Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Ahmadipour, Moein
Doctor of Philosophy - PhD
University of British Columbia
Wood has been a key construction material in North America and around the world for decades for the principal residence of the public. It is gaining constant attention due to its renewability nature, cost-effectiveness and lightweight compared to other major construction materials. I-joists are engineered timber beams widely used in the building industry as main structural components. They have gained popularity over conventional sawn lumber beams, thanks to their effectiveness and convenience of use. These beams are mainly meant to carry the loads in roof and floor systems. During construction, however, these beams are often drilled in their web or notched through their flange to pass service, electrical or mechanical facilities through. Even though this makes them more favorable for construction, cutting the beam through can reduce its capacity to a significant extent. This raises serious concerns for the safety of the structure and the public. It is vital to notify the responsible sectors of the importance of this fact and have guidelines to evaluate the extent of the decadence and design the joists accordingly. The current wood design code in Canada does not have direct instructions for designing I-joists with openings. In the absence of verified test results, it leaves the allowability of web openings with I-joist manufacturers and does not allow flange cuts. In practice, however, it has been seen that these limits are at times contravened, and joists are retrofitted improperly using unsupported methods to compensate for their reduced capacity. Having a guideline for openings in I-joists will lead to safer and more efficient building designs. Such guidelines will provide structural engineers with directions to determine the sufficiency of I-joists considering their opening’s shape, location and size. It may also help designers find remedies to reinforce the joists by retrofitting them using approved means. In this PhD dissertation, the stiffness of simply-supported timber I-joists with a web hole and those with a flange notch is examined. The susceptibility of these joists to openings is studied through experimental and numerical study. As a first step, wood specimens are tested in order to obtain the material properties of the timber used in the I-joists. Three-dimensional large-scale finite element models of the beams are then developed. In the next step, a sensitivity analysis is conducted to study the effects of the shape, size, and location of the opening, and find the important factors affecting the performance of the joist. Using the response surface method, expressions are then developed to predict the response characteristics of the joists, including the stiffness of the joist for given serviceability criteria. Furthermore, an additional study is performed to assess the effectiveness of two retrofit types in reinforcing joists with an opening. Characteristic parameters of the retrofits are also studied, such as the type and length of the retrofit used. Finally, guidelines are proposed for designing I-joists with a web hole and those with a flange notch, assimilating the routines of the current Canadian wood design standard, CSA O86-19. In addition, guidelines are also proposed to suggest retrofits to bring the joists back to their expected serviceability criteria.
LIMIT STATE DESIGN OF TIMBER I-JOISTS WITH WEB OPENINGS AND FLANGE NOTCHES by Moein Ahmadipour B.A.Sc., Shiraz University, 2010 M.A.Sc., Shiraz University, 2012 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE COLLEGE OF GRADUATE STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) August 2023 © Moein Ahmadipour, 2023 ii The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis/dissertation entitled: LIMIT STATE DESIGN OF TIMBER I-JOISTS WITH WEB OPENINGS AND FLANGE NOTCHES submitted by Moein Ahmadipour in partial fulfillment of the requirements of the degree of Doctor of Philosophy . Dr. Shahria Alam, School of Engineering Supervisor Dr. Abbas S. Milani, School of Engineering Supervisory Committee Member Dr. Rudolf Seethaler, School of Engineering Supervisory Committee Member Dr. Thomas Johnson, School of Engineering University Examiner Dr. Ying Hei Chui, Department of Civil and Environmental Engineering, University of Alberta External Examiner iii ABSTRACT Wood has been a key construction material in North America and around the world for decades for the principal residence of the public. It is gaining constant attention due to its renewability nature, cost-effectiveness and lightweight compared to other major construction materials. I-joists are engineered timber beams widely used in the building industry as main structural components. They have gained popularity over conventional sawn lumber beams, thanks to their effectiveness and convenience of use. These beams are mainly meant to carry the loads in roof and floor systems. During construction, however, these beams are often drilled in their web or notched through their flange to pass service, electrical or mechanical facilities through. Even though this makes them more favorable for construction, cutting the beam through can reduce its capacity to a significant extent. This raises serious concerns for the safety of the structure and the public. It is vital to notify the responsible sectors of the importance of this fact and have guidelines to evaluate the extent of the decadence and design the joists accordingly. The current wood design code in Canada does not have direct instructions for designing I-joists with openings. In the absence of verified test results, it leaves the allowability of web openings with I-joist manufacturers and does not allow flange cuts. In practice, however, it has been seen that these limits are at times contravened, and joists are retrofitted improperly using unsupported methods to compensate for their reduced capacity. Having a guideline for openings in I-joists will lead to safer and more efficient building designs. Such guidelines will provide structural engineers with directions to determine the sufficiency of I-joists considering their opening’s shape, location and size. It may also help designers find remedies to reinforce the joists by retrofitting them using approved means. In this PhD dissertation, the stiffness of simply-supported timber I-joists with a web hole and those with a flange notch is examined. The susceptibility of these joists to openings is studied through experimental and numerical study. As a first step, wood specimens are tested in order to obtain the material properties of the timber used in the I-joists. Three-dimensional large-scale finite element models of the beams are then developed. In the next step, a sensitivity analysis is conducted to study the effects of the shape, size, and location of the opening, and find the important factors affecting the performance of the joist. Using the response surface method, expressions are then iv developed to predict the response characteristics of the joists, including the stiffness of the joist for given serviceability criteria. Furthermore, an additional study is performed to assess the effectiveness of two retrofit types in reinforcing joists with an opening. Characteristic parameters of the retrofits are also studied, such as the type and length of the retrofit used. Finally, guidelines are proposed for designing I-joists with a web hole and those with a flange notch, assimilating the routines of the current Canadian wood design standard, CSA O86-19. In addition, guidelines are also proposed to suggest retrofits to bring the joists back to their expected serviceability criteria. v LAY SUMMARY Wood is a key construction material in North America thanks to its renewable nature, cost-effectiveness, and light weight. I-joists are engineered timber beams widely used in buildings. These beams can be drilled in their web or notched through their flange to pass electrical or mechanical facilities through. This, however, can reduce their capacity to a significant extent, which arises concerns for the safety of the structure and the public. The current wood design code in Canada does not have direct instructions for designing I-joists with openings. Having such guideline will lead to safer and more efficient building designs. In this PhD dissertation, timber I-joists with an opening are examined to study the effects of the shape, size and location of the opening. Additional study is performed to assess the effectiveness of two retrofit types in reinforcing these joists. Finally, guidelines are suggested for designing and retrofitting I-joists with an opening. vi PREFACE The research in this dissertation has been conducted as part of a project in Dr. Shahria Alam’s research group. It is published or will be submitted for publications in a number of journals and conference proceedings. Chapter 2 is based on the following peer-reviewed journals, corresponding to experimental research conducted by Md. Shahnewaz, M. Shahidul Islam and Moein Ahmadipour, and reviewed by Drs. Thomas Tannert and Shahria Alam: • Shahnewaz, M., Islam, M. S., Ahmadipour, M., Tannert, T., & Alam, M. S. (2017). Reinforced Wood I-Joists with Web Openings. Journal of Structural Engineering, 143(6), 11. https://doi.org/10.1061/(ASCE)ST.1943-541X.0001747 • Islam, M. S., Shahnewaz, M., & Alam, M. S. (2015). Structural capacity of timber I-joist with flange notch: Experimental evaluation. Construction and Building Materials, 79, 290–300. https://doi.org/http://dx.doi.org/10.1016/j.conbuildmat.2015.01.017 The research in Chapter 4 is presented in the 6th International Conference on Engineering Mechanics and Materials, Canadian Society for Civil Engineering,2017 and was awarded 2nd place for the Best Student Paper in Engineering Mechanics and Materials. The research was performed by Moein Ahmadipour and reviewed by Dr. Shahria Alam: • Ahmadipour, Moein, and M. Shahria Alam. 2017. “Finite Element Modeling of Timber I-Joists with Web Holes and Flange Notches.” In 2017 Canadian Society for Civil Engineering Annual Conference (CSCE 2017), Vancouver, BC, Canada. The research in Chapters 5, 6 and 8 is in the submission process for journal publications. Moein Ahmadipour is preparing the following articles under the supervision of Dr. Shahria Alam: • Ahmadipour, Moein, and M. Shahria Alam. “Finite Element Simulation of Composite Timber I-Joists with Web Hole and Flange Notches.” • Ahmadipour, Moein, and M. Shahria Alam. “Response Sensitivity of Timber I-Joists to Web and Flange Openings using RSM.” vii TABLE OF CONTENTS ABSTRACT ................................................................................................................................... iii LAY SUMMARY ........................................................................................................................... v PREFACE ...................................................................................................................................... vi TABLE OF CONTENTS .............................................................................................................. vii LIST OF TABLES ......................................................................................................................... xi LIST OF FIGURES ...................................................................................................................... xv ACKNOWLEDGEMENTS ......................................................................................................... xxi DEDICATION .......................................................................................................................... xxiii Chapter 1: INTRODUCTION AND ORGANIZATION OF THIS DISSERTATION .................. 1 1.1. General ................................................................................................................................. 1 1.2. Objectives ............................................................................................................................ 2 1.3. Organization of this dissertation .......................................................................................... 4 Chapter 2: LITERATURE REVIEW AND REFERENCE EXPERIMENTS ............................... 6 2.1. I-joists in the timber building industry................................................................................. 6 2.2. I-Joists with openings .......................................................................................................... 8 2.3. Reference experiments (experimental study) ..................................................................... 11 Chapter 3: CHARACTERIZATION OF FLANGE MATERIAL ................................................ 16 3.1. General ............................................................................................................................... 16 3.2. Experimental test procedures ............................................................................................. 16 3.2.1. General test requirements ........................................................................................... 17 3.2.2. Compression parallel to the grain ............................................................................... 18 3.2.3. Compression perpendicular to the grain ..................................................................... 20 3.2.4. Tension parallel to the grain ....................................................................................... 20 viii 3.2.5. Tension perpendicular to the grain ............................................................................. 21 3.2.6. Shear parallel to grain ................................................................................................. 22 3.3. Experiments ....................................................................................................................... 24 3.3.1. Design of experiment (DoE) ....................................................................................... 24 3.3.2. Preparation of the raw material ................................................................................... 25 3.3.3. Test equipment ............................................................................................................ 28 3.3.4. Tension-parallel-to-grain test results .......................................................................... 28 3.3.5. Tension-perpendicular-to-grain test ............................................................................ 35 3.3.6. Compression parallel to grain test............................................................................... 42 3.3.7. Compression perpendicular to grain test..................................................................... 48 3.3.8. Shear tests ................................................................................................................... 51 3.4. Summary ............................................................................................................................ 55 Chapter 4: FINITE ELEMENT MODELING OF TIMBER I-JOISTS WITH WEB HOLES OR FLANGE NOTCHES ................................................................................................................... 56 4.1. General ............................................................................................................................... 56 4.1.1. Constitutive equation .................................................................................................. 56 4.1.2. Definition of isotropic, orthotropic and anisotropic materials .................................... 57 4.1.3. Relationship between tensorial and engineering constants ......................................... 59 4.2. Finite element modeling of the test specimens .................................................................. 62 4.3. Finite element modeling of timber I-joists......................................................................... 66 4.4. Summary ............................................................................................................................ 70 Chapter 5: SENSITIVITY ANALYSIS OF TIMBER I-JOISTS WITH WEB HOLES OR FLANGE NOTCHES ................................................................................................................... 72 5.1. General ............................................................................................................................... 72 5.2. Summary of experimental tests .......................................................................................... 73 5.3. Design of experiment ......................................................................................................... 74 ix 5.4. I-joists with web holes ....................................................................................................... 74 5.5. I-joists with flange notches ................................................................................................ 79 5.6. Summary ............................................................................................................................ 84 Chapter 6: PREDICTIVE EQUATIONS FOR I-JOISTS WITH WEB HOLES OR FLANGE NOTCHES .................................................................................................................................... 86 6.1. General ............................................................................................................................... 86 6.2. Response Surface Methodology ........................................................................................ 86 6.3. Method, factors and response variables ............................................................................. 87 6.4. Results and response surface metamodels ......................................................................... 93 6.4.1. Metamodels for I-joists with web hole ....................................................................... 95 6.4.2. Metamodels for I-joists with flange notch .................................................................. 98 6.5. Verification of the response surface models .................................................................... 102 6.6. Summary .......................................................................................................................... 108 Chapter 7: PERFORMANCE EVALUATION OF TIMBER I-JOISTS WITH OPENINGS RETROFITTED WITH WEB AND FLANGE REINFORCEMENTS ..................................... 110 7.1. General ............................................................................................................................. 110 7.2. Retrofit details .................................................................................................................. 111 7.3. Finite element model of the retrofitted I-joists ................................................................ 112 7.4. Design of experiment ....................................................................................................... 113 7.5. Results .............................................................................................................................. 115 7.6. Predictive equations for retrofitted I-joists ...................................................................... 120 7.6.1. Retrofitted joists with web holes............................................................................... 121 7.6.2. Retrofitted joists with flange notches ....................................................................... 128 7.7. Verification of the predictive models .............................................................................. 135 7.8. Summary .......................................................................................................................... 141 x Chapter 8: LIMIT STATE DESIGN OF TIMBER I-JOISTS WITH OPENINGS IN FLOOR SYSTEMS................................................................................................................................... 143 8.1. General ............................................................................................................................. 143 8.1.1. Scope ......................................................................................................................... 144 8.2. Design of I-joists as per CSA O86 ................................................................................... 144 8.2.1. Introduction to specified strengths and design modification factors ........................ 144 8.2.2. Design considerations for I-joists with openings ...................................................... 147 8.2.3. Design considerations for serviceability ................................................................... 147 8.2.4. Design considerations for strength............................................................................ 148 8.3. Design modification factors for I-joists with openings .................................................... 149 8.4. Design examples of a floor joist with opening ................................................................ 164 8.5. Summary .......................................................................................................................... 165 Chapter 9: SUMMARY, CONCLUSIONS AND RECOMMENDED FUTURE STUDY ....... 167 9.1. Summary .......................................................................................................................... 167 9.2. Conclusions ...................................................................................................................... 168 9.2.1. Characterization of timber material .......................................................................... 168 9.2.2. Development of finite element models ..................................................................... 168 9.2.3. Sensitivity analysis of the performance of I-joists with openings ............................ 169 9.2.4. Predictive equations for I-joists with openings ......................................................... 170 9.2.5. Evaluation of retrofitting of I-joists with openings ................................................... 170 9.2.6. Design guideline development for I-joists with openings with and without retrofits171 9.3. Contributions of this research .......................................................................................... 172 9.4. Limitations of the study and recommendations for future research ................................ 172 Bibliography ............................................................................................................................... 175 xi LIST OF TABLES Table 2-1. I-joist experiment specimens ....................................................................................... 13 Table 3-1. Specimen details for compression-parallel-to-grain test ............................................. 19 Table 3-2. Specimen details for compression-perpendicular-to-grain test ................................... 20 Table 3-3. Specimen dimensions for tension-parallel-to-grain tests ............................................ 21 Table 3-4. Specimen dimensions for tension-perpendicular-to-grain tests .................................. 22 Table 3-5. Nominal dimensions of shear test specimens .............................................................. 23 Table 3-6. Tests summary ............................................................................................................. 24 Table 3-7. Design of experiment for material characterization tests ............................................ 25 Table 3-8. Tested specimens’ actual dimensions and moisture content ....................................... 31 Table 3-9. Summary of the test results for replications 1 to 9 for parallel-to-grain tension tests . 33 Table 3-10. Summary of the test results for replications 10 to 18 for parallel-to-grain tension tests....................................................................................................................................................... 34 Table 3-11. Specimen dimensions and moisture content for radial-tension test specimens ......... 37 Table 3-12. Specimen dimensions and moisture content for tangential-tension test specimens .. 38 Table 3-13. Stress-strain test data summary for radial-direction test specimens used for determining material properties and verification .......................................................................... 40 Table 3-14. Stress-strain test data summary for tangential-direction test specimens used for determining material properties and verification .......................................................................... 41 Table 3-15. Specimen dimensions and moisture content for compression-parallel-to-grain test specimens ...................................................................................................................................... 45 Table 3-16. Stress-strain test data summary for parallel-to-grain compression test specimens used for determining material properties and validation ....................................................................... 47 Table 3-17. Specimen dimensions and moisture content for radial-compression test specimens 48 Table 3-18. Specimen dimensions and moisture content for tangential-compression test specimens....................................................................................................................................................... 48 Table 3-19. Stress-strain test data summary for radial-compression test specimens used for determining material characteristics ............................................................................................. 50 Table 3-20. Stress-strain test data summary for tangential-compression test specimens used for determining material characteristics ............................................................................................. 50 xii Table 3-21. Specimen dimensions and moisture content for longitudinal-radial shear test specimens ...................................................................................................................................... 53 Table 3-22. Specimen dimensions and moisture content for longitudinal-tangential shear test specimens ...................................................................................................................................... 53 Table 3-23. Specimen dimensions and moisture content for tangential-radial shear test specimens....................................................................................................................................................... 53 Table 3-24. Stress-strain test data summary for longitudinal-radial shear test specimens used for determining material properties and verification .......................................................................... 54 Table 3-25. Stress-strain test data summary for longitudinal-tangential shear test specimens used for determining material properties and verification .................................................................... 54 Table 3-26. Stress-strain test data summary for tangential-radial shear test specimens used for determining material properties .................................................................................................... 54 Table 4-1. Moduli of elasticity for timber material (MPa) ........................................................... 63 Table 4-2. Moduli of elasticity for OSB material (MPa) (Islam, 2017) ....................................... 67 Table 5-1. Experimental test results summary – P: Average maximum load at joist failure, Δ: Average maximum deflection at joint failure ............................................................................... 73 Table 5-2. Design of experiment and parameter levels for I-joists with web openings ............... 75 Table 5-3. Design of experiment and parameter levels for I-joists with flange notches .............. 79 Table 6-1. Input factors for joists with web holes ........................................................................ 88 Table 6-2. Input factors for joists with flange notches ................................................................. 88 Table 6-3. RSM designs for 12- and 20-ft I-joists with web hole ................................................ 90 Table 6-4. RSM designs for 12-ft I-joists with flange notch ........................................................ 91 Table 6-5. RSM designs for 20-ft I-joists with flange notch ........................................................ 92 Table 6-6. Capacity reduction percentage for 12-ft I-joists with web hole .................................. 94 Table 6-7. Capacity reduction percentage for 20-ft I-joists with web hole .................................. 94 Table 6-8. Capacity reduction percentage for 12-ft I-joists with flange notch ............................. 94 Table 6-9. Capacity reduction percentage for 20-ft I-joists with flange notch ............................. 95 Table 6-10. R2 and adjusted-R2 values for tested models for I-joists with web holes .................. 96 Table 6-11. Summary of ANOVA analysis for tested models for I-joists with web holes .......... 97 Table 6-12. R2 and adjusted-R2 values for tested models for I-joists with flange notches ........... 99 Table 6-13. Summary of ANOVA analysis for tested models for I-joists with web holes ........ 100 xiii Table 6-14. Fit measurements of the suggested prediction models ............................................ 103 Table 6-15. Verification models for joists with web holes ......................................................... 107 Table 6-16. Verification models for joists with flange notches .................................................. 107 Table 7-1. Input factors for retrofitted joists with web holes ..................................................... 114 Table 7-2. Input factors for retrofitted joists with flange notches .............................................. 114 Table 7-3. Capacity increase percentage for 12-ft I-joists with web holes and Type I retrofit .. 121 Table 7-4. Capacity increase percentage for 12-ft I-joists with web holes and Type II retrofit . 121 Table 7-5. Capacity increase percentage for 20-ft I-joists with web holes and Type I retrofit .. 121 Table 7-6. Capacity increase percentage for 20-ft I-joists with web holes and Type II retrofit . 122 Table 7-7. Fit measurements for web-holed joists with Type I retrofit ...................................... 123 Table 7-8. Fit measurements for web-holed joists with Type II retrofit ..................................... 123 Table 7-9. Summary of ANOVA analysis for tested models for I-joists with web holes and Retrofit Type I .......................................................................................................................................... 127 Table 7-10. Summary of ANOVA analysis for tested models for I-joists with web holes and Retrofit Type II ........................................................................................................................... 127 Table 7-11. Capacity increase percentage for 12-ft I-joists with flange notches and Type I retrofit..................................................................................................................................................... 128 Table 7-12. Capacity increase percentage for 12-ft I-joists with flange notches and Type II retrofit..................................................................................................................................................... 128 Table 7-13. Capacity increase percentage for 20-ft I-joists with flange notches and Type I retrofit..................................................................................................................................................... 128 Table 7-14. Capacity increase percentage for 20-ft I-joists with flange notches and Type II retrofit..................................................................................................................................................... 129 Table 7-15. Fit measurements for flange-notched joists with Type I retrofit ............................. 129 Table 7-16. Fit measurements for flange-notched joists with Type II retrofit ............................ 130 Table 7-17. Summary of ANOVA analysis for tested models for I-joists with flange notches and Retrofit Type I............................................................................................................................. 134 Table 7-18. Summary of ANOVA analysis for tested models for I-joists with flange notches and Retrofit Type II ........................................................................................................................... 134 Table 7-19. Verification models for joists with a web hole and retrofit Type I ......................... 136 Table 7-20. Verification models for joists with a web hole and retrofit Type II ........................ 137 xiv Table 7-21. Verification models for joists with a flange notch and retrofit Type I .................... 138 Table 7-22. Verification models for joists with a flange notch and retrofit Type II ................... 139 Table 8-1. Opening Factors for 12-ft I-joists with a web hole.................................................... 150 Table 8-2. Opening Factors for 20-ft I-joists with a web hole.................................................... 150 Table 8-3. Opening Factors for 12-ft I-joists with a flange notch .............................................. 