-0.4-1-33 99 17 SPF3-0.4-2-38 21 3 SPF3-0.4-1-34 16 18 SPF3-0.4-2-39 60 4 SPF3-0.4-1-35 3 19 SPF3-0.4-2-40 6 5 SPF3-0.4-1-36 2 20 SPF3-0.4-2-41 11 6 SPF3-0.4-1-37 19 21 SPF3-0.4-3-32 38 7 SPF3-0.4-1-38 9 22 SPF3-0.4-3-33 31 8 SPF3-0.4-1-39 7 23 SPF3-0.4-3-34 163 9 SPF3-0.4-1-40 29 24 SPF3-0.4-3-35 4 10 SPF3-0.4-1-41 7 25 SPF3-0.4-3-36 25 11 SPF3-0.4-2-32 16 26 SPF3-0.4-3-37 5 12 SPF3-0.4-2-33 14 27 SPF3-0.4-3-38 9 13 SPF3-0.4-2-34 37 28 SPF3-0.4-3-39 12 14 SPF3-0.4-2-35 2 29 SPF3-0.4-3-40 48 15 SPF3-0.4-2-36 212 30 SPF3-0.4-3-41 119 224 Appendix G Peak Failure Torque Loads in Five-Layer CLT in Torque Loading Test Table G.1 Peak failure torque loads No. Notation of specimen Peak failure torque load (N mm) No. Notation of specimen Peak failure torque load (N mm) 1 SPF5-0.4-1-01 125880 17 SPF5-0.4-2-07 129695 2 SPF5-0.4-1-02 77435 18 SPF5-0.4-2-08 83160 3 SPF5-0.4-1-03 93075 19 SPF5-0.4-2-09 93840 4 SPF5-0.4-1-04 84685 20 SPF5-0.4-2-10 92315 5 SPF5-0.4-1-05 109860 21 SPF5-0.4-2-11 94600 6 SPF5-0.4-1-06 92695 22 SPF5-0.4-3-01 104900 7 SPF5-0.4-1-07 115580 23 SPF5-0.4-3-02 94600 8 SPF5-0.4-1-08 99940 24 SPF5-0.4-3-03 93075 9 SPF5-0.4-1-09 96125 25 SPF5-0.4-3-04 72475 10 SPF5-0.4-1-10 116725 26 SPF5-0.4-3-05 141900 11 SPF5-0.4-2-01 112910 27 SPF5-0.4-3-06 102610 12 SPF5-0.4-2-02 131220 28 SPF5-0.4-3-07 111005 13 SPF5-0.4-2-03 120160 29 SPF5-0.4-3-08 86590 14 SPF5-0.4-2-04 117075 30 SPF5-0.4-3-09 85445 15 SPF5-0.4-2-05 105665 31 SPF5-0.4-3-10 114820 16 SPF5-0.4-2-06 123590 225 Appendix H Peak Failure Torque Loads in Three-Layer CLT in Torque Loading Test Table H.1 Peak failure torque loads No. Notation of specimen Peak failure torque load (N mm) No. Notation of specimen Peak failure torque load (N mm) 1 SPF3-0.4-1-01 101085 16 SPF3-0.4-2-10 78960 2 SPF3-0.4-1-02 102610 17 SPF3-0.4-3-01 86970 3 SPF3-0.4-1-03 101085 18 SPF3-0.4-3-02 74765 4 SPF3-0.4-1-04 92695 19 SPF3-0.4-3-04 91930 5 SPF3-0.4-1-05 75910 20 SPF3-0.4-3-05 88115 6 SPF3-0.4-1-06 89260 21 SPF3-0.4-3-06 68660 7 SPF3-0.4-2-01 79725 22 SPF3-0.4-3-07 98415 8 SPF3-0.4-2-02 80490 23 SPF3-0.4-3-08 86210 9 SPF3-0.4-2-03 64850 24 SPF3-0.4-3-10 67520 10 SPF3-0.4-2-04 125500 25 SPF3-0.4-3-11 79345 11 SPF3-0.4-2-05 80870 12 SPF3-0.4-2-06 67520 13 SPF3-0.4-2-07 66755 14 SPF3-0.4-2-08 83160 15 SPF3-0.4-2-09 81630 226 Appendix I Rolling Shear Stress Calculation in Different Beam Theories for Three-Layer and Five-Layer CLT Assume the applied centre-point load is 𝑃𝑙𝑜𝑎𝑑 = 1𝑘𝑁 on the three-layer and five-layer CLT beam. Figure I. 1 Shear stress calculation for three-layer CLT 1) Layered beam theory for three-layer CLT Parameters and material properties: 𝐸1 = 381𝑁 𝑚𝑚2⁄ 𝐸2 = 11430𝑁 𝑚𝑚2⁄ ℎ1 = 17𝑚𝑚 ℎ2 = 34𝑚𝑚 𝑤 = 50.8𝑚𝑚 Shear force in the section: 𝑉 =𝑃𝑙𝑜𝑎𝑑2= 0.5𝑘𝑁 Considering the section is symmetric, the moment of inertia of the transformed cross section: neutral axis173450,8h1h2wE2E1ABCLoad=1kNneutral axis173450,8h1h2wE2E1ABCCLT beamlayer 2layer 1three-layer CLT227 𝐼′ = 2 ×𝑤𝐸2× [𝐸1 × (ℎ1)312+ (𝐸1 × ℎ1) × (ℎ12)2+𝐸2 × (ℎ2)312+ (𝐸2 × ℎ2) × (ℎ22+ ℎ1)2] The area moment for point A, B and C: 𝑄𝐴′ =𝑤𝐸2× [(𝐸2 × ℎ2) × (ℎ1 +ℎ22) + (𝐸1 × ℎ1) × (ℎ12)] 𝑄𝐵′ =𝑤𝐸2× [(𝐸2 × ℎ2) × (ℎ1 +ℎ22)] = 𝑤 × ℎ2 × (ℎ1 +ℎ22) 𝑄𝐶′ =𝑤𝐸2× [(𝐸2 ×ℎ22) × (ℎ1 + ℎ2 −ℎ24)] =𝑤 × ℎ22× (ℎ1 + ℎ2 −ℎ24) Therefore, the shear stress for point A, B and C: 𝜏𝐴 =𝑉𝑄𝐴′𝐼′𝑤= {0.5 × 1000 ×50.811430× [(11430 × 34) × (17 +342) + (381 × 17) × (172)]}÷ {2 ×50.811430× [381 × (17)312+ (381 × 17) × (172)2+11430 × (34)312+ (11430 × 34)× (342+ 17)2] × 50.8} = 0.134𝑀𝑃𝑎 Similarly, 𝜏𝐵 =𝑉𝑄𝐵′𝐼′𝑤= 0.133𝑀𝑃𝑎 𝜏𝐶 =𝑉𝑄𝐶′𝐼′𝑤= 0.083𝑀𝑃𝑎 2) Gamma beam theory for three-layer CLT The test span is 𝐿 = 612𝑚𝑚. Rolling shear modulus is 𝐺𝑅 = 66.6𝑁 𝑚𝑚2⁄ 228 The connection efficiency factor: 𝛾1 = 𝛾2 =1𝜋2 × 𝐸2 × 𝑤 × ℎ2 × 2ℎ1𝐺𝑅 ×𝑤 × 𝐿2+ 1= 0.160567938 The effective stiffness of the beam 𝐸𝐼𝑒𝑓𝑓 =∑(𝐸𝑖𝐼𝑖2𝑖=1+ 𝛾𝑖𝐸𝑖𝐴𝑖𝑎𝑖2) = 2 × [𝐸2 ×𝑤ℎ2312+ 𝛾1𝐸2 × 𝑤ℎ2 × (ℎ1 +ℎ22)2] Therefore: 𝜏𝐴 =𝑉 × [𝛾1 × 𝐸2 ×𝑤 × ℎ2 × (ℎ1 +ℎ22 ) + 𝐸1 × 𝑤 × ℎ1 × (ℎ12 )]𝐸𝐼𝑒𝑓𝑓 × 𝑤=0.5 × 1000 × [0.160567938 × 11430 × 50.8 × 34 × (17 +342 ) + 381 × 50.8 × 17 × (172 )]2 × [11430 ×50.8 × 34312 + 0.160567938 × 11430 × 50.8 × 34 × (17 +342 )2] × 50.8= 0.098𝑀𝑃𝑎 𝜏𝑐 =𝑉 × [𝛾1 × 𝐸2 ×𝑤 × ℎ2 × (ℎ1 +ℎ22 )]𝐸𝐼𝑒𝑓𝑓 × 𝑤= 0.095𝑀𝑃𝑎 3) Shear analogy theory for three-layer CLT The bending stiffness of Beam A and Beam B are calculated by: 𝐵𝐴 =∑𝐸𝑖𝐼𝑖𝑛𝑖=1= 2 × 𝐸2 ×𝑤ℎ2312+𝐸1 × 𝑤 × (ℎ1)312 𝐵𝐵 =∑𝐸𝑖𝐴𝑖𝑎𝑖2𝑛𝑖=1= 2 × 𝐸2 × 𝑤ℎ2 × (ℎ1 +ℎ22)2 𝑉𝐴 = 𝑉 ×𝐵𝐴𝐵𝐴 + 𝐵𝐵 229 𝑉𝐵 = 𝑉 − 𝑉𝐴 𝜏𝐴,1 =1.5𝑉𝐴ℎ1 ∙ 𝑤∙𝐸1𝐼1𝐵𝐴=1.5𝑉𝐴ℎ1 ∙ 𝑤∙𝐸1 ×𝑤 × (ℎ1)312𝐵𝐴 𝜏𝐵,1 =𝑉𝐵𝐵𝐵𝑤∙∑𝐸𝑗𝐴𝑗𝑎𝑗3𝑗=2=𝑉𝐵𝐵𝐵𝑤∙ 𝐸2𝑤ℎ2 × (ℎ1 +ℎ22) Therefore, the shear stress for point A in Figure I. 1 in the middle cross layer: 𝜏𝐴′ = 𝜏𝐴,1 + 𝜏𝐵,1 =1.5𝑉𝐴ℎ1 ∙ 𝑤∙𝐸1 ×𝑤(ℎ1)312𝐵𝐴+𝑉𝐵𝐵𝐵𝑤∙ 𝐸2𝑤ℎ2 × (ℎ1 +ℎ22) = 0.001 + 0.133= 0.134𝑀𝑃𝑎 Figure I. 2 Shear stress calculation for five-layer CLT neutral axis193450,8h2h3wE2E1ABDlayer 2layer 1layer 334h1E3CE9.5Load=1kNCLT beamfive-layer CLTneutral axis193450,8h2h3wE2E1ABDlayer 2layer 1layer 317E3CE9.5230 4) Layered beam theory for five-layer CLT Parameters and material properties: 𝐸1 = 11430𝑁 𝑚𝑚2⁄ 𝐸2 = 381𝑁 𝑚𝑚2⁄ 𝐸3 = 𝐸1 = 11430𝑁 𝑚𝑚2⁄ ℎ1 = 34𝑚𝑚 ℎ2 = 19𝑚𝑚 ℎ3 = ℎ1 = 34𝑚𝑚 𝑤 = 50.8𝑚𝑚 Shear force in the section: 𝑉 =𝑃𝑙𝑜𝑎𝑑2= 0.5𝑘𝑁 Considering the section is symmetric, the moment of inertia of the transformed cross section: 𝐼′ =𝑤 × (ℎ1)312+ 2 ×𝑤𝐸1× [𝐸2 × (ℎ2)312+ (𝐸2 × ℎ2) × (ℎ1 + ℎ22)2+𝐸3 × (ℎ3)312+ (𝐸3 × ℎ3)× (ℎ3 + ℎ12+ ℎ2)2] The area moment for point A, B C and D: 𝑄𝐴′ =𝑤𝐸1× [(𝐸1 ×ℎ12) × (ℎ14) + (𝐸2 × ℎ2) × (ℎ1 + ℎ22) + (𝐸3 × ℎ3) × (ℎ1 + ℎ32+ ℎ2)] 𝑄𝐵′ =𝑤𝐸1× [(𝐸2 × ℎ2) × (ℎ1 + ℎ22) + (𝐸3 × ℎ3) × (ℎ1 + ℎ32+ ℎ2)] 𝑄𝐶′ =𝑤𝐸1× [(𝐸3 × ℎ3) × (ℎ1 + ℎ32+ ℎ2)] 𝑄𝐷′ =𝑤𝐸1× [(𝐸3 ×ℎ32) × (ℎ1 + ℎ32+ ℎ2 +ℎ34)] Therefore, the shear stress for point A, B C and D: 231 𝜏𝐴 =𝑉𝑄𝐴′𝐼′𝑤= {0.5 × 1000 ×50.811430× [(11430 ×342) × (344) + (381 × 19) × (34 + 192) + (11430 × 34)× (34 + 342+ 19)]}÷ {50.8 × (34)312+ 2 ×50.811430× [381 × (19)312+ (381 × 19) × (34 + 192)2+11430 × (34)312+ (11430 × 34) × (34 + 342+ 19)2]} ÷ 50.8 = 0.096𝑀𝑃𝑎 Similarly, 𝜏𝐵 =𝑉𝑄𝐵′𝐼′𝑤= 0.089𝑀𝑃𝑎 𝜏𝐶 =𝑉𝑄𝐶′𝐼′𝑤= 0.088𝑀𝑃𝑎 𝜏𝐷 =𝑉𝑄𝐷′𝐼′𝑤= 0.051𝑀𝑃𝑎 5) Gamma beam theory for five-layer CLT The test span is 𝐿 = 840𝑚𝑚. Rolling shear modulus is 𝐺𝑅 = 66.6𝑁 𝑚𝑚2⁄ The connection efficiency factor: 𝛾1 = 𝛾3 =1𝜋2 × 𝐸1 × 𝑤 × ℎ1 × ℎ2𝐺𝑅 ×𝑤 × 𝐿2+ 1= 0.392039423 𝛾2 = 1 232 The effective stiffness of the beam 𝐸𝐼𝑒𝑓𝑓 =∑(𝐸𝑖𝐼𝑖3𝑖=1+ 𝛾𝑖𝐸𝑖𝐴𝑖𝑎𝑖2)=𝐸1 ×𝑤 × (ℎ1)312+ 2 × [𝐸3 ×𝑤 × (ℎ3)312+ 𝛾1𝐸3 ×𝑤ℎ3 × (ℎ3 + ℎ12+ ℎ2)2] Therefore: 𝜏𝐴 =𝑉 [𝛾1𝐸3𝑤ℎ3 × (ℎ32 + ℎ2 +ℎ12 ) + 𝐸2𝑤ℎ2 × (ℎ22 +ℎ12 ) + 𝛾2𝐸1𝑤 × (ℎ12 ) × (ℎ14 )]𝐸𝐼𝑒𝑓𝑓 ×𝑤= {0.5 × 1000× [0.392039423 × 11430 × 50.8 × 34 × (342+ 19 +342) + 381 × 50.8 × 19× (192+342) + 11430 × 50.8 × (342) × (344)]}÷ {11430 × 50.8 × (34)312+ 2× [11430 ×50.8 × (34)312+ 0.392039423 × 11430 × 50.8 × 34× (34 + 342+ 19)2]} ÷ 50.8 = 0.101𝑀𝑃𝑎 𝜏𝐸 =𝑉 × [𝛾1 × 𝐸3 × 𝑤 × ℎ3 × (ℎ32 + ℎ2 +ℎ12 ) + 𝐸2 ×𝑤 × ℎ2 × (ℎ22 +ℎ12 )]𝐸𝐼𝑒𝑓𝑓 ×𝑤= 0.084𝑀𝑃𝑎 6) Shear analogy theory for five-layer CLT The bending stiffness of Beam A and Beam B are calculated by: 233 𝐵𝐴 =∑𝐸𝑖𝐼𝑖𝑛𝑖=1= 2 × 𝐸3 ×𝑤ℎ3312+ 2 × 𝐸2 ×𝑤ℎ2312+𝐸1 × 𝑤 × (ℎ1)312 𝐵𝐵 =∑𝐸𝑖𝐴𝑖𝑎𝑖2𝑛𝑖=1= 2 × 𝐸2 × 𝑤ℎ2 × (ℎ12+ℎ22)2+ 2 × 𝐸3 × 𝑤ℎ3 × (ℎ12+ ℎ2 +ℎ32)2 𝑉𝐴 = 𝑉 ×𝐵𝐴𝐵𝐴 + 𝐵𝐵 𝑉𝐵 = 𝑉 − 𝑉𝐴 𝜏𝐴,2 =1.5𝑉𝐴ℎ2 ∙ 𝑤∙𝐸2𝐼2𝐵𝐴=1.5𝑉𝐴ℎ2 ∙ 𝑤∙ 𝐸2 ×𝑤ℎ2312𝐵𝐴 𝜏𝐵,2 =𝑉𝐵𝐵𝐵𝑤∙∑𝐸𝑗𝐴𝑗𝑎𝑗3𝑗=3=𝑉𝐵𝐵𝐵𝑤∙ 𝐸3𝑤ℎ3 × (ℎ12+ ℎ2 +ℎ32) Therefore, the shear stress for point E in the Figure I. 2 in the middle cross layer: 𝜏𝐸′ = 𝜏𝐴,2 + 𝜏𝐵,2 =1.5𝑉𝐴ℎ2 ∙ 𝑤∙ 𝐸2 ×𝑤ℎ2312𝐵𝐴+𝑉𝐵𝐵𝐵𝑤∙ 𝐸3𝑤ℎ3 × (ℎ12+ ℎ2 +ℎ32) = 0.000 + 0.085= 0.085𝑀𝑃𝑎 234 Appendix J Further Investigation on the Shear Failure Mode in Torque Loading Tests Test results show that the longitudinal shear strength of Spruce-Pine-Fir (SPF), a Canadian softwood species group, ranges from 6.94 MPa to 8.08 MPa (Lam et al., 1997). SPF species comprises timber from white spruce, engelmann spruce, lodgepole pine, and alpine fir species in western Canada (British Columbia and Alberta states). According to the Wood Handbook (FPL, 2010), the longitudinal shear strength of lodgepole pine grown in Canada is 8.50 MPa in 12% moisture content, and the white spruce grown in Canada has the lowest longitudinal shear strength (mean value of 6.80 MPa in 12% moisture content) among SPF species. There is also research informing the average longitudinal shear strength (mean value of 7.27 MPa) of MPB-afflicted lodgepole pine, based on the test data (Uyema, 2012). Therefore, the values of 6.80 MPa and 7.27 MPa were selected in the following simulation process. In Eurocode 5 (Eurocode 5, 2004), a characteristic rolling shear strength value of 1.0 MPa is used for wood independent of its strength class. Rolling shear strength typically is between 18% and 28% (about 1/5 to 1/3) of parallel-to-grain shear strength (Blass and Görlacher, 2003). In this appendix, a Monte Carlo simulation procedure was performed as follows to investigate the probability of different shear failure modes in torque loading tests considering the randomness of the rolling shear strength and longitudinal shear strength in wood material, and this process was as follows: 235 1) The mean values of rolling shear strength 𝜏𝑅𝑆𝑚 (for example, 𝜏𝑅𝑆𝑚 = 1.0 𝑀𝑃𝑎) and longitudinal shear strength 𝜏𝐿𝑆𝑚 (for example, a lower longitudinal shear mean strength 𝜏𝐿𝑆𝑚 =6.80 𝑀𝑃𝑎) were chosen, and a coefficient of variation of 18% was also chosen. 2) Assume the rolling shear strength and longitudinal shear strength were lognormally distributed, and based on the initial distribution parameters chosen in step 1), a random sample size of 𝑁𝑅𝑛𝑑 = 1000 sets of 𝜏𝑅𝑆 and 𝜏𝐿𝑆 was generated. 𝜏𝑅𝑆 and 𝜏𝐿𝑆 represent the rolling shear strength and longitudinal shear strength in wood material of this given random set. 3) For each random set, the ratio (i.e., 𝑅𝑎𝑡𝑖𝑜𝑅𝑎𝑛𝑑𝑜𝑚) between 𝜏𝑅𝑆 and 𝜏𝐿𝑆 was calculated as 𝑅𝑎𝑡𝑖𝑜𝑅𝑎𝑛𝑑𝑜𝑚 =𝜏𝑅𝑆𝜏𝐿𝑆⁄ . 4) As investigated in Section 4.4.3, if 𝑅𝑎𝑡𝑖𝑜𝑅𝑎𝑛𝑑𝑜𝑚 < 𝑚𝑖𝑛(RShearThree−layer, RShearFive−layer), where: RShearThree−layer = 0.55 ≈ 1 1.82⁄ , RShearFive−layer = 0.47 ≈ 1 2.13⁄ there is rolling shear failure occurred in this random set; otherwise, the failure mode will be longitudinal shear failure. 5) The probability of rolling shear failure 𝑝𝑓𝑅𝑆 can be evaluated based on the number of sets with rolling shear failure mode. In the simulation, different variables were also included and investigated, considering the variable mean values of rolling shear strength 𝜏𝑅𝑆𝑚 (1.0 𝑀𝑃𝑎, 1.5 𝑀𝑃𝑎, 2.0 𝑀𝑃𝑎, 2.5 𝑀𝑃𝑎 𝑎𝑛𝑑 3.0 𝑀𝑃𝑎) and longitudinal shear strength 𝜏𝐿𝑆𝑚 (6.80 𝑀𝑃𝑎 𝑎𝑛𝑑 7.27 𝑀𝑃𝑎), and the coefficient of variation (16% and 18%). 