150 Table 8-4. Opening Factors for 20-ft I-joists with a flange notch .............................................. 151 Table 8-5. Opening Factors for 12-ft I-joists with a web hole, Type I retrofitted, R=1 ............. 152 Table 8-6. Opening Factors for 20-ft I-joists with a web hole, Type I retrofitted, R=1 ............. 152 Table 8-7. Opening Factors for 12-ft I-joists with a web hole, Type I retrofitted, R=2 ............. 153 Table 8-8. Opening Factors for 20-ft I-joists with a web hole, Type I retrofitted, R=2 ............. 153 Table 8-9. Opening Factors for 12-ft I-joists with a web hole, Type I retrofitted, R=3 ............. 154 Table 8-10. Opening Factors for 20-ft I-joists with a web hole, Type I retrofitted, R=3 ........... 154 Table 8-11. Opening Factors for 12-ft I-joists with a web hole, Type II retrofitted, R=1 .......... 155 Table 8-12. Opening Factors for 20-ft I-joists with a web hole, Type II retrofitted, R=1 .......... 155 Table 8-13. Opening Factors for 12-ft I-joists with a web hole, Type II retrofitted, R=2 .......... 156 Table 8-14. Opening Factors for 20-ft I-joists with a web hole, Type II retrofitted, R=2 .......... 156 Table 8-15. Opening Factors for 12-ft I-joists with a web hole, Type II retrofitted, R=3 .......... 157 Table 8-16. Opening Factors for 20-ft I-joists with a web hole, Type II retrofitted, R=3 .......... 157 Table 8-17. Opening Factors for 12-ft I-joists with a flange notch, Type I retrofitted, R=1 ...... 158 Table 8-18. Opening Factors for 20-ft I-joists with a flange notch, Type I retrofitted, R=1 ...... 158 Table 8-19. Opening Factors for 12-ft I-joists with a flange notch, Type I retrofitted, R=2 ...... 159 Table 8-20. Opening Factors for 20-ft I-joists with a flange notch, Type I retrofitted, R=2 ...... 159 Table 8-21. Opening Factors for 12-ft I-joists with a flange notch, Type I retrofitted, R=3 ...... 160 Table 8-22. Opening Factors for 20-ft I-joists with a flange notch, Type I retrofitted, R=3 ...... 160 Table 8-23. Opening Factors for 12-ft I-joists with a flange notch, Type II retrofitted, R=1 .... 161 Table 8-24. Opening Factors for 20-ft I-joists with a flange notch, Type II retrofitted, R=1 .... 161 Table 8-25. Opening Factors for 12-ft I-joists with a flange notch, Type II retrofitted, R=2 .... 162 Table 8-26. Opening Factors for 20-ft I-joists with a flange notch, Type II retrofitted, R=2 .... 162 Table 8-27. Opening Factors for 12-ft I-joists with a flange notch, Type II retrofitted, R=3 .... 163 Table 8-28. Opening Factors for 20-ft I-joists with a flange notch, Type II retrofitted, R=3 .... 163 xv LIST OF FIGURES Figure 2-1. Schematic (a) side view and (b) cross-section of I-joists ........................................... 11 Figure 2-2. Schematic drawing of beams with (a) web openings and (b) flange notches ............ 12 Figure 2-3. Load-deflection curves for control 12-ft (left) and 20-ft (right) joists ....................... 13 Figure 2-4. Load-deflection curves for 12-ft joists with web hole ............................................... 14 Figure 2-5. Load-deflection curves for 20-ft joists with web hole ............................................... 14 Figure 2-6. Load-deflection curves for 12-ft joists with flange notch .......................................... 15 Figure 2-7. Load-deflection curves for 20-ft joists with flange notch .......................................... 15 Figure 3-1. Schematic view of compression-parallel-to-grain test specimens: (a) 3D view and (b) cross-section .................................................................................................................................. 19 Figure 3-2. Possible failure scenarios in compression (adapted from (ASTM International, 2014))....................................................................................................................................................... 19 Figure 3-3. Schematic view of compression-perpendicular-to-grain test specimens ................... 20 Figure 3-4. Specimens for tension-parallel-to-grain-tests: (a) front view and (b) side view ........ 21 Figure 3-5. Specimens for tension-perpendicular-to-grain tests: (a) front view, (b) side view and (c) top view ................................................................................................................................... 22 Figure 3-6. Sketch of shear test specimens ................................................................................... 23 Figure 3-7. Shear test setup ........................................................................................................... 24 Figure 3-8. Longitudinal and transverse directions in wood specimens ....................................... 25 Figure 3-9. Raw material used to make timber material specimens, taken un-planed from lumber production line .............................................................................................................................. 26 Figure 3-10. Raw material before (left) and after (right) being planed ........................................ 26 Figure 3-11. Lumber pieces dried in an oven (left) and cracks observed in some timber pieces after drying (right) ................................................................................................................................. 27 Figure 3-12. Wood moisture meter ............................................................................................... 27 Figure 3-13. Universal testing machines used for the material tests (a) Instron 3385H and (b) MTS Landmark ...................................................................................................................................... 28 Figure 3-14. Preparing the 3-dimensional dog-bone specimens by CNC machine ...................... 29 Figure 3-15. A completed parallel-to-grain tension test specimen with shoulders (top) and without shoulders (bottom) ........................................................................................................................ 30 xvi Figure 3-16. Steel fixture for applying tensile load to parallel-to-grain tension test samples and its dimensions (mm) .......................................................................................................................... 30 Figure 3-17. Failure modes of parallel-to-grain tension tests: (a) brittle tension, (b) splintering tension, (c) combined shear and tension, and (d) shear ................................................................ 31 Figure 3-18. Stress-strain test results for replications 1 to 9 for parallel-to-grain tension tests ... 32 Figure 3-19. Stress-strain test results for replications 10 to 18 for parallel-to-grain tension tests 33 Figure 3-20. Determined equivalent stress-strain curve and modulus of elasticity ...................... 35 Figure 3-21. Manufacturing of perpendicular-to-grain tension test specimens ............................ 35 Figure 3-22. Aluminum fixture for applying displacement to perpendicular-to-grain tension specimens (dimensions in mm) ..................................................................................................... 36 Figure 3-23. Stress-strain curves for radial-tension test specimens, replications 1 to 10 used for determining material properties .................................................................................................... 39 Figure 3-24. Stress-strain curves for radial-tension test specimens, replications 11 to 20 used for validation of material properties ................................................................................................... 39 Figure 3-25. Stress-strain curves for tangential-tension test specimens, replications 1 to 10 used for determining material properties .............................................................................................. 40 Figure 3-26. Stress-strain curves for tangential-tension test specimens, replications 11 to 20 used for validation of material properties ............................................................................................. 41 Figure 3-27. Dominant failure modes observed in tension-perpendicular-to-grain tests: radial-tension (left) and tangential-tension (right) .................................................................................. 42 Figure 3-28. Sample compression-parallel-to-grain test specimens with strain gauge attached .. 43 Figure 3-29. Specimen preparation and strain gauge attachment for compression-parallel-to-grain specimens ...................................................................................................................................... 44 Figure 3-30. Compression parallel to grain test setup .................................................................. 44 Figure 3-31. Stress-strain curves for parallel-to-grain compression test specimens, replications 1 to 7 used for determining material properties ............................................................................... 46 Figure 3-32. Stress-strain curves for parallel-to-grain compression test specimens, replications 8 to 14 used for validation of material properties ............................................................................ 46 Figure 3-33. Observed failure modes in compression parallel to grain: (a) crushing, (b) shearing, (c) end-rolling or brooming, and (d) splitting due to internal defect ............................................ 47 xvii Figure 3-34. Stress-strain curves for radial-compression test specimens, left: replications 1 to 5 used for determining material properties, right: replications 6 to 10 used for validation of material properties....................................................................................................................................... 49 Figure 3-35. Stress-strain curves for tangential-compression test specimens, left: replications 1 to 5 used for determining material properties, right: replications 6 to 10 used for validation of material properties....................................................................................................................................... 50 Figure 3-36. Dominant failure modes for perpendicular-to-grain compression tests ................... 51 Figure 3-37. Shear test specimens ................................................................................................ 51 Figure 3-38. Shear test set-up in universal testing machine ......................................................... 52 Figure 4-1. Planar stress situation ................................................................................................. 57 Figure 4-2. Finite element model of the tensile test specimen ..................................................... 62 Figure 4-3. Support condition of the tensile test specimen ........................................................... 63 Figure 4-4. Validation of tensile test specimen simulation results ............................................... 64 Figure 4-5. Contour plot of stress distribution in longitudinal direction for tensile test specimens....................................................................................................................................................... 65 Figure 4-6. Stress distribution pattern in tensile test specimen with shoulders ............................ 65 Figure 4-7. Comparison of stress-strain response of tensile test specimens with and without shoulders ....................................................................................................................................... 66 Figure 4-8. Finite element model of 12-ft (a) web-holed and (b) flange-notched I-joists (c) symmetry condition ...................................................................................................................... 68 Figure 4-9. Load-deflection curves for specimens (a) 12-Control (control joist), (b) 12-5-8⅜, (c) 12-5-6, (d) 12-5-4 and (e) 12-4.5-4x6 ........................................................................................... 69 Figure 4-10. Load-deflection curves for specimens (a) 20-Control (control joist), (b) 20-7-8⅜, (c) 20-9-6, (d) 20-9-4x4 and (e) 20-8.5-4x6 ....................................................................................... 70 Figure 5-1. Average load-deflection plots for experimental tests, 12-ft I-joists (left) and 20-ft I-joists (right) ................................................................................................................................... 74 Figure 5-2. Schematic of sensitivity analysis parameters on I-joists with web holes ................... 75 Figure 5-3. Opening size effect plots. Top: Load-deflection curves for 12-ft I-joists with web holes at 0-, 3- and 5-ft distances from the mid-span (left to right). Bottom: Load-deflection curves for 20-ft I-joists with web holes at 0-, 5- and 9-ft distances from the mid-span (left to right) .......... 76 xviii Figure 5-4. Opening location effect plots. Top: Load-deflection curves for 12-ft I-joists with 4-, 6- and 8-in. diameter web holes (left to right). Bottom: Load-deflection curves for 20-ft I-joists with 4-, 6- and 8-in. diameter web holes (left to right) ......................................................................... 77 Figure 5-5. 3-D effect plot for 12-ft I-joists with web openings, effect of opening diameter vs. location .......................................................................................................................................... 78 Figure 5-6. 3-D effect plot for 20-ft I-joists with web openings, effect of opening diameter vs. location .......................................................................................................................................... 78 Figure 5-7. Schematic of sensitivity analysis parameters on I-joists with web holes ................... 79 Figure 5-8. Notch size effect plots. Top: Load-deflection curves for 12-ft I-joists with flange notches at 0-, 3- and 5-ft distances from the mid-span (left to right). Bottom: Load-deflection curves for 20-ft I-joists with flange notches at 0-, 5- and 9-ft distances from the mid-span (left to right) .............................................................................................................................................. 80 Figure 5-9. Notch location effect plots. Top: Load-deflection curves for 12-ft I-joists with 4x4, 6x6 and 8x8 flange notches (left to right). Bottom: Load-deflection curves for 20-ft I-joists with 4x4, 6x6 and 8x8 flange notches (left to right) ............................................................................. 81 Figure 5-10. 3-D effect plots for 12-ft I-joists with flange notches, (top) effect of notch depth vs. location and (bottom) effect of notch length vs. location ............................................................. 82 Figure 5-11. 3-D effect plots for 20-ft I-joists with flange notches, (top) effect of notch depth vs. location and (bottom) effect of notch length vs. location ............................................................. 83 Figure 6-1.General load-deflection characteristic curve for I-joists ............................................. 89 Figure 6-2. Response surface plots for 12-ft joists (left) and 20-ft joists (right) with web holes . 98 Figure 6-3. Response surface plots for 12-ft I-joists with flange notches, showing the effects of notch depth and length (top left), notch location and depth (top right) and notch location and length (bottom)....................................................................................................................................... 101 Figure 6-4. Response surface plots for 20-ft I-joists with flange notches, showing the effects of notch depth and length (top left), notch location and depth (top right) and notch location and length (bottom)....................................................................................................................................... 102 Figure 6-5. Normal probability plots of the residuals for: (a) 12-ft I-joists with web holes, (b) 20-ft I-joists with web holes, (c) 12-ft I-joists with flange notches, and (d) 20-ft I-joists with flange notches ........................................................................................................................................ 104 Figure 6-6. Diagnostic plots for 12-ft I-joists with web holes .................................................... 105 xix Figure 6-7. Diagnostic plots for 20-ft I-joists with web holes .................................................... 105 Figure 6-8. Diagnostic plots for 12-ft I-joists with flange notches ............................................. 106 Figure 6-9. Diagnostic plots for 20-ft I-joists with flange notches ............................................. 106 Figure 6-10. Comparison of actual response versus predicted response for I-joists with web holes..................................................................................................................................................... 108 Figure 6-11. Comparison of actual response versus predicted response for I-joists with flange notches ........................................................................................................................................ 108 Figure 7-1. Schematics of I-joist retrofit types. Left: I-joists with web holes, and right: I-joists with flange notches. (a): Type I retrofit, (b): Type II retrofit ............................................................. 111 Figure 7-2. Retrofit dimensions and details ................................................................................ 111 Figure 7-3. Finite element model for web holed I-joists with retrofits. Left: Type I retrofit. Right: Type II retrofit............................................................................................................................. 113 Figure 7-4. Finite element model for flange notched I-joists with retrofits. Left: Type I retrofit. Right: Type II retrofit.................................................................................................................. 113 Figure 7-5. (a) Average stiffness increase with respect to no retrofit condition (left) and (b) relative stiffness (right) versus retrofit extension ratio for web-holed I-joists with Type I and Type II retrofits ........................................................................................................................................ 116 Figure 7-6. Average stiffness increase of retrofitted web-holed joists versus (a) opening location measured from mid-span of the joist (left) and (b) opening size (right) ..................................... 117 Figure 7-7. (a) Average stiffness increase with respect to no retrofit condition (left) and (b) relative stiffness (right) versus retrofit extension ratio for flange-notched I-joists with Type I and Type II retrofits ........................................................................................................................................ 119 Figure 7-8. Average stiffness increase of retrofitted flange-notched joists versus (a) notch location measured from mid-span of the joist (left) and (b) notch size (right) ......................................... 120 Figure 7-9. Response surface curves for 12-ft web-holed joists with Type I retrofit ................. 125 Figure 7-10. Response surface curves for 12-ft web-holed joists with Type II retrofit .............. 125 Figure 7-11. Response surface curves for 20-ft web-holed joists with Type I retrofit ............... 126 Figure 7-12. Response surface curves for 20-ft web-holed joists with Type II retrofit .............. 126 Figure 7-13. Response surface curves for 12-ft flange-notched joists with Type I retrofit ........ 132 Figure 7-14. Response surface curves for 12-ft flange-notched joists with Type II retrofit ...... 132 Figure 7-15. Response surface curves for 20-ft flange-notched joists with Type I retrofit ........ 133 xx Figure 7-16. Response surface curves for 20-ft flange-notched joists with Type II retrofit ...... 133 Figure 7-17. Actual vs predicted response for joists with a web hole and retrofit Type I: 12-ft (left) and 20-ft (right) ........................................................................................................................... 140 Figure 7-18. Actual vs predicted response for joists with a web hole and retrofit Type II: 12-ft (left) and 20-ft (right) ........................................................................................................................... 140 Figure 7-19. Actual vs predicted response for joists with a flange notch and retrofit Type I: 12-ft (left) and 20-ft (right) .................................................................................................................. 141 Figure 7-20. Actual vs predicted response for joists with a flange notch and retrofit Type II: 12-ft (left) and 20-ft (right) .................................................................................................................. 141 Figure 8-1. Schematic plot of a parallel support system for System Factor ............................... 146 xxi ACKNOWLEDGEMENTS First, I am profoundly indebted to the Almighty God who has been with me in every step of my work that has been accomplished. I would like to deeply thank my supervisor, Prof. M. Shahria Alam, for his patient guidance, moral support and warm friendship. During my course of education as his student, he has been a firm supporter and has had a great contribution to my learning. In addition to his expertise and knowledge, I have learned from him some key aspects including leadership, thoughtfulness and humaneness. Without his help and guidance, my research would have not been successful. My sincere thanks go to my thesis committee members, Prof. Abbas S. Milani and Prof. Rudolf Seethaler. I deeply appreciate their invaluable guidance and support throughout my course of Ph.D. study. I feel privileged to have had the opportunity of working with S-FRAME Software Inc. for the most part of my study as an NSERC fellow and Mitacs intern. This led to advancements in my research and my professional development and helped me gain experience to advance my professional career. I would like to deeply appreciate their continued technical and financial support, without which it would have not been possible to carry out this research. I would like to gratefully thank my research group in conducting this research. Particularly, M. Shahidul Islam, Md. Shahnewaz, Farshad Hedayati Dezfuli, and Kader Siddiquee, who contributed and helped in carrying out this research in experimental and numerical phases. I would also like to extend my thanks to the great staff at UBC Okanagan, in Wood Shop and Machine Shop, particularly Kim Nordstrom, Alec Smith, Durwin Bossy, Marc Nadeau, Praveen Rajan, David Zinz, Tim Giesbrecht and Luis Diaz, as well as UBC’s Survive and Thrive Applied Research (STAR) for kindly letting me use their facility for fabrication of my test specimens. In addition, I would also like to extend my gratuity to UBC Okanagan’s School of Engineering, its administrative staff, the Office of Research Services and the TUUM EST initiative fund for their financial support to attend conferences to present this research. I would like to acknowledge the support I have received from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Mitacs Accelerates Program to xxii perform this research. I would also acknowledge the support of AcuTruss Industries Ltd. for providing joist specimens and letting us conduct experimental tests in their facilities, and Tolko Industries Ltd. for providing the raw material for specimen tests. Last but not least, I am truly grateful for the unconditional support of my family, and my wife, for being side-by-side with me throughout the course of this research. Without their support, sacrifice and enduring love, it would have been impossible to undertake this journey. xxiii I dedicate this thesis to: my family, who have wholeheartedly supported me in every aspect of my life to learn, succeed and be the person I am today, and my wife, who has been with me every step of this way, through good times and bad, and has been my source of hope and happiness with all her unconditional love and support. 1 Chapter 1: INTRODUCTION AND ORGANIZATION OF THIS DISSERTATION 1.1. General Timber I-joists are engineered timber beams that are meant to mainly carry loads from structural and non-structural elements in roof and floor systems. These important building elements are seen to be cut into the web or flanges, in order to pass through electrical and mechanical facilities during construction. Despite the convenience and benefits that this approach provides, it causes significant reductions in the structural capacity of these important elements. Even though some manufacturers have attempted to offer instructions on allowable limits of openings in the web of I-joists, a comprehensive code-accepted provision for these openings is not yet available. Furthermore, openings in flanges are completely prohibited due to the lack of knowledge on the extent of the effects of these openings. Limited research has been performed to suggest the level of destruction these openings cause and to recommend ways to reinforce these beams. The current wood design code in Canada (Canadian Standards Association, 2019) does not have direct instructions for designing I-joists with openings. In the absence of verified test results, it leaves the allowability of web openings with I-joist manufacturers and does not allow flange cuts. In practice, however, it has been seen that these limits are sometimes contravened, and joists are retrofitted using methods not backed up by the design code to compensate for their reduced capacity. Having a guideline for openings in I-joists will lead to safer and more efficient building designs. Such guidelines will provide structural engineers with directions to determine the sufficiency of I-joists considering their opening’s shape, location, and size. It may also help designers find remedies to reinforce the joists by retrofitting them using approved means. This research intends to comprehensively evaluate the effects of the existence of a web hole or a flange notch on the structural performance of prefabricated I-joists. This is done by studying a 2 large number of joists with different opening configurations and obtaining their performance in terms of their load-deflection response using experimental and numerical approaches. The effects of shape, size, and location of the opening are evaluated, and methods are suggested to predict the effects of any possible configuration within the limits of the study. Furthermore, an additional study is performed to assess the effectiveness of two retrofit types in reinforcing joists with an opening. Characteristic parameters of the retrofits are also studied, such as the type and length of the retrofit used. Finally, guidelines are proposed for designing I-joists with a web hole and those with a flange notch, assimilating the routines of the current Canadian wood design standard, CSA O86-19. Additionally, guidelines are also proposed to suggest retrofits to bring the joists back to their expected serviceability criteria. 1.2. Objectives The overall objective of this research is to assess the performance of timber I-joists with a web hole and those with a flange notch. Furthermore, it is aimed at providing guidelines for designing these joists with respect to the shape, size, and location of the opening. Results of experimental and numerical studies are used in order to achieve this goal. To this end, first, timber specimens are tested experimentally to obtain the material behavior of timber. Using the results of these tests, real-scale three-dimensional finite element models are developed to numerically simulate the behavior of the considered I-joists. The load-deflection responses obtained are then validated using experimental results. Using this finite element approach, a full-factorial sensitivity analysis is conducted to answer the following questions: • How the structural performance of I-joists is affected by openings? • What are the effects of the characteristics of the opening, including its shape, size, and location? • What are the most important input factors, and what are the characteristic responses of the beams, including their load-deflection relationship and structural stiffness? The finite element results, along with those of the sensitivity analysis are used in the next step to develop equations to predict the response characteristics of the I-joists with an opening. In the next stage of the research, the effectiveness of two suggested retrofit types is assessed through finite element analysis. A sensitivity analysis is performed to find the effect of the length 3 (amount) of retrofit, and find out if the retrofit is sufficient to bring the I-joist back to expected serviceability level. Using the response surface method, predictive equations are also derived to predict response characteristics of retrofitted web-holed and flange-notched joists, including their load-deflection behavior and stiffness. In the last stage, the sensitivity analyses and predictive equations are used to propose a design guideline for I-joists with a web hole and those with a flange notch. This suggested approach is designed to assimilate the existing design approach in the current Canadian wood design code (Canadian Standards Association, 2019). Lastly, a procedure is recommended to propose an efficient retrofit to bring the joist with opening back to its expected serviceability level. The objectives of this research can be summarized as follows: • Characterizing timber material behavior and proposing a robust finite element model for timber I-joists, • Identifying the effects of opening characteristics, including its shape, size, and location, on limit state behavior of I-joists, • Evaluating the effectiveness of retrofits on the performance of I-joists with openings, • Developing predictive equations for the response characteristics of I-joists with an opening, with and without retrofits, and • Proposing a guideline for designing I-joists with a web hole or a flange notch, and recommending a retrofit for the joist to regain its performance. This research is critical as it helps to achieve a more profound understanding of the performance of I-joists with openings. The available research in this area is restricted to a limited range of opening configurations achieved through experimental tests. Efforts are made to develop a robust 3D finite element for real-scale I-joists. The finite element approach suggested in this study paves the way to expand the applicability of the research to a wider range of joists. In addition to finite element techniques, the significance of this research lies in a large number of opening configurations evaluated. This minimizes the errors, increases the reliability of the results, and helps to evaluate the contribution of a large number of factors on the performance of I-joists. The predictive equations obtained in this research will help researchers and engineers to predict the response of I-joists, alleviating the need to carry out physical tests. Furthermore, as the wood 4 design code in Canada currently lacks a design approach for I-joists with openings, it is vital to perform this research to propose a guideline for practicing engineers to design safer structures more confidently. The study conducted on retrofitted I-joists also helps to suggest reinforcements to the joists, which will ease the design procedure, while making the design more efficient. 