236 It needs to be emphasized that the selected longitudinal shear strength category of 6.80 𝑀𝑃𝑎 has already been an underestimation, considering the longitudinal shear strength of lodgepole pine grown in Canada is 8.50 MPa, and considering that the lodgepole pine, rather than white spruce species, was adopted in the related CLT manufacture process (Chen, 2011). Table J. 1 shows the probability of rolling shear failure 𝑝𝑓𝑅𝑆 based on different ratios (τRSm τLSm⁄ ) between the chosen mean values of rolling shear strength 𝜏𝑅𝑆𝑚 and longitudinal shear strength 𝜏𝐿𝑆𝑚. Although the criterion was strict in the longitudinal shear strength simulation, in most cases it still shows 𝑝𝑓𝑅𝑆 > 94%, and this gives more confidence on the observed rolling shear failure mode in tube specimens. The cases with 𝑝𝑓𝑅𝑆 ≤ 94% only occurred when the ratio between 𝜏𝑅𝑆𝑚 and 𝜏𝐿𝑆𝑚 was larger than 1/3. This is not realistic considering rolling shear strength is recognized to be between 18% and 28% (about 1/5 to 1/3) of parallel-to-grain shear strength. When the ratio τRSm τLSm⁄ is between 1/5 and 1/3, the 𝑝𝑓𝑅𝑆 > 97%. Even when the ratio τRSm τLSm⁄ is impractically larger than 1/3 (τRSm τLSm⁄ = 1/2.4 and with assumed rolling shear strength to be 3 MPa), 𝑝𝑓𝑅𝑆 is still in a high confidence level (75.4%). 237 Table J. 1 Results for the Monte Carlo simulation 𝛕𝐑𝐒𝐦𝛕𝐋𝐒𝐦⁄ Rolling shear mean strength 𝛕𝐑𝐒𝐦 Longitudinal shear mean strength 𝛕𝐋𝐒𝐦 𝐂𝐎𝐕 Probability of rolling shear failure 𝐩𝐟𝐑𝐒 1/6.8 1.00 6.80 18% 100.0% 1/4.8 1.50 7.27 16% 100.0% 1/4.8 1.50 7.27 18% 100.0% 1/4.5 1.50 6.80 16% 100.0% 1/4.5 1.50 6.80 18% 100.0% 1/3.6 2.00 7.27 16% 99.6% 1/3.6 2.00 7.27 18% 99.1% 1/3.4 2.00 6.80 16% 98.2% 1/3.4 2.00 6.80 18% 97.3% 1/2.9 2.50 7.27 16% 93.3% 1/2.9 2.50 7.27 18% 90.3% 1/2.7 2.50 6.80 16% 88.9% 1/2.7 2.50 6.80 18% 85.9% 1/2.4 3.00 7.27 16% 75.4% 238 Appendix K Introduction of the Fitting Process and Adjusted Parameter Values in Finite Element Beam Modeling for Short-Term Ramp Loading The finite element beam model fitting process for short-term load-displacement behaviour (in Figure 5.1 and Figure 5.2) and the corresponding adjusted parameter values (i.e., elastic properties in Table 3.1 and the multi-linear stress-strain relationship simulating timber mechanical behaviour) are given in this appendix. In the model, the elastic properties from Table 3.1 were first scaled by the factor SCE, to achieve approximately the same initial stiffness compared with the test data (as shown in Figure 5.1 and Figure 5.2). Then, parameters for the multi-linear stress-strain relationship (σ1, ε1, σ2 and ε2 as shown in Figure 3.6 and Figure K. 1) simulating the timber mechanical behaviour were adjusted within a reasonable range, to achieve a good agreement between the model and test results. Corresponding to the results in Figure 5.1 and Figure 5.2, the values of the factor SCE and the adjusted parameters are given in Table K. 1 and Figure K. 1. The scaled factor SCE was not mechanics based, and it was chosen to fit the simulated and observed load-deformation results of the short-term static bending tests. In the finite element model, the consideration of 1 mm wide gaps in cross layers can lead to a significant lower estimate of the stiffness of the member. Also, the finite element estimation of the deformation from the simply supported member did not consider the difference between local deformation at the pin supports and the contribution of the bearing steel plates in the real test. The local deformation issues can lead to model prediction with around 5% reduction in stiffness. Finally 239 there is no verification of the assumed elastic material property values given in Table 3.2, which can lead to differences in the predicted load-deformation response of the beams. Although using the scaled elasticity property cannot be fully justified mechanistically, the impact on the estimation of rolling shear stress is small, as shown in Table K.2. For example, with or without considering the scaled factor, the evaluated rolling shear stresses have small differences for three-layer CLT (between 1.13 MPa and 1.17 MPa - a difference of 3.4%). Therefore, either the scaled or non-scaled finite element models can be used to estimate rolling shear stresses in the thesis. The adjusted values were also given in Table K. 1. For example, for three-layer CLT in Figure 5.2, SCE = 1.20. 1) For tension parallel to grain behaviour with the chosen tension parallel to grain strength σ2 = 65.00 MPa, σ1 = 65.00 × 0.75 = 48.75 MPa, ε1 =σ1EL=48.7511.43×103×1.2= 3.55 × 10−3. Then σ2 = 65.00 MPa, and ε2 = ε1 + 1.5 ×(σ2−σ1)EL= 3.55 × 10−3 + 1.5 ×16.2511.43×103×1.2=5.33 × 10−3. 2) Similarly for compression parallel to grain behaviour with the chosen compression parallel to grain strength σ2 = 43.10 MPa, σ1 = 43.10 × 0.75 = 32.33 MPa, ε1 =σ1EL=32.3311.43×103×1.2=2.36 × 10−3. Then σ2 = 43.10 MPa, and ε2 = ε1 + 1.5 ×(σ2−σ1)EL= 2.36 × 10−3 + 1.5 ×10.7711.43×103×1.2= 3.54 × 10−3. 240 3) For longitudinal shear behaviour (zero stiffness in the end) with the chosen longitudinal shear strength σ1 = 7.27 MPa, ε1 =σ1GLT=7.270.667×103×1.2= 9.09 × 10−3. 4) For rolling shear behaviour (zero stiffness in the end) in three-layer CLT, the definition starts from SCE = 1.20 and the chosen rolling shear strength σ2 = 1.00 MPa. Then σ1 = 0.20 × σ2 =0.20 MPa, and ε1 =σ1GRT=0.2066.67×1.2= 2.50 × 10−3. ε2 = ε1 + 2 ×(σ2−σ1)GRT= 2.50 × 10−3 + 2 ×0.866.67×1.2= 2.25 × 10−2. 241 Table K. 1 Adjusted parameters in finite element beam modeling for short-term ramp loading CLT Mechanical behaviour 𝛔𝟏 (𝑴𝑷𝒂) 𝛆𝟏 𝛔𝟐 (𝑴𝑷𝒂) 𝛆𝟐 Scaled factor 𝐒𝐂𝐄 Stress-strain relation Three-layer in Figure 5.2 Tension parallel to grain 48.75 3.55× 10−3 65.00 5.33× 10−3 1.20 Bi-linear Compression parallel to grain 32.32 2.36× 10−3 43.10 3.54× 10−3 Bi-linear Longitudinal shear (Zero stiffness in the end) 7.27 9.09× 10−3 Bi-linear Rolling shear (Zero stiffness in the end) 0.20 2.50× 10−3 1.00 2.25× 10−2 Multi-linear Five-layer in Figure 5.1 Tension parallel to grain 48.75 2.03× 10−3 65.00 3.05× 10−3 2.10 Bi-linear Compression parallel to grain 32.32 1.35× 10−3 43.10 2.02× 10−3 Bi-linear Longitudinal shear (Zero stiffness in the end) 7.27 5.20× 10−3 Bi-linear Rolling shear (Zero stiffness in the end) 0.22 1.57× 10−3 1.10 1.42× 10−2 Multi-linear 242 Table K. 2 Evaluated maximum rolling shear stresses with or without scaled factors in finite element models Evaluated maximum rolling shear stresses Scaled factor 𝐒𝐂𝐄 Stress with scaled factor (𝑴𝑷𝒂) Stress without scaled factor (𝑴𝑷𝒂) Three-layer CLT under 10.33 kN load level Elastic finite element analysis 1.20 1.13 1.17 Nonlinear finite element analysis 1.00 1.00 Five-layer CLT under 17.69 kN load level Elastic finite element analysis 2.10 1.58 1.61 Nonlinear finite element analysis 1.09 1.10 Figure K. 1 Stress-strain relationship 0 Multi-linear 0 Bi-linear (Zero stiffness in the end) 0 Linear 0 Bi-linear 243 Appendix L Introduction of the Fitting Process and Adjusted Parameter Values in Finite Element Beam Modeling for Long-Term Trapezoidal Loading In order to investigate and model the recorded displacement measurements related to creep behaviour (as shown from Figure 5.6 to Figure 5.14), the modified five-parameter rheological model was embedded into the developed Ansys finite element model. The displacement of specimens with a lower span-to-depth ratio is mainly attributed to the shear deformation in the CLT element (Fellmoser and Blass, 2004), and considering the computational efficiency and the objective of investigation of rolling shear, only the wood material in the cross layer was included in this creep modeling. In the Ansys platform, there is no defined creep behaviour in the library corresponding to the modified five-parameter rheological model, so the User programmable features in the ANSYS v14.0 platform (SAS, 2011) was adopted to provide the rheological constitutive relationship for wood by coding customized Fortran subroutines, i.e., the constitutive relationship based on the modified five-parameter rheological model in Equation ( 3.4 ) and ( 3.5 ). This allows the model to calculate and estimate the creep in the program. In the subroutines, the constant characteristic strength value in Equation ( 3.4 ) and ( 3.5 ) was 𝜎?̅? = 1.41𝑀𝑃𝑎 for three-layer CLT, and 𝜎?̅? = 1.76𝑀𝑃𝑎 for five-layer CLT (to be consistent with the mean strength results in Table 5.1). According to the literature (SAS, 2011), the creep calculation in the Ansys platform is based on the creep equations integrated with an explicit Euler forward algorithm. By coding and calculating the stress, the creep strain, the creep strain increment and the strain converted tensor, 244 the creep behaviour can be evaluated under the different loading protocols. The further calculation details can be found in the literature (SAS, 2011). The finite element beam model fitting process for long-term displacement records (from Figure 5.6 to Figure 5.14) and the corresponding adjusted parameter values (i.e., elastic properties in Table 3.1, the multi-linear stress-strain relationship simulating timber mechanical behaviour and the parameters of the rheological model) are also given in this appendix. In the model, to achieve approximately the same initial stiffness compared with the test data (from Figure 5.6 to Figure 5.14), the elastic properties from Table 3.1 were first scaled by the factor SCE. Then, to achieve a good agreement between the model and test results, only rolling shear multi-linear stress-strain relationship was adjusted among the previous adjusted parameters of Appendix K (σ1, ε1, σ2 and ε2 which simulated the timber mechanical behaviour in short-term ramp loading tests, as shown in Figure 3.6, Figure K. 1 and Table K. 1). Corresponding to the results from Figure 5.6 to Figure 5.14, the values of the factor SCE and the adjusted parameters, including the five parameters of the embedded rheological model, are given in Table L. 1 and Table L. 2. The search for the values of the five parameters in the rheological model is also introduced as follows: 1) The initial distribution parameters (mean value and COV) for the five parameters (β1 to β5) were chosen, and the timber rolling shear multi-linear stress-strain relation was also defined. 245 2) Assume the five parameters (β1 to β5) were lognormally distributed, and based on the initial distribution parameters chosen in step 1), a random sample size of 𝑁𝑅𝑛𝑑 = 30 sets of the five parameters was generated. 3) Each random set was the input for a finite element model calculation in the long-term displacement investigation. 4) Then, based on the model displacement output in time history, by comparing the test data and the model simulation, the random parameter set showing small differences from the test results was recorded. This random parameter set was slightly adjusted and was sent back to step 1) as the initial chosen mean value of the distribution parameters. 