1.3. Organization of this dissertation This dissertation is organized into nine chapters. Following the introduction presented in this chapter, the research in the succeeding chapters are as follows: • Chapter 2 presents a literature review on the use of timber I-joists with openings in the construction industry and the research conducted towards them. Furthermore, the reference experiments conducted on I-joists are described in this chapter, along with an introduction to the test configurations. • Chapter 3 focuses on characterizing timber material to be used in the next steps of the study. Test procedures for obtaining material behavior are first described. The results of tests on a large number of test specimens made and tested in the facilities of the University of British Columbia are then presented in this chapter. • Chapter 4 is devoted to the development of a robust finite element model for I-joists. In this part of the research, efforts are made to create three-dimensional real-scale finite element models for the I-joists. The models are then validated against experimental test results. The finite element approach presented in this chapter is intended to numerically study these beams. • In Chapter 5, the numerical model introduced in the preceding chapter is used to conduct a sensitivity analysis on the performance of I-joists. A full-factorial experiment is designed, and near 200 beam configurations are examined to evaluate the effects of the input factors on the response of the beam. The input factors include the shape of the opening (web hole and flange notch), size of the opening and its location. The responses measured are the load-deflection curves of the beam and its overall structural stiffness. The effects of the input factors are measured, and the important ones are recognized. • Numerical models and results presented in earlier chapters are used to develop predictive equations of I-joist responses in Chapter 6. Response Surface Method (RSM) is used to derive equations to predict the stiffness of the joists using the significant input parameters 5 found from the sensitivity analysis. The findings of this chapter are employed to introduce procedures for limit state design of joists with a web opening and those with a flange notch. • In Chapter 7, the effectiveness of two suggested retrofit types for I-joists with openings is evaluated. Factorial analysis is designed to find the response of the joists with different opening configurations reinforced using the two retrofit types. The effectiveness of the retrofits on openings with different sizes and at different locations of the beam is first evaluated. The effects of the type and length of the retrofit are also assessed, and the change in the stiffness of the joists is measured as response characteristics. Furthermore, predictive equations are derived for obtaining the response of the beam given the opening configuration, retrofit type, and length. • Chapter 8 delivers a guideline for designing I-joists with a web opening or a flange notch based on the results of this study. This design approach is premeditated to fit into the structure of the current Canadian wood design code. New design parameters, named Opening Modification Factors, are introduced, and tables are developed suggesting these factors for two common joist spans, 12 and 20 feet. The opening factors are meant to adjust the specified capacity of the joists and are given for openings with various applicable sizes and locations along the length of the beam. Furthermore, the design approach is expanded to include the use of retrofits at opening locations. Modification factors are also suggested for retrofitted joists, and a procedure is introduced to suggest a retrofit configuration (type and length) to bring the joist back to expected serviceability levels. • Chapter 9, lastly, summarizes the major findings of this research, along with suggestions for future research in this area. 6 Chapter 2: LITERATURE REVIEW AND REFERENCE EXPERIMENTS 2.1. I-joists in the timber building industry Wood has been a key construction material in North America and around the world for decades for the principal residence of the public (Kisternaya et al., 2016; Kouris & Kappos, 2014; Reynolds & Casagrande, 2016; Shahnewaz et al., 2018). It has gained constant attention due to its renewability nature, cost-effectiveness and lighter weight compared to other construction materials such as steel and concrete. In addition, its availability and ease of use have given rise to its popularity, especially for short residential and commercial buildings. Not to mention, thanks to the improvements in manufacturing and building technology, timber is now becoming a favorite construction material for mid-rise buildings with six to twenty storeys. The British Columbia Building Code (Building Safety Standards Branch, 2018), for instance, will be the leader to allow 12-storey buildings in Canada in the near future. As a result, it is essential for wood design codes to address common practice issues adequately for safer and more efficient buildings. Timber I-joists are engineered timber beams that were introduced to the timber construction industry in the 1970s. Since then, they have been widely used in the timber building industry as main structural components in roof and floor systems, growingly replacing conventional sawn lumber joists. They have mostly been used in low- and mid-rise residential- and commercial-use buildings, and their popularity has been increasing over the past few decades, thanks to their advantages over conventional sawn lumber beams (American Forest & Paper Association Inc, 2001; Lam, 2009). Even though I-joists are allowed to be custom-built, they are widely used as prefabricated proprietary products. The joists need to meet the requirements of the ASTM D5055 standard (ASTM International, 2019). These I-joists are designed to mainly carry the bending moment and vertical shear forces in roof and flooring diaphragms. They are designed so that the weight of the beam is mostly concentrated away from the neutral axis so that the extreme fibers take the moments more efficiently. The OSB 7 web, however, possesses enough cross-sectional area to take the shear forces. Such design makes these beams lighter and more effective structural components than conventional timber beams (Leichti & Tang, 1983). Besides, timber I-joists are pre-fabricated in pre-defined sizes and span lengths. This fact helps to reduce the on-site preparations as well as to reduce the construction waste, making them easier to work with and more environmentally friendly. I-joists consist of oriented-strand board (OSB) webs and timber flanges. The flanges are made of different wood species or structural composite lumber, such as laminated veneer lumber (LVL). Spruce-Pine-Fir (SPF), as an example, is commonly used lumber for I-joist flanges in Canada and the United States. It is a combination of Canadian Spruces, Pines, and Firs growing in different regions (Canada Wood Group, 2017). The OSB web is an engineered wood panel, usually made of Aspen or Poplar strands (Zhou, 1990). The strands are bonded under pressure at high temperatures using water-resistant adhesives according to the requirements of CSA O112 and CSA O325 (Canadian Standards Association, 2012, 2014b). CSA has put requirements in place for the fabrication of the two major plywood types in Canada, Douglas-Fir plywood (DFP) and Canadian softwood plywood (CSP) (Canadian Standards Association, 2013a, 2013b, 2014a). In addition to the materials and the adhesive, the web-flange connection also plays an important role in the overall performance of the I-joist. As an example, Zhang et al. researched different joint arrangements such as tapered groove, routed groove, Y-groove, and double-flange connections, and proved the importance of the length and depth of the joint (Zhang et al., 2011). More research has been conducted on I-joists, examples of which can be found in (Chen et al., 2015; Chui et al., 2005; Clinch, 1993; Damery et al., 2007; Davids et al., 2011; Grandmont et al., 2010b, 2010a; Mark et al., 2010; Pirzada et al., 2008; Polocoser, 2012; Polocoser et al., 2013; Wisniewski & Manbeck, 2003; Wu, 2014; Zhu et al., 2007; Zhu, Guan, et al., 2005a; Zhu, Pope, et al., 2005). During the construction phase, these beams are often drilled in the web or notched through the flange for passing through service or electrical/mechanical facilities. This, on one hand, lowers the floor panel depth, which is favorable for construction purposes. However, on the other hand, cutting the beam can lead to significant reductions in its load-carrying capacity. The current Canadian and US wood design codes do not have clear guidelines for introducing openings to the joists and rely on manufacturer-provided instructions if any (American Wood Council, 2015; Canadian Standards Association, 2019). Some manufacturers have specified the allowable extents of these cuts. However, in practice, it has been seen that, at times, these extents have been exceeded 8 or the manufacturer guidelines are contravened (Kirkham, 2018). This can be due to undermining the importance of the load-carrying capacity of every single I-joist in the structural system of the building, or the belief that the effect of the cuts can be compensated by on-site retrofits. The latter, however, could be true, if done with regards to proven instructions obtained by validated research or verified test results. Despite timber I-joists’ popularity, not an adequate degree of research has been conducted on their behavior when cut or notched. In the following sections, a literature review of the research conducted in this field will be presented. 2.2. I-Joists with openings An experimental evaluation was made by Afzal et al. in 2006 to investigate the effects of web openings in timber I-joists of two different depths (Afzal et al., 2006). They tested two shapes, including square and round, and three sizes of openings, and used a three-point shear test for the evaluations. It was found that, due to stress concentration effects, the square openings show a higher capacity reduction than the round-shape ones. The effect of the square-shape openings was found to be as significant as that of a larger-size round-shape opening, encompassing the square. The researchers attributed the significance of the capacity reductions to the size of the openings and independent of the cross-sectional depth of the beam. They also investigated the effect of the existence of two openings with a diameter of ¾ depth at the same time and suggested a minimum clear spacing of twice the opening diameter. A study performed at Purdue University by Fergus was performed on I-joists with web holes and stiffeners to determine their structural performance (Fergus, 1979). In the mentioned research, joists were categorized based on their major failure mode: moment or shear. It was observed that for beams with moment failure mode, web openings of up to seventy percent of the total height of the beam would not significantly impact the performance of the beam. For those with shear failure, web buckling was seen to play an important role. I-joists with web materials with higher shear moduli performed better, as a higher shear modulus results in a delayed web buckling. On the contrary, in a study performed by Canadian Forest Services, it was shown that the shear capacity of I-joists with web holes can reduce by about eighty percent (Wang & Cheng, 1995). 9 Zhu et al. investigated the effect openings on the load-carrying capacity of I-joists (Zhu, Guan, et al., 2005b). They examined beams with multiple holes at different locations along the beam’s length. Their research showed that the opening regions of the web are more vulnerable to fracture, especially the areas subject to tension. Cracks initiate in the tension areas and spread to the flanges, causing the beam to fail. Besides, the capacity of the joist was shown to be inversely proportional to the size of the opening. Guan and Zhu continued this research to find the effects of round and square openings. In agreement with several previous researchers, they found that square openings, in general, have twenty percent less capacity compared to joists with round holes (Guan & Zhu, 2004). Several different retrofit methods have been introduced and researched. These techniques use additional structural panels such as plywood and OSB, steel plates sized to the web of the joist, or other materials such as glass fibre-reinforced polymer (Islam, 2017; Metwood Building Solutions, 2019). The on-site retrofit methods have generally shown to restore only up to 75 percent of the original capacity of the beam (Shahnewaz et al., 2019). A study conducted by Hindman and Loferski examined the effectiveness of screwed steel plates in restoring the capacity of I-joists with flange notches (Loferski & Hindman, 2008). It showed that this technique can increase the capacity of I-joist with flange notch by up to fifty percent. However, this is only sixty percent of the original capacity of the joist without the cut, since flange notches impact the joists significantly. In a research project at Villanova University, Villanova, Pennsylvania, a total of 20 I-joists with different web opening configurations were experimentally tested (Dinehart & Morrissey, 2006). The joists tested include beams with no openings, beams with the maximum allowed opening size, beams with openings following manufacturer recommendations, as well as a number of them violating manufacturer instructions, but retrofitted with steel angles. It was found in this research that all the beams that followed the manufacturer's rules possessed a capacity of at least 3 times the manufacturer’s proposed minimum capacity. It was observed that longer-span beams mostly failed at defects at the tension-side flange. However, those with openings tend to fail in different failure modes, depending on the opening size and opening location. These failure modes include failure in shear and buckling in the compression-side flange. A web-flange connection failure mode was also observed in some test cases. It was also observed that using steel retrofits the failure moves to locations that normally handle higher stresses. 10 These researchers expanded their work by testing I-joists with two different beam depths with openings of round and square shapes, retrofitted with steel angles at opening locations (Morrissey et al., 2009). The test configurations were also numerically modeled using finite element simulation to investigate the effect of the openings and the proposed retrofits on beams’ stiffness and failure mode. Also, this numerical simulation was used to study the stress distribution near the hole locations. Interestingly, it was observed that the opening size could change the failure mode. As opposed to regular beams failing at the bottom (tension-side) flange, beams with openings tend to fail in shear on the web, mostly at locations near the openings. It was also observed that beams with square openings possess approximately 10% more reduction in the load-carrying capacity than the beams with round openings. Polocoser et al. performed a study to evaluate reinforcement techniques on timber I-joists with a single web hole in them (Polocoser et al., 2013). In their study, real-size joists with three different depths, different flange widths, and two spans were tested. This study showed that having an opening in the web changes the rupture mode of the joist to shear. They found OSB panels effective retrofit means, given their relatively inexpensive price and simplicity of application. In a comparable study, it was found that using OSB panels as reinforcements can prevent the brittle rupture seen in I-joists with a web hole, and restore the capacity of cut joists by up to 27% (Shahnewaz et al., 2017). In 2011, Harte and Baylor proposed a castellated timber I-joist which is intended to ease the passage of service ducts without having to cut the web during or after the building construction phase (Harte & Baylor, 2011). This also helps to increase the moment-resisting capacity of the joist without increasing the weight of the beam. The castellation takes place by cutting the OSB web in the desired pattern and re-connecting the upper and lower segments at an offset, a similar approach to the manufacturing process of castellated steel I-beams. They proposed the manufacturing process of such beams and conducted experimental research on a total of 23 full-size joists of two different sizes. They found the shear capacity of their proposed beams identical to those of beams with equivalent sizes. Later, the same researchers evaluated the performance of castellated timber I-joists numerically, using finite element simulations, to have a more in-depth understanding of the structural behavior of the joists and to optimize the design parameters (Baylor & Harte, 2013). In a comparison 11 between experimental and numerical results, a difference ranging from -20% to +30% was observed. Using the numerical results, they made a comparison between bending stiffness of the castellated beams and solid beams, for two different cross-sections of I-joists with depths of 241 and 305 mm. It was observed that the castellated beams were having around 16% and 15% more deflection, respectively, at the same maximum loads. 2.3. Reference experiments (experimental study) A total of 180 I-joist specimens were experimentally tested. The specimens were produced and prepared in the facilities of AcuTruss Industries Ltd., Kelowna, Canada1. They came in two spans of 12 and 20 feet, with various web hole and flange notch configurations described later in this section. These span lengths would represent a wide range of applications of I-joists and are used to reference the performance of short and medium span lengths. The beams consist of a 9.5-millimeter oriented strand board (OSB) web and 64x38-millimetre No. 2 SPF lumber flanges. The OSB used for the web is made conforming to the requirements of CSA O325 and Performance Standard 2 (Canadian Standards Association, 2012; National Institute of Standards and Technology, 2010). The manufacturer has limited the span length of such cross-section to 20 feet (NASCOR, 2010). A simply-supported pin-roller condition was provided as end supports and, to control the lateral buckling, 10 and 16 lateral supports were used for 12- and 20-ft joists, respectively, on the top flanges. Two-point loads were applied at 1/3 and 2/3 of beams’ length to simulate bending, conforming to ASTM D5055 (ASTM International, 2019) and WIMJA (Wood I-Joist Manufacturers Association, 2008). Figure 2-1 illustrates the configuration of this test setup. Figure 2-1. Schematic (a) side view and (b) cross-section of I-joists 1 AcuTruss Industries Ltd., 1913 Kent Rd. Kelowna, BC V1Y 7S6, Canada 12 Figure 2-2 shows the schematic configuration of beams with openings. In this figure, L is used to denote the length of the beam, and XWH and XFN represent the location of the web hole or the flange notch, respectively, measured from the mid-span of the beam. D denotes the web hole’s diameter, and LFN and DFN are used to show the length and depth of the flange notch, respectively. Figure 2-2. Schematic drawing of beams with (a) web openings and (b) flange notches Specifications of the test specimens, including joist length and opening shape, size and location are listed in Table 2-1 below. The test on each specimen was replicated 10 times to account for the errors and variabilities in the experiment. The applied force was measured by the built-in pressure gauge of the test set-up and calibrated according to the device manufacturer’s manual. Also, the deflection of the beam was measured using a digital extensometer and verified through video processing using three HD cameras located at 1/4, 1/2 and 3/4 lengths of the beam span. For details of the test set-up, please refer to (Islam et al., 2015). As a result of this study, the load-carrying response of each I-joist is found, and their load-displacement graph has been obtained, up to the failure point of the beam. Joists were typically seen to fail due to the existence of a knot causing the flange to crack, breakage of the bond between web and flange, flexural failure of the flanges, shearing of the web, and combined shear and flexural failure. Observed failure modes in the test specimens are reported in (Islam, 2017; Shahnewaz, M. et al, 2017). Figure 2-3 presents the load-deflection results for control joists with no opening. Figure 2-4 and Figure 2-6 illustrate these results for 12-ft joists with a web hole and a flange notch, respectively. The results of tests on 20-ft joists are plotted in Figure 2-5 and Figure 2-7. 13 Table 2-1. I-joist experiment specimens Length (ft) Specimen Label Opening Type Opening Size (in. or in.xin.) Opening Location* (ft) 12 12-Control No Opening - - 12-5-8⅜ Web Hole 8⅜ 5 12-4-8⅜ 8⅜ 4 12-5-6 6 5 12-5-4 4 5 12-5-4x4 Flange Notch 4x4 5 12-4.5-4x4 4x4 4.5 12-4-4x4 4x4 4 12-4.5-4x6 4x6 4.5 20 20-Control No Opening - - 20-9-8⅜ Web Hole 8⅜ 9 20-7-8⅜ 8⅜ 7 20-9-6 6 9 20-9-4 4 9 20-9-4x4 Flange Notch 4x4 9 20-8.5-4x4 4x4 8.5 20-8-4x4 4x4 8 20-8.5-4x6 4x6 8.5 *Opening location measured from mid-point of the beam span to center of the opening Figure 2-3. Load-deflection curves for control 12-ft (left) and 20-ft (right) joists 0510152025303540450 10 20 30 40 50P (kN)Δ (mm)051015202530350 25 50 75 100 125P (kN)Δ (mm)Replicate 1Replicate 2Replicate 3Replicate 4Replicate 5Replicate 6Replicate 7Replicate 8Replicate 9Replicate 10Mean Curve14 Figure 2-4. Load-deflection curves for 12-ft joists with web hole Figure 2-5. Load-deflection curves for 20-ft joists with web hole 051015200 10 20 30P (kN)Δ (mm) Specimen:…051015200 5 10 15 20P (kN)Δ (mm) Specimen: 12-4-8⅜Replicate 1Replicate 2Replicate 3Replicate 4Replicate 5Replicate 6Replicate 7Replicate 8Replicate 9Replicate 10Mean Curve0510152025300 10 20 30P (kN)Δ (mm) Specimen: 12-5-6010203040500 20 40 60P (kN)Δ (mm) Specimen: 12-5-405101520250 20 40 60 80P (kN)Δ (mm) Specimen: 20-9-8⅜05101520250 25 50 75 100P (kN)Δ (mm) Specimen: 20-7-8⅜Replicate 1Replicate 2Replicate 3Replicate 4Replicate 5Replicate 6Replicate 7Replicate 8Replicate 9Replicate 10Mean Curve0510152025300 25 50 75 100P (kN)Δ (mm) Specimen: 20-9-60510152025300 50 100 150P (kN)Δ (mm) Specimen: 20-9-415 Figure 2-6. Load-deflection curves for 12-ft joists with flange notch Figure 2-7. Load-deflection curves for 20-ft joists with flange notch 051015200 10 20 30P (kN)Δ (mm) Specimen: 12-5-4x40246810120 5 10 15 20P (kN)Δ (mm) Specimen: 12-4.5-4x4Replicate 1Replicate 2Replicate 3Replicate 4Replicate 5Replicate 6Replicate 7Replicate 8Replicate 9Replicate 10Mean Curve024680 5 10 15P (kN)Δ (mm) Specimen: 12-4-4x40246810120 5 10 15 20P (kN)Δ (mm) Specimen: 12-4.5-4x602468101214160 20 40 60 80P (kN)Δ (mm) Specimen: 20-9-4x40246810120 20 40 60P (kN)Δ (mm) Specimen: 20-8.5-4x4Replicate 1Replicate 2Replicate 3Replicate 4Replicate 5Replicate 6Replicate 7Replicate 8Replicate 9Replicate 10Mean Curve02468100 10 20 30 40P (kN)Δ (mm) Specimen: 20-8-4x40123456780 10 20 30 40P (kN)Δ (mm) Specimen: 20-8.5-4x616 Chapter 3: CHARACTERIZATION OF FLANGE MATERIAL 3.1. General The first step to study the behavior of I-joists is to understand the material properties of their components. In our study, the I-joists are made up of timber flanges and an oriented strand board (OSB) web. The OSB material used in the I-joists was experimentally characterized in a previous study at UBC Okanagan (Islam, 2017). However, the timber material used in I-joist flanges needed to be tested to obtain its material properties. To this end, experiments are planned to test the timber material in different directions. The goal of these tests is to obtain moduli of elasticity and shear for the flange timber material. This will allow us to characterize wood material and provide us with the means to define accurate numerical models. It should be noted that strength prediction is not pursued in this research and, thus, timber strength properties are not obtained from these tests. This chapter will focus on finding the material properties of wood through physical experiments and finding the appropriate characters to be used for numerical modeling in the next chapters. 3.2. Experimental test procedures The very fundamental step of every finite element (FE) simulation is to assign correct and realistic material properties to each element. I-joists are made up of two main parts: an oriented-strand board (OSB) web and two timber flanges. Experimental tests are planned in this chapter to obtain the material properties of the timber parts to be able to capture the true behavior of the beam. Wood possesses different properties in three directions. This characteristic makes wood an orthotropic material. As a result, a series of tests in three directions are designed to construct the elastic material matrices. To do so, clear specimens of the material are made and tested following the instructions as described in the following paragraphs. Due to the wide range of varieties of wood and the differences in manufacturing processes, the need to test each specific material continues to exist. In this research, I-joist flanges are made of 17 No. 2 visually-stress-graded S-P-F2 lumber, which is tested following the ASTM International3 material test standards as explained below. ASTM D143 test procedure is used to obtain material properties of timber used in flanges of the I-joists (ASTM International, 2014). A general primary specimen with a cross-section size of 2x2 in2. is suggested for most methods of this test. However, a smaller specimen, referred to as secondary, is also introduced which shall have a cross-section of 1x1 in2. The latter is offered for cases in which making or finding the larger specimen is not possible due to either availability or test limitations. The primary specimen size is designed in such a way that it is large enough to represent a reasonable portion of the mother material and, at the same time, small enough to ease the handling of the test. Details of specimens used in each method are presented in the following sections, and the results are discussed in section 3.3. 3.2.1. General test requirements In addition to the specific instructions to be followed in each test series, a number of general requirements have to be met in all tests. These requirements are presented as follows. Data acquisition facilities: The use of electronic data recording systems is highly encouraged, where available. This includes the application of loads or displacements, measuring the input or read data and recording the results. It should be noted that the use of such systems is conditional upon not affecting the framework of the test and not lowering their accuracy. Moisture content: Due to the sensitivity of wood to moisture, the moisture content of the specimen is required to be noted at the time of the test. The specimens shall be in an air-dry condition. This condition is obtained by drying the specimen out to gain constant weight before the test. Temperature: Temperature interacts with the relative humidity to affect the strength of wood. The room temperature shall be recorded while testing and reported when preparing the results. 2 Spruce-Pine-Fir species 3 American Society for Testing and Materials 18 3.2.2. Compression parallel to the grain The instructions of section 9 of the test standard ASTM D143 are followed in carrying out the tests in this section (ASTM International, 2014). In this test, the capacity of the sample wood under compression in the direction of its grains (the longitudinal direction) is measured. The primary specimen size is preferred for this test. However, the smaller secondary test specimen is also acceptable. Even though the specimen dimensions are given in the test standard, it is of significant importance to measure the actual dimensions of each specimen at the time of the test and report them along with the results. The suggested dimensions for both primary and secondary method tests are listed in Table 3-1. Figure 3-1 shows a schematic view of the specimen. The secondary test specimens are used in this research. To ensure that the capacity is being measured in the right direction, i.e. parallel to the grain, it is important to make sure that the end grain surfaces are at right angles to the grains. Also, they should be made parallel to one another. The rate at which the compressive load is applied shall be set to 0.003 mm (or in.) per unit length of nominal length of the specimen per minute – that is 0.6 mm/min (0.024 in./min) for primary specimens and 0.3 mm/min (0.012 in./min) for secondary specimens. The loading rate is maintained constant during the entire test. This specified test speed should be maintained within a 25% tolerance; otherwise, the actual loading rate needs to be reported along with the test results. The gauge length for measuring the compression is 150 mm (6 in.) for primary and 50 mm (2 in.) for secondary method specimens. The force shall be increased until the proportional limit is obviously passed and the load-deflection curve shall be continuously recorded and plotted afterward. All failure symptoms observed shall be reported in the order they happened. Some possible failure types include crushing, wedge split, shearing, splitting, shearing parallel to grain and brooming, also known as end-rolling. Also, the report shall be followed by a sketch of the failure pattern. Figure 3-2 illustrates such possible failure scenarios. 19 Figure 3-1. Schematic view of compression-parallel-to-grain test specimens: (a) 3D view and (b) cross-section Table 3-1. Specimen details for compression-parallel-to-grain test Test method Specimen shape Cross-section dimensions AB mm x mm (in. x in.) Nominal length L mm (in.) Gauge length mm (in.) Primary Rectangular block 50x50 (2x2) 200 (8) 150 (6) Secondary Rectangular block 25x25 (1x1) 100 (4) 50 (2) Figure 3-2. Possible failure scenarios in compression (adapted from (ASTM International, 2014)) 20 3.2.3. Compression perpendicular to the grain The capacity of the sample wood under compression perpendicular to the grain is obtained using this test. Unlike the compression-parallel-to-grain test, only one specimen size is suggested for this test as described in Table 3-2. Figure 3-3 shows a schematic view of this test’s specimen. For reporting the results, the actual specimen dimensions need to be measured and reported. Table 3-2. Specimen details for compression-perpendicular-to-grain test Test method Specimen shape Cross-section dimensions AB mm x mm (in. x in.) Nominal length L mm (in.) Gauge length mm (in.) Primary Rectangular block 50x50 (2x2) 150 (6) 50 (2) Figure 3-3. Schematic view of compression-perpendicular-to-grain test specimens A metal bearing plate with a width of 50 mm (2 in.) should be used to transfer the load from the device to the sample. It should be noted that the actual width of this bearing plate should be included in the report, as it affects the pressure transferred to the specimen. The compressive load is applied at a rate of 0.305 mm/min or, equivalently, 0.012 in./min. The gauge length applicable to this test over which the deflection shall be measured is the distance between the two moving heads of the loading device. 3.2.4. Tension parallel to the grain The characteristics of wood, like any other material, under tension is an important property that defines a major part of the overall behavior of the material. ASTM D143 (ASTM International, 2014) is followed to capture the tensile properties of timber parts. Three-dimensional dog-bone specimens are used for this test. The shape of the test specimens is shown in Figure 3-4 and corresponding dimensions are given in Table 3-3. 21 Figure 3-4. Specimens for tension-parallel-to-grain-tests: (a) front view and (b) side view Table 3-3. Specimen dimensions for tension-parallel-to-grain tests A1 A2 B C D1 D2 E F1 F2 G mm 100 115.9 63 25 4.8 9.5 12.4 40.53 82.6 57.9 in. 4 4 9 16⁄ 2 1 2⁄ 1 3 16⁄ 3 8⁄ 1 2⁄ 1 19 32⁄ 3 1 4⁄ 2 9 32⁄ The orientation of the specimen should be set in a way that the annual rings be oriented perpendicular to the larger dimension of the specimen’s cross-section at ends. It should not be forgotten to record the actual dimensions of the tested specimens to be used in the calculation of stresses and strains. The rate of separation of heads of the loading machine is 1 mm or, equivalently, 0.05 in. per minute. The gauge length or the length over which the extension is to be measured is 50 mm (or equivalently 2 in.) at the centre of the specimen. It has to be made sure to avoid any slippage between the grips and the specimen so that the entire loading be transferred to the specimen. The load-displacement data should be recorded at all times and the failure type shall be sketched in the test report. 3.2.5. Tension perpendicular to the grain The tension-perpendicular-to-grain test should also be conducted using a certified tensile test machine. The shape of the test specimens, following the specifications of Section 17 of ASTM 22 D143, is illustrated in Figure 3-5 followed by its dimensions in Table 3-4 (ASTM International, 2014). Just like any other test in this series, the actual dimensions of each specimen should be recorded and used for further calculations. The loading rate used for this test is 2.5 mm/min (equivalently 0.10 in./min) which represents the rate of separation of the two grips of the test machine. In this test, the deflection is read from the relative displacement of the machine grips. Figure 3-5. Specimens for tension-perpendicular-to-grain tests: (a) front view, (b) side view and (c) top view Table 3-4. Specimen dimensions for tension-perpendicular-to-grain tests A B C D E mm 63 50 50 13 6 in. 2 1 2⁄ 2 2 1 2⁄ 1 4⁄ 3.2.6. Shear parallel to grain This section describes the details for obtaining the shear properties of timber in parallel-to-grain direction. Instructions of Clause 14 of ASTM D143-14 are used to determine these properties in this section. The test specimen in this method is a wood block of an overall dimension of 50x50x63 23 mm (2x2x2½in.), with details given in Table 3-5 below. Figure 3-6 shows a sketch of the shear test specimen. The actual dimensions of the specimen should be measured and reported along with the test results, as well as the moisture content of the specimen at the time of testing. The shearing surface area should be specifically noted. Figure 3-6. Sketch of shear test specimens Table 3-5. Nominal dimensions of shear test specimens A B C D E mm 63 50 30 13 20 in. 2½ 2 2-3/16 ½ 3/4 The shearing load shall be applied to the specimens using a force- or displacement-controlled hydraulic device. The advised rate of loading is 0.6 mm/min (0.024 in./min). The specimen should be placed in the device as per Figure 3-7. Caution should be used to carefully align the specimen in the setup with an offset of 3 mm (1/8 in.) between the adjacent edges of the supporting plate and loading device. The maximum load on the specimen is to be recorded and reported. The load-deflection curve of the experiment shall be used to obtain the shear modulus of sample material. A sketch of the occurred failure should also be provided. 24 Figure 3-7. Shear test setup A summary of the test’s information is given in Table 3-6 below. Table 3-6. Tests summary Test Specimen Loading rate mm/min (in./min) Gauge length mm (in.) Compression parallel to grain Primary method Figure 3-1 0.6 (0.024) 150 (6) Secondary method 0.3 (0.012) 50 (2) Compression perpendicular to grain Figure 3-3 0.305 (0.012) Machine heads distance Tension parallel to grain Figure 3-4 1 (0.05) 50 (2) Tension perpendicular to grain Figure 3-5 2.5 (0.10) Machine heads distance Shear parallel to grain Figure 3-6 0.6 (0.024) Machine head distance 3.3. Experiments This section presents the results of material characterization tests. A summary of the test plan is presented first, followed by the design of experiment, including type and number of replications in each test. The results of each test series are then presented in separate sections. 3.3.1. Design of experiment (DoE) Experimental tests conducted for characterization of timber include tensile and compression tests in the longitudinal direction (parallel to grain) and two transverse directions (perpendicular to the grain). The perpendicular-to-grain directions include tangential and radial directions, as shown in Figure 3-8. Tests were planned in accordance with ASTM standards in both parallel- and 25 perpendicular-to-grain directions. Table 3-7 shows the experiment design for the conducted tests. The number of replications is purposed to allow for characterizing and verification of the test results. Figure 3-8. Longitudinal and transverse directions in wood specimens Table 3-7. Design of experiment for material characterization tests Test Direction Specimens ID Number of Replications Tension - Parallel to grain Longitudinal TL 26 Tension - Perpendicular to grain Radial TR 21 Tension - Perpendicular to grain Tangential TT 23 Compression - Parallel to grain Longitudinal CL 22 Compression - Perpendicular to grain Radial CR 10 Compression - Perpendicular to grain Tangential CT 10 Shear - Parallel to grain Longitudinal GLR and GLT 21 Shear - Perpendicular to grain Transverse GRT and GTR 11 3.3.2. Preparation of the raw material The timber material used in studied I-joists was Spruce-Pine-Fir (SPF) grade No. 2. Despite its wide availability, the raw material for making the timber specimens was procured directly from the facilities of Tolko Industries Ltd. in Vernon, BC. The reason was that Tolko products were used in the manufacturing of the I-joists that were physically tested in our research. Besides, the regularly-available SPF products in retail stores were already milled down to specific sizes that were not big enough to accommodate some of the material test specimens according to ASTM standards. 26 The material was taken raw and un-planed, directly from the production line, shown in Figure 3-9. The supplied material could not be used right away, as it was wet and un-planed. The nominal section size of these lumber pieces was 4" x 4". However, the actual dimensions were smaller. The first step to prepare the lumbers to work with was to dry them up and plane their outer surfaces to make smooth surfaces and square-angle edges. This led to even smaller actual dimensions to work with. Figure 3-10 shows a sample of the lumbers before and after being planed. The next step was to chop the lumbers down to several shorter pieces (18-, 21- and 25-in. long pieces), according to the material tests plan. Figure 3-9. Raw material used to make timber material specimens, taken un-planed from lumber production line Figure 3-10. Raw material before (left) and after (right) being planed 27 The moisture content present in the timber could result in changes in lumber dimensions after being cut and getting dried out. On the other hand, the rapid loss of moisture causes high-rate shrinkages that result in both surface and deep cracks in the wood. Thus, the supplied material was initially kept at room temperature for three days for the major part of the moisture to be lost. Next, shortened and planed lumber pieces were kept in an oven at a temperature of 90 ᵒC for another two days to become dry enough. Figure 3-11 shows the arrangement. Even though rapid moisture loss was avoided, some minor cracks could still be observed in the timber after being dried, as seen in this figure. Some timber pieces even experienced long and deep cracks, which were discarded and not used in the experiments. Drying out the wood was required also for stabilizing the dimensions of the specimens. The instantaneous moisture content of each specimen was measured at the time of the experiment, using a moisture meter device specific to wood material, a photo of which is shown in Figure 3-12. Figure 3-11. Lumber pieces dried in an oven (left) and cracks observed in some timber pieces after drying (right) Figure 3-12. Wood moisture meter 28 3.3.3. Test equipment Instron 3385H and MTS Landmark universal testing machines were used for the experiments, as specified below. The Instron universal testing machine was used for all tensile tests, the shear tests, and compression-perpendicular-to-grain tests. The MTS Landmark testing machine was used for the compression-parallel-to-grain tests since it could be synchronized with the strain data-measuring system. The testing machines are shown in Figure 3-13. Figure 3-13. Universal testing machines used for the material tests (a) Instron 3385H and (b) MTS Landmark 3.3.4. Tension-parallel-to-grain test results The most important material property of timber is its behavior in tension in the direction of the grains. The modulus of elasticity, which dictates the stiffness of the member, is also determined using the slope of the stress-strain curve of the timber in tension, which is later on used in the numerical modeling as well. Three-dimensional dog-bone specimens were made using a CNC cutting machine in the School of Engineering’s wood shop at UBC Okanagan. The specimens were designed following the ASTM 29 D143-14 instructions for testing clear wood specimens in tension. The specimens were modelled in Fusion 360 software. Dimensions of this specimen type are given in Table 3-3 in Section 3.2.4. The raw material was first sized down to 460-mm long rectangular bars and then shaped to the desired dimensions using the CNC cutting machine. Figure 3-14 shows the preparation steps of the dog-bone specimens being cut by the CNC machine. A specific jig was made to hold the specimens. It was important to fix the raw bar tightly in the jig, to prevent the specimen from breaking as it becomes slender. Figure 3-14. Preparing the 3-dimensional dog-bone specimens by CNC machine Figure 3-15 shows a completed dog-bone specimen for parallel-to-grain tension tests. The exact dimensions of each test sample were measured and reported in its test data sheet at the time of the test. As per ASTM instructions, two steel fixtures were made to hold the specimen and mount it in the testing machine’s jaws for applying tension. Figure 3-16 shows the fixture along with its schematic drawing and dimensions. The tensile testing machine’s heads could not travel far enough to accommodate the length of the sample and the fixtures. As a result, the end shoulders of our dog-bone specimens had to be trimmed and fit directly in the jaws of the testing machine. A pair of rough-surface jaw grips were used to ensure there is minimal to no slippage of the wood specimens in the grips. 30 Figure 3-15. A completed parallel-to-grain tension test specimen with shoulders (top) and without shoulders (bottom) Figure 3-16. Steel fixture for applying tensile load to parallel-to-grain tension test samples and its dimensions (mm) Table 3-8 shows the details of all specimens tested, along with their exact dimensions and moisture content at the time of the test. A total of 26 specimens were made and tested, eight of which were discarded because of reasons such as specimen sudden breaks, distortion of the specimen, inaccurate dimensions and existence of defects within the critical section of the specimens. The 18 acceptable tests that were relied on are presented in Table 3-8. The values presented in this table are the actual dimensions of the specimens and the instantaneous moisture content at the time of the test and are presented in the order that the tests are conducted. The tests were conducted up to the complete failure of the specimens. The observed failure types were brittle tension, combined shear and tension, splintering tension, and shear. Among the accepted specimens, only one experienced failure in shear, splintering tension was observed the 31 most and brittle shear and combined shear and tension were observed equally. Also, one specimen was observed to break in brittle tension at a defect that existed near the clear area. Figure 3-17 shows an illustration of each failure mode that was observed. Figure 3-17. Failure modes of parallel-to-grain tension tests: (a) brittle tension, (b) splintering tension, (c) combined shear and tension, and (d) shear Table 3-8. Tested specimens’ actual dimensions and moisture content Replication Specimen ID Critical Section Width (mm) Critical Section Depth (mm) Gauge Length (mm) Moisture Content (%) 1 TL-1 9.60 7.06 50.47 5.4 2 TL-2 9.66 4.84 50.16 5.4 3 TL-3 9.42 5.10 50.66 5.0 4 TL-4 9.59 6.48 50.65 5.5 5 TL-5 9.56 6.02 49.80 5.5 6 TL-6 9.52 4.85 50.57 5.4 7 TL-7 9.49 3.93 50.85 5.5 8 TL-8 9.60 4.37 49.41 5.7 9 TL-9 9.39 5.62 51.43 5.6 10 TL-10 9.24 5.15 51.09 6.2 11 TL-11 9.45 3.80 50.29 5.9 12 TL-12 9.67 4.82 51.02 5.5 13 TL-13 9.15 4.03 50.00 6.3 14 TL-14 9.53 6.82 48.81 6.0 15 TL-15 9.43 5.83 50.16 6.0 16 TL-16 9.38 4.66 49.2 5.8 17 TL-17 9.62 6.11 50.54 5.8 18 TL-18 9.53 5.99 50.14 6.0 32 Among the acceptable tested specimens, half of them were used to determine the target material properties and the other half was used for verification of the observed results. The basis for selecting the specimens of each group was just the order the tests were conducted in. The observed load-displacement of the material in the longitudinal direction was consistently linear. The target material properties in this series of tests were the modulus of elasticity in the longitudinal direction and the maximum strength, which were taken as the slope of the obtained stress-strain curve and the maximum stress found at the break for each specimen, respectively. The load-displacement curve was then re-plotted using the found material properties and response patterns. Figure 3-18 plots the results of parallel-to-grain tension tests for the first half of the tests in the form of stress-strain curves, and Table 3-9 summarizes the obtained material properties extracted from the gathered data. Accordingly, Figure 3-19 and Table 3-10 present the results and material properties of the second half of the tests. A good match is observed between the two groups of tests. The deviation of properties observed within each group can be attributed to the variabilities and the non-homogeneity in wood material and possible specimen fabrication errors, including the exact direction of grains in the specimens. We chose to test a relatively high number of specimens to compensate for this variability and increase the level of confidence for the determined properties. Figure 3-18. Stress-strain test results for replications 1 to 9 for parallel-to-grain tension tests 0204060801001201401600 0.2 0.4 0.6 0.8 1 1.2 1.4Stress (MPa)Strain (%)TL-1TL-2TL-3TL-4TL-5TL-6TL-7TL-8TL-933 Table 3-9. Summary of the test results for replications 1 to 9 for parallel-to-grain tension tests Replication Specimen ID Max Stress (MPa) Max Strain (%) E (MPa) 1 TL-1 67.5 0.82% 8378.8 2 TL-2 129.6 0.87% 14979.8 3 TL-3 73.0 1.03% 7536.2 4 TL-4 79.5 1.26% 6491.7 5 TL-5 35.0 0.34% 10334.3 6 TL-6 117.4 0.95% 12830.5 7 TL-7 135.5 1.12% 13340.7 8 TL-8 135.2 0.99% 13742.0 9 TL-9 69.8 0.96% 7308.9 Average 93.6 0.93% 10549.2 σ 34.4 0.24% 3048.1 Figure 3-19. Stress-strain test results for replications 10 to 18 for parallel-to-grain tension tests 0204060801001201401600 0.2 0.4 0.6 0.8 1 1.2 1.4Stress (MPa)Strain (%)TL-10TL-11TL-12TL-13TL-14TL-15TL-16TL-17TL-1834 Table 3-10. Summary of the test results for replications 10 to 18 for parallel-to-grain tension tests Replication Specimen ID Max Stress (MPa) Max Strain (%) E (MPa) 10 TL-10 102.7 0.82% 13099.7 11 TL-11 136.9 0.86% 16231.1 12 TL-12 110.7 0.74% 15058.2 13 TL-13 69.6 0.53% 12176.4 14 TL-14 73.8 0.96% 7564.4 15 TL-15 75.9 1.03% 7591.6 16 TL-16 68.6 1.01% 7232.9 17 TL-17 78.4 1.15% 7362.0 18 TL-18 89.9 1.04% 8689.3 Average 89.6 0.91% 10556.2 σ 21.7 0.18% 3400.0 In determining the material properties, the average of maximum strengths is chosen to represent the breakpoint of the material, and the average of moduli of elasticity at different strain levels is chosen to represent the modulus of elasticity of the material. Even though the material showed a consistent linear load-deflection behavior, the average modulus of elasticity was calculated both graphically and numerically. First, a graphical approach was used to find the modulus of elasticity that best matches the curve’s slope. Additionally, an algorithm was written in MATLAB compiler (The MathWorks Inc., 2011) to average the moduli of elasticity at all strain levels. Because of the linearity of the material, a very close match was observed between the two approaches, whose results were presented in Table 3-9 and Table 3-10 above. Figure 3-20 plots all 18 experimental test results along with the averaged material curve and modulus of elasticity. The black thick curve shows the mean material stress-strain curve and the red dashed line shows the slope calculated to determine the equivalent modulus of elasticity. The mean curve had to be plotted only up to the minimum available strength, because of lack of data for some of the tests beyond this point. This figure demonstrates the reasonability of our choice for the determined equivalent material behavior and properties. 35 Figure 3-20. Determined equivalent stress-strain curve and modulus of elasticity 3.3.5. Tension-perpendicular-to-grain test The next series of tests is the tensile test in the directions perpendicular to material’s grain – that is, in the radial and transverse directions. Clear specimens of wood in such directions can hardly be found with appropriate sizes big enough to accommodate specimen types similar to those for parallel-to-grain tests. Therefore, a specimen shape introduced in Section 3.2.5 is used for this series. To make these specimens, after being planed and dried, the raw wood material was cut down to bars with cross-sectional dimensions of 22½ in. and about 21 inches long, using a chop saw and band saw. The bar was initially made slightly oversized to account for later dimension corrections and minor dimension changes due to sanding and moisture content variations. The bar was then shaped to get the right cross-section (illustrated in Figure 3-5) using a router table, by running a round bit through the length of the bar. Finally, the bar was cut down to 2 in. long specimens, as desired. Figure 3-21 shows the manufacturing process of these specimens. Figure 3-21. Manufacturing of perpendicular-to-grain tension test specimens 0204060801001201401600 0.2 0.4 0.6 0.8 1 1.2 1.4Stress (MPa)Strain (%)TL-1TL-2TL-3TL-4TL-5TL-6TL-7TL-8TL-9TL-10TL-11TL-12TL-13TL-14TL-15TL-16TL-17TL-18MeanE36 In order to apply the tension force to such specimens, two aluminum fixtures were made, as shown in Figure 3-22. The dimensions of these fixtures are given in the mentioned figure. These parts were made using a waterjet machine with very small tolerances, to fit into the specimens tightly and broaden the contact area with the specimens as much as possible. Widening the contact surfaces lowers the stress concentrations at contact locations and prevents the specimen from sudden breaking due to the unwanted stress concentrations. Figure 3-22. Aluminum fixture for applying displacement to perpendicular-to-grain tension specimens (dimensions in mm) A total of 44 specimens were tested in tangential and radial directions, 4 of which were discarded due to deficiencies in them or sudden breaks during the tests, leaving 20 samples available for each direction. All real specimen dimensions were measured at the time of the test, along with the instantaneous moisture content of the specimens. Table 3-11 presents the actual specimen dimensions and moisture content for radial-tension specimens and Table 3-12 presents those for tangential-tension specimens. Half of the specimens in each group were chosen for determining the material properties and the other half were used to validate the results. The basis for selecting the specimens was the order they were tested in. 37 Table 3-11. Specimen dimensions and moisture content for radial-tension test specimens Rep. Specimen ID Width (mm) Height (mm) Depth (mm) Critical Section Width (mm) Critical Section Depth (mm) Gauge Length (mm) Moisture Content (%) 1 TR-1 64.62 50.68 50.97 24.92 50.97 25.68 5.1 2 TR-2 64.48 50.62 50.87 24.98 50.87 25.62 5.3 3 TR-4 66.42 51.20 51.09 25.92 51.09 26.20 5.3 4 TR-5 65.64 51.37 50.95 25.72 50.95 26.37 5.2 5 TR-6 65.81 65.81 51.09 25.64 51.09 25.00 5.2 6 TR-7 65.39 51.15 51.12 25.29 51.12 26.15 5.8 7 TR-9 64.88 50.86 51.28 25.26 51.28 25.86 5.5 8 TR-10 65.21 50.95 50.77 24.82 50.77 25.95 5.2 9 TR-11 63.05 51.93 51.27 25.13 51.27 26.93 5.7 10 TR-12 63.16 51.58 51.21 25.34 51.21 26.58 5.9 11 TR-13 63.03 52.00 51.14 25.16 51.14 27.00 5.3 12 TR-14 63.50 52.20 51.39 25.48 51.39 27.20 5.5 13 TR-15 63.32 51.92 51.23 25.37 51.23 26.92 5.7 14 TR-16 63.44 50.90 49.23 25.90 49.23 25.90 5.7 15 TR-17 62.56 49.81 49.35 25.24 49.35 24.81 5.5 16 TR-18 62.31 49.67 49.39 25.25 49.39 24.67 5.6 17 TR-19 63.33 51.64 50.60 25.36 50.60 26.64 5.5 18 TR-20 63.13 51.82 51.03 25.25 51.03 26.82 5.2 19 TR-21 62.86 51.54 51.10 25.30 51.10 26.54 5.4 20 TR-22 62.93 49.88 49.69 25.50 49.69 24.88 5.8 38 Table 3-12. Specimen dimensions and moisture content for tangential-tension test specimens Rep. Specimen ID Width (mm) Height (mm) Depth (mm) Critical Section Width (mm) Critical Section Depth (mm) Gauge Length (mm) Moisture Content (%) 1 TT-1 65.80 51.75 51.22 26.67 51.22 26.75 5.3 2 TT-2 65.22 51.13 51.00 26.33 51.00 26.13 5.0 3 TT-3 64.88 50.95 51.04 26.10 51.04 25.95 5.3 4 TT-4 64.82 51.46 51.06 26.06 51.06 26.46 5.2 5 TT-5 64.63 50.89 50.92 26.04 50.92 25.89 5.4 6 TT-6 64.47 51.44 51.12 25.89 51.12 26.44 5.3 7 TT-7 64.43 50.44 50.98 25.68 50.98 25.44 5.3 8 TT-8 64.35 50.50 50.79 25.81 50.79 25.50 5.4 9 TT-9 64.36 50.49 50.73 26.00 50.73 25.49 5.3 10 TT-10 63.99 50.81 51.64 25.91 51.64 25.81 5.2 11 TT-11 64.39 51.49 51.06 26.90 51.06 26.49 5.4 12 TT-12 63.01 51.05 49.73 25.17 49.73 26.05 5.6 13 TT-14 62.90 51.10 49.48 25.18 49.48 26.10 5.6 14 TT-15 64.17 51.30 51.95 26.44 51.95 26.30 5.3 15 TT-16 64.57 52.15 51.59 26.57 51.59 27.15 5.5 16 TT-17 53.12 50.70 49.52 25.30 49.52 25.70 5.7 17 TT-18 64.89 52.63 48.96 26.69 48.96 27.63 5.5 18 TT-19 62.94 51.23 49.82 25.23 49.82 26.23 5.7 19 TT-20 63.07 51.15 49.87 25.58 49.87 26.15 5.6 20 TT-21 62.98 50.92 49.52 25.41 49.52 25.92 5.2 The specimens were tested using Instron universal testing machine and test instructions introduced in section 3.2.5. The tests were continued up to the complete rupture of the specimen. Figure 3-23 and Figure 3-24 present the recorded stress-strain curves for test specimens used for determining and verification of material properties, respectively, and Table 3-13 summarizes the extracted material properties for radial-direction tension (TR) specimens. Figure 3-25, Figure 3-26 and Table 3-14 present the same data for tangential-direction tension (TT) specimens. 39 Figure 3-23. Stress-strain curves for radial-tension test specimens, replications 1 to 10 used for determining material properties Figure 3-24. Stress-strain curves for radial-tension test specimens, replications 11 to 20 used for validation of material properties 00.10.20.30.40.50.60.70.80.90 2 4 6 8 10 12 14Stress (MPa)Strain (%)TR-1 (1A)TR-2 (2A)TR-4 (4A)TR-5 (5A)TR-6 (6A)TR-7 (7A)TR-9 (9A)TR-10 (10A)TR-11 (11A)TR-12 (12A)Mean00.10.20.30.40.50.60.70.80.90 2 4 6 8 10 12 14 16Stress (MPa)Strain (%)TR-13 (13A)TR-14 (14A)TR-15 (15A)TR-16 (16A)TR-17 (17A)TR-18 (18A)TR-19 (19A)TR-20 (20A)TR-21 (21A)TR-22 (22A)Mean40 Table 3-13. Stress-strain test data summary for radial-direction test specimens used for determining material properties and verification Rep. Specimen ID Max Stress (MPa) Max Strain (%) E (MPa) Rep. Specimen ID Max Stress (MPa) Max Strain (%) E (MPa) 1 TR-1 0.59 11.77% 2.65 11 TR-13 0.73 13.36% 1.86 2 TR-2 0.79 13.33% 3.53 12 TR-14 0.61 11.09% 2.48 3 TR-4 0.56 9.53% 2.45 13 TR-15 0.29 11.11% 1.72 4 TR-5 0.69 10.07% 2.29 14 TR-16 0.61 11.82% 1.53 5 TR-6 0.34 8.92% 2.40 15 TR-17 0.51 11.27% 2.12 6 TR-7 0.38 8.85% 2.70 16 TR-18 0.36 12.50% 0.95 7 TR-9 0.64 11.34% 4.37 17 TR-19 0.73 11.98% 3.96 8 TR-10 0.47 9.63% 3.32 18 TR-20 0.81 12.69% 1.86 9 TR-11 0.57 11.03% 1.91 19 TR-21 0.60 11.00% 2.63 10 TR-12 0.55 11.05% 2.73 20 TR-22 0.64 13.97% 2.61 Average 0.56 10.55% 2.83 Average 0.59 12.08% 2.17 σ 0.13 1.34% 0.68 σ 0.15 0.98% 0.77 Figure 3-25. Stress-strain curves for tangential-tension test specimens, replications 1 to 10 used for determining material properties 00.20.40.60.811.21.40 2 4 6 8 10Stress (MPa)Strain (%)TT-1 (1B)TT-2 (2B)TT-3 (3B)TT-4 (4B)TT-5 (5B)TT-6 (6B)TT-7 (7B)TT-8 (8B)TT-9 (9B)TT-10 (10B)Mean41 Figure 3-26. Stress-strain curves for tangential-tension test specimens, replications 11 to 20 used for validation of material properties Table 3-14. Stress-strain test data summary for tangential-direction test specimens used for determining material properties and verification Rep. Specimen ID Max Stress (MPa) Max Strain (%) E (MPa) Rep. Specimen ID Max Stress (MPa) Max Strain (%) E (MPa) 1 TT-1 0.25 8.61% 0.75 11 TT-11 0.46 7.13% 0.79 2 TT-2 1.29 9.31% 1.00 12 TT-12 0.51 13.44% 0.94 3 TT-3 1.02 6.42% 0.88 13 TT-14 0.51 12.20% 0.78 4 TT-4 1.00 6.38% 0.75 14 TT-15 0.62 9.93% 0.86 5 TT-5 0.91 5.97% 0.96 15 TT-16 0.66 11.85% 1.24 6 TT-6 0.94 5.58% 0.89 16 TT-17 0.54 13.62% 1.08 7 TT-7 1.23 6.86% 0.83 17 TT-18 0.69 9.68% 0.61 8 TT-8 1.06 8.50% 0.71 18 TT-19 0.51 10.40% 0.72 9 TT-9 1.12 7.76% 0.74 19 TT-20 0.52 10.20% 0.83 10 TT-10 0.25 4.62% 0.93 20 TT-21 0.30 9.39% 0.45 Average 0.91 7.00% 0.84 Average 0.53 10.78% 0.83 σ 0.35 1.42% 0.10 σ 0.11 1.90% 0.21 As the names of the tests imply, the tests were designed to apply the tension force in radial and tangential directions of the specimens. For radial tension (TR) tests, cracks started to initiate in the direction along the annual rings of the specimen, trying to split the specimen at the annual rings’ position. For the tangential tension (TT) tests, however, the cracks were observed to initiate and 00.20.40.60.811.21.40 2 4 6 8 10 12 14 16Stress (MPa)Strain (%)TT-11 (11B)TT-12 (12B)TT-13 (13B)TT-14 (14B)TT-15 (15B)TT-16 (16B)TT-17 (17B)TT-18 (18B)TT-19 (19B)TT-20 (20B)Mean42 grow in the direction across (perpendicular to) the annual rings, trying to split the specimen by ripping through the grain. Figure 3-27 shows the major failure mode observed in the two mentioned sets of experiments. In both directions, the specimens underwent a sudden rupture, which demonstrates a dominantly brittle failure mode for these directions. In very few specimens, a gradual rupture was observed, which could be attributed to the accuracy of the direction relative to grain in which the tension load was applied, since, due to the size and shape of the specimens, it was hardly possible to perfectly fabricate the specimens in the exact radial or tangential directions. Figure 3-27. Dominant failure modes observed in tension-perpendicular-to-grain tests: radial-tension (left) and tangential-tension (right) 3.3.6. Compression parallel to grain test The parallel-to-grain direction material properties play significantly important roles in defining the material properties of wood. In section 3.3.4, the results of tension tests in this direction were presented and discussed. This section will present the compression properties in this direction. Section 3.2.2 explains the instructions to prepare the test specimens and conduct the compression-parallel-to-grain test. 43 To fabricate the specimens, the planed and dried wood material was first sized down to rectangular-section bars using a band saw and chop saw. Then, the bars were cut down to slightly oversized specimens to account for dimension corrections using sanding machine and dimension changes due to further drying and potential shrinkages. Figure 3-28 shows samples of ready-to-test compression-parallel-to-grain test specimens. Figure 3-28. Sample compression-parallel-to-grain test specimens with strain gauge attached For this test, MTS Landmark universal testing machine in UBC Okanagan’s facilities was used. This machine is capable of applying a compression force of up to 500kN. The forces were measured and recorded by the in-house load cell of the machine. The deflection of the specimens was measured using external strain gauges attached to the specimens directly and recorded using the LabVIEW program (National Instruments, 2016). The strain gauges were installed on each specimen by following the manufacturer’s instructions carefully. To do so, the specimen’s surface on which the strain gauge was going to be installed was cleaned. Afterward, the surface was sanded using sandpaper to make the surface a bit rougher to increase the effectiveness of the bond between the strain gauge and the specimen. The surface was sanded in different directions to increase the roughness evenly throughout the surface. Next, the specimen’s surface was immaculately cleaned using rubbing alcohol and clean cloth. The appropriate type of adhesive was then applied to the surface and the strain gauge was placed at the desired location accurately. The specimens with the strain gauges were left at room temperature overnight to let the adhesive cure and dry to reach its maximum effectiveness. Figure 3-29 illustrates the strain gauge preparation and installation, and Figure 3-30 shows the full test set-up with the specimen in it. 44 Figure 3-29. Specimen preparation and strain gauge attachment for compression-parallel-to-grain specimens Figure 3-30. Compression parallel to grain test setup A total of 22 specimens were made and tested in this series. Since external strain gauges were used, some specimens did not provide robust results and had to be discarded. Five specimens were discarded due to warping or detachment of the strain gauges during the test, one had internal defects and two were considered as outliers that were removed from the results. Similar to other 45 tests, the actual dimensions of the specimens were measured along with specimens’ moisture content at the time of the test are reported accordingly. Table 3-15 presents this information. Table 3-15. Specimen dimensions and moisture content for compression-parallel-to-grain test specimens Rep. Specimen ID Width (mm) Height (mm) Depth (mm) Critical Section Width (mm) Critical Section Depth (mm) Gauge Length (mm) Moisture Content (%) 1 CL-1 25.70 104.18 25.59 25.70 25.59 60.00 6.9 2 CL-2 25.23 104.44 23.53 25.23 23.53 60.00 7.0 3 CL-3 65.67 100.88 25.24 65.67 25.24 60.00 7.1 4 CL-5 25.79 100.92 25.49 25.79 25.49 60.00 6.2 5 CL-6 26.19 98.96 25.42 26.19 25.42 60.00 6.0 6 CL-7 25.16 102.39 23.46 25.16 23.46 60.00 6.6 7 CL-8 25.98 102.58 25.35 25.98 25.35 60.00 6.4 8 CL-9 25.70 98.94 26.34 25.70 26.34 60.00 6.0 9 CL-10 25.20 100.56 25.15 25.20 25.15 60.00 6.3 10 CL-11 25.72 101.70 25.62 25.72 25.62 60.00 6.3 11 CL-16 25.56 105.04 25.93 25.56 25.93 60.00 6.5 12 CL-19 25.32 100.45 23.74 25.32 23.74 60.00 5.8 13 CL-20 25.84 100.85 25.66 25.84 25.66 60.00 6.3 14 CL-22 25.28 100.64 25.61 25.28 25.61 60.00 7.0 The first half of the working specimens were used to determine the material characteristics and the other half was used for validation of the determined characteristics. The tests were conducted as per ASTM standard and were continued up to a point that failure was completely observed in the specimen. Figure 3-31 and Figure 3-32 below plot the stress-strain curves for the two categories of the tests, respectively, and Table 3-16 summarizes the obtained results. A good match is observed between the results of the two groups of specimens. The observed difference can be attributed to the use of external strain gauges for measuring strains. The accuracy of the strain gauge placement, the good adhesion of the strain gauge to the surface below it and proper cleaning of the surface are some of the contributing factors. 46 Figure 3-31. Stress-strain curves for parallel-to-grain compression test specimens, replications 1 to 7 used for determining material properties Figure 3-32. Stress-strain curves for parallel-to-grain compression test specimens, replications 8 to 14 used for validation of material properties 0510152025300 0.2 0.4 0.6 0.8 1 1.2 1.4Stress (MPa)Strain (%)CL-1CL-2CL-3CL-5CL-6CL-7CL-805101520250 0.2 0.4 0.6 0.8 1 1.2 1.4Stress (MPa)Strain (%)CL-9CL-10CL-11CL-16CL-19CL-20CL-2247 Table 3-16. Stress-strain test data summary for parallel-to-grain compression test specimens used for determining material properties and validation Rep. Specimen ID Max Stress (MPa) Max Strain (%) E (MPa) Rep. Specimen ID Max Stress (MPa) Max Strain (%) E (MPa) 1 CL-1 19.9 0.38% 3846.2 8 CL-9 16.5 1.02% 4562.3 2 CL-2 22.1 0.61% 7724.0 9 CL-10 19.3 0.38% 5914.7 3 CL-3 5.4 0.21% 2691.4 10 CL-11 19.3 0.55% 7680.0 4 CL-5 21.7 0.65% 6114.8 11 CL-16 18.4 0.54% 4052.4 5 CL-6 18.4 1.01% 3155.8 12 CL-19 20.4 0.55% 8033.0 6 CL-7 18.2 0.39% 5227.0 13 CL-20 19.2 1.05% 3500.2 7 CL-8 24.0 0.77% 6101.9 14 CL-22 22.1 0.57% 12210.8 Average 18.5 0.58% 4980.2 Average 19.3 0.67% 6564.8 σ 6.2 0.27% 1912.8 σ 1.3 0.21% 1600.7 The dominant failure mode observed in this series of experiments was the crushing of the specimens under compression. Shearing was also observed in one specimen and a few experienced end-rolling. The mentioned failure modes are illustrated in Figure 3-33. Splitting of the specimen happened in two cases which was due to the existence of internal defects prior to testing. These specimens were discarded and not used in determining the material curves and properties. Figure 3-33. Observed failure modes in compression parallel to grain: (a) crushing, (b) shearing, (c) end-rolling or brooming, and (d) splitting due to internal defect 48 3.3.7. Compression perpendicular to grain test Another step of the experimental tests are to understand the behavior of the material in compression in the directions perpendicular to the grain, radial, and tangential directions. The procedures for these two sets are identical and explained in section 3.2.3 of this document. After drying the raw lumber, it was planed and sized down to 50x50x150 mm rectangular-shaped blocks using a band saw, chop saw and sanding machine. The actual dimensions and moisture content of the specimens are measured at the time of the tests and reported in Table 3-17 and Table 3-18 for radial- and tangential-compression specimens, respectively. Table 3-17. Specimen dimensions and moisture content for radial-compression test specimens Rep. Specimen ID Width (mm) Height (mm) Depth (mm) Critical Section Width (mm) Critical Section Depth (mm) Gauge Length (mm) Moisture Content (%) 1 CR-1 148.48 50.40 49.49 50.56 49.49 50.40 6.2 2 CR-2 15.15 48.57 48.72 50.56 48.72 48.57 6.2 3 CR-3 149.95 49.05 49.24 50.56 49.24 49.05 6.5 4 CR-4 152.07 49.79 48.73 50.56 48.73 49.79 6.4 5 CR-5 150.92 49.24 48.38 50.56 48.38 49.24 6.2 6 CR-6 150.61 49.20 48.09 50.56 48.09 49.20 6.1 7 CR-7 148.86 49.37 47.81 50.56 47.81 49.37 5.9 8 CR-8 149.03 49.20 49.90 50.56 49.90 49.20 6.7 9 CR-9 150.69 50.50 50.08 50.56 50.08 50.50 5.8 10 CR-10 150.84 48.72 48.66 50.56 48.66 48.72 6.3 Table 3-18. Specimen dimensions and moisture content for tangential-compression test specimens Rep. Specimen ID Width (mm) Height (mm) Depth (mm) Critical Section Width (mm) Critical Section Depth (mm) Gauge Length (mm) Moisture Content (%) 1 CT-1 150.37 48.12 50.00 50.56 50.00 48.12 6.1 2 CT-2 148.52 48.36 48.92 50.56 48.92 48.36 6.3 3 CT-3 147.47 46.92 49.65 50.56 49.65 46.92 6.4 4 CT-4 147.32 48.90 50.15 50.56 50.15 48.90 6.0 5 CT-5 149.20 47.03 49.96 50.56 49.96 47.03 6.1 6 CT-6 148.94 47.99 50.21 50.56 50.21 47.99 6.2 7 CT-7 148.97 49.18 49.08 50.56 49.08 49.18 5.7 8 CT-8 149.22 49.74 49.73 50.56 49.73 49.74 6.2 9 CT-9 149.53 47.88 50.06 50.56 50.06 47.88 5.7 10 CT-10 150.87 49.43 49.19 50.56 49.19 49.43 6.1 49 A total of 10 specimens were tested for each test directions (radial and tangential). The tests were continued well beyond the initial failure and ASTM standard expectations. Each test was run for 50 minutes which, with a rate of 0.305 mm/min, provides results for up to 30% compressive strain. This was done because the exact point of failure was not explicitly detectible. Furthermore, the entire post-yield behavior of the material could be captured. As the results for these experiments were more consistent compared to other directions, the number of specimens tested was enough to produce reliable material characteristics. Similar to other tests, the first half of the specimens were used to determine the characteristics and the second half was used for validation purposes. Figure 3-34 and Figure 3-35 plot the stress-strain data curves for radial and tangential perpendicular-to-grain compression tests, respectively. Summaries of the extracted material properties are presented inTable 3-19 and Table 3-20. Figure 3-36 shows a sample specimen under loading, along with the typical failure mode the specimens experienced. As the compressive deformation continues to increase in the specimen, cracks start to initiate in the direction perpendicular to the direction of loading and the specimen starts to rip off from the sides. Even though all specimens experienced cracks in the horizontal direction, the radial-compression samples had more cracks in the radial direction, showing the tendency of more crack growths in this direction. The mentioned figure illustrates this scenario. Figure 3-34. Stress-strain curves for radial-compression test specimens, left: replications 1 to 5 used for determining material properties, right: replications 6 to 10 used for validation of material properties 024681012140 10 20 30 40Stress (MPa)Strain (%)CR-1CR-2CR-3CR-4CR-5024681012140 10 20 30 40Stress (MPa)Strain (%)CR-6CR-7CR-8CR-9CR-1050 Table 3-19. Stress-strain test data summary for radial-compression test specimens used for determining material characteristics Rep. Specimen ID Max Stress (MPa) Max Strain (%) E (MPa) Rep. Specimen ID Max Stress (MPa) Max Strain (%) E (MPa) 1 CR-1 6.5 29.5% 101.4 6 CR-6 11.7 30.4% 252.4 2 CR-2 8.7 30.9% 145.5 7 CR-7 12.5 30.3% 277.8 3 CR-3 7.2 30.6% 139.6 8 CR-8 8.0 30.4% 139.0 4 CR-4 12.8 29.1% 226.5 9 CR-9 7.5 29.6% 138.1 5 CR-5 10.9 31.2% 231.3 10 CR-10 9.4 30.7% 139.3 Average 9.2 30.2% 168.8 Average 9.8 30.3% 189.3 σ 2.3 0.8% 51.3 σ 2.0 0.4% 62.4 Figure 3-35. Stress-strain curves for tangential-compression test specimens, left: replications 1 to 5 used for determining material properties, right: replications 6 to 10 used for validation of material properties Table 3-20. Stress-strain test data summary for tangential-compression test specimens used for determining material characteristics Rep. Specimen ID Max Stress (MPa) Max Strain (%) E (MPa) Rep. Specimen ID Max Stress (MPa) Max Strain (%) E (MPa) 1 CT-1 12.7 38.2% 230.1 6 CT-6 11.3 31.2% 225.7 2 CT-2 11.7 31.0% 209.6 7 CT-7 8.9 30.5% 172.7 3 CT-3 8.6 32.0% 154.6 8 CT-8 6.8 30.1% 139.9 4 CT-4 10.6 30.7% 221.2 9 CT-9 8.2 31.3% 101.5 5 CT-5 8.6 31.9% 91.8 10 CT-10 6.5 30.3% 141.4 Average 10.4 32.7% 181.4 Average 8.3 30.7% 156.2 σ 1.6 2.8% 51.9 σ 1.7 0.5% 41.4 024681012140 10 20 30 40Stress (MPa)Strain (%)CT-1CT-2CT-3CT-4CT-5024681012140 10 20 30 40Stress (MPa)Strain (%)CT-6CT-7CT-8CT-9CT-1051 Figure 3-36. Dominant failure modes for perpendicular-to-grain compression tests 3.3.8. Shear tests Section 3.2.6 explains the instructions for preparation of test specimens and test set-up for shear tests. This section presents the results of the shear tests conducted. Specimens for these tests were fabricated in UBC Okanagan’s Wood Shop facility, and were tested using universal testing machine. Figure 3-37 shows the fabricated specimens and Figure 3-38 show the test set-up. Figure 3-37. Shear test specimens 52 Figure 3-38. Shear test set-up in universal testing machine A total of 26 specimens were tested for shear tests in three perpendicular directions. Two of the specimens were found defective and were discarded, and 24 were used for determining and verifying the shear modulus. The number of tests in this set was limited due to unavailability of raw material. For longitudinal direction series, the first half of the tests, in the order they were tested, were used for determining the shear modulus and the second half were used for verification. For the tangential-radial test set, due to limited raw material available, only 5 specimens were tested and all were used for obtaining the material property. Actual specimen dimensions and moisture content of each specimen are presented in Table 3-21, Table 3-22 and Table 3-23. These are followed by Table 3-24, Table 3-25 and Table 3-26, which summarize the obtained shear modulus, as well as maximum stress and strain recorded for the specimens from the tests. 