5) In an iteration process, the final set of the adjusted parameter values (in Table L. 1 and Table L. 2) was selected, when the model outputs and test results agreed well (as shown from Figure 5.6 to Figure 5.14). It is not clear about the physical meaning behind the searched values of these five parameters in the rheological model (except for their conceptual definitions (Wang, 2010) as given in Section 2.5 and Section 3.4). However, the adjusted parameters given in Table L. 1 and Table L. 2 can represent the test data well. Since this introduced creep modeling work is only an attempt to predict creep response and to track the recorded displacement measurements related to creep behaviour, no further investigation was performed. In other words, no more information was produced from these developed codes and models, such as the different model behaviour in serviceability under different loading protocols. Therefore, more test data and detailed modeling investigation, evaluating the creep behaviour and the physical meaning of these model parameters, are recommended in the future research. 246 Table L. 1 Adjusted parameters in finite element SPF3 beam model for long-term trapezoidal loading Three-layer CLT Rheological model SPF3-Minmum Mechanical behaviour 𝛔𝟏 (𝑴𝑷𝒂) 𝛆𝟏 𝛔𝟐 (𝑴𝑷𝒂) 𝛆𝟐 Stress-strain relation Rolling shear (Zero stiffness in the end) 0.32 4.07× 10−3 1.60 3.66× 10−2 Multi-linear In Figure 5.12 𝛃𝟏 𝛃𝟐 𝛃𝟑 𝛃𝟒 𝛃𝟓 Scaled factor 𝐒𝐂𝐄 SPF3-Min 9.005× 10−6 1.287× 10−6 13.647 9.266× 10−8 0.368 1.650 Rheological model SPF3-Maximum Mechanical behaviour 𝛔𝟏 (𝑴𝑷𝒂) 𝛆𝟏 𝛔𝟐 (𝑴𝑷𝒂) 𝛆𝟐 Stress-strain relation Rolling shear (Zero stiffness in the end) 0.28 4.81× 10−3 1.41 4.33× 10−2 Multi-linear In Figure 5.14 𝛃𝟏 𝛃𝟐 𝛃𝟑 𝛃𝟒 𝛃𝟓 Scaled factor 𝐒𝐂𝐄 SPF3-Max 2.032× 10−5 4.361× 10−6 13.647 3.324× 10−7 0.368 1.180 247 Table L. 2 Adjusted parameters in finite element SPF5 beam model for long-term trapezoidal loading Five-layer CLT Rheological model SPF5-Minmum Mechanical behaviour 𝛔𝟏 (𝑴𝑷𝒂) 𝛆𝟏 𝛔𝟐 (𝑴𝑷𝒂) 𝛆𝟐 Stress-strain relation Rolling shear (Zero stiffness in the end) 0.38 2.59× 10−3 1.90 2.33× 10−2 Multi-linear In Figure 5.7 𝛃𝟏 𝛃𝟐 𝛃𝟑 𝛃𝟒 𝛃𝟓 Scaled factor 𝐒𝐂𝐄 SPF5-Min 8.208× 10−8 3.821× 10−8 10.965 9.291× 10−9 0.504 2.200 Rheological model SPF5-Maximum Mechanical behaviour 𝛔𝟏 (𝑴𝑷𝒂) 𝛆𝟏 𝛔𝟐 (𝑴𝑷𝒂) 𝛆𝟐 Stress-strain relation Rolling shear (Zero stiffness in the end) 0.36 3.42× 10−3 1.80 3.08× 10−2 Multi-linear In Figure 5.9 𝛃𝟏 𝛃𝟐 𝛃𝟑 𝛃𝟒 𝛃𝟓 Scaled factor 𝐒𝐂𝐄 SPF5-Max 4.752× 10−8 2.080× 10−8 10.965 8.802× 10−8 0.504 1.580 248 Appendix M Introduction of the Evaluation of the Rolling Shear Strength by Short-Term Ramp Loading Based on the cumulative distributions of rolling shear failure load data from short-term ramp loading tests, as shown in Figure 4.18, thirty rolling shear failure load data points (fitted to the same distribution) corresponding to the respective rank (the 𝑖𝑡ℎ data point is corresponding to the cumulative probability of 𝑖 31⁄ ) are generated to represent the rolling shear failure load distributions in Figure 4.18 (ASTM D2915-10, 2011). Different fitting techniques, such as non-parametric fitting, Weibull fitting and Lognormal fitting, are available for this fitting process (Foschi, 1989; Madsen, 1992). The non-parametric fitting technique, which generates data points by linear interpolation from test data, was adopted. The coefficients of variation of the test data were 24.3% for three-layer CLT and 12.6% for five-layer CLT. The fitting process would introduce slight differences in the coefficients of variation, and the coefficients of variation with the small differences were 23.3% for three-layer CLT and 12.2% for five-layer CLT. These thirty generated data points for each CLT configuration are listed in Table M. 1 and Table M. 2. Each one of these thirty generated rolling shear failure loads was then inputted into the finite element beam model for a linear elastic finite element analysis. The maximum rolling shear stress (i.e., the rolling shear strength) was evaluated in the cross layer, as shown in Table M. 1 and Table M. 2. 249 Table M. 1 Rolling shear failure load fitting data and rolling shear strength for three-layer CLT No. Rolling shear failure loads (kN) Cumulative probability rank Rolling shear strength (MPa) No. Rolling shear failure loads (kN) Cumulative probability rank Rolling shear strength (MPa) 1 7.24 0.03 0.82 16 12.56 0.52 1.42 2 7.96 0.06 0.90 17 12.79 0.55 1.45 3 8.50 0.10 0.96 18 13.20 0.58 1.49 4 8.89 0.13 1.00 19 13.40 0.61 1.51 5 9.12 0.16 1.03 20 13.62 0.65 1.54 6 9.71 0.19 1.10 21 14.02 0.68 1.58 7 10.00 0.23 1.13 22 14.23 0.71 1.61 8 10.33 0.26 1.17 23 14.74 0.74 1.67 9 10.83 0.29 1.22 24 14.92 0.77 1.69 10 11.00 0.32 1.24 25 15.42 0.81 1.74 11 11.10 0.35 1.25 26 15.81 0.84 1.79 12 11.29 0.39 1.28 27 15.97 0.87 1.80 13 11.48 0.42 1.30 28 16.52 0.90 1.87 14 11.55 0.45 1.31 29 17.38 0.94 1.96 15 11.91 0.48 1.35 30 18.51 0.97 2.09 250 Table M. 2 Rolling shear failure load fitting data and rolling shear strength for five-layer CLT No. Rolling shear failure loads (kN) Cumulative probability rank Rolling shear strength (MPa) No. Rolling shear failure loads (kN) Cumulative probability rank Rolling shear strength (MPa) 1 13.71 0.03 1.25 16 19.90 0.52 1.81 2 14.74 0.06 1.34 17 19.97 0.55 1.82 3 15.39 0.10 1.40 18 20.29 0.58 1.85 4 16.30 0.13 1.48 19 20.65 0.61 1.88 5 16.62 0.16 1.51 20 20.77 0.65 1.89 6 17.02 0.19 1.55 21 20.84 0.68 1.90 7 17.42 0.23 1.59 22 20.90 0.71 1.90 8 17.69 0.26 1.61 23 21.08 0.74 1.92 9 18.08 0.29 1.65 24 21.32 0.77 1.94 10 18.56 0.32 1.69 25 21.44 0.81 1.95 11 19.18 0.35 1.75 26 21.67 0.84 1.97 12 19.57 0.39 1.78 27 21.86 0.87 1.99 13 19.66 0.42 1.79 28 22.07 0.90 2.01 14 19.76 0.45 1.80 29 22.33 0.94 2.03 15 19.82 0.48 1.80 30 23.22 0.97 2.11 251 Appendix N Introduction of the Calculated Ratio 𝑻𝑽 The information of the ratio 𝑇𝑉 in Equation ( 6.3 ) is introduced in this Appendix. 