53 Table 3-21. Specimen dimensions and moisture content for longitudinal-radial shear test specimens Rep. Specimen ID Width (mm) Height (mm) Depth (mm) Critical Section Width (mm) Critical Section Depth (mm) Gauge Length (mm) Moisture Content (%) 1 GLR-1 53.73 64.31 40.14 49.21 40.14 49.21 6.1 2 GLR-2 54.46 64.81 40.08 49.40 40.08 49.40 6.8 3 GLR-3 53.23 65.02 40.31 49.78 40.31 49.78 6.2 4 GLR-4 53.23 65.02 40.31 49.78 40.31 49.78 6.2 5 GLR-5 53.54 64.94 39.68 49.68 39.68 49.68 6.3 6 GLR-6 53.29 65.02 39.85 49.43 39.85 49.43 6.2 7 GLR-7 54.52 64.55 39.52 49.49 39.52 49.49 5.9 8 GLR-8 53.32 64.92 39.96 49.30 39.96 49.30 6.2 9 GLR-10 50.03 63.43 49.53 48.66 49.53 48.66 6.0 10 GLR-11 53.53 64.48 40.08 49.56 40.08 49.56 6.4 Table 3-22. Specimen dimensions and moisture content for longitudinal-tangential shear test specimens Rep. Specimen ID Width (mm) Height (mm) Depth (mm) Critical Section Width (mm) Critical Section Depth (mm) Gauge Length (mm) Moisture Content (%) 1 GLT-1 51.96 65.30 35.55 48.50 35.55 48.50 6.1 2 GLT-3 52.53 65.89 35.79 49.44 35.79 49.44 5.6 3 GLT-4 52.68 65.96 34.45 49.63 34.45 49.63 6.2 4 GLT-5 52.81 65.88 36.47 49.46 36.47 49.46 6.1 5 GLT-6 52.90 66.13 35.36 49.51 35.36 49.51 5.9 6 GLT-7 50.05 65.10 49.73 49.41 49.73 49.41 5.6 7 GLT-8 52.82 66.00 36.32 49.20 36.32 49.20 5.4 8 GLT-9 51.80 66.16 36.01 49.71 36.01 49.71 6.0 9 GLT-10 51.90 65.79 35.66 49.57 35.66 49.57 6.1 Table 3-23. Specimen dimensions and moisture content for tangential-radial shear test specimens Rep. Specimen ID Width (mm) Height (mm) Depth (mm) Critical Section Width (mm) Critical Section Depth (mm) Gauge Length (mm) Moisture Content (%) 1 GTR-1 50.79 57.40 38.24 41.66 38.24 41.66 7.0 2 GTR-2 50.20 57.77 38.17 41.79 38.17 41.79 7.2 3 GTR-3 50.69 58.19 38.45 51.94 38.45 51.94 6.0 4 GTR-4 50.36 57.61 38.74 41.26 38.74 41.26 5.6 5 GTR-5 50.57 57.48 38.86 42.02 38.86 42.02 6.2 54 Table 3-24. Stress-strain test data summary for longitudinal-radial shear test specimens used for determining material properties and verification Rep. Specimen ID Max Stress (MPa) Max Strain (%) G (MPa) Rep. Specimen ID Max Stress (MPa) Max Strain (%) G (MPa) 1 GLR-1 0.71 1.85% 90.28 6 GLR-6 3.90 1.84% 251.69 2 GLR-2 1.88 2.13% 146.47 7 GLR-7 1.33 1.37% 143.70 3 GLR-3 1.02 1.74% 72.89 8 GLR-8 2.28 1.96% 204.57 4 GLR-4 3.33 1.44% 252.66 9 GLR-10 0.44 1.69% 65.35 5 GLR-5 4.34 2.35% 227.16 10 GLR-11 2.38 2.04% 137.60 Average 2.26 1.90% 157.9 Average 2.06 1.78% 160.58 σ 1.39 0.31% 71.7 σ 1.15 0.24% 63.43 Table 3-25. Stress-strain test data summary for longitudinal-tangential shear test specimens used for determining material properties and verification Rep. Specimen ID Max Stress (MPa) Max Strain (%) G (MPa) Rep. Specimen ID Max Stress (MPa) Max Strain (%) G (MPa) 1 GLT-1 6.73 3.69% 255.87 5 GLT-6 2.89 1.48% 261.11 2 GLT-3 6.41 3.43% 294.92 6 GLT-7 4.42 2.93% 216.31 3 GLT-4 3.43 2.08% 204.06 7 GLT-8 4.68 2.66% 265.16 4 GLT-5 3.34 2.00% 206.09 8 GLT-9 5.87 2.73% 298.61 9 GLT-10 4.05 2.28% 242.62 Average 4.98 2.80% 240.2 Average 4.38 2.42% 256.76 σ 1.59 0.77% 37.8 σ 0.96 0.51% 27.12 Table 3-26. Stress-strain test data summary for tangential-radial shear test specimens used for determining material properties Rep. Specimen ID Max Stress (MPa) Max Strain (%) G (MPa) 1 GTR-1 0.80 1.41% 63.70 2 GTR-2 0.62 1.50% 49.84 3 GTR-3 0.75 1.42% 52.49 4 GTR-4 0.87 1.87% 55.64 5 GTR-5 0.87 2.11% 46.79 Average 0.78 1.66% 53.7 σ 0.09 0.28% 5.8 55 3.4. Summary In this chapter, the experimental study conducted on the timber material used in the studied I-joists was presented. The test procedures were first described, according to ASTM standards, for testing wood in three dimensions in tension and compression. The process of procurement and preparation of the raw test material was then presented, followed by the manufacturing process for the test specimens. The timber material is tested as per the ASTM standard to obtain its mechanical properties. The tests are conducted in three directions using uniaxial tensile-and-compressive testing machines, in tension and compression, and replicated to take into account the effects of non-homogeneity of wood and variability of the specimens. Specimen specifications for each test are provided, and test results have been presented. 56 Chapter 4: FINITE ELEMENT MODELING OF TIMBER I-JOISTS WITH WEB HOLES OR FLANGE NOTCHES 4.1. General This chapter focuses on developing finite element models for timber I-joist tests. This is used to further evaluate the performance of these joists under different working conditions. A brief introduction to the basics of the numerical model is first given. Numerical models are then developed to examine the response of the models. To this end, a material model is first adopted for wood, and physical experiments are simulated complying with the procedures introduced in the previous section. The results of the simulations are compared to those of experiments to validate the numerical models. The material model is then used to model the I-joists in a finite element environment. In the scope of this research, material failure is not pursued, and numerical models are evaluated using serviceability limits and load-deflection response. The developed model will be used in the following chapters to analyze the effect of web and flange openings on the stiffness and serviceability performance of the joists. 4.1.1. Constitutive equation Constitutive equations express the connection between applied stresses or forces and strains or deformations. It forms an inevitable part of any material behavior study. It is also known as the generalized form of Hooke’s law. Eq. (4.1) shows the general form of this equation. In this equation, 𝜎 represents stress, 𝐶𝑖𝑗𝑘𝑙’s are the elastic constants and 𝜀 represents the strain corresponding to the applied stress. The elastic constants are also sometimes referred to as elastic moduli or stiffness. Indexes i, j, k, and l could represent x, y and z directions. 𝜎𝑖𝑗 = 𝐶𝑖𝑗𝑘𝑙𝜀𝑘𝑙 (4.1) 57 This equation can be rewritten to express the stress-strain relation in compliance form. It is derived by inverting the equation, as follows, where 𝑆𝑖𝑗𝑘𝑙’s are inverses of the elastic constants – i.e. 𝑆𝑖𝑗𝑘𝑙 =𝐶𝑖𝑗𝑘𝑙−1 . 𝜀𝑖𝑗 = 𝑆𝑖𝑗𝑘𝑙𝜎𝑘𝑙 (4.2) The constitutive equation can be written between stresses and strains in any direction in three dimensions. If stresses in one direction, including normal and shear stresses, are neglected and, therefore, set to be zero, it forms a planar stress condition. It should be noted that zero normal and shear stresses in this direction do not necessitate their strains to be zero. As an example, a planar stress condition is drawn in Figure 4-1. In this example, normal and shear stresses in the z-direction, respectively noted by 𝜎𝑧𝑧, 𝜎𝑥𝑧 and 𝜎𝑦𝑧 are assumed to be negligible, resulting in a planar stress condition in the x-y plane. In this case, there will exist deformations in z-direction and strains in this direction, 𝜀𝑧𝑧, 𝜀𝑥𝑧 and 𝜀𝑦𝑧, can be calculated using the constitutive equations. In a planar strain case, on the contrary, normal and shear strains in one direction are neglected and corresponding stresses can be found using constitutive relations. Figure 4-1. Planar stress situation 4.1.2. Definition of isotropic, orthotropic and anisotropic materials Material’s response to applied loads is generally measured individually in each direction. Materials whose behavior is not direction-dependent are known as isotropic – i.e., isotropic materials will exhibit similar characteristics in all directions. On the contrary, those that respond differently in different directions are called anisotropic. Orthotropic materials, on the other hand, are those who 58 have two orthogonal planes of symmetry. A general representation of an anisotropic material’s elastic matrix is given in Eq. (4.3). This matrix is symmetric and involves 21 constants which are dependent on the moduli of elasticity and Poisson’s ratios in all directions. 𝐶𝑖𝑗 =[ 𝐶11 𝐶12 𝐶13 𝐶14 𝐶15 𝐶16𝐶22 𝐶23 𝐶24 𝐶25 𝐶26𝐶33 𝐶34 𝐶35 𝐶36𝐶44 𝐶45 𝐶46𝐶55 𝐶56𝐶66] (4.3) Conventions: In this matrix, the first index of each component denotes the plane’s normal vector on which the property is defined, and the second index denotes the direction of the property. For instance, 𝜎𝑖𝑗 or 𝜀𝑖𝑗 denote, respectively, the stress or strain in j direction on the plane whose normal vector is parallel to i. For normal stresses and strains, tension is considered as the positive direction, meaning the force or deflection is pointing out of the plane on which it is acting. The shear stresses and strains, on the other hand, are considered positive when both i and j are in the same direction and negative when they are in opposite directions. If the material has one plane of symmetry, the elastic matrix reduces to the one given in Eq. (4.4). In this equation, the plane of symmetry is the one normal to the 3-direction. Such material is called monoclinic. 𝐶𝑖𝑗 =[ 𝐶11 𝐶12 𝐶13 0 0 𝐶16𝐶22 𝐶23 0 0 𝐶26𝐶33 0 0 𝐶36𝐶44 𝐶45 0𝐶55 0𝐶66] (4.4) If the material exhibits two orthogonal planes of symmetry, this matrix reduces to the one shown in Eq. (4.5). As observed, the number of constants involved is only 9, which is very important in significant numerical operations such as finite element analyses. Such material is known as an orthotropic material. It can be proven that in the case of two orthogonal planes of symmetry, the third mutually orthogonal plane will also form a plane of symmetry. 59 𝐶𝑖𝑗 =[ 𝐶11 𝐶12 𝐶13 0 0 0𝐶22 𝐶23 0 0 0𝐶33 0 0 0𝐶44 0 0𝐶55 0𝐶66] (4.5) A material that exhibits transverse isotropy in a plane, plane 𝑥2-𝑥3 for instance, will have only 5 independent elastic constants, given in Eq. (4.6). Continuing to isotropic materials, exhibiting identical responses in all directions, the matrix elements reduce to only two independent constants – that is, the response of the material to applied loads can be determined using only two independent properties. The elastic matrix of an isotropic material is shown in Eq. (4.7). 𝐶𝑖𝑗 =[ 𝐶11 𝐶12 𝐶12 0 0 0𝐶22 𝐶23 0 0 0𝐶22 0 0 012(𝐶22−𝐶23) 0 0𝐶66 0𝐶66] (4.6) 𝐶𝑖𝑗 =[ 𝐶11 𝐶12 𝐶12 0 0 0𝐶11 𝐶12 0 0 0𝐶11 0 0 012(𝐶11−𝐶12) 0 012(𝐶11−𝐶12) 012(𝐶11−𝐶12)] (4.7) 4.1.3. Relationship between tensorial and engineering constants Material properties are normally obtained from experimental tests. In this case, engineering constants need to be converted to tensorial form to be implemented in constitutive equations. The engineering strains, denoted by 𝛾𝑖𝑗, are related to tensorial strains, denoted by 𝜀𝑖𝑗, using Eq. (4.8) below. Eq. (4.9) shows this relationship in a matrix format. The stress tensor, on the other hand, is shown in Eq. (4.10), in which 𝜎 and 𝜏 represent normal and shear stresses, respectively. 𝛾𝑖𝑗 = 2𝜀𝑖𝑗 (4.8) 60 [𝜀] = [𝜀𝑥𝑥 𝜀𝑥𝑦 𝜀𝑥𝑧𝜀𝑦𝑥 𝜀𝑦𝑦 𝜀𝑦𝑧𝜀𝑧𝑥 𝜀𝑧𝑦 𝜀𝑧𝑧] = [𝜀𝑥𝑥𝛾𝑥𝑦2𝛾𝑥𝑧2𝛾𝑦𝑥2𝜀𝑦𝑦𝛾𝑦𝑧2𝛾𝑧𝑥2𝛾𝑧𝑦2𝜀𝑧𝑧] (4.9) [𝜎] = [𝜎𝑥𝑥 𝜏𝑥𝑦 𝜏𝑥𝑧𝜏𝑦𝑥 𝜎𝑦𝑦 𝜏𝑦𝑧𝜏𝑧𝑥 𝜏𝑧𝑦 𝜎𝑧𝑧] (4.10) Wood shows different material properties in different directions. As a result, an anisotropic or orthotropic material model is often used for its constitutive model. To construct the constitutive relation for this material, the stiffness matrix introduced in Eq. (4.3) and Eq. (4.5) are used. To obtain the material characteristics, data from experimental tests need to be adjusted to meet the requirements of these material models. Since the engineering constants can be used more directly in the compliance matrix, the compliance matrix is first introduced, and the stiffness matrix is then derived. Eq. (4.11) shows the compliance matrix for an orthotropic material obtained with the help of engineering constants. In this equation, 𝐸𝑖’s, 𝐺𝑖’s and 𝜈𝑖𝑗’s represent the moduli of elasticity, shear moduli and Poisson’s ratios in different directions, respectively. The first two (𝐸𝑖 and 𝐺𝑖) are slopes of the stress-strain curves and 𝜈’s are slopes of strain-strain curves. 𝑆𝑖𝑗 =[ 𝑆11 𝑆12 𝑆13 0 0 0𝑆21 𝑆22 𝑆23 0 0 0𝑆31 𝑆32 𝑆33 0 0 0𝑆44 0 0𝑆55 0𝑆66] =[ 1𝐸1− 𝜈21𝐸2− 𝜈31𝐸30 0 0− 𝜈12𝐸11𝐸2− 𝜈32𝐸30 0 0− 𝜈13𝐸1− 𝜈23𝐸21𝐸30 0 01𝐺230 01𝐺3101𝐺12] (4.11) In this equation, the Poisson’s ratio is defined as 𝜈𝑖𝑗 = −𝜀𝑗𝜀𝑖 and shows the ratio of the extension in j direction to that in i direction, when the applied stress is in the i direction. They are obtained by having zero stresses in all other directions. It was previously mentioned that an orthotropic material’s stiffness and compliance matrices are symmetric and, thus, it has nine independent components. Applying this to Eq. (4.11) with the engineering constants plugged in, it will be concluded that: 61 𝜈𝑖𝑗𝐸𝑖=𝜈𝑗𝑖𝐸𝑗 𝑖 ≠ 𝑗 (4.12) which means only three of the six Poisson’s ratios (one per direction set) need to be further considered. This is known as a compliance symmetry condition. As a result, since the moduli of elasticity in all directions are available, in this mentioned equation only 𝜈21, 𝜈31 and 𝜈32 (or alternatively 𝜈12, 𝜈13 and 𝜈23) will be required. Since the stiffness matrix is the inverse of the compliance matrix, its components can be found using the following set of equations: 𝐶11 =𝑆22𝑆33 − 𝑆232𝑆 𝐶22 =𝑆33𝑆11 − 𝑆132𝑆 𝐶33 =𝑆11𝑆22 − 𝑆122𝑆 𝐶12 =𝑆13𝑆23 − 𝑆12𝑆33𝑆 𝐶13 =𝑆12𝑆23 − 𝑆13𝑆22𝑆 𝐶23 =𝑆12𝑆13 − 𝑆23𝑆11𝑆 𝐶44 =1𝑆44 𝐶55 =1𝑆55 𝐶66 =1𝑆66 (4.13) where: 𝑆 = 𝑆11𝑆22𝑆33 − 𝑆11𝑆232 − 𝑆22𝑆132 − 𝑆33𝑆122 + 2𝑆12𝑆23𝑆13 (4.14) This results in having the stiffness matrix components in the following form in terms of the engineering constants: 𝐶11 =1 − 𝜈23𝜈32𝐸2𝐸3∆ 𝐶22 =1 − 𝜈13𝜈31𝐸1𝐸3∆ 𝐶33 =1 − 𝜈12𝜈21𝐸1𝐸2∆ 𝐶12 =𝜈21 + 𝜈31𝜈23𝐸2𝐸3∆=𝜈12 + 𝜈32𝜈13𝐸1𝐸3∆ 𝐶23 =𝜈32 + 𝜈12𝜈31𝐸1𝐸3∆=𝜈23 + 𝜈21𝜈13𝐸1𝐸2∆ 𝐶13 =𝜈31 + 𝜈21𝜈32𝐸2𝐸3∆=𝜈13 + 𝜈12𝜈23𝐸1𝐸2∆ 𝐶44 = 𝐺23 𝐶55 = 𝐺31 𝐶66 = 𝐺12 (4.15) where: ∆=1 − 𝜈12𝜈21 − 𝑣23𝜈32 − 𝜈31𝜈13 − 2𝜈21𝜈32𝜈13𝐸1𝐸2𝐸3 (4.16) It should be noted that the constructed stiffness and compliance matrices should meet the requirements as follows. Firstly, the total work done by all stresses on the element should be positive. Additionally, these matrices relating forces to deformations need to possess positive 62 principal values, meaning that these matrices should be positive definite. To satisfy this condition, the diagonal components of the compliance matrix should be positive: 𝑆11, 𝑆22, 𝑆33, 𝑆44, 𝑆55 and 𝑆66 > 0. It will suffice to have positive moduli of elasticity and shear moduli to make this happen (𝐸1, 𝐸2, 𝐸3, 𝐺23, 𝐺31 and 𝐺12 > 0). Mutually, the diagonal elements of the stiffness matrix should be positive, too, which results in meeting the following conditions: 1 − 𝜈23𝜈32 > 0 1 − 𝜈13𝜈31 > 0 1 − 𝜈12𝜈21 > 0 ∆> 0 (4.17) 4.2. Finite element modeling of the test specimens To verify the applicability of the material model to the tests, finite element models of the tensile test specimen was modeled and analyzed in ANSYS finite element software (ANSYS Inc., 2012). The geometry of the specimen was precisely modeled within ANSYS, complying with that shown in Figure 3-4. The dimensions of the model are listed in Table 3-3. The model also includes the exact 3D curvatures in both transverse directions. Figure 4-2 shows the created model. Figure 4-2. Finite element model of the tensile test specimen 63 As shown in the figure above, the mesh is designed so that the density of mesh near the necking area (gauge length) of the specimen is higher, with a reasonable mesh density at other parts. A sensitivity analysis was conducted on the size and number of elements to find their effect on the results. It was observed that, given the linearity of the material and the fact that the specimen is under simple tension, this effect is not significant, and a relatively coarse mesh can be used to capture the stress-strain response of the model. Table 4-1 presents the moduli of elasticity and shear of the timber material, obtained from experimental tests. Table 4-1. Moduli of elasticity for timber material (MPa) EL ET ER GLR GLT GTR 𝝂𝑳𝑹 𝝂𝑳𝑻 𝝂𝑻𝑹 Timber 10549.2 0.84 2.83 157.9 240.2 53.7 0.370 0.470 0.250 SOLID 64 element is used in the specimen modeling, which is compatible with material behaviors in solid structures (ANSYS Inc, 2006). This element possesses three degrees of freedom in the form of nodal translations at each of its eight nodes. It is a good choice to model wood specimens since it can also undergo large deflections. Nodes at one end within a length equal to the length of the specimen supported in the tensile test machine grips were fixed to model the bottom end of the specimen (Figure 4-3). It is assumed, in this approach, that there is negligible slippage between the specimen and test machine jaws. Similarly, the nodes falling within the grip length at the other end of the specimen were subjected to the displacement, as prescribed in the test procedure. Figure 4-3. Support condition of the tensile test specimen To compare the simulation results to those of the experiment, it is required to plot the stress-strain curve for the specimen model. To this end, the necking area of the specimen was looked at to calculate the strain at each solution step. The elongation was calculated as the difference between 64 the displacements of the start and end nodes of the gauge length at every time step. Subsequently, strain at each time step was calculated as the elongation divided by the original gauge length. The stress, on the other hand, was obtained by summing up the forces, in a longitudinal direction, at supported nodes and dividing it by the neck area’s cross-section. The results of the simulation of the test specimen using the mean, lower-bound and higher-bound material properties collected from the first half of the specimens (TL-1 to TL-9) are plotted against those of the second half (TL-10 to TL-18) in Figure 4-4. It is observed that the results are in a very good agreement with those of the experiments. This proves that the model can be reliably used for the used timber material. Figure 4-5 illustrates a contour plot of stress distribution in the longitudinal direction for this specimen. It is observed that the maximum stress is observed near the gauge length marks at the necking area ends. This is where most specimens experienced failure in experimental tests. Figure 4-4. Validation of tensile test specimen simulation results 65 Figure 4-5. Contour plot of stress distribution in longitudinal direction for tensile test specimens Furthermore, additional models were developed and run to assess the effect of removing the shoulders of the specimens. An excellent match between the specimens with and without shoulders was found, in terms of their stress-strain response curves. The stress distribution in the specimen was observed to be slightly different, because of the presence of the shoulders in those parts. However, the stress distribution in the necking area was found to be similar, with the maximum stress happening at the same location in the specimen. Figure 4-7 plots the stress-strain curve comparison for the two specimen types, and Figure 4-6 depicts the stress distribution pattern in the specimen with shoulders. Figure 4-6. Stress distribution pattern in tensile test specimen with shoulders 66 Figure 4-7. Comparison of stress-strain response of tensile test specimens with and without shoulders 4.3. Finite element modeling of timber I-joists Despite a large number of specimens that were tested in the experimental phase, some limitations still exist in the research. This is mainly because of the limitations on the number and cost of specimens and includes considering the effect of various sizes of web holes and flange notches, the location of the cuts and the interaction between them. Furthermore, the real loading condition, a uniformly-distributed load, cannot be tested experimentally effortlessly. This section presents the finite element simulation of timber I-joists to predict the load-deflection behavior of these members with respect to the size and location of the web hole and flange cut. The numerical study presented in this section is pursued to pave the way to analyze these beams without the need for further physical tests. Mechanical APDL tool in ANSYS finite element program (ANSYS Inc., 2012) is implemented for this purpose and to produce more data to add to the experimental ones. The challenges involved in this study are reviewed, including the web-flange connectivity, numerical model meshing, computational precision, and material properties. The numerical models are validated against experimental research and the data is used to obtain the load-deflection relationship for considered beams. It has been demonstrated that the proposed finite element simulation can capture the behavior of the beams with a very reasonable tolerance. 0102030405060700 0.1 0.2 0.3 0.4 0.5 0.6Stress (MPa)Strain (%)Specimen w/o shouldersSpecimen w/ shoulders67 Detailed full-scale I-joists were modeled in ANSYS in real dimensions. The geometry and dimensions are so chosen that they match real-scale I-joists. These follow the tested joist configurations explained in section 2.3. The connectivity between the web and flanges is considered to be a perfect bonding. This assumption is evaluated by verifying the results of the FE simulation against experimental ones, brought later in this section. Material properties for the flange material are obtained from experimental properties as stated in Table 4-1, and those for the web are obtained from (Islam, 2017), resulting from experiments conducted in ALAMS lab in the University of British Columbia, as summarized in Table 4-2. Table 4-2. Moduli of elasticity for OSB material (MPa) (Islam, 2017) EL ET ER GLR GLT GTR 𝝂𝑳𝑹 𝝂𝑳𝑻 𝝂𝑻𝑹 OSB 5322.60 3230.68 3703.3 1495.2 1560.6 1832.0 0.248 0.309 0.146 In order to validate the finite element model with the experimental results, the loading and boundary conditions are also set to represent real test scenarios. The exact two-point bending loading used in the experiments was simulated. The loads act at 1/3 points of beam’s length (as illustrated in Figure 2-1), transferred over a 4-inch bearing plate to distribute the pressure and avoid stress concentration. A simply-supported ends condition is introduced by constraining the movement of support nodes in the vertical direction at both supports and prohibiting the movement of only one support in the longitudinal direction of the beam. Lateral movements are prevented using 10 lateral supports for 12-ft and 16 for 20-ft specimens at both side faces of the top flange of the joist. These supports play a significantly important role, as they prevent the torsion in cut beams. Cut beams, specifically those with flange notches, are highly susceptible to torsion due to the stiffness asymmetry in longitudinal direction. Preventing joists against torsion is a realistic assumption, as the transverse blocking members and the sheathing intend to support the joist laterally in roof or flooring systems (Canadian Standards Association, 2019). Because of the large size of the specimens, meshing is an important challenge in this study. If not appropriately created, it can result in an expensive simulation in terms of time and computational efforts. In our finite element study, we conducted a sensitivity analysis to find out the appropriate element distribution leading to acceptable results explained in the following sections. It is important to have adequately dense meshes in regions susceptible to stress concentrations such as web hole and flange notch areas. A meshing block with an adjustable size is created around the 68 cuts to control the mesh density in this region and relate its finer mesh to the coarser mesh surrounding it. This method has helped to capture the full behavior of the sensitive regions while saving computational costs in other regions. Another significant step in reducing the number of elements and, as a result, the computational efforts was introducing a symmetry condition at the middle plane of the beam. This symmetry plane cuts the joist’s width in half throughout the entire length of the joist, which in turn cuts the computational time and efforts by half. This means the nodes on the symmetry plane have zero displacements normal to the plane – in the z-direction, in our model. For all parts, mapped meshing with an ordered pattern is implemented, which makes the model more consistent. Figure 4-8 shows sample developed models for full-scale 12-ft I-joist with web hole and flange notches. Light blue color represents the OSB material and purple represents the flanges made of timber. Figure 4-8. Finite element model of 12-ft (a) web-holed and (b) flange-notched I-joists (c) symmetry condition Ten specimens of the experimental study described earlier were selected to validate the numerical study. To this end, these specimens were modeled with all details according to the simulation specifications described. The numerical models were subjected to bending loads identical to those in the experimental study, shown in Figure 2-1. The deflection of the joist is measured at the center point of the beam, at the mid-span. It is obtained by averaging the vertical displacements over nodes located within one inch of the mid-span. The analysis has been continued to the maximum deflection observed in the experiments, to capture an adequate response. 69 Figure 4-9. Load-deflection curves for specimens (a) 12-Control (control joist), (b) 12-5-8⅜, (c) 12-5-6, (d) 12-5-4 and (e) 12-4.5-4x6 Figure 4-9 and Figure 4-10 illustrate the load-deflection curves for 12-ft and 20-ft specimens, respectively. A very good match is observed between the experimental and numerical results, which shows the robustness of the proposed simulation approach. This allows us to use the proposed numerical model to base our further studies on. This numerical model is used in the following chapters to conduct a sensitivity analysis on the type, size, and position of the cut or opening in the I-joist. 70 Figure 4-10. Load-deflection curves for specimens (a) 20-Control (control joist), (b) 20-7-8⅜, (c) 20-9-6, (d) 20-9-4x4 and (e) 20-8.5-4x6 4.4. Summary This chapter focused on developing finite element models to further evaluate the I-joists under study. The data gathered from test specimens explained in the previous chapter was analyzed and a model for the tested timber material was developed. First, 3D finite element models of the tested specimens were developed to evaluate the suggested numerical model. A sensitivity analysis was conducted on the meshing of the models to examine the effects of the mesh density on the results. It was shown that, given the fact that the specimen 71 was under nearly pure tension, and since the specimen geometry was designed for tensile tests, a relatively coarse mesh pattern can be used, without compromising the accuracy of the results. The stress-strain curves resulted from the simulations were then compared against those obtained from the experiments. A very good match was found between the results, which showed the applicability of the developed model to the study. In the next step, 3D finite element models of full timber I-joists under study were developed, starting with control joists with no openings, in two span lengths. Solid elements were used in this modeling. The boundary conditions, including the end supports and lateral supports, as well as loading conditions, were modeled to replicate the real testing conditions. Models of I-joists with web holes and flange notches were then developed, as per joist configurations used in the reference experimental study. A mesh sensitivity study was also conducted for these full-scale specimens to find the balance between the precision of the results and computational efforts. A higher mesh density was used in the areas near the openings (i.e. web holes and flange notches) to more closely capture the behavior and the effects of stress concentration in these areas. The load-deflection response obtained from the simulation of the full-scale joists were compared against those obtained from the reference experimental tests. It was observed that the proposed finite element model can capture the behavior of the joists under study within a reasonable tolerance. This demonstrates that the proposed model can be used to predict the behavior of a wider range of joist configurations that have not been experimentally tested. This will help facilitate the study of the effect of the presence of web holes and flange notches in I-joists, as well as the effect of their size and location, on the stiffness of these joists. The proposed models explained in this chapter are used later in this study to conduct a sensitivity analysis on the opening specifications, develop predictive equations, assess the effectiveness of some retrofit techniques, and analyze the design procedure of these joists. 72 Chapter 5: SENSITIVITY ANALYSIS OF TIMBER I-JOISTS WITH WEB HOLES OR FLANGE NOTCHES 5.1. General Earlier in this thesis, the significance of the existence of openings and cuts in I-joists was highlighted. The effect of these openings was assessed through experimental tests and it was shown that such openings can reduce the load-carrying capacity of these beams by up to 80%. The beams were modeled numerically in ANSYS finite element software whose results were validated against the presented experimental work. Even though the importance of the effects of such openings is known to many, a thorough investigation of these effects has not yet been performed in the literature. In this chapter, a comprehensive sensitivity analysis will be conducted to further expand this evaluation. The objective of this sensitivity analysis is to measure the impacts of type, size, and location of the opening or cut on the bending moment capacity of the joists with two span lengths of 12 and 20 ft., acting primarily as beams. Two types of openings are examined: (1) web holes and (2) flange notches. Three opening sizes for the web hole and nine different sizes for the flange notch are considered in the sensitivity analysis, varying the diameter of the hole, and length and depth of the notch. The sizes are determined to cover the most applicable ranges in the industry. The effects of notch length and depth are reported individually. The location of the opening is varied at 1-ft increments to see the importance of the location of the web hole or the flange notch and the results are reported. Using a design-of-experiment (DoE) approach, a full factorial experiment has been designed with each factor having three levels. This design of the experiment requires a total of 48 cases for beams with web holes and 144 cases for beams with flange notches to be studied. The results of this sensitivity analysis will be used in the next chapters to derive predictive equations to predict the capacity of the beam based on the type, size, location, and length of the beam. The predictive 73 equations will also be employed to propose opening modification factors for the limit state design of such joists. The following sections present the details of the sensitivity analysis. 5.2. Summary of experimental tests As explained in Table 2-1, 18 scenarios were tested experimentally, each having 10 replications. A consistent linear behavior was detected for all I-joists, including the ones with and without openings. Taking advantage of this linearity, we used the load-deflection curve slope as an identification to measure the overall structural stiffness of the joists and for comparison purposes. Table 5-1 below summarizes the average maximum load-carrying capacity and deflection of the above-referenced experiments, along with their so-called structural stiffness. These values have been used as references for the end load-carrying point to continue the simulations up to. It can be observed that the existence of the openings has a considerable effect on the stiffness of the joists, the extent of which depends on the shape, size, and location of the opening. In the following sections, these effects are investigated, and the results are analyzed. Figure 5-1 plots average load-displacement curves for the joists in the experimental study. An algorithm was written in Matlab to generate these curves by finding the average deflection of the joist at a given load for all replications of each specimen. Table 5-1. Experimental test results summary – P: Average maximum load at joist failure, Δ: Average maximum deflection at joint failure 12-ft 20-ft Opening Type Specimen Label Max Load, P (kN) Max Deflec-tion, Δ (mm) Average Linear Stiffness (kN/mm) Specimen Label Max Load, P (kN) Max Deflec-tion, Δ (mm) Average Linear Stiffness (kN/mm) No Opening 12-Control 31.4 27.21 1.155 20-Control 21.9 77.77 0.282 Web Hole 12-5-8⅜ 15.2 15.42 0.985 20-9-8⅜ 17.4 50.81 0.342 12-4-8⅜ 14.5 14.24 1.021 20-7-8⅜ 16.9 54.13 0.312 12-5-6 22.2 21.26 1.044 20-9-6 23.6 69.84 0.339 12-5-4 30.6 27.95 1.095 20-9-4 20.8 71.99 0.290 Flange Notch 12-5-4x4 14.9 16.99 0.878 20-9-4x4 12.3 43.19 0.285 12-4.5-4x4 9.2 12.98 0.712 20-8.5-4x4 9.7 37.14 0.261 12-4-4x4 6.2 8.89 0.700 20-8-4x4 6.7 25.67 0.262 12-4.5-4x6 9.2 12.98 0.712 20-8.5-4x6 5.5 26.81 0.204 74 Figure 5-1. Average load-deflection plots for experimental tests, 12-ft I-joists (left) and 20-ft I-joists (right) 5.3. Design of experiment Using the finite element model explained in previous chapters, we designed sets of simulation experiments consisting of a series of numerical tests to measure the effects of opening characteristics on the stiffness of the I-joists. The factors studied include type, size, and location of the opening being tested on two beam span lengths. For this purpose, web holes and flange notches are tested, the opening sizes are varied in applicable ranges, and the location of the opening is varied at 1-ft increments starting from the mid-span of the beam towards the supports. This design of the experiment, thus, will make 48 cases for I-joists with web holes and 144 cases for I-joists with flange cuts. A sensitivity analysis is conducted using a full-factorial experiment with regards to the defined factors. A total of 192 configurations are tested to find the stiffness of the considered beams. Section 5.4 and 5.5 below present the results of the conducted simulations on I-joists with web holes and I-joists with flange cuts, respectively. The details of the design of the experiment and parameter levels for each case is presented, and the effect factor plots are illustrated. 5.4. I-joists with web holes In our sensitivity analysis, a total of 48 configurations of I-joists with web holes of different sizes and locations were tested. Hole size levels, i.e. hole diameters, are chosen to cover the most applicable ranges in the real-world timber building industry. Besides, the chosen range covers the 051015202530350 10 20 30P (kN)Δ…12-Control12-5-8⅜12-4-8⅜12-5-612-5-412-5-4x412-4.5-4x412-4-4x412-4.5-4x605101520250 25 50 75 100P (kN)Δ (mm)20-Control20-9-8⅜20-7-8⅜20-9-620-9-420-9-4x420-8.5-4x420-8-4x420-8.5-4x675 widest possible area of the web. Hole location is varied, at 1-ft increments, starting from the mid-span point, spanning throughout the length of the beam. This helps to fully understand how the hole location will affect the strength of the beam. Table 5-2 shows the design of the experiment and parameter levels for this sensitivity analysis. Figure 5-2 illustrates the explained parameters (web opening diameter and location) along with a schematic of the joist. Table 5-2. Design of experiment and parameter levels for I-joists with web openings Beam Span (ft.) Web Hole Diameter (in.) Web Hole Location (ft., from mid-point) 12 4, 6 and 8 0, 1, 2, 3, 4 and 5 20 4, 6 and 8 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 Figure 5-2. Schematic of sensitivity analysis parameters on I-joists with web holes It was observed, from the numerical analyses, that the joists with web openings still follow a linear load-deflection behavior under the considered loading. However, the stiffness of the beam gets affected with respect to the size and location of the opening. Figure 5-3 and Figure 5-4 below show the load-deflection behavior of a selection of I-joists with web holes to observe the effect of location and size of the I-joists. For all opening sizes, it was observed that the effect of the opening size is the least at the mid-point of the beam span and increases as the opening moves away from the mid-span. Figure 5-3 demonstrates this fact. This can be attributed to the fact that the beam is loaded in pure bending and the web is meant to carry the shear forces. Smaller-size openings (i.e. 4-in. holes) showed to have insignificant effect at all locations. The leftmost plot in Figure 5-4 illustrates this fact. This rationalizes the fact that most manufacturers allow small openings throughout the length of their I-joist. The effect of the location of the opening, however, was seen to be more significant for 76 larger openings. A maximum of 6.7% and 2.7% reduction in capacity was found for 8-in-diameter holes in 12-ft and 20-ft beams, respectively. This is a significant capacity reduction compared to 1.7% and 1.0% for 4-in holes in 12-ft and 20-ft beams, respectively. Figure 5-3. Opening size effect plots. Top: Load-deflection curves for 12-ft I-joists with web holes at 0-, 3- and 5-ft distances from the mid-span (left to right). Bottom: Load-deflection curves for 20-ft I-joists with web holes at 0-, 5- and 9-ft distances from the mid-span (left to right) 024681012141618200 10 20P (kN)Δ (mm)12-0-D412-0-D612-0-D812-Control 024681012141618200 10 20P (kN)Δ (mm)12-3-D412-3-D612-3-D812-Control 024681012141618200 10 20P (kN)Δ (mm)12-5-D412-5-D612-5-D812-Control01234567890 10 20 30P (kN)Δ (mm)20-0-D420-0-D620-0-D820-Control 01234567890 10 20 30P (kN)Δ (mm)20-5-D420-5-D620-5-D820-Control 01234567890 10 20 30P (kN)Δ (mm)20-9-D420-9-D620-9-D820-Control77 Figure 5-4. Opening location effect plots. Top: Load-deflection curves for 12-ft I-joists with 4-, 6- and 8-in. diameter web holes (left to right). Bottom: Load-deflection curves for 20-ft I-joists with 4-, 6- and 8-in. diameter web holes (left to right) 3-D effect plots are produced to better observe the interactive effects of location and size of the hole. Figure 5-5 and Figure 5-6 below illustrate the aforementioned effect plots for 12-ft and 20-ft beams, respectively. 024681012141618200 10 20P (kN)Δ (mm)12-0-D412-1-D412-2-D412-3-D412-4-D412-5-D412-Control 024681012141618200 10 20P (kN)Δ (mm)12-0-D612-1-D612-2-D612-3-D612-4-D612-5-D612-Control 024681012141618200 10 20P (kN)Δ (mm)12-0-D812-1-D812-2-D812-3-D812-4-D812-5-D812-Control01234567890 10 20 30P (kN)Δ (mm)20-0-D420-1-D420-2-D420-3-D420-4-D420-5-D420-6-D420-7-D420-8-D420-9-D420-Control01234567890 10 20 30P (kN)Δ (mm)20-0-D620-1-D620-2-D620-3-D620-4-D620-5-D620-6-D620-7-D620-8-D620-9-D620-Control 01234567890 10 20 30P (kN)Δ (mm)20-0-D820-1-D820-2-D820-3-D820-4-D820-5-D820-6-D820-7-D820-8-D820-9-D820-Control78 Figure 5-5. 3-D effect plot for 12-ft I-joists with web openings, effect of opening diameter vs. location Figure 5-6. 3-D effect plot for 20-ft I-joists with web openings, effect of opening diameter vs. location 79 5.5. I-joists with flange notches A total of 144 configurations of joists with flange notches were tested in a full-factorial sensitivity analysis. This includes beams with flange notches with varying lengths and depths, at locations starting from the mid-point of the beam spanning throughout the length of the beam at 1-ft increments. Table 5-3 shows the design of experiment and parameter levels for this sensitivity analysis. Figure 5-7 illustrates the explained parameters (notch length, depth, and location), along with a schematic of the joist. Table 5-3. Design of experiment and parameter levels for I-joists with flange notches Beam Span (ft.) Notch Length (in.) Notch Depth (in.) Notch Location (ft., from mid-point) 12 4, 6 and 8 4, 6 and 8 0, 1, 2, 3, 4 and 5 20 4, 6 and 8 4, 6 and 8 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 Figure 5-7. Schematic of sensitivity analysis parameters on I-joists with web holes Similar to beams without openings and with web holes, those with flange notches still tend to show an overall linear structural behaviour. On the contrary to web holes, flange notches show to have a more considerable impact on the stiffness of the beam. It is observed that flange notches reduce the beam’s stiffness by up to 92% and 85% in 12-ft and 20-ft beams, respectively. The maximum reduction was detected when the notch was located at the mid-span of the beam, getting less as the notch moves toward the support. This can be attributed to the lack of integrity in the moment-carrying components of the beams – flanges. Figure 5-10 and Figure 5-11 illustrate the 3-D effect plots showing the effects of notch depth and length, respectively, for 12- and 20-ft joists. 80 Figure 5-8. Notch size effect plots. Top: Load-deflection curves for 12-ft I-joists with flange notches at 0-, 3- and 5-ft distances from the mid-span (left to right). Bottom: Load-deflection curves for 20-ft I-joists with flange notches at 0-, 5- and 9-ft distances from the mid-span (left to right) 024681012141618200 10 20P (kN)Δ (mm)12-0-4x412-0-6x412-0-8x412-0-4x612-0-6x612-0-8x612-0-4x812-0-6x812-0-8x812-Control024681012141618200 10 20P (kN)Δ (mm)12-3-4x412-3-6x412-3-8x412-3-4x612-3-6x612-3-8x612-3-4x812-3-6x812-3-8x812-Control024681012141618200 10 20P (kN)Δ (mm)12-5-4x412-5-6x412-5-8x412-5-4x612-5-6x612-5-8x612-5-4x812-5-6x812-5-8x812-Control01234567890 10 20 30P (kN)Δ (mm)20-0-4x420-0-6x420-0-8x420-0-4x620-0-6x620-0-8x620-0-4x820-0-6x820-0-8x820-Control01234567890 10 20 30P (kN)Δ (mm)20-5-4x420-5-6x420-5-8x420-5-4x620-5-6x620-5-8x620-5-4x820-5-6x820-5-8x820-Control01234567890 10 20 30P (kN)Δ (mm)20-9-4x420-9-6x420-9-8x420-9-4x620-9-6x620-9-8x620-9-4x820-9-6x820-9-8x820-Control81 Figure 5-9. Notch location effect plots. Top: Load-deflection curves for 12-ft I-joists with 4x4, 6x6 and 8x8 flange notches (left to right). Bottom: Load-deflection curves for 20-ft I-joists with 4x4, 6x6 and 8x8 flange notches (left to right) 024681012141618200 10 20P (kN)Δ (mm)12-0-4x412-1-4x412-2-4x412-3-4x412-4-4x412-5-4x412-Control024681012141618200 10 20P (kN)Δ (mm)12-0-6x612-1-6x612-2-6x612-3-6x612-4-6x612-5-6x612-Control024681012141618200 10 20P (kN)Δ (mm)12-0-8x812-1-8x812-2-8x812-3-8x812-4-8x812-5-8x812-Control01234567890 10 20 30P (kN)Δ (mm)20-0-4x420-1-4x420-2-4x420-3-4x420-4-4x420-5-4x420-6-4x420-7-4x420-8-4x420-9-4x420-Control01234567890 10 20 30P (kN)Δ (mm)20-0-6x620-1-6x620-2-6x620-3-6x620-4-6x620-5-6x620-6-6x620-7-6x620-8-6x620-9-6x620-Control01234567890 10 20 30P (kN)Δ (mm)20-0-8x820-1-8x820-2-8x820-3-8x820-4-8x820-5-8x820-6-8x820-7-8x820-8-8x820-9-8x820-Control82 Figure 5-10. 3-D effect plots for 12-ft I-joists with flange notches, (top) effect of notch depth vs. location and (bottom) effect of notch length vs. location 83 Figure 5-11. 3-D effect plots for 20-ft I-joists with flange notches, (top) effect of notch depth vs. location and (bottom) effect of notch length vs. location 84 5.6. Summary In this chapter, the study was continued by assessing the effect of web holes and flange notches on the stiffness of timber I-joists. To this end, numerous timber I-joists were numerically studied, with various opening configurations. A design of experiment was developed to assess the effect of the size and location of the web hole or the flange notch. In this full-factorial design, 48 cases of I-joists with web hole were modeled and analyzed, having the hole at different locations with 1-foot increments, with varying sizes (4, 6 and 8 in. diameter). The load-deflection curves for these models were plotted and their structural stiffness was calculated. Analysis of the results showed that the existence of the hole can significantly reduce the stiffness of the joist, depending on the size and location of the hole. Smaller holes (4-in. diameter) had a negligible effect on the stiffness of the beams. For these holes, the location also had very little effect, up to 1.8% for 12-ft joists and 1% for 20-ft joists. As the diameter of the hole increased, however, the role of the hole in decreasing the strength of the joist became more significant. It was also observed that the location of the hole becomes more important for larger holes. Besides, it was observed that, for all sizes of the hole, those near supports contributed to decreasing the beam’s stiffness more. For instance, 8-in. holes could cause a 5% more decrease in the capacity when closer to the supports. 3D plots were created to simultaneously illustrate the effects of the size and location of the opening. 144 configurations of joists with flange notches were also modeled and analyzed in this sensitivity analysis. The dimensions of the notch were varied at 2 in. increments and the notch was considered to be located anywhere along the beam span at 1 ft increments. The effect of the existence of a flange notch was seen to be much more significant than that of a web hole. For 12-ft and 20-ft joists, respectively, a maximum stiffness reduction of 92% and 85% was observed. Unlike web holes, notches closer to the mid-span of the beam had a more significant role in decreasing its capacity. The results showed that, for beams with flange notches, the depth of the notch affected the capacity reduction 7.6% more, compared to the length. It was also observed that the effect of the size of the notch increases as the notch moves away from joist’s supports, and it is the most at mid-span. Overall, even though the size of the notch could play an important role, the location was seen to be a more important factor in affecting the capacity. 3D effect plots were created to better see the effect of studied factors simultaneously. 85 In general, the existence of a web hole in joists was observed to have a much less significant effect on the performance of the beam than flange notches. This promotes the idea of retrofitting joists with web holes to bring them back to their expected performance level. This idea will be assessed later in this thesis. The significant stiffness decrease seen in joists with flange notches, on the other hand, supports the idea of them not being permitted by a wide variety of I-joist manufacturers. 86 Chapter 6: PREDICTIVE EQUATIONS FOR I-JOISTS WITH WEB HOLES OR FLANGE NOTCHES 6.1. General Following the analysis performed in the previous chapter, the parameters of the sensitivity study, as well as the results of the analysis, are used for further study in this chapter. These factors, whose effects were shown to be significant on the stiffness of the beam, are: location of the opening (web hole or flange notch), diameter of the web hole, and size (length and depth) of the flange notch. The objective of this chapter is to derive predictive equations for response characteristics of considered I-joists, including effective structural stiffness Keff and load at span/180 deflection 𝑃180. Response Surface Methodology (RSM) is implemented to obtain the predictive equations for the response characteristics. The produced equations will be employed to introduce capacity modification factors suggested for limit state design of I-joists with openings. 6.2. Response Surface Methodology Response Surface Methodology (RSM) is implemented, to define the relationship between beam characteristics and studied factors. RSM employs a combination of mathematical and statistical approaches and is used in situations where the response is a factor of multiple input parameters. The RSM is mainly used for the following purposes (Myers et al., 2009): 1. Producing a three-dimensional surface illustrating response variable(s) versus multiple input variables. This can be used for predicting the response of the model for un-tested design points. 2. Optimization of the model. 3. Finding regions of interest that meet certain conditions, taking into account the effects of more than one factor at a time. In this method, the response factors are usually shown by yi and input factors are denoted by Xi. The relationship between the two sets of factors is usually presented as yi = f (X0, X1, …, Xi) + ϵ, 87 in which ϵ serves as a random error. Where a linear model can be accepted as the relationship between the factors, a first-order polynomial regression model can be used as follows in Eq. (6.1): 𝑦𝑖 = 𝛽0 +∑𝛽𝑖𝑋𝑖𝑖+∈ (6.1) βi’s in the preceding equation are constant coefficients. For models not following linear patterns, however, higher-order regression models should be used to represent the curvature in the relationship between the response and input factors. The following equation represents a second-order (quadratic) model, which consists of second-order effects of the input variables, as well as the interaction between them (Eq. (6.2)): 𝑦𝑖 = 𝛽0 +∑𝛽𝑖𝑋𝑖𝑖+∑𝛽𝑖𝑖𝑋𝑖2𝑖+∑∑𝛽𝑖𝑗𝑋𝑖𝑋𝑗𝑗>𝑖𝑖+∈ (6.2) The goal in such an analysis is to find constant coefficients, βi’s. This can be done in a number of different ways. One popular way is to find βi’s by minimizing the random error using the least-square method (Montgomery, 2012). That is, minimizing the sum of squares of the vertical distances between the predicted response values and actual response values, ?̂?𝑖 and yi, respectively. This is also known as minimizing the random error being ∈𝑖= 𝑦𝑖 − ?̂?𝑖 and sum of squares of errors being ∑ ∈𝑖2𝑖 . This term accounts for experiment inconsistencies in physical experiments. In numerical simulations, however, this is attributed to simplifying assumptions and variables that are assumed negligible enough to be excluded from being considered in the simulation. This error can also account for the interactions between the variables. Another way to minimize this is to include more interaction terms in the design-of-experiment analysis (Modirzadeh et al., 2012). In the following sections, the input design factors and response variables are described, and the results of the RSM are given, including the regression models found and the response surfaces for considered factors. 6.3. Method, factors and response variables In order to come up with a good sample domain, it is required to use a design-of-experiment method. A sample domain is known as a combination of input factors in the experiment. An experiment is referred to as computing a target parameter (usually known as a response) using a 88 certain input configuration. In most cases, an effective selection of the sample domain increases the efficiency of the experiment. In experimental results, replications of the same input combinations might not yield the same results due to instrumental, observational and environmental errors. In numerical simulations, like the case of computer modeling, however, such errors do not exist, and the errors are more of a theoretical type. Replications of the same simulation in such a scenario result in similar findings. Therefore, not all designs suit all types of experiments. Factorial design and composite design are examples of design types that better suit the former and I-optimal design is an example of design types that are more suitable for the latter (Montgomery, 2012; Simpson et al., 2001). According to the finite element and sensitivity analyses conducted in Chapter 4 and Chapter 5, respectively, all studied factors found to be significantly affecting the response characteristics of the studied I-joists. The factors, along with their respective levels, are presented in Table 6-1 and Table 6-2 below. Table 6-1. Input factors for joists with web holes Factor Symbol No. of Levels Levels Units Opening Diameter A 4 0, 4, 6 and 8 in. Opening Location (for 12-ft joists) B 6 0, 1, 2, 3, 4 and 5 ft Opening Location (for 20-ft joists) B 10 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 ft Table 6-2. Input factors for joists with flange notches Factor Symbol No. of Levels Levels Units Notch Depth A 4 0, 4, 6 and 8 in. Notch Length B 4 0, 4, 6 and 8 in. Notch Location (for 12-ft joists) C 6 0, 1, 2, 3, 4 and 5 ft Notch Location (for 20-ft joists) C 10 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 ft The so-called overall structural stiffness of the joist (K) was chosen as the main response characteristic. This can then be interpreted as load at any deflection by multiplying the stiffness by the deflection, thanks to the linearity of the load-deflection curves. The span/180 deflection (Δ180) is used to calculate the load at this deflection. In most of studied cases, this can be used as the 89 maximum allowable load for the Δ180 serviceability criteria. However, since this study has not considered material failure, this load shall be used with caution and only as an estimate where Δ180 is a proper deflection limit for the joist. Figure 6-1 illustrates these factors. Figure 6-1.General load-deflection characteristic curve for I-joists Design-Expert software was used to generate the list of factor combinations for the simulations (Stat-Ease Inc, 2018). A total of 64 and 256 design models for I-joists with web hole and flange notch, respectively, were numerically modeled and analyzed. These design models are listed in Table 6-3 for the former and in Table 6-4 and Table 6-5 for the latter. 90 Table 6-3. RSM designs for 12- and 20-ft I-joists with web hole L = 12 ft Factor L = 20 ft Factor A B K P A B K P Model in. ft kN/mm kN Model in. ft kN/mm kN 1 0.0 0.0 0.8662 17.6009 1 0.0 0.0 0.2517 8.5247 2 0.0 1.0 0.8662 17.6009 2 0.0 1.0 0.2517 8.5247 3 0.0 2.0 0.8662 17.6009 3 0.0 2.0 0.2517 8.5247 4 0.0 3.0 0.8662 17.6009 4 0.0 3.0 0.2517 8.5247 5 0.0 4.0 0.8662 17.6009 5 0.0 4.0 0.2517 8.5247 6 0.0 5.0 0.8662 17.6009 6 0.0 5.0 0.2517 8.5247 7 4.0 0.0 0.8669 17.6154 7 0.0 6.0 0.2517 8.5247 8 4.0 1.0 0.8534 17.3405 8 0.0 7.0 0.2517 8.5247 9 4.0 2.0 0.8531 17.3346 9 0.0 8.0 0.2517 8.5247 10 4.0 3.0 0.8512 17.2962 10 0.0 9.0 0.2517 8.5247 11 4.0 4.0 0.8521 17.3149 11 4.0 0.0 0.2501 8.4689 12 4.0 5.0 0.8518 17.3093 12 4.0 1.0 0.2493 8.4415 13 6.0 0.0 0.8659 17.5944 13 4.0 2.0 0.2493 8.4429 14 6.0 1.0 0.8402 17.0728 14 4.0 3.0 0.2492 8.4387 15 6.0 2.0 0.8379 17.0263 15 4.0 4.0 0.2503 8.4763 16 6.0 3.0 0.8363 16.9931 16 4.0 5.0 0.2502 8.4734 17 6.0 4.0 0.8372 17.0124 17 4.0 6.0 0.2462 8.3367 18 6.0 5.0 0.8404 17.0772 18 4.0 7.0 0.2504 8.4810 19 8.0 0.0 0.8626 17.5283 19 4.0 8.0 0.2497 8.4548 20 8.0 1.0 0.8167 16.5962 20 4.0 9.0 0.2502 8.4722 21 8.0 2.0 0.8094 16.4463 21 6.0 0.0 0.2498 8.4607 22 8.0 3.0 0.8082 16.4224 22 6.0 1.0 0.2480 8.3986 23 8.0 4.0 0.8094 16.4469 23 6.0 2.0 0.2479 8.3946 24 8.0 5.0 0.8171 16.6028 24 6.0 3.0 0.2478 8.3932 25 6.0 4.0 0.2490 8.4313 26 6.0 5.0 0.2498 8.4593 27 6.0 6.0 0.2486 8.4189 28 6.0 7.0 0.2491 8.4363 29 6.0 8.0 0.2483 8.4103 30 6.0 9.0 0.2489 8.4308 31 8.0 0.0 0.2492 8.4396 32 8.0 1.0 0.2454 8.3094 33 8.0 2.0 0.2450 8.2960 34 8.0 3.0 0.2450 8.2968 35 8.0 4.0 0.2462 8.3367 36 8.0 5.0 0.2471 8.3674 37 8.0 6.0 0.2460 8.3326 38 8.0 7.0 0.2465 8.3482 39 8.0 8.0 0.2458 8.3231 40 8.0 9.0 0.2471 8.3698 91 Table 6-4. RSM designs for 12-ft I-joists with flange notch Factor Factor A B C K P A B C K P Model in. in. ft kN/mm kN Model in. in. ft kN/mm kN 1 0.0 0.0 0.0 0.8662 17.6009 49 6.0 0.0 0.0 0.8662 17.6009 2 0.0 0.0 1.0 0.8662 17.6009 50 6.0 0.0 1.0 0.8662 17.6009 3 0.0 0.0 2.0 0.8662 17.6009 51 6.0 0.0 2.0 0.8662 17.6009 4 0.0 0.0 3.0 0.8662 17.6009 52 6.0 0.0 3.0 0.8662 17.6009 5 0.0 0.0 4.0 0.8662 17.6009 53 6.0 0.0 4.0 0.8662 17.6009 6 0.0 0.0 5.0 0.8662 17.6009 54 6.0 0.0 5.0 0.8662 17.6009 7 0.0 4.0 0.0 0.8662 17.6009 55 6.0 4.0 0.0 0.1361 2.7652 8 0.0 4.0 1.0 0.8662 17.6009 56 6.0 4.0 1.0 0.1460 2.9671 9 0.0 4.0 2.0 0.8662 17.6009 57 6.0 4.0 2.0 0.2315 4.7050 10 0.0 4.0 3.0 0.8662 17.6009 58 6.0 4.0 3.0 0.3588 7.2913 11 0.0 4.0 4.0 0.8662 17.6009 59 6.0 4.0 4.0 0.5615 11.4096 12 0.0 4.0 5.0 0.8662 17.6009 60 6.0 4.0 5.0 0.7767 15.7824 13 0.0 6.0 0.0 0.8662 17.6009 61 6.0 6.0 0.0 0.1318 2.6772 14 0.0 6.0 1.0 0.8662 17.6009 62 6.0 6.0 1.0 0.1427 2.8989 15 0.0 6.0 2.0 0.8662 17.6009 63 6.0 6.0 2.0 0.2229 4.5294 16 0.0 6.0 3.0 0.8662 17.6009 64 6.0 6.0 3.0 0.3484 7.0790 17 0.0 6.0 4.0 0.8662 17.6009 65 6.0 6.0 4.0 0.5456 11.0857 18 0.0 6.0 5.0 0.8662 17.6009 66 6.0 6.0 5.0 0.7526 15.2918 19 0.0 8.0 0.0 0.8662 17.6009 67 6.0 8.0 0.0 0.1203 2.4451 20 0.0 8.0 1.0 0.8662 17.6009 68 6.0 8.0 1.0 0.1397 2.8378 21 0.0 8.0 2.0 0.8662 17.6009 69 6.0 8.0 2.0 0.2132 4.3327 22 0.0 8.0 3.0 0.8662 17.6009 70 6.0 8.0 3.0 0.3365 6.8383 23 0.0 8.0 4.0 0.8662 17.6009 71 6.0 8.0 4.0 0.5293 10.7559 24 0.0 8.0 5.0 0.8662 17.6009 72 6.0 8.0 5.0 0.7414 15.0660 25 4.0 0.0 0.0 0.8662 17.6009 73 8.0 0.0 0.0 0.8662 17.6009 26 4.0 0.0 1.0 0.8662 17.6009 74 8.0 0.0 1.0 0.8662 17.6009 27 4.0 0.0 2.0 0.8662 17.6009 75 8.0 0.0 2.0 0.8662 17.6009 28 4.0 0.0 3.0 0.8662 17.6009 76 8.0 0.0 3.0 0.8662 17.6009 29 4.0 0.0 4.0 0.8662 17.6009 77 8.0 0.0 4.0 0.8662 17.6009 30 4.0 0.0 5.0 0.8662 17.6009 78 8.0 0.0 5.0 0.8662 17.6009 31 4.0 4.0 0.0 0.2232 4.5363 79 8.0 4.0 0.0 0.0714 1.4504 32 4.0 4.0 1.0 0.2455 4.9894 80 8.0 4.0 1.0 0.0812 1.6508 33 4.0 4.0 2.0 0.3679 7.4749 81 8.0 4.0 2.0 0.1375 2.7950 34 4.0 4.0 3.0 0.5059 10.2794 82 8.0 4.0 3.0 0.2293 4.6586 35 4.0 4.0 4.0 0.6763 13.7425 83 8.0 4.0 4.0 0.4278 8.6919 36 4.0 4.0 5.0 0.8168 16.5965 84 8.0 4.0 5.0 0.7173 14.5765 37 4.0 6.0 0.0 0.2190 4.4509 85 8.0 6.0 0.0 0.0715 1.4526 38 4.0 6.0 1.0 0.2407 4.8917 86 8.0 6.0 1.0 0.0799 1.6240 39 4.0 6.0 2.0 0.3544 7.2015 87 8.0 6.0 2.0 0.1322 2.6868 40 4.0 6.0 3.0 0.4931 10.0194 88 8.0 6.0 3.0 0.2248 4.5671 41 4.0 6.0 4.0 0.6635 13.4829 89 8.0 6.0 4.0 0.4133 8.3982 42 4.0 6.0 5.0 0.8014 16.2841 90 8.0 6.0 5.0 0.6800 13.8178 43 4.0 8.0 0.0 0.2108 4.2825 91 8.0 8.0 0.0 0.0663 1.3478 44 4.0 8.0 1.0 0.2374 4.8233 92 8.0 8.0 1.0 0.0780 1.5850 45 4.0 8.0 2.0 0.3412 6.9322 93 8.0 8.0 2.0 0.1252 2.5444 46 4.0 8.0 3.0 0.4765 9.6822 94 8.0 8.0 3.0 0.2152 4.3729 47 4.0 8.0 4.0 0.6500 13.2080 95 8.0 8.0 4.0 0.3992 8.1117 48 4.0 8.0 5.0 0.7970 16.1941 96 8.0 8.0 5.0 0.6576 13.3629 92 Table 6-5. RSM designs for 20-ft I-joists with flange notch Factor Factor Factor A B C K P A B C K P A B C K P Model in. in. ft kN/mm kN Model in. in. ft kN/mm kN Model in. in. ft kN/mm kN 1 0 0 0 0.2517 8.5247 55 4 4 4 0.1321 4.4751 109 6 6 8 0.2231 7.5559 2 0 0 1 0.2517 8.5247 56 4 4 5 0.1615 5.4685 110 6 6 9 0.2423 8.2072 3 0 0 2 0.2517 8.5247 57 4 4 6 0.1848 6.2577 111 6 8 0 0.0393 1.3303 4 0 0 3 0.2517 8.5247 58 4 4 7 0.2145 7.2657 112 6 8 1 0.0426 1.4432 5 0 0 4 0.2517 8.5247 59 4 4 8 0.2370 8.0250 113 6 8 2 0.0537 1.8200 6 0 0 5 0.2517 8.5247 60 4 4 9 0.2477 8.3898 114 6 8 3 0.0678 2.2972 7 0 0 6 0.2517 8.5247 61 4 6 0 0.0685 2.3214 115 6 8 4 0.0863 2.9230 8 0 0 7 0.2517 8.5247 62 4 6 1 0.0728 2.4658 116 6 8 5 0.1136 3.8487 9 0 0 8 0.2517 8.5247 63 4 6 2 0.0923 3.1256 117 6 8 6 0.1457 4.9359 10 0 0 9 0.2517 8.5247 64 4 6 3 0.1094 3.7063 118 6 8 7 0.1844 6.2463 11 0 4 0 0.2517 8.5247 65 4 6 4 0.1332 4.5104 119 6 8 8 0.2214 7.4977 12 0 4 1 0.2517 8.5247 66 4 6 5 0.1574 5.3322 120 6 8 9 0.2403 8.1376 13 0 4 2 0.2517 8.5247 67 4 6 6 0.1833 6.2077 121 8 0 0 0.2517 8.5247 14 0 4 3 0.2517 8.5247 68 4 6 7 0.2138 7.2419 122 8 0 1 0.2517 8.5247 15 0 4 4 0.2517 8.5247 69 4 6 8 0.2355 7.9742 123 8 0 2 0.2517 8.5247 16 0 4 5 0.2517 8.5247 70 4 6 9 0.2458 8.3231 124 8 0 3 0.2517 8.5247 17 0 4 6 0.2517 8.5247 71 4 8 0 0.0668 2.2620 125 8 0 4 0.2517 8.5247 18 0 4 7 0.2517 8.5247 72 4 8 1 0.0723 2.4496 126 8 0 5 0.2517 8.5247 19 0 4 8 0.2517 8.5247 73 4 8 2 0.0882 2.9865 127 8 0 6 0.2517 8.5247 20 0 4 9 0.2517 8.5247 74 4 8 3 0.1061 3.5943 128 8 0 7 0.2517 8.5247 21 0 6 0 0.2517 8.5247 75 4 8 4 0.1272 4.3069 129 8 0 8 0.2517 8.5247 22 0 6 1 0.2517 8.5247 76 4 8 5 0.1546 5.2364 130 8 0 9 0.2517 8.5247 23 0 6 2 0.2517 8.5247 77 4 8 6 0.1828 6.1915 131 8 4 0 0.0235 0.7960 24 0 6 3 0.2517 8.5247 78 4 8 7 0.2111 7.1502 132 8 4 1 0.0248 0.8393 25 0 6 4 0.2517 8.5247 79 4 8 8 0.2346 7.9435 133 8 4 2 0.0328 1.1119 26 0 6 5 0.2517 8.5247 80 4 8 9 0.2446 8.2851 134 8 4 3 0.0426 1.4427 27 0 6 6 0.2517 8.5247 81 6 0 0 0.2517 8.5247 135 8 4 4 0.0563 1.9055 28 0 6 7 0.2517 8.5247 82 6 0 1 0.2517 8.5247 136 8 4 5 0.0796 2.6956 29 0 6 8 0.2517 8.5247 83 6 0 2 0.2517 8.5247 137 8 4 6 0.1096 3.7117 30 0 6 9 0.2517 8.5247 84 6 0 3 0.2517 8.5247 138 8 4 7 0.1571 5.3189 31 0 8 0 0.2517 8.5247 85 6 0 4 0.2517 8.5247 139 8 4 8 0.2086 7.0632 32 0 8 1 0.2517 8.5247 86 6 0 5 0.2517 8.5247 140 8 4 9 0.2416 8.1833 33 0 8 2 0.2517 8.5247 87 6 0 6 0.2517 8.5247 141 8 6 0 0.0227 0.7682 34 0 8 3 0.2517 8.5247 88 6 0 7 0.2517 8.5247 142 8 6 1 0.0243 0.8214 35 0 8 4 0.2517 8.5247 89 6 0 8 0.2517 8.5247 143 8 6 2 0.0325 1.1017 36 0 8 5 0.2517 8.5247 90 6 0 9 0.2517 8.5247 144 8 6 3 0.0414 1.4037 37 0 8 6 0.2517 8.5247 91 6 4 0 0.0419 1.4196 145 8 6 4 0.0572 1.9360 38 0 8 7 0.2517 8.5247 92 6 4 1 0.0438 1.4833 146 8 6 5 0.0779 2.6372 39 0 8 8 0.2517 8.5247 93 6 4 2 0.0581 1.9686 147 8 6 6 0.1080 3.6561 40 0 8 9 0.2517 8.5247 94 6 4 3 0.0718 2.4333 148 8 6 7 0.1542 5.2207 41 4 0 0 0.2517 8.5247 95 6 4 4 0.0916 3.1028 149 8 6 8 0.2045 6.9253 42 4 0 1 0.2517 8.5247 96 6 4 5 0.1195 4.0482 150 8 6 9 0.2366 8.0142 43 4 0 2 0.2517 8.5247 97 6 4 6 0.1489 5.0434 151 8 8 0 0.0213 0.7224 44 4 0 3 0.2517 8.5247 98 6 4 7 0.1902 6.4403 152 8 8 1 0.0237 0.8034 45 4 0 4 0.2517 8.5247 99 6 4 8 0.2261 7.6564 153 8 8 2 0.0304 1.0295 46 4 0 5 0.2517 8.5247 100 6 4 9 0.2457 8.3199 154 8 8 3 0.0400 1.3560 47 4 0 6 0.2517 8.5247 101 6 6 0 0.0407 1.3776 155 8 8 4 0.0530 1.7948 48 4 0 7 0.2517 8.5247 102 6 6 1 0.0432 1.4622 156 8 8 5 0.0758 2.5665 49 4 0 8 0.2517 8.5247 103 6 6 2 0.0567 1.9193 157 8 8 6 0.1069 3.6207 50 4 0 9 0.2517 8.5247 104 6 6 3 0.0700 2.3722 158 8 8 7 0.1511 5.1163 51 4 4 0 0.0733 2.4836 105 6 6 4 0.0917 3.1040 159 8 8 8 0.2017 6.8315 52 4 4 1 0.0759 2.5690 106 6 6 5 0.1165 3.9448 160 8 8 9 0.2324 7.8713 53 4 4 2 0.0930 3.1501 107 6 6 6 0.1469 4.9735 54 4 4 3 0.1114 3.7711 108 6 6 7 0.1877 6.3578 93 6.4. Results and response surface metamodels Using the design input models mentioned in the previous section, a total of 320 models were analyzed. The response factors of the models are mentioned in Table 6-3 to Table 6-5. Load-deflection response curves of a selection of the models are plotted in Figure 5-3 and Figure 5-4 for joists with web hole and Figure 5-8 and Figure 5-9 for those with flange notch. It is observed, from the tables and figures, that the effect of the opening on the stiffness of the joist is highly dependent on the location and size of the opening. For instance, in the case of joists with web holes, it is seen that the effect of the location of the notch for 4-in. diameter holes is negligible for both 12- and 20-ft joists. However, the location effect is highlighted more as the hole grows in size. On the other hand, the size effect is shown to be small for web holes at the mid-span of the beam. This effect starts to increase as the hole is moved away from the mid-span point of the beam. A similar manner is generally observed in both beam lengths considered. Table 6-6 and Table 6-7 summarize the capacity reduction percentage for 12- and 20-ft beams with web holes, respectively. In contrast to joists with web holes, joists with flange notches were observed to be sensitive to the location of the notch for all sizes of notches. Generally, notches closer to the mid-span of the beam caused a larger decrease in the stiffness of the beam. However, the impact of those closer to supports was directly dependent on the size of the notch. A relatively similar percentage of overall capacity reduction was observed for both 12- and 20-ft long joists. We examined the effect of the size of the notch by varying the length and depth of the notch independently. The load at span/180 deflection and structural stiffness of the beam were meaningfully seen to be inversely affected by the area of the notch. Meanwhile, the depth of the notch was seen to be more destructive than the length of the notch. A continuous decrease in the capacity was seen, as the length of the notch was increased. However, the same amount of change in notch’s depth was seen to cause a sudden capacity decline. As the notch is moved closer to the supports, load-deflection curves show that this decline decreases, and the capacity is less affected by the increased notch depth. At the same time, the effect of the length of the notch was slightly increased as the notch is moved away from the beam’s mid-point. This trend was observed in both span lengths studied. Table 6-8 and Table 6-9 summarize the capacity reduction percentage for 12- and 20-ft beams with flange notches, respectively. 