𝑇𝑉 is defined as the ratio between the sectional load-carrying capacity calculated from different beam theories (i.e., the layered beam theory, the gamma beam theory and the shear analogy theory) and the shear stress value. For each beam theory, the relationship between the sectional load-carrying capacity and the corresponding calculated shear stress value is introduced in Appendix I and Section 2.3, as shown in Equations ( 2.1 ), ( 2.2 ) and ( 2.4 ). The calculated 𝑇𝑉 values for the three-layer and five-layer CLT groups are shown in Table N. 1. For example, in the layered beam theory, assuming the applied centre-point load is 𝑃𝑙𝑜𝑎𝑑 = 1𝑘𝑁 on the three-layer CLT, the calculated shear stress value in the middle of the cross layer is 𝜏 = 𝜏𝐴 = 0.134𝑀𝑃𝑎 (sectional point A is as shown in Figure I. 1). The ratio 𝑇𝑉 =𝑃𝑙𝑜𝑎𝑑𝜏⁄ =10.134⁄ = 7.46𝑘𝑁𝑀𝑃𝑎⁄ . Table N. 1 Summary of the calculated TV values for the three-layer and five-layer CLT groups CLT 𝑇𝑉 from Layered beam 𝑇𝑉 from Gamma beam 𝑇𝑉 from Shear analogy Three-layer 7.46 10.20 7.46 Five-layer 11.24 11.90 11.76 252 Appendix O Introduction of Snow Load in Halifax and Vancouver The snow load information for different locations (Quebec City, Ottawa, Saskatoon, Halifax and Vancouver) is introduced in this Appendix. This snow load information comes from the statistics on the maximum annual snow depth, the snow duration and the ground-to-roof snow conversion factors provided by the National Research Council of Canada (Foschi, 1989). In Section 6.5.1 for the reliability analysis of short-term rolling shear strength of CLT, based on the First Order Reliability Method (FORM), the statistics on the dead load random variable d in Equation ( 6.6 ) is given below: 𝑑 =𝐷𝐷𝑛= 1.0 + 𝑉𝐷𝑅𝑛 where 𝑅𝑛 is a standard normal random variable. The coefficient of variation 𝑉𝐷 has been taken equal to 𝑉𝐷 = 0.10. The statistics on the live load (i.e., snow load) random variable q in Equation ( 6.6 ) is given as follows: 𝑞 =𝑄𝑄𝑛= 𝑟 ∙ 𝑔 where 𝑟 (i.e., the roof load ratio) is taken to be lognormally distributed with the statistical parameters given in Appendix P, and this 𝑟 introduces the variability between actual roof load and that predicted using the adjusting coefficient from the design code; and, 𝑔 is one random variable which is an Extreme Type I distribution (i.e., Gumbel distribution) expressed as follows: 253 𝑔 = 𝐵∗ −ln(− ln(𝑝))𝐴∗ 𝐵∗ =𝐴 ∙ 𝐵 + ln(30)𝐴 ∙ 𝐵 + 3.3843 𝐴∗ = 𝐴 ∙ 𝐵 + 3.3843 where 𝑝 is an uniform random number between 0.0 and 1.0; and, the distribution parameters 𝐴 and 𝐵 are known for different locations across Canada, as shown in Appendix P. In Section 6.5.2 for the reliability analysis of CLT rolling shear strength under thirty-year snow load, based on the Monte Carlo simulation procedure, the simulation strategy for combined dead load plus snow load was as follows (Foschi, 1989): 1) A value of 𝑁𝑆 was chosen, and this 𝑁𝑆 referred to the number of snow load segments per winter. 2) To calculate the dead load 𝑑, a standard normal random number was chosen and was taken as constant for the whole service life. 3) For each live load (i.e., snow load) segment, a uniform random number 𝑝0 between 0.0 and 1.0 was selected. If 𝑝0 > 𝑝𝑒, where: 𝑝𝑒 = 1.0 − exp [−1𝑁𝑆∙ exp(𝐴 ∙ 𝐵)] there was no snow in that segment, otherwise, another uniform random number was chosen to calculate live load 𝑞. 4) The dead load and live load were combined according to the following paragraph, and the detailed introduction can be found in the literature (Foschi, 1989). 254 Considering the live load Q is snow load, a sequence of total load for NY years (NY=30) with 𝑁𝑆 load segment per winter, will include 𝑁𝑆 × 𝑁𝑌 independent random live load variables 𝑄 plus the random dead load variable 𝐷. The total load at any segment during the winter will be the sum of 𝐷 + 𝑄, and 𝐷 + 𝑄 = 𝑄𝑛(𝑑𝑟 + 𝑞). 𝑄𝑛 is the design live load, and: 𝑟 =𝐷𝑛𝑄𝑛 𝑑 =𝐷𝐷𝑛 𝑞 =𝑄𝑄𝑛 Based on the ultimate strength limit state design equation from the design code (introduced in Section 6.5.1): 1.25𝐷𝑛 + 1.50𝑄𝑛 = 𝜙𝑅𝑆(0.05)𝑇𝑉 The segment load can be expressed as: 𝐷 + 𝑄 =𝜙𝑅𝑆(0.05)𝑇𝑉(1.25𝑟 + 1.50)(𝑑𝑟 + 𝑞) ( O.1 ) During summer months, of course, 𝑄 = 0 and only dead load is present. Then the stress history in each segment is calculated as: 𝜎(𝑡) =𝐷 + 𝑄𝑇𝑉=𝜙𝑅𝑆(0.05)(1.25𝑟 + 1.50)(𝑑𝑟 + 𝑞) For any applied stress history 𝜎(𝑡), Equation ( 3.23 ) can be integrated to calculate the damage parameter 𝛼 at any time 𝑇. For an arbitrary load history, such as thirty-year snow load case, closed-form integration of Equation ( 3.23 ) is not easy to perform. Therefore, an approximate load model is adopted, which subdivides the load history into time intervals of duration Δt, each with a constant applied stress. The detailed introduction can be found in the literature (Foschi, 1989). Consider the i-th interval in the load history, which has at its beginning damage 𝛼𝑖−1. If 255 the constant applied stress during the i-th interval is 𝜎𝑖, the cumulative damage 𝛼𝑖 at the end of the interval is then given by (Foschi, 1989): 𝛼𝑖 = 𝛼𝑖−1𝐾𝑖 + 𝐿𝑖 where 𝐾𝑖 = 𝑒𝑥𝑝[𝑐(𝜎𝑖 − 𝜏0𝜎s)𝑛Δt] 𝐿𝑖 =𝑎𝑐(𝜎𝑖 − 𝜏0𝜎s)𝑏−𝑛(𝐾𝑖 − 1) In Section 6.5.3 for the reliability analysis of CLT rolling shear strength under thirty-year dead load only case, the segment load can be expressed as: 𝐷 + 𝑄 =𝜙𝑅𝑆(0.05)𝑇𝑉(1.40𝑟 + 1.50)(𝑑𝑟 + 𝑞) where 𝑟 = 1000. Then the stress history in each segment is calculated as: 𝜎(𝑡) =𝐷 + 𝑄𝑇𝑉=𝜙𝑅𝑆(0.05)(1.40𝑟 + 1.50)(𝑑𝑟 + 𝑞) 256 Appendix P Statistics on Random Variables and Parameters for Reliability Analysis under Snow Load Table