94 Table 6-6. Capacity reduction percentage for 12-ft I-joists with web hole Opening Location*, ft (mm) Opening Size, in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.08 0.04 0.41 1 (304.8) 1.48 3.00 5.71 2 (609.6) 1.51 3.26 6.56 3 (914.4) 1.73 3.45 6.70 4 (1219.2) 1.62 3.34 6.56 5 (1524.0) 1.66 2.98 5.67 * Measured from mid-span of the beam Table 6-7. Capacity reduction percentage for 20-ft I-joists with web hole Opening Location*, ft (mm) Opening Size, in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.66 0.75 1.00 1 (304.8) 0.98 1.48 2.53 2 (609.6) 0.96 1.53 2.68 3 (914.4) 1.01 1.54 2.67 4 (1219.2) 0.57 1.10 2.21 5 (1524.0) 0.60 0.77 1.85 6 (1828.8) 2.21 1.24 2.25 7 (2133.6) 0.51 1.04 2.07 8 (2438.4) 0.82 1.34 2.36 9 (2743.2) 0.62 1.10 1.82 * Measured from mid-span of the beam Table 6-8. Capacity reduction percentage for 12-ft I-joists with flange notch Notch Location*, ft (mm) D*** ↓ L** → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 74.23 74.71 75.67 84.29 84.79 86.11 91.76 91.75 92.34 1 (304.8) 71.65 72.21 72.60 83.14 83.53 83.88 90.62 90.77 90.99 2 (609.6) 57.53 59.08 60.61 73.27 74.27 75.38 84.12 84.73 85.54 3 (914.4) 41.60 43.07 44.99 58.57 59.78 61.15 73.53 74.05 75.16 4 (1219.2) 21.92 23.40 24.96 35.18 37.02 38.89 50.62 52.29 53.91 5 (1524.0) 5.71 7.48 7.99 10.33 13.12 14.40 17.18 21.49 24.08 * Measured from mid-span of the beam ** Notch length, in. (mm) *** Notch depth, in. (mm) 95 Table 6-9. Capacity reduction percentage for 20-ft I-joists with flange notch Notch Location*, ft (mm) D*** ↓ L** → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 70.87 72.77 73.46 83.35 83.84 84.39 90.66 90.99 91.53 1 (304.8) 69.86 71.07 71.27 82.60 82.85 83.07 90.15 90.36 90.58 2 (609.6) 63.05 63.33 64.97 76.91 77.49 78.65 86.96 87.08 87.92 3 (914.4) 55.76 56.52 57.84 71.46 72.17 73.05 83.08 83.53 84.09 4 (1219.2) 47.50 47.09 49.48 63.60 63.59 65.71 77.65 77.29 78.95 5 (1524.0) 35.85 37.45 38.57 52.51 53.72 54.85 68.38 69.06 69.89 6 (1828.8) 26.59 27.18 27.37 40.84 41.66 42.10 56.46 57.11 57.53 7 (2133.6) 14.77 15.05 16.12 24.45 25.42 26.73 37.61 38.76 39.98 8 (2438.4) 5.86 6.46 6.82 10.19 11.36 12.05 17.14 18.76 19.86 9 (2743.2) 1.58 2.37 2.81 2.40 3.73 4.54 4.01 5.99 7.67 * Measured from mid-span of the beam ** Notch length, in. (mm) *** Notch depth, in. (mm) Using the analysis results presented earlier in this chapter and the previous chapter, Design-Expert software was used to generate the response surface curves and obtain regression equations. The regression equations, also called metamodels, are generated based on least-square regression analysis and are used to predict the response variable for input variables that have not existed in the design inputs. The main response variable in this study is chosen to be the overall structural stiffness of the I-joist, which can be used to calculate the load at span/180 deflection. The next sections present the explained metamodels. 6.4.1. Metamodels for I-joists with web hole In order to find metamodels for the response variable of the models with a web hole, a number of different types of polynomial equations were tested to find the most suitable model. The tested models include (from low to high polynomial order): 1. Two-factor interaction model (2FI) 2. Quadratic model – including second-order effects 3. Cubic model – including third-order effects 4. Quartic model – including fourth-order effects Notations given in Table 6-3 were used in the generated metamodel equations to represent opening diameter, opening location and the response variables. For each model, ANOVA results were reviewed to assess the significance of each term in the polynomial. Even though a higher-order 96 polynomial is generally expected to result in a better fit, the analysis results and their diagnostics should carefully be reviewed to ensure the right model is being used. While watching the metamodels for accuracy, we tried to select the model with lower-order terms to keep the model as simple as possible. To this end, the terms with p-values of greater than 0.1 were excluded and, with each attempt, R2 and adjusted-R2 were closely watched. Table 6-10 is made to draw a comparison between the tested models in terms of the accuracy of prediction of the response variable for 12- and 20-ft long joists. Table 6-10. R2 and adjusted-R2 values for tested models for I-joists with web holes L = 12 ft L = 20 ft Model R2 Adjusted-R2 Model R2 Adjusted-R2 Quartic 0.9961 0.9910 Quartic 0.8901 0.8351 Cubic 0.9765 0.9614 Cubic 0.8499 0.8048 Quadratic 0.9015 0.8741 Quadratic 0.8152 0.7880 2FI 0.7445 0.7062 2FI 0.7779 0.7594 Modified (reduced cubic) 0.9169 0.8875 Modified (reduced cubic) 0.8256 0.7939 R2 is used as an indication of how close the metamodel can predict the response to the actual values. Adjusted-R2, in the meantime, takes into account the number of parameters in the model relative to the number of points in the design. It helps to make sure the model is better in terms of prediction, rather than just because of its high order. It is observed in the table above that higher-order models are returning higher values of R2. For both span lengths, in the quartic model (including fourth-order terms), despite the high R2 values, some terms were aliased with other terms or were not suitable to be included in the regression calculations. Further, it would result in too many terms (14) in the metamodel, which was not desired. The two-factor interaction and cubic models, on the other hand, resulted in relatively lower R2 values. In an attempt, a modified cubic model in which higher-order terms with p-values of higher than 0.1 were removed was also tested. Since the difference it made was not significant, the modified model was chosen to be used to generate the predictive equations. Eq. (6.3) and (6.4) below show the fitted response surface equations for the overall structural stiffness and load at the maximum deflection of 12-ft joists with web hole using a reduced cubic model: 97 𝐾 = 874.987 + 2.658𝐴 − 21.839𝐵 − 0.853𝐴𝐵 − 0.767𝐴2 + 8.106𝐵2− 0.765𝐵3 𝑁/𝑚𝑚 (6.3) 𝑃180 = 17.780 + 0.054𝐴 − 0.444𝐵 − 0.017𝐴𝐵 − 0.016𝐴2 + 0.165𝐵2− 0.016𝐵3 𝑘𝑁 (6.4) Eq. (6.5) and (6.6) below are predictive equations for the same response characteristics for 20-ft joists with web hole: 𝐾 = 252.382 − 0.286𝐴 − 0.891𝐵 − 0.001𝐴𝐵 − 0.045𝐴2 + 0.203𝐵2− 0.012𝐵3 𝑁/𝑚𝑚 (6.5) 𝑃180 = 8.547 − 0.010𝐴 − 0.030𝐵 − 0.002𝐴2 + 0.007𝐵2 𝑘𝑁 (6.6) In the above equations, A is the opening diameter in inches and B denotes the location of the opening measured from the mid-span of the joist in feet. Table 6-11 summarizes the ANOVA results for this analysis. In this table, A represents the opening diameter and B represents the opening location. Table 6-11. Summary of ANOVA analysis for tested models for I-joists with web holes L = 12 ft L = 20 ft Source Sum of Squares df Mean Square F-value p-value Source Sum of Squares df Mean Square F-value p-value Model 0.009 6 0.0015 31.25 < 0.0001 Model 0.0002 6 0 26.03 < 0.0001 A 0.0065 1 0.0065 135.26 < 0.0001 A 0.0001 1 0.0001 150.97 < 0.0001 B 7.7E-06 1 7.7E-06 0.1601 0.694 B 1.6E-06 1 1.6E-06 1.68 0.2044 AB 0.0004 1 0.0004 9.23 0.0074 AB 5.8E-09 1 5.8E-09 0.006 0.9387 A² 0.0007 1 0.0007 14.74 0.0013 A² 4.0E-06 1 4.0E-06 4.15 0.0498 B² 0.0008 1 0.0008 17.36 0.0006 B² 2.8E-06 1 2.8E-06 2.91 0.0976 B³ 0.0002 1 0.0002 3.15 0.094 B³ 1.9E-06 1 1.9E-06 1.96 0.1705 Residual 0.0008 17 0 Residual 0 33 9.6E-07 Cor Total 0.0099 23 Cor Total 0.0002 39 Figure 6-2 shows the 3-D response surface plots of K for 12- and 20-ft I-joists with a web hole. In these plots, the surface depicts the prediction for the defined range of inputs. Dark red dots show the actual values that are higher than the predicted value, and pink dots show the actual values that 98 are lower than their predicted value. As seen in the plots, the predictive equations are well capable of predicting the design input responses. It can be observed that the effect of smaller holes can be neglected or potentially be compromised with minor retrofits around them. The effect of bigger ones, however, is dependent on the location of the hole. It is also observed that, for shorter joists, the effect of the hole near mid-span is minimal regardless of the size of the hole – a conclusion than can also be made from actual input points of the study. For longer joists, however, the effect of the size of the opening is more dominant. The ANOVA analysis also suggests that the interaction between opening size and location is less for longer joists. Figure 6-2. Response surface plots for 12-ft joists (left) and 20-ft joists (right) with web holes 6.4.2. Metamodels for I-joists with flange notch This section presents the metamodels constructed for I-joists with flange notches of different sizes and locations. Notations given in Table 6-3 and Table 6-4 were used in the generated metamodel equations to represent notch depth, notch length, notch location, and the response variables. Similar to the approach used for those with web holes, we tested several polynomial models to find one with reasonable accuracy and yet less complicacy. Since there are three contributing factors in this series, the order of the analysis would highly affect the number of terms included in it. Table 6-12 lists the R2 and adjusted-R2 values for the tested models. In all four models, relatively high R2 values were found. In the quartic model, however, aliased terms were found which made the model unsuitable. The two-factor interaction model, which was the simplest model tested, resulted in a relatively low R2 value. 99 Table 6-12. R2 and adjusted-R2 values for tested models for I-joists with flange notches L = 12 ft L = 20 ft Model R2 Adjusted-R2 Model R2 Adjusted-R2 Quartic 0.9972 0.9957 Quartic 0.9967 0.9958 Cubic 0.9851 0.9814 Cubic 0.9836 0.9814 Quadratic 0.9159 0.9071 Quadratic 0.9122 0.9069 2FI 0.8374 0.8264 2FI 0.8481 0.8421 Modified (reduced cubic) 0.9726 0.9678 Modified (reduced cubic) 0.9729 0.9703 The quadratic and cubic models had acceptable R2 values. However, the R2 values of the former were meaningfully lower than those of the latter. As a result, the cubic model was chosen for further analysis. By looking at the ANOVA analysis of the cubic model, we noticed that some of the factors have higher p-values. This indicated that the influence of these terms is less than that of the others. We removed these terms from the model and re-analyzed it. The reduced cubic model in the table shows the results of this modified model. Since the resulted R2 values were close to the cubic model and this model had fewer terms, we chose it to generate the predictive equations. Eq. (6.7) and (6.8) below show the fitted response surface equations for the overall structural stiffness and load at the maximum deflection of 12-ft joists with flange notch using a reduced cubic model: 𝐾 = 1013.358 − 98.368𝐴 − 141.578𝐵 − 98.158𝐶 − 30.529𝐴𝐵 + 24.982𝐴𝐶+ 24.624𝐵𝐶 + 11.701𝐴2 + 30.948𝐵2 + 13.884𝐶2 + 1.940𝐴𝐵𝐶− 2.852𝐴2𝐶 + 2.043𝐴𝐵2 − 2.811𝐵2𝐶 − 1.837𝐵3 𝑁/𝑚𝑚 (6.7) 𝑃180 = 20.591 − 1.999𝐴 − 2.877𝐵 − 1.995𝐶 − 0.620𝐴𝐵 + 0.508𝐴𝐶 + 0.500𝐵𝐶+ 0.238𝐴2 + 0.629𝐵2 + 0.282𝐶2 + 0.039𝐴𝐵𝐶 − 0.058𝐴2𝐶+ 0.042𝐴𝐵2 − 0.057𝐵2𝐶 − 0.037𝐵3 𝑘𝑁 (6.8) Eq. (6.9) and (6.10) below are predictive equations for the same response characteristics for 20-ft joists with flange notch: 𝐾 = 292.943 − 25.154𝐴 − 40.234𝐵 − 14.633𝐶 − 8.526𝐴𝐵 + 3.846𝐴𝐶 + 4.352𝐵𝐶+ 2.907𝐴2 + 8.580𝐵2 + 1.091𝐶2 + 0.402𝐴𝐵𝐶 − 0.425𝐴2𝐶+ 0.538𝐴𝐵2 − 0.507𝐵2𝐶 − 0.487𝐵3 𝑁/𝑚𝑚 (6.9) 100 𝑃180 = 9.921 − 0.852𝐴 − 1.363𝐵 − 0.496𝐶 − 0.289𝐴𝐵 + 0.130𝐴𝐶 + 0.147𝐵𝐶+ 0.098𝐴2 + 0.291𝐵2 + 0.037𝐶2 + 0.014𝐴𝐵𝐶 − 0.014𝐴2𝐶+ 0.018𝐴𝐵2 − 0.017𝐵2𝐶 − 0.016𝐵3 𝑘𝑁 (6.10) In the above equations, A and B are the notch depth and length, respectively, in inches, and C denotes the location of the notch measured from the mid-span of the joist in feet. Table 6-13 below presents the ANOVA results for the tested joists with flange notch. Table 6-13. Summary of ANOVA analysis for tested models for I-joists with web holes L = 12 ft L = 20 ft Source Sum of Squares df Mean Square F-value p-value Source Sum of Squares df Mean Square F-value p-value Model 8.74 14 0.6245 205.26 < 0.0001 Model 1.1 14 0.0789 371.6 < 0.0001 A 1.49 1 1.49 491.13 < 0.0001 A 0.1705 1 0.1705 802.61 < 0.0001 B 0.0067 1 0.0067 2.22 0.1403 B 0.0006 1 0.0006 2.86 0.0931 C 0.7746 1 0.7746 254.58 < 0.0001 C 0.1145 1 0.1145 539.14 < 0.0001 AB 0.6313 1 0.6313 207.48 < 0.0001 AB 0.0703 1 0.0703 331.11 < 0.0001 AC 0.2315 1 0.2315 76.07 < 0.0001 AC 0.0467 1 0.0467 220.04 < 0.0001 BC 0.2299 1 0.2299 75.57 < 0.0001 BC 0.0402 1 0.0402 189.19 < 0.0001 A² 0.1009 1 0.1009 33.15 < 0.0001 A² 0.008 1 0.008 37.47 < 0.0001 B² 0.4041 1 0.4041 132.8 < 0.0001 B² 0.0453 1 0.0453 213.25 < 0.0001 C² 0.1151 1 0.1151 37.84 < 0.0001 C² 0.0101 1 0.0101 47.33 < 0.0001 ABC 0.0807 1 0.0807 26.51 < 0.0001 ABC 0.0163 1 0.0163 76.94 < 0.0001 A²C 0.1145 1 0.1145 37.64 < 0.0001 A²C 0.012 1 0.012 56.46 < 0.0001 AB² 0.1764 1 0.1764 57.96 < 0.0001 AB² 0.0204 1 0.0204 95.9 < 0.0001 B²C 0.1113 1 0.1113 36.57 < 0.0001 B²C 0.0171 1 0.0171 80.32 < 0.0001 B³ 0.0271 1 0.0271 8.92 0.0037 B³ 0.0032 1 0.0032 14.97 0.0002 Residual 0.2465 81 0.003 Residual 0.0308 145 0.0002 Cor Total 8.99 95 Cor Total 1.14 159 Analysis of the predictive equations in terms of coded factors shows that the interaction between notch dimension (either depth or length) and notch location is almost half of the interaction between notch dimensions for 12-ft joists. For 20-ft joists, however, the interactions are of the same order. It is also noticed that, for both span lengths, second-order effects of the notch location and third-order effects become less significant. Figure 6-3 shows the 3-D response surface plots of K for 12-ft I-joists with a flange notch, and Figure 6-4 plots those for 20-ft ones. In these plots, the surface depicts the prediction for the 101 defined range of inputs. Since there are three input parameters involved, in each plot, the predicted value is shown for a pair of parameters and is average over the third. The plots show the predicted values in a good agreement with the design input responses. Figure 6-3. Response surface plots for 12-ft I-joists with flange notches, showing the effects of notch depth and length (top left), notch location and depth (top right) and notch location and length (bottom) Unlike web openings discussed in the previous section, flange cuts are found to have much higher impacts on the stiffness of the joists, and almost no flange cut can be easily neglected. Supporting the findings in previous chapters, notch size has a direct impact on decreasing the stiffness of the joist. As well, the location plays a significant role. Notches closer to the supports have less impact on the stiffness. However, those at the mid-span decrease the stiffness the most. 102 Figure 6-4. Response surface plots for 20-ft I-joists with flange notches, showing the effects of notch depth and length (top left), notch location and depth (top right) and notch location and length (bottom) 6.5. Verification of the response surface models High R2 and adjusted-R2 values reported in the previous sections convey the impression that the suggested predictive equations can be used to obtain the response characteristics with a good accuracy. However, considering R2 alone may not always be sufficient. In order to verify the applicability of the generated equations, more diagnostic measures are looked at in this section. 103 Even though increasing the order of the model (using higher-order polynomials) leads to an increased R2, it might cause an overfitted model, in which the data random noise is also captured. The adjusted-R2 is R2 that is adjusted for the order of the model being used, meaning the number of predictors in it. In Table 6-10 and Table 6-12, it is observed that both R2 values increase as the order of the model is increased. This confirms that the model is better not only due to the higher number of predictors in it, but also because it provides a better fit. Another measure of the acceptancy of the model is predicted-R2 which helps us to determine how well the fitted model can predict the output for new input data sets. This is a key measurement to prevent a model from being overfitted. The value of predicted-R2 is always lower than the adjusted-R2. A difference of 0.2 or less between the two indicates that the adjusted- and predicted-R2 are in reasonable agreement and the model is acceptable. If a sudden drop is observed in this difference, it indicates that there are more-than-enough terms in the model. In addition to this factor, the adequacy of the precision is assessed using the signal-to-noise ratio. This compares the range of the predicted values at the design points to the average prediction error. A ratio of four or higher shows an adequate precision. Table 6-14 lists the adjusted-R2, predicted-R2, the difference between the two, and the signal-to-noise ratio (adequate precision) for the used models. It is observed that the mentioned factors are in acceptable ranges, which helps to confirm the acceptability of the models. Table 6-14. Fit measurements of the suggested prediction models Series R2 Adjusted R2 Predicted R2 Δ* Adequate Precision Web Hole (12-ft) 0.9169 0.8875 0.7498 0.1377 17.1013 Web Hole (20-ft) 0.8256 0.7939 0.7389 0.0550 15.4775 Flange Notch (12-ft) 0.9726 0.9678 0.9542 0.0136 48.0230 Flange Notch (20-ft) 0.9729 0.9703 0.9642 0.0061 68.9094 *|Adjusted-R2 – Predicted-R2| In addition to the fit factors discussed above, we assessed other diagnostic measures including the normal probability plots and the residual plots. The normal probability plot is a measure to help identify any unwanted patterns in the residuals. It is used to confirm that the residuals follow a normal distribution. Figure 6-5 shows the normal probability plot of the used models. Since they reasonably follow a straight line and no specific pattern is observed, it depicts that an appropriate transformation of the response has been used. 104 Figure 6-5. Normal probability plots of the residuals for: (a) 12-ft I-joists with web holes, (b) 20-ft I-joists with web holes, (c) 12-ft I-joists with flange notches, and (d) 20-ft I-joists with flange notches Figure 6-6 to Figure 6-9 show residual plots of the used models, generated by Design-Expert program. The first graph is a plot of the residuals against the predicted values sorted ascending. The second graph shows the residuals versus the run order. In physical experiments, this is used to identify any time-related issues affecting the measured responses. In computer simulations, it can account for any error propagations or potential data accumulation mistakes. The third graph plots predicted values against the actual values of the response variable. It helps to identify the data that are not well predicted by the model. Lastly, the graphs are followed by plots of the residuals against the input factor levels. The residual plots are expected to be in the form of random scatters. Any pattern in these plots may be an indication of a trend in the experiments that influences the response variables (Vining, 2010). 105 Figure 6-6. Diagnostic plots for 12-ft I-joists with web holes Figure 6-7. Diagnostic plots for 20-ft I-joists with web holes 106 Figure 6-8. Diagnostic plots for 12-ft I-joists with flange notches Figure 6-9. Diagnostic plots for 20-ft I-joists with flange notches 107 In addition to the diagnostic measures above, in order to further assess the accuracy of the predictive equations, twenty additional models (five per series) were randomly chosen and modeled. The input characteristics of these models were randomly generated, including the location and size of the opening – web hole diameter and flange notch dimensions, independently. Their corresponding finite element models were made, and the obtained response characteristics were compared against those resulted from the predictive equations. Table 6-15 lists the verification models for I-joists with a web hole and Table 6-16 lists those for I-joists with a flange notch. Table 6-15. Verification models for joists with web holes Model A: Opening Diameter B: Opening Location Stiffness (Actual) Stiffness (Predicted) 12-3.92-D2.93 2.93 3.92 0.862 0.859 12-2.41-D6.73 6.73 2.41 0.833 0.828 12-1.07-D5.23 5.23 1.07 0.847 0.848 12-3.90-D4.21 4.21 3.90 0.859 0.851 12-3.65-D6.77 6.77 3.65 0.832 0.828 20-4.51-D2.50 2.50 4.51 0.253 0.250 20-5.23-D7.06 7.06 5.23 0.248 0.247 20-6.40-D4.67 4.67 6.40 0.250 0.249 20-4.61-D3.83 3.83 4.61 0.251 0.250 20-5.82-D5.61 5.61 5.82 0.251 0.249 Table 6-16. Verification models for joists with flange notches Model A: Notch Depth B: Notch Length C: Notch Location Stiffness (Actual) Stiffness (Predicted) 12-3.13-4.57x6.46 6.46 4.57 3.13 0.345 0.373 12-4.00-5.89x6.09 6.09 5.89 4.00 0.544 0.533 12-1.09-6.99x4.83 4.83 6.99 1.09 0.201 0.235 12-3.86-3.13x2.44 2.44 3.13 3.86 0.733 0.709 12-2.34-7.96x3.44 3.44 7.96 2.34 0.422 0.465 20-3.49-4.72x4.32 4.32 4.72 3.49 0.114 0.120 20-5.75-6.93x7.60 7.60 6.93 5.75 0.108 0.122 20-8.62-7.76x3.01 3.01 7.76 8.62 0.246 0.249 20-6.95-2.14x6.26 6.26 2.14 6.95 0.188 0.188 20-2.25-3.63x2.29 2.29 3.63 2.25 0.137 0.155 108 Figure 6-10 and Figure 6-11 plot the actual stiffness versus predicted stiffness as the results of this comparison. It is observed that the R2 values for these comparisons are above 0.90. This suggests a sufficient level of accuracy for the proposed equations and state that the equations can be used to predict the response characteristics of new models with input parameters within the range of our study. Figure 6-10. Comparison of actual response versus predicted response for I-joists with web holes Figure 6-11. Comparison of actual response versus predicted response for I-joists with flange notches 6.6. Summary The key objective of this chapter has been to derive predictive equations for the response of timber I-joists with web holes or flange cuts. Two input factors were considered for joists with web hole: R² = 0.95750.8200.8330.8450.8580.8700.820 0.833 0.845 0.858 0.870Actual (kN/mm)Predicted (kN/mm)R² = 0.90140.2470.2490.2500.2520.2530.247 0.249 0.250 0.252Actual (kN/mm)Predicted (kN/mm)R² = 0.99240.1000.3000.5000.7000.100 0.300 0.500 0.700Actual (kN/mm)Predicted (kN/mm)R² = 0.98720.0500.1000.1500.2000.2500.3000.050 0.100 0.150 0.200 0.250 0.300Actual (kN/mm)Predicted (kN/mm)109 opening diameter and opening location. Three input factors were studied for joists with flange notch: notch length, notch depth, and notch location. These factors were chosen since they were found to considerably impact the bending capacity of the beam in previous chapters. The response of the joists was measured in terms of overall structural stiffness of the joist and its load at span/180 deflection. A total of 64 and 256 models were used to define the design input data for joists with web holes and flange notches, respectively. Response surface methodology was employed to develop the regression models used for predicting the response at new observations falling in the range of the input parameters. Predictive equations were developed for the structural stiffness and load at span/180 deflection of the joists. To evaluate the acceptability of the predicted models, diagnostic measures were looked at, including normal probability plots of the residuals, residual-versus-predicted value plots, residual-versus-run plots, and residual-versus-factor plots. The predicted-versus-actual response plots were also made and investigated. Finally, twenty additional runs were randomly chosen and performed to further confirm the suitability of the suggested predictive models. It was shown that the suggested predictive models were capable of predicting the response for new observations in the range of the input data, with an acceptable accuracy. These equations will be used in the following chapters to suggest a limit state design approach for timber I-joists with web openings and those with flange notches. 110 Chapter 7: PERFORMANCE EVALUATION OF TIMBER I-JOISTS WITH OPENINGS RETROFITTED WITH WEB AND FLANGE REINFORCEMENTS 7.1. General Previous chapters of this study have shown that the existence of openings has a significant impact on the structural stiffness of the I-joist. It was observed that an opening in the web of the joist can reduce its stiffness by 6.7%, depending on the size and location of the opening. A flange notch, on the other hand, was observed to have a highly more significant impact. The maximum stiffness reduction caused by a flange notch was seen to be as high as 92%, which happened in the case of notches closer to the mid-span of the joist. The Canadian wood design code has restricted the use of notching or cutting of the flanges of timber I-joists only to acceptable notching details supported by documented test data (Canadian Standards Association, 2019). Most manufacturers of prefabricated I-joists do not allow the use of such notches, due to the high capacity reduction they cause. In the case of web openings, the Canadian Standards Association has left the allowances for the existence of such openings to be set out by the manufacturers of I-joists. Even though not all manufacturers agree on web hole allowances, these openings are most often allowed up to a certain size and at locations far enough from the supports of the joist. This idea is also supported by the results of this research presented in previous chapters, as the joist is more vulnerable to openings closer to the supports. In this chapter, retrofitting of the I-joists around openings using OSB panels is examined. Finite element analysis is used for this purpose. New finite element models have been developed to simulate the retrofitting configurations. Up to three layers of retrofit is considered for the openings. For web holes, two layers of OSB panel fitted to the web of the joist are considered as the main configuration. For flange-notched joists, the retrofit is considered to span long enough across the notch, and from the bottom of the bottom flange to top of the top flange, in addition to the web 111 fillers. It should be noted that, in this research, the bond between the web or flange of the joist and the retrofitting pieces is assumed to be perfect and the effectiveness of the bonding means (nails, adhesive, etc.) has been beyond the scope of this study. Figure 7-1 shows schematics of the studied retrofit types for I-joists with web holes and flange notches. Figure 7-1. Schematics of I-joist retrofit types. Left: I-joists with web holes, and right: I-joists with flange notches. (a): Type I retrofit, (b): Type II retrofit 7.2. Retrofit details The studied retrofit techniques use pieces of OSB panels equivalent to those used for the web of the I-joist. Two retrofit arrangements are used for each opening type. The first arrangement thickens the web, covering it completely on one side, and spanning between inner edges of the flanges of the joist. Two layers of retrofitting pieces are used in this arrangement, which are meant to cross-cover the seams of the OSB pieces and maintain the integrity of the proposed retrofit. This arrangement is referred to as Type I or RFT12 in this text. The second arrangement adds to the first retrofit type a wider OSB panel which spans between the extreme top and bottom edges of the joist vertically, which is meant to connect the flanges for a more integrated moment-carrying capacity. This arrangement, in this text, is referred to as Type II or RFT123. Figure 7-2 illustrates the details of these two retrofit types. Figure 7-2. Retrofit dimensions and details 112 In the figure above, R denotes the ratio of the extent of the retrofit in lengthwise direction on either side of the opening. It is defined as the ratio of extension of the retrofit on each side of the opening to the dimension of the opening in the lengthwise direction. That is, a retrofit with R = 1 extends across the opening and further on each side of it equal to the span of the opening. 7.3. Finite element model of the retrofitted I-joists Sets of new finite element models were developed for I-joists with retrofits. In these models, the retrofitting panel pieces are modeled as solid elements given the same thickness as the joist web. A very controlled mesh is applied to the retrofit panels so that their mesh lines up perfectly with the adjacent solids of the joist itself. Similar to the joists with no retrofit, the mesh near the opening is made finer to account for any singularities and effects of concentrations around the opening. For areas farther from the mesh, however, a coarser mesh is introduced to reduce the computational efforts, while maintaining a good accuracy of the model. Furthermore, the mesh size and density are adjusted with respect to the location and size of the opening to maintain consistency between the models. Unlike joists with no retrofits, retrofitted joists could not take advantage of symmetry about the vertical mid-plane, since the retrofitting pieces are applied only to one side of the joist. Due to this fact, all retrofitted joists were modelled in full to capture the true behavior of the beam. Figure 7-3 and Figure 7-4 show the finite element models developed for I-joists with different retrofit arrangements, for web holed and flange notched joists, respectively. The results of the analyses are presented and discussed in the following sections. 113 Figure 7-3. Finite element model for web holed I-joists with retrofits. Left: Type I retrofit. Right: Type II retrofit Figure 7-4. Finite element model for flange notched I-joists with retrofits. Left: Type I retrofit. Right: Type II retrofit 7.4. Design of experiment Using the finite element models explained above, design-of-experiment (DoE) set was designed consisting of a series of retrofitted I-joists to find the effect of retrofit characteristics on the load-carrying properties of the joists. The characteristics include the type and length of the OSB retrofitting boards used. In this DoE set, the two types of retrofits (Type I and Type II) are applied to joists with web holes and flange notches with different sizes and at different locations. Besides, the length of the retrofit is varied with respect to the size of the opening with extension ratios ranging from one to three (R = 1,2,3), as defined in section 7.2. The load at span/180 deflection of the joists is obtained in terms of the structural stiffness of the beam at span/180 deflection. The 114 results are then used to obtain metamodels predicting the stiffness of a joist with a web hole or a flange notch retrofitted using the studied retrofit types with their given length. The obtained capacity of the retrofitted joists is also compared with that of those with no retrofit, examining the effectiveness of the retrofits. According to the studies conducted in previous chapters, all input factors were found to be significantly affecting the response characteristics. Due to a large number of possible configurations, a fractional factorial design approach is adopted. The number and distribution of tests were chosen with respect to the sensitivity of the responses to the input variables, as well as the applicability of the size of the retrofit to the location of the opening. The input factors, along with their respective levels, are presented in Table 7-1 and Table 7-2 below. Table 7-1. Input factors for retrofitted joists with web holes Factor Symbol No. of Levels Levels Units Opening Diameter A 4 0, 4, 6 and 8 in. Opening Location (for 12-ft joists) B 6 0, 1, 2, 3, 4 and 5 ft Opening Location (for 20-ft joists) B 10 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 ft Retrofit Extension Ratio D 3 1, 2 and 3 - Retrofit Type - 2 Type I and Type II - Table 7-2. Input factors for retrofitted joists with flange notches Factor Symbol No. of Levels Levels Units Notch Depth A 4 0, 4, 6 and 8 in. Notch Length B 4 0, 4, 6 and 8 in. Notch Location (for 12-ft joists) C 6 0, 1, 2, 3, 4 and 5 ft Notch Location (for 20-ft joists) C 10 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 ft Retrofit Extension Ratio D 3 1, 2 and 3 - Retrofit Type - 2 Type I and Type II - Design-Expert software is used to conduct the design of the experiment set and analyze the data. To obtain the input data for the experiment set, simulation models explained above were developed and run. In our DoE, a total of 270 models are studied, comprising of 154 cases for I-joists with a web hole and 116 cases for those with a flange notch. 