P. 1 Summary of the random variables and parameters Random variables or parameters Category Mean value or constant value COV (if applicable) The roof load ratio 𝑟 All location, sheltered flat roofs 0.600 0.450 Vancouver, sloping roofs 0.800 0.000 𝐴 Quebec City 2.350 Not applicable Ottawa 2.256 Saskatoon 3.551 Halifax 2.151 Vancouver 2.047 𝐵 Quebec City 2.340 Ottawa 1.000 Saskatoon 0.740 Halifax 0.930 Vancouver 0.240 257 Appendix Q Reliability Analysis Results under Thirty-Year Snow Load Cases from Three Different Locations: Quebec City, Ottawa and Saskatoon This appendix is giving the calculated values as shown from Figure 6.21 to Figure 6.26. Following the same reliability analysis process in Sections 6.5.1 and 6.5.2, the reliability results for the rolling shear strength adjustment factors are summarized in Table 6.9 and Table 6.10 for another three different locations: Quebec City, Ottawa and Saskatoon. Curves with the same information in these tables are shown from Figure 6.21 to Figure 6.26. For reliability analysis on five-layer CLT under the thirty-year snow load case: Table Q. 1 Results between the reliability index and the performance factor in the five-layer CLT in curve one (without DOL effect) Five-layer Using the 𝑇𝑉 from Layered beam Using the 𝑇𝑉 from Gamma beam Using the 𝑇𝑉 from Shear analogy Average 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 Quebec City 0.6 3.780 0.6 3.661 0.6 3.687 0.6 3.709 0.7 3.460 0.7 3.337 0.7 3.364 0.7 3.387 0.8 3.174 0.8 3.047 0.8 3.075 0.8 3.099 0.9 2.915 0.9 2.786 0.9 2.814 0.9 2.838 1.0 2.678 1.0 2.546 1.0 2.575 1.0 2.600 1.1 2.459 1.1 2.325 1.1 2.354 1.1 2.379 1.2 2.256 1.2 2.119 1.2 2.149 1.2 2.175 1.3 2.066 1.3 1.926 1.3 1.957 1.3 1.983 258 Figure Q. 1 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Five-layer/ Quebec City, curve one) Table Q. 2 Curve one and curve two results between the reliability index and the performance factor in the five-layer CLT Five-layer Curve one (without DOL effect) Curve two (with DOL effect) 𝜙 𝛽 𝜙 𝛽 Quebec City 0.6 3.709 0.4 2.748 0.7 3.387 0.45 2.512 0.8 3.099 0.5 2.257 0.9 2.838 1.0 2.600 1.1 2.379 1.2 2.175 1.3 1.983 259 Table Q. 3 Results between the reliability index and the performance factor in the five-layer CLT in curve one (without DOL effect) Five-layer Using the 𝑇𝑉 from Layered beam Using the 𝑇𝑉 from Gamma beam Using the 𝑇𝑉 from Shear analogy Average 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 Ottawa 0.6 3.567 0.6 3.452 0.6 3.477 0.6 3.499 0.7 3.257 0.7 3.139 0.7 3.165 0.7 3.187 0.8 2.982 0.8 2.861 0.8 2.888 0.8 2.910 0.9 2.734 0.9 2.611 0.9 2.638 0.9 2.661 1.0 2.508 1.0 2.382 1.0 2.410 1.0 2.433 1.1 2.300 1.1 2.171 1.1 2.200 1.1 2.224 1.2 2.106 1.2 1.975 1.2 2.004 1.2 2.028 1.3 1.925 1.3 1.792 1.3 1.821 1.3 1.846 Figure Q. 2 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Five-layer/ Ottawa, curve one) 260 Table Q. 4 Curve one and curve two results between the reliability index and the performance factor in the five-layer CLT Five-layer Curve one (without DOL effect) Curve two (with DOL effect) 𝜙 𝛽 𝜙 𝛽 Ottawa 0.6 3.499 0.35 2.911 0.7 3.187 0.4 2.590 0.8 2.910 0.5 2.101 0.9 2.661 1.0 2.433 1.1 2.224 1.2 2.028 1.3 1.846 261 Table Q. 5 Results between the reliability index and the performance factor in the five-layer CLT in curve one (without DOL effect) Five-layer Using the 𝑇𝑉 from Layered beam Using the 𝑇𝑉 from Gamma beam Using the 𝑇𝑉 from Shear analogy Average 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 Saskatoon 0.6 3.604 0.6 3.488 0.6 3.514 0.6 3.535 0.7 3.293 0.7 3.174 0.7 3.200 0.7 3.222 0.8 3.016 0.8 2.894 0.8 2.921 0.8 2.944 0.9 2.766 0.9 2.641 0.9 2.669 0.9 2.692 1.0 2.538 1.0 2.411 1.0 2.439 1.0 2.463 1.1 2.327 1.1 2.198 1.1 2.227 1.1 2.251 1.2 2.132 1.2 2.000 1.2 2.029 1.2 2.054 1.3 1.949 1.3 1.815 1.3 1.845 1.3 1.870 Figure Q. 3 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Five-layer/ Saskatoon, curve one) 262 Table Q. 6 Curve one and curve two results between the reliability index and the performance factor in the five-layer CLT Five-layer Curve one (without DOL effect) Curve two (with DOL effect) 𝜙 𝛽 𝜙 𝛽 Saskatoon 0.6 3.535 0.35 3.090 0.7 3.222 0.4 2.652 0.8 2.944 0.5 2.409 0.9 2.692 0.6 1.774 1.0 2.463 1.1 2.251 1.2 2.054 1.3 1.870 For reliability analysis on three-layer CLT under the thirty-year snow load case: 263 Table Q. 7 Results between the reliability index and the performance factor in the three-layer CLT in curve one (without DOL effect) Three-layer Using the 𝑇𝑉 from Layered beam Using the 𝑇𝑉 from Gamma beam Using the 𝑇𝑉 from Shear analogy Average 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 Quebec City 0.6 3.562 0.6 3.138 0.6 3.562 0.6 3.421 0.7 3.358 0.7 2.912 0.7 3.358 0.7 3.209 0.8 3.173 0.8 2.706 0.8 3.173 0.8 3.017 0.9 3.003 0.9 2.516 0.9 3.003 0.9 2.841 1.0 2.845 1.0 2.338 1.0 2.845 1.0 2.676 1.1 2.697 1.1 2.170 1.1 2.697 1.1 2.521 1.2 2.557 1.2 2.012 1.2 2.557 1.2 2.375 1.3 2.423 1.3 1.861 1.3 2.423 1.3 2.236 Figure Q. 4 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Three-layer/ Quebec City, curve one) 264 Table Q. 8 Curve one and curve two results between the reliability index and the performance factor in the three-layer CLT Three-layer Curve one (without DOL effect) Curve two (with DOL effect) 𝜙 𝛽 𝜙 𝛽 Quebec City 0.6 3.421 0.4 2.652 0.7 3.209 0.5 2.457 0.8 3.017 0.55 2.257 0.9 2.841 0.6 1.960 1.0 2.676 0.7 1.685 1.1 2.521 1.2 2.375 1.3 2.236 265 Table Q. 9 Results between the reliability index and the performance factor in the three-layer CLT in curve one (without DOL effect) Three-layer Using the 𝑇𝑉 from Layered beam Using the 𝑇𝑉 from Gamma beam Using the 𝑇𝑉 from Shear analogy Average 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 Ottawa 0.6 3.485 0.6 3.056 0.6 3.485 0.6 3.342 0.7 3.279 0.7 2.828 0.7 3.279 0.7 3.129 0.8 3.092 0.8 2.620 0.8 3.092 0.8 2.935 0.9 2.920 0.9 2.428 0.9 2.920 0.9 2.756 1.0 2.760 1.0 2.250 1.0 2.760 1.0 2.590 1.1 2.610 1.1 2.082 1.1 2.610 1.1 2.434 1.2 2.469 1.2 1.925 1.2 2.469 1.2 2.288 1.3 2.336 1.3 1.775 1.3 2.336 1.3 2.149 Figure Q. 5 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Three-layer/ Ottawa, curve one) 266 Table Q. 10 Curve one and curve two results between the reliability index and the performance factor in the three-layer CLT Three-layer Curve one (without DOL effect) Curve two (with DOL effect) 𝜙 𝛽 𝜙 𝛽 Ottawa 0.6 3.342 0.4 2.576 0.7 3.129 0.5 2.512 0.8 2.935 0.6 2.014 0.9 2.756 0.7 1.530 1.0 2.590 1.1 2.434 1.2 2.288 1.3 2.149 267 Table Q. 11 Results between the reliability index and the performance factor in the three-layer CLT in curve one (without DOL effect) Three-layer Using the 𝑇𝑉 from Layered beam Using the 𝑇𝑉 from Gamma beam Using the 𝑇𝑉 from Shear analogy Average 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 𝜙 𝛽 Saskatoon 0.6 3.499 0.6 3.071 0.6 3.499 0.6 3.356 0.7 3.293 0.7 2.843 0.7 3.293 0.7 3.143 0.8 3.107 0.8 2.636 0.8 3.107 0.8 2.950 0.9 2.935 0.9 2.444 0.9 2.935 0.9 2.771 1.0 2.776 1.0 2.266 1.0 2.776 1.0 2.606 1.1 2.626 1.1 2.098 1.1 2.626 1.1 2.450 1.2 2.485 1.2 1.940 1.2 2.485 1.2 2.303 1.3 2.352 1.3 1.791 1.3 2.352 1.3 2.165 Figure Q. 6 Curves between the reliability index and the performance factor without considering the duration-of-load effect (Three-layer/ Saskatoon, curve one) 268 Table Q. 12 Curve one and curve two results between the reliability index and the performance factor in the three-layer CLT Three-layer Curve one (without DOL effect) Curve two (with DOL effect) 𝜙 𝛽 𝜙 𝛽 Saskatoon 0.6 3.356 0.45 2.576 0.7 3.143 0.5 2.512 0.8 2.950 0.6 2.197 0.9 2.771 0.7 1.812 1.0 2.606 1.1 2.450 1.2 2.303 1.3 2.165
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Duration-of-load and size effects on the rolling shear strength of cross laminated timber Li, Yuan 2015
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Title | Duration-of-load and size effects on the rolling shear strength of cross laminated timber |
Creator |
Li, Yuan |
Publisher | University of British Columbia |
Date Issued | 2015 |
Description | In the beginning of the twenty-first century, the largest mountain pine beetle (MPB) outbreak ever recorded struck western Canada. A huge volume of MPB-attacked lodgepole pine is expected to hit the BC forest industry in the next decade. Technologies to convert MPB-attacked lumber into engineering wood products are urgently required. Cross laminated timber (CLT) is a technology that can produce massive timber members as an engineered wood product for timber structures. In this study, the duration-of-load and size effects on the rolling shear strength of CLT manufactured from MPB-afflicted lumber were evaluated. The study of the duration-of-load effect on the strength properties of wood products is typically challenging; and, additional complexity exists with the duration-of-load effect on the rolling shear strength of CLT, given the necessary consideration of crosswise layups of wood boards, existing gaps and glue bonding between layers. In this research, short-term ramp loading tests and long-term trapezoidal fatigue loading tests (damage accumulation tests) were used to study the duration-of-load behaviour of the rolling shear strength of CLT. In the ramp loading test, three-layer CLT products showed a relatively lower rolling shear load-carrying capacity. Torque loading tests on CLT tubes were also performed. The finite element method was adopted to simulate the structural behaviour of CLT specimens. Evaluation of the rolling shear strength based on test data was discussed. The size effect on the rolling shear strength was investigated. A stress-based damage accumulation theory was used to evaluate the duration-of-load effect on CLT rolling shear. The model was first calibrated against the test data and used to investigate the long-term CLT rolling shear strength. A reliability-based method was then applied to assess the CLT rolling shear performance. Finally, a duration-of-load adjustment factor for CLT rolling shear strength was established. The duration-of-load adjustment factor for the three-layer and five-layer CLT is different from that for lumber. The results suggest that the rolling shear duration-of-load strength adjustment factor for CLT is more severe than the general duration-of-load adjustment factor for lumber, and this difference should be considered in the introduction of CLT into the building codes for engineered wood design. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2015-03-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 2.5 Canada |
DOI | 10.14288/1.0135686 |
URI | http://hdl.handle.net/2429/52397 |
Degree |
Doctor of Philosophy - PhD |
Program |
Forestry |
Affiliation |
Forestry, Faculty of |
Degree Grantor | University of British Columbia |
GraduationDate | 2015-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/2.5/ca/ |
AggregatedSourceRepository | DSpace |
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