115 7.5. Results Using the design of experiment described in the previous section, the effect of retrofitting of web-holed or flange-notched I-joists is examined. Results of the finite element analysis indicate that the retrofitted joists still follow an almost perfect linear force-deflection pattern. This can be attributed to the linear material behavior of its individual components and materials, including the web and flange. Simulation results of the retrofitted web-holed I-joists show that the retrofit can be effective in increasing the stiffness of the joist. The results show that the retrofit is generally more effective for 12-ft joists. It is observed that Type I retrofit can introduce an average of 6% increase in the stiffness relative to the web-holed joist for 12-ft joists. Type II retrofit, for the same group of samples, has introduced an 8% increase in the stiffness. The highest stiffness improvements due to retrofits are observed in the cases where the hole has had the most influence in lowering the stiffness. As mentioned in previous chapters, this is where the hole is closer to the supports, where the web’s shear load capacity plays an important role in the load-carrying behavior of the beam. The results of this section also attest to this fact. The amount of stiffness improvement in 20-ft I-joists is seen to be relatively lower compared to 12-ft joists. This has been about a 2% increase for Type I and 3.21% for Type II retrofits. Like in shorter span joists, the stiffness is observed to be increasing with the retrofit extension ratio in longer joists. In cases retrofitted with Type I retrofit, it is seen that the retrofit has compensated for the stiffness loss of the joist, bringing the stiffness back to its original amount, equaling to that of the control joist. However, it has not caused a significant increase relative to that of the control joist. However, Type II retrofit has caused the stiffness to increase beyond the original amount of the control I-joist. This can be attributed to the extra layer of retrofit connecting the top and bottom flanges, thus introducing a higher stiffness to the beam. Figure 7-5(a) and (b) draw a comparison between the results of web-holed I-joists retrofitted with Type I and Type II retrofits, as well as between their effect on 12-ft and 20-ft joists. It can be concluded that both retrofit types Type I and Type II are effective in increasing the stiffness of the web-holed joists for both span lengths. However, it can be seen that the retrofits have been more effective on 12-ft I-joists in comparison with the 20-ft ones. Figure 7-5(b) compares the average 116 stiffness of the retrofitted joists relative to the control joist as if no web hole existed. It is observed that both retrofit types, even Type I, have been able to compensate for the decreased capacity of the beam due to the opening. Figure 7-5. (a) Average stiffness increase with respect to no retrofit condition (left) and (b) relative stiffness (right) versus retrofit extension ratio for web-holed I-joists with Type I and Type II retrofits Figure 7-6(a) plots the average stiffness of the two series of joists with the two retrofit techniques versus the location of the opening, measured from the mid-span of the beam. It is found that the retrofits have been more effective for 12-ft joists, increasing the stiffness of the web-holed joist by an average of 2.30% to 9.69% using Type I retrofit and 4.62% to 10.98% using Type II retrofit. Furthermore, it can be observed that the retrofits have been most effective for web holes closer to the support, where the stiffness drop due to the opening had been the largest. Figure 7-6(b) plots the same results against the size of the web hole opening. For both retrofit techniques and joist span lengths, it is seen that the retrofits are more effective for larger holes. This is where the joist has lost its shear bearing capacity the most, which the retrofit can compensate for the most. This finding is in line with manufacturers and the CSA allowing openings not close to joist supports. As seen in Figure 7-6(b), the stiffness of the retrofitted joists is higher than a joist without an opening, in some cases near 10% higher. 0%2%4%6%8%10%12%0 1 2 3 4% Average Stiffness IncreaseR: Retrofit Extension RatioType I - 12-ftType I - 20-ftType II - 12-ftType II - 20-ft0.981.001.021.041.061.081.100 1 2 3 4K/K_ControlR: Retrofit Extension RatioType I - 12-ftType I - 20-ftType II - 12-ftType II - 20-ftControl117 Figure 7-6. Average stiffness increase of retrofitted web-holed joists versus (a) opening location measured from mid-span of the joist (left) and (b) opening size (right) For flange-notched I-joists, it is observed that the retrofits can increase the stiffness to a much higher extent compared to web-holed I-joists. Retrofit Type I is observed to cause an average of 43% and 35% increase for 12-ft and 20-ft joists, respectively. The average stiffness increase for joists having Type II retrofit, on the other hand, is seen to be 57% and 48% for 12-ft and 20-ft joists, respectively, relative to the notched joists. This can be attributed to the higher amount of initial stiffness reduction in the joist due to the flange cuts. Unlike web-holed joists, it is seen that, in joists with a flange notch, the stiffness increase increases with the extension ratio of the retrofit. This has been seen to be true for both Type I and Type II retrofits. In contrast to web-holed joists, it has also been observed that the retrofits have been effective the most for joists with notches closer to the mid-span. This is because, in such simply-supported beams, the moment demand is the highest at the mid-span, where the flanges become important in carrying these moments. Because of the same reason, retrofit Type II has been found to introduce an average of 50% more increase to the stiffness compared to Type I, which should be attributed to the connectivity the former establishes between the top and bottom flanges, making a more integrated moment-carrying capability. It should be noted that, despite the very high amount of stiffness increase relative to notched joists, in very few arrangements the retrofit has been able to bring back the initial stiffness to the joist. In most cases, the retrofitted flange-notched joist was observed to still have a much lower capacity 0%2%4%6%8%10%12%0 2 4 6 8 10% Average Stiffness IncreaseOpening Location (ft)Type I - 12-ftType II - 12-ftType I - 20-ftType II - 20-ft0%2%4%6%8%10%12%2 4 6 8 10% Average Stiffness IncreaseOpenineg Diameter (in.)Type I - 12-ftType II - 12-ftType I - 20-ftType II - 20-ft118 than the control joist. This indicates that neither retrofit type is sufficient or satisfactory to compensate for the flange notch. The only situation in which the retrofit is effective is when the notch is closer to the supports and the size of the notch is small. In this situation, the moment demand of the joist at the notch location is not too high, and the initial stiffness reduction of the beam had been small, making the relative increase outstand more. Figure 7-7(a) demonstrates the percentage increase of the stiffness due to the retrofits for both 12-ft and 20-ft joists. It can be seen that, since there had been a higher initial drop in the stiffness due to the flange notch, the amount of stiffness increase is higher relative to joists with web holes. It can also be observed that, like on web-holed joists, the retrofits have had higher effectiveness on the shorter span beams. Despite the higher increase in the stiffness, it is illustrated in Figure 7-7(b) that neither retrofit type has been sufficient to bring back the initial stiffness to the beam. In this figure, the average stiffness of the retrofitted joists relative to the control joist is illustrated. This can be attributed to the importance of the flanges in carrying the moments, and lack of moment-carrying capacity due to the existence of cuts in the flange. This conclusion well supports the fact that neither I-joist manufacturers nor the CSA allows flange notches anywhere along the joist, even retrofitted. Figure 7-8(a) plots the average stiffness of the two series of joists with the two retrofit techniques versus the location of the notch, measured from the mid-span of the beam. It is found that the retrofits have increased the stiffness of the notched 12-ft joists by an average of 25% to 58% using Type I retrofit and 30% to 80% using Type II retrofit. The stiffness increases for 20-ft joists are 6% to 54% and 6% to 72% using Type I and Type II retrofits, respectively. Furthermore, it can be observed that the retrofits have been most effective for notches closer to the mid-span of the beam, where the contribution of the flange is expected to be maximum. 119 Figure 7-7. (a) Average stiffness increase with respect to no retrofit condition (left) and (b) relative stiffness (right) versus retrofit extension ratio for flange-notched I-joists with Type I and Type II retrofits Figure 7-8(b) plots the same results against the size of the notch. For both retrofit techniques and joist span lengths, it is seen that the retrofits become more effective for larger notches. This is where the joist has lost its capacity the most and the retrofit becomes helpful. Studying this figure and Figure 7-7(b) simultaneously, it is found that despite the significant improvement in the stiffness relative to the notched joists, it is still remarkably lower than the stiffness of joists without a notch. This gives rise to the importance of avoiding notches that cut through joist flanges. Reviewing the results for retrofitted web-holed and flange-notched joists, it can be concluded that the retrofit techniques used do a good job of compensating for the cut-out opening from the web. This is where the beam lacks shear bearing area, which the retrofits can be effective in contributing to. The retrofits, however, do not provide the joist with a good connectivity between the two segments of the cut flange to the sides of the notch. This is required for the flange to provide the beam with the needed moment-carrying capacity. 0%10%20%30%40%50%60%70%0 1 2 3 4% Average Stiffness IncreaseR: Retrofit Extension RatioType I - 12-ftType I - 20-ftType II - 12-ftType II - 20-ft0.400.500.600.700.800.901.001.100 1 2 3 4K/K_ControlR: Retrofit Extension RatioType I - 12-ftType I - 20-ftType II - 12-ftType II - 20-ftControl120 Figure 7-8. Average stiffness increase of retrofitted flange-notched joists versus (a) notch location measured from mid-span of the joist (left) and (b) notch size (right) 7.6. Predictive equations for retrofitted I-joists It was shown in preceding sections that the retrofit techniques introduced can have significant effects on the stiffness of the joists with openings. A DoE set was designed to study these effects based on which a total of 270 retrofitted joists were simulated and examined. It was concluded that the degree of effectiveness of the retrofits highly depends on the type of opening, as well as its location and dimension. In this section, the results of the designed DoE will be analyzed to quantitatively find the contribution of the retrofits. The numerical results are presented to which ANOVA technique has been applied to find the effects and fit a metamodel. The predictive equations are then obtained and presented. Using the results presented in the following two sections, Design-Expert software has been used to generate the response surface curves as well as the predictive regression equations. The regression equations will be used to obtain the response results for input scenarios that have not been used in generating the predictive model. The overall structural stiffness of the joist has been considered as the main response parameter in this study. Load at span/180 deflection is also calculated as an estimate for allowable serviceability load in applicable cases. 0%10%20%30%40%50%60%70%80%90%0 2 4 6 8 10% Average Stiffness IncreaseOpening Location (ft)Type I - 12-ftType II - 12-ftType I - 20-ftType II - 20-ft0%10%20%30%40%50%60%70%80%4x4 6x6 8x8% Average Stiffness IncreaseOpenineg Size (in.xin.)Type I - 12-ftType II - 12-ftType I - 20-ftType II - 20-ft121 7.6.1. Retrofitted joists with web holes Tables below present the configurations of the retrofitted joists, with their opening and retrofit details. Table 7-3 to Table 7-6 present the stiffness results for 12-ft and 20-ft web-holed joists with retrofit types I and II. In some joists with the opening close to their supports, larger retrofits would exceed the length of the joist, which is not a feasible scenario. In these cases, an opening one foot farther from the support with the same retrofit extension ratio is instead modeled and tested. In the tables, the opening location is measured from the mid-span of the beam. Table 7-3. Capacity increase percentage for 12-ft I-joists with web holes and Type I retrofit R = 1 R = 2 R = 3 Opening Location, ft Opening diameter, in. Opening Location, ft Opening Diameter, in. Opening Location, ft Opening Diameter, in. 4 6 8 4 6 8 4 6 8 0 0.59 1.19 1.31 0 1.18 2.38 2.61 0 1.90 3.92 5.58 2 2.52 4.24 6.47 2 4.20 6.42 9.45 2 5.52 8.61 12.81 4 4.35 7.42 11.14 4 6.52 10.63 15.80 4 8.70 12.98 Table 7-4. Capacity increase percentage for 12-ft I-joists with web holes and Type II retrofit R = 1 R = 2 R = 3 Opening Location, ft Opening diameter, in. Opening Location, ft Opening Diameter, in. Opening Location, ft Opening Diameter, in. 4 6 8 4 6 8 4 6 8 0 1.18 2.38 2.97 0 2.61 4.76 5.82 0 4.03 7.37 10.45 2 3.12 5.21 8.08 2 5.16 8.36 12.44 2 7.19 11.76 17.79 4 4.71 8.28 12.69 4 7.25 11.99 18.26 4 9.66 14.96 Table 7-5. Capacity increase percentage for 20-ft I-joists with web holes and Type I retrofit R = 1 R = 2 R = 3 Opening Location, ft Opening diameter, in. Opening Location, ft Opening Diameter, in. Opening Location, ft Opening Diameter, in. 4 6 8 4 6 8 4 6 8 0 0.45 0.45 0.90 0 0.90 1.35 2.26 0 1.80 2.25 3.17 2 0.45 0.90 0.45 2 1.36 1.36 2.26 2 1.81 2.26 3.17 4 0.90 1.82 2.76 4 1.81 2.73 4.15 4 2.71 3.64 5.07 6 0.90 1.36 2.76 6 1.36 2.27 4.15 6 2.26 3.18 5.53 8 0.90 1.36 2.76 8 3.62 2.27 4.15 8 2.26 2.73 122 Table 7-6. Capacity increase percentage for 20-ft I-joists with web holes and Type II retrofit R = 1 R = 2 R = 3 Opening Location, ft Opening diameter, in. Opening Location, ft Opening Diameter, in. Opening Location, ft Opening Diameter, in. 4 6 8 4 6 8 4 6 8 0 0.90 1.35 2.26 0 1.80 3.15 4.52 0 3.15 4.95 6.79 1 0.90 1.81 1.81 1 2.26 3.17 4.52 1 3.62 4.98 6.33 3 0.90 1.81 2.27 3 2.26 3.17 4.55 3 3.17 4.98 6.82 4 1.36 2.27 3.69 4 2.71 3.64 5.53 4 3.62 5.00 7.37 6 0.90 1.82 3.23 6 1.81 3.18 5.07 6 2.71 4.09 6.91 7 0.90 1.82 2.76 7 1.81 2.73 4.61 7 2.26 3.64 8.29 9 0.89 0.45 8 2.26 3.18 Several polynomial models were tested to find the most suitable metamodel for the results presented above. These include linear model, two-factor interaction model, quadratic model, cubic model, and quartic model. While watching the results and accuracy of the models, we tried to eliminate the factors that were not found significant in the prediction model to make the model simpler and more practical to use. This led to using reduced versions of the mentioned models. For each model, the R2 factor, the adjusted R2, the predicted R2, the difference between the latter two, as well as the adequate precision were examined to make sure the model is capable of representing the sample domain with minimal errors. Table 7-7 and Table 7-8 present the results of the mentioned factors for web-holed joists with Type I and Type II retrofits, respectively. The criteria explained in Section 6.4 was used to assess the adequacy of these factors. 123 Table 7-7. Fit measurements for web-holed joists with Type I retrofit Model R2 Adjusted R2 Predicted R2 Adequate Precision Δ* L = 12 ft Quartic 0.9998 0.9981 0.9723 86.0219 0.0258 Cubic 0.9971 0.9919 0.9574 48.7932 0.0345 Quadratic 0.9891 0.9829 0.9671 44.4044 0.0158 2FI 0.9823 0.9767 0.9651 43.5025 0.0116 Linear 0.8777 0.8610 0.8139 24.1684 0.0471 Reduced Quadratic 0.9877 0.9838 0.9726 54.9649 0.0112 L = 20 ft Quartic 0.9438 0.8577 0.4545 11.8692 0.4032 Cubic 0.8712 0.7869 0.5952 11.0135 0.1917 Quadratic 0.8345 0.7906 0.7139 13.6619 0.0767 2FI 0.8211 0.7920 0.7412 16.3563 0.0508 Linear 0.7731 0.7561 0.7213 19.1157 0.0348 Reduced Quartic 0.8824 0.8315 0.7232 14.099 0.1083 *|Adjusted-R2 – Predicted-R2| Table 7-8. Fit measurements for web-holed joists with Type II retrofit Model R2 Adjusted R2 Predicted R2 Adequate Precision Δ* L = 12 ft Quartic 0.9999 0.9988 0.9809 116.3450 0.0179 Cubic 0.9981 0.9946 0.9728 62.4279 0.0218 Quadratic 0.9928 0.9887 0.9789 54.5208 0.0098 2FI 0.9894 0.9860 0.9793 57.6780 0.0067 Linear 0.9005 0.8869 0.8487 26.3558 0.0382 Reduced 2FI 0.9886 0.9857 0.9795 61.1021 0.0062 L = 20 ft Cubic 0.9723 0.9605 0.9216 33.8445 0.0389 Quadratic 0.9636 0.9568 0.9402 44.6207 0.0166 2FI 0.9606 0.9560 0.9441 51.9176 0.0119 Linear 0.8809 0.8743 0.8576 40.6758 0.0167 Reduced Quadratic 0.9635 0.9584 0.9441 51.0438 0.0143 *|Adjusted-R2 – Predicted-R2| Equations (7.1) to (7.8) have been derived below, with respect to the fitted responses, to predict the stiffness and load at span/180 deflection of web-holed joists with Type I and Type II retrofits. In these equations, D is the hole’s diameter in inches, X is the location of the hole, measured from joist’s mid-span in ft, and R is the extension ratio of the retrofit as defined in Section 7.2. These parameters can be used interchangeably with symbols introduced in Table 7-1 in the design of experiment. 124 Predictive equations for 12-ft I-joists with a web hole and Type I retrofit: 𝐾 = 835.253 + 5.442𝐷 + 0.242𝑋 − 9.121𝑅 + 3.485𝐷𝑅 + 3.318𝑋𝑅− 0.769𝐷2 𝑁𝑚𝑚 (7.1) 𝑃180 = 16.972 + 0.111𝐷 + 0.005𝑋 − 0.185𝑅 + 0.071𝐷𝑅 + 0.067𝑋𝑅− 0.016𝐷2 𝑘𝑁 (7.2) Predictive equations for 20-ft I-joists with a web hole and Type I retrofit: 𝐾 = 219.836 + 0.768𝐷 − 1.212𝑋 − 0.385𝑅 − 0.393𝐷𝑋 + 0.276𝐷𝑅 + 0.624𝑋𝑅− 0.075𝐷2 + 1.248𝑋2 + 0.199𝑅2 + 0.029𝐷2𝑋 − 0.158𝑋𝑅2− 0.241𝑋3 + 0.014𝑋4 𝑁/𝑚𝑚 (7.3) 𝑃180 = 7.445 + 0.026𝐷 − 0.041𝑋 − 0.013𝑅 − 0.013𝐷𝑋 + 0.009𝐷𝑅 + 0.021𝑋𝑅− 0.003𝐷2 + 0.042𝑋2 + 0.007𝑅2 + 0.001𝐷2𝑋 − 0.005𝑋𝑅2− 0.008𝑋3 + 0.0005𝑋4 𝑘𝑁 (7.4) Predictive equations for 12-ft I-joists with a web hole and Type II retrofit: 𝐾 = 859.922 − 3.727𝐷 + 0.606𝑋 − 10.608𝑅 + 5.441𝐷𝑅 + 2.275𝑋𝑅 𝑁/𝑚𝑚 (7.5) 𝑃180 = 17.474 − 0.076𝐷 + 0.012𝑋 − 0.216𝑅 + 0.111𝐷𝑅 + 0.046𝑋𝑅 𝑘𝑁 (7.6) Predictive equations for 20-ft I-joists with a web hole and Type II retrofit: 𝐾 = 221.431 − 0.122𝐷 + 0.565𝑋 + 0.219𝑅 − 0.059𝐷𝑋 + 0.626𝐷𝑅 − 0.172𝑋𝑅− 0.030𝑋2 𝑁/𝑚𝑚 (7.7) 𝑃180 = 7.499 − 0.004𝐷 + 0.019𝑋 + 0.007𝑅 − 0.002𝐷𝑋 + 0.021𝐷𝑅 − 0.006𝑋𝑅− 0.001𝑋2 𝑘𝑁 (7.8) Figure 7-9 and Figure 7-10 show the 3-D response surface plots for 12-ft web-holed joists with Type I and Type II retrofits, respectively. Figure 7-11 and Figure 7-12 illustrate the response surface plots for 20-ft web-holed joists in the same order. The figures plot the stiffness of the beam against two factors simultaneously, averaged over the third factor. The figures prove that the stiffness of the beam increases with the retrofit extension ratio. It is seen that the retrofits have more contribution in joists with larger openings, which need more shear bearing area. It is also 125 observed that the retrofits have generally been of more help for openings closer to supports, which is where the shear demand in joists is the highest. Figure 7-9. Response surface curves for 12-ft web-holed joists with Type I retrofit Figure 7-10. Response surface curves for 12-ft web-holed joists with Type II retrofit 126 Figure 7-11. Response surface curves for 20-ft web-holed joists with Type I retrofit Figure 7-12. Response surface curves for 20-ft web-holed joists with Type II retrofit Table 7-9 and Table 7-10 summarize the ANOVA results for joists with web holes for Type I and Type II retrofit types, respectively. In these tables, A, B and C represent opening diameter, opening location and retrofit length ratio, respectively. 127 Table 7-9. Summary of ANOVA analysis for tested models for I-joists with web holes and Retrofit Type I L = 12 ft L = 20 ft Source Sum of Squares df Mean Square F-value p-value Source Sum of Squares df Mean Square F-value p-value Model 0.0087 6 0.0015 254.77 < 0.0001 Model 0.0001 13 0 17.32 < 0.0001 A 0.0007 1 0.0007 117.58 < 0.0001 A 7.1E-06 1 7.1E-06 11.21 0.0022 B 0.0031 1 0.0031 548.69 < 0.0001 B 1.4E-06 1 1.4E-06 2.24 0.145 C 0.0056 1 0.0056 984.07 < 0.0001 C 0.0001 1 0.0001 184.86 < 0.0001 AC 0.0005 1 0.0005 90.13 < 0.0001 AB 1.5E-06 1 1.5E-06 2.32 0.1378 BC 0.0005 1 0.0005 81.71 < 0.0001 AC 5.6E-06 1 6.0E-06 8.81 0.0058 A² 0.0001 1 0.0001 9.68 0.0057 BC 1.3E-08 1 1.3E-08 0.0209 0.8861 Residual 0.0001 19 5.7E-06 A² 2.8E-07 1 2.8E-07 0.4344 0.5149 Cor Total 0.0088 25 B² 3.7E-06 1 3.7E-06 5.89 0.0215 C² 1.8E-06 1 1.8E-06 2.89 0.0995 A²B 1.1E-06 1 1.1E-06 1.67 0.2067 BC² 1.9E-06 1 1.9E-06 3.02 0.0923 B³ 6.9E-07 1 6.9E-07 1.09 0.3045 B⁴ 4.0E-06 1 4.0E-06 6.24 0.0182 Residual 0 30 6.4E-07 Cor Total 0.0002 43 Table 7-10. Summary of ANOVA analysis for tested models for I-joists with web holes and Retrofit Type II L = 12 ft L = 20 ft Source Sum of Squares df Mean Square F-value p-value Source Sum of Squares df Mean Square F-value p-value Model 0.0154 5 0.0031 345.73 < 0.0001 Model 0.0007 7 0.0001 188.55 < 0.0001 A 0.0034 1 0.0034 380.67 < 0.0001 A 0.0001 1 0.0001 200.07 < 0.0001 B 0.0018 1 0.0018 197.6 < 0.0001 B 0.0001 1 0.0001 103.99 < 0.0001 C 0.0117 1 0.0117 1314.07 < 0.0001 C 0.0004 1 0.0004 769.81 < 0.0001 AC 0.0013 1 0.0013 141.04 < 0.0001 AB 3.6E-06 1 3.6E-06 7.15 0.0101 BC 0.0002 1 0.0002 24.64 < 0.0001 AC 0 1 0 78.97 < 0.0001 Residual 0.0002 20 8.9E-06 BC 9.1E-06 1 9.1E-06 17.98 < 0.0001 Cor Total 0.0156 25 B² 2.0E-06 1 2.0E-06 3.96 0.0519 Residual 0 50 5.1E-07 Cor Total 0.0007 57 128 7.6.2. Retrofitted joists with flange notches Table 7-11 to Table 7-14 present the stiffness results for 12-ft and 20-ft flange-notched joists with retrofit types I and II. Similar to the joists with web holes, for notches close to the support with high retrofit extension ratios, a notch one foot farther from the support with the same retrofit extension ratio is instead modeled and tested. In the tables, the notch location is measured from the mid-span of the beam. Table 7-11. Capacity increase percentage for 12-ft I-joists with flange notches and Type I retrofit R = 1 R = 2 R = 3 Notch Location, ft Notch Size, in. Notch Location, ft Notch Size, in. Notch Location, ft Notch Size, in. 4x4 6x6 8x8 4x4 6x6 8x8 4 6 8 0 37.00 58.08 54.22 0 54.00 70.66 57.83 0 60.00 74.25 57.83 2 23.47 42.43 45.88 2 33.40 52.96 50.00 2 37.84 56.58 51.18 4 12.22 23.69 30.61 4 17.44 30.60 35.36 4 20.27 32.84 Table 7-12. Capacity increase percentage for 12-ft I-joists with flange notches and Type II retrofit R = 1 R = 2 R = 3 Notch Location, ft Notch Size, in. Notch Location, ft Notch Size, in. Notch Location, ft Notch Size, in. 4x4 6x6 8x8 4x4 6x6 8x8 4x4 6x6 8x8 0 46.67 78.44 77.11 0 69.33 97.01 81.93 0 78.33 101.80 83.13 2 28.33 54.28 62.94 2 40.17 67.76 69.41 2 46.09 73.68 71.76 4 14.31 29.29 40.11 4 20.42 37.50 46.17 4 23.85 40.86 Table 7-13. Capacity increase percentage for 20-ft I-joists with flange notches and Type I retrofit R = 1 R = 2 R = 3 Notch Location, ft Notch Size, in. Notch Location, ft Notch Size, in. Notch Location, ft Notch Size, in. 4x4 6x6 8x8 4x4 6x6 8x8 4x4 6x6 8x8 0 31.25 54.55 53.57 0 45.83 65.45 57.14 0 50.00 69.09 57.14 3 24.58 44.59 47.62 3 34.75 55.41 50.00 3 38.98 56.76 52.38 6 10.11 20.57 27.55 6 14.04 25.53 29.59 6 15.73 26.95 30.61 9 1.34 1.80 9 2.68 8 4.63 7.32 129 Table 7-14. Capacity increase percentage for 20-ft I-joists with flange notches and Type II retrofit R = 1 R = 2 R = 3 Notch Location, ft Notch Size, in. Notch Location, ft Notch Size, in. Notch Location, ft Notch Size, in. 4x4 6x6 8x8 4x4 6x6 8x8 4x4 6x6 8x8 0 39.58 72.73 75.00 0 57.29 89.09 78.57 0 64.58 92.73 82.14 3 29.66 56.76 176.19 3 42.37 70.27 66.67 3 47.46 72.97 69.05 6 11.80 25.53 35.71 6 16.85 31.21 39.80 6 18.54 33.33 40.82 9 1.34 2.25 9 4.02 8 5.09 8.78 The results of the simulations for retrofitted flange-notched joists were analyzed to find the most suitable prediction model to predict the stiffness of the retrofitted joists. The analysis is done in Design-Expert software. It is also used to plot the response surface curves for these joists. More than six models were tested, and finally, reduced models were selected to represent the response of the problem. The fit statistics of the models are presented in Table 7-15 and Table 7-16 for flange-notched joists with Type I and Type II retrofits, respectively. Table 7-15. Fit measurements for flange-notched joists with Type I retrofit Model R2 Adjusted R2 Predicted R2 Adequate Precision Δ* L = 12 ft Quartic 1.0000 0.9999 0.9987 381.906 0.0012 Cubic 0.9999 0.9997 0.9984 216.1284 0.0013 Quadratic 0.9923 0.988 0.9783 50.9188 0.0097 2FI 0.9875 0.9826 0.9727 47.237 0.0099 Linear 0.982 0.9796 0.9745 61.8267 0.0051 Reduced 2FI 0.9854 0.9808 0.9716 48.153 0.0092 L = 20 ft Quartic 1.0000 0.9999 NA 335.7372 NA Cubic 0.9997 0.9994 0.9982 177.9659 0.0012 Quadratic 0.9837 0.977 0.9654 41.9861 0.0116 2FI 0.9708 0.9623 0.947 35.7829 0.0153 Linear 0.9564 0.9518 0.9419 45.0348 0.0099 Reduced Quartic 0.9801 0.9732 0.9598 40.9091 0.0134 *|Adjusted-R2 – Predicted-R2| 130 Table 7-16. Fit measurements for flange-notched joists with Type II retrofit Model R2 Adjusted R2 Predicted R2 Adequate Precision Δ* L = 12 ft Quartic 1.0000 0.9999 0.999 436.4529 0.0009 Cubic 0.9998 0.9994 0.9973 161.6438 0.0021 Quadratic 0.9926 0.9884 0.9787 52.421 0.0097 2FI 0.9883 0.9838 0.9742 49.5396 0.0096 Linear 0.9756 0.9722 0.9648 53.5965 0.0074 Reduced 2FI 0.9838 0.9797 0.9722 51.2136 0.0075 L = 20 ft Quartic 0.9946 0.9664 NA 20.1092 NA Cubic 0.9644 0.9212 0.7338 15.7007 0.1874 Quadratic 0.9206 0.8881 0.8234 18.3324 0.0647 2FI 0.9143 0.8893 0.8501 20.0357 0.0392 Linear 0.8591 0.844 0.811 23.6627 0.033 Reduced Cubic 0.9356 0.9002 0.7755 17.3867 0.1247 *|Adjusted-R2 – Predicted-R2| With respect to the fitted responses, the predictive equations for the model have been derived. The predictive equations are brought below. Equations (7.9) and (7.10) present the predictive equation for stiffness and maximum load capacity of 12-ft I-joists with a web hole and retrofit Type I, and equations (7.11) and (7.12) represent those for Type II retrofit. Equations (7.13) to (7.16) present the same for 20-ft joists in the same order. In these equations, L and D are notch length and depth, respectively, in inches, X is the location of the notch, measured from the mid-span, in ft, and R is the extension ratio of the retrofit as defined in previous sections. These parameters can be used interchangeably with symbols introduced in Table 7-2 in the design of experiment. Predictive equations for 12-ft I-joists with a flange notch and Type I retrofit: 𝐾 = 674.295 − 71.907𝐷 + 76.043𝑋 + 59.636𝑅 + 2.659𝐷𝑋 − 7.058𝐷𝑅+ 0.567𝑋𝑅 𝑁/𝑚𝑚 (7.9) 𝑃180 = 13.702 − 1.461𝐷 + 1.545𝑋 + 1.212𝑅 + 0.054𝐷𝑋 − 0.143𝐷𝑅+ 0.012𝑋𝑅 𝑘𝑁 (7.10) Predictive equations for 20-ft I-joists with a flange notch and Type I retrofit: 𝐾 = 214.351 − 22.609𝐷 − 3.858𝑋 + 15.385𝑅 + 1.699𝐷𝑋 − 1.650𝐷𝑅 + 0.176𝑋𝑅+ 0.946𝑋2 − 0.348𝑅2 𝑁/𝑚𝑚 (7.11) 131 𝑃180 = 7.259 − 0.766𝐷 − 0.131𝑋 + 0.521𝑅 + 0.058𝐷𝑋 − 0.056𝐷𝑅 + 0.006𝑋𝑅+ 0.032𝑋2 − 0.012𝑅2 𝑘𝑁 (7.12) Predictive equations for 12-ft I-joists with a flange notch and Type II retrofit: 𝐾 = 712.799 − 74.826𝐷 + 59.591𝑋 + 75.292𝑅 + 5.487𝐷𝑋 − 8.444𝐷𝑅 𝑁/𝑚𝑚 (7.13) 𝑃180 = 14.484 − 1.520𝐷 + 1.211𝑋 + 1.530𝑅 + 0.111𝐷𝑋 − 0.172𝐷𝑅 𝑘𝑁 (7.14) Predictive equations for 20-ft I-joists with a flange notch and Type II retrofit: 𝐾 = 224.208 − 18.347𝐷 + 30.857𝑋 − 38.333𝑅 − 3.750𝐷𝑋 + 2.364𝐷𝑅− 3.773𝑋𝑅 − 3.404𝑋2 + 18.860𝑅2 + 0.607𝐷𝑋𝑅 + 0.684𝐷𝑋2− 1.984𝐷𝑅2 𝑁/𝑚𝑚 (7.15) 𝑃180 = 7.593 − 0.621𝐷 + 1.045𝑋 − 1.298𝑅 − 0.127𝐷𝑋 + 0.080𝐷𝑅 − 0.128𝑋𝑅− 0.115𝑋2 + 0.639𝑅2 + 0.021𝐷𝑋𝑅 + 0.023𝐷𝑋2 − 0.067𝐷𝑅2 𝑘𝑁 (7.16) Figure 7-13 and Figure 7-14 show the 3-D response surface plots for 12-ft flange-notched joists with Type I and Type II retrofits, respectively. Figure 7-15 and Figure 7-16, in turn, illustrate the response surface plots for 20-ft joists in the same order. The figures plot the stiffness of the beam against two factors simultaneously, averaged over the other two factors. It was seen previously that, compared to non-retrofitted joists, the existence of the retrofit significantly increases their capacity. However, it can be seen in the figures below that, among the retrofitted joists, the increase in the length of the retrofit does not significantly contribute to the stiffness. It is also observed that the location of the notch still highly affects the stiffness of the beam. Those closer to the mid-span play an important role in decreasing the stiffness significantly. This is where the contribution of the flanges becomes important in carrying the moment demands. Needless to say, mid-span is where joists are expected to experience the highest bending moment. From the figures below, it is also understood that increases in notch dimensions can also adversely affect the stiffness of the joists. 132 Table 7-17 and Table 7-18 summarize the ANOVA results for joists with flange notches for Type I and Type II retrofit types, respectively. In these tables, A, C and D denote notch depth, notch location and retrofit length ratio, respectively. Figure 7-13. Response surface curves for 12-ft flange-notched joists with Type I retrofit Figure 7-14. Response surface curves for 12-ft flange-notched joists with Type II retrofit 133 Figure 7-15. Response surface curves for 20-ft flange-notched joists with Type I retrofit Figure 7-16. Response surface curves for 20-ft flange-notched joists with Type II retrofit 134 Table 7-17. Summary of ANOVA analysis for tested models for I-joists with flange notches and Retrofit Type I L = 12 ft L = 20 ft Source Sum of Squares df Mean Square F-value p-value Source Sum of Squares df Mean Square F-value p-value Model 1.1 6 0.1836 213.79 < 0.0001 Model 0.1072 8 0.0134 141.65 < 0.0001 A 0.4257 1 0.4257 495.69 < 0.0001 A 0.0222 1 0.0222 234.62 < 0.0001 C 0.5668 1 0.5668 660.04 < 0.0001 C 0.0468 1 0.0468 495.27 < 0.0001 D 0.0055 1 0.0055 6.46 0.0199 D 0.0005 1 0.0005 5.23 0.0318 AC 0.0012 1 0.0012 1.37 0.2561 AC 0.0017 1 0.0017 17.98 0.0003 AD 0.0021 1 0.0021 2.42 0.1366 AD 0.0001 1 0.0001 1.55 0.2264 CD 0 1 0 0.0156 0.9019 CD 6.1E-06 1 6.1E-06 0.064 0.8025 Residual 0.0163 19 0.0009 C² 0.0016 1 0.0016 16.5 0.0005 Cor Total 1.12 25 D² 8.3E-07 1 8.3E-07 0.0087 0.9264 Residual 0.0022 23 0.0001 Cor Total 0.1094 31 Table 7-18. Summary of ANOVA analysis for tested models for I-joists with flange notches and Retrofit Type II L = 12 ft L = 20 ft Source Sum of Squares df Mean Square F-value p-value Source Sum of Squares df Mean Square F-value p-value Model 1.1 5 0.2205 242.29 < 0.0001 Model 0.0845 11 0.0077 26.41 < 0.0001 A 0.4313 1 0.4313 473.83 < 0.0001 A 0.0069 1 0.0069 23.79 < 0.0001 C 0.5662 1 0.5662 622.04 < 0.0001 C 0.032 1 0.032 109.93 < 0.0001 D 0.01 1 0.01 11.02 0.0034 D 0.0002 1 0.0002 0.639 0.4334 AC 0.0051 1 0.0051 5.61 0.0281 AC 0.0052 1 0.0052 17.92 0.0004 AD 0.003 1 0.003 3.32 0.0834 AD 0.0004 1 0.0004 1.34 0.26 Residual 0.0182 20 0.0009 CD 3.2E-06 1 3.2E-06 0.0109 0.918 Cor Total 1.12 25 C² 0.0007 1 0.0007 2.42 0.1355 D² 0.0003 1 0.0003 1.11 0.3056 ACD 0.0002 1 0.0002 0.5529 0.4658 AC² 0.0018 1 0.0018 6.15 0.0222 AD² 0.0001 1 0.0001 0.2485 0.6236 Residual 0.0058 20 0.0003 Cor Total 0.0904 31 135 7.7. Verification of the predictive models In the previous section, predictive equations were obtained to find the stiffness of the retrofitted joists with respect to the input parameters, including openings size, opening location and the length of the retrofit. The fit statistics presented in Table 7-7and Table 7-8 for web-holed joists and Table 7-15 and Table 7-16 for flange-notched joists are found to meet the criteria to fall within a satisfactory range. The criteria start with high values of R2 and adjusted-R2. These values for the obtained models are in most cases well close to 1.0, which is a preliminary indication of a suitable fit. Besides, the trend of change in both R2 values is traced and is found that with the increase in the order of the predictive model, these values also increase, which confirms that the model is better not only due to the higher number of predictors in it, but also because it provides a better fit. Another key measurement is to examine the predicted-R2 and its distance from the adjusted-R2, shown by Δ in fit measurement tables. A difference of larger than 0.2 is an indication of an over-fitted model. In all models, this difference is found to be very small, demonstrating a good agreement between the two mentioned R2 values. Furthermore, the signal-to-noise ratios of the models are also checked for assessing their adequacy of precision. All models have ratios of way higher than four, which meets the requirement of an adequate precision. In addition to the fit measurements mentioned above, other diagnostic measures of the models are also assessed, including the normal probability and residual plots. These are used to identify any unwanted patterns in the residuals and to confirm that they follow a normal distribution. These plots have been reviewed and found not to show any indication of unwanted patterns in the experiment set. The review of the statistics explained above validates the accuracy of the models. To confirm this and to further validate the models, forty additional joist configurations (five per predictive model) were simulated and analyzed. The input characteristics of the models were randomly generated. These include the location of the opening, the extension ratio of the retrofit and the size of the opening, which represents the diameter of the opening in joists with a web hole, and length and depth of the notch in joists with a flange notch. Table 7-19 to Table 7-22 list the verification models explained. 136 In these tables, the name of the web hole models is formatted as the length of the joist, location and the diameter of the opening, respectively. The model name is then followed by the retrofit type and its extension ratio. The name of the flange notch models is formatted as the length of the joist, location of the notch, notch length and notch depth, and then followed by the retrofit details in the same format as the web hole models. As far as the units are concerned, all joist lengths and opening locations are in feet, opening dimensions are in inches and stiffness values are in kN/mm. Figure 7-17 to Figure 7-20 plot the actual stiffness found from simulation results versus the predicted stiffness calculated using the obtained prediction models, in the same order as presented in the preceding tables. The pattern and high R2 values for these comparisons suggest a sufficient level of accuracy. This verifies that the obtained models are acceptable and capable of predicting the responses with minimal errors. Table 7-19. Verification models for joists with a web hole and retrofit Type I Model A: Opening Diameter B: Opening Location D: Retrofit Extension Ratio Stiffness (Actual) Stiffness (Predicted) 12-3.45-D7.53 Type I-R2.33 7.53 3.45 2.33 0.901 0.998 12-0.33-D6.89 Type I-R1.67 6.89 0.33 1.67 0.857 0.903 12-2.38-D7.63 Type I-R2.50 7.63 2.38 2.50 0.888 0.977 12-3.82-D4.06 Type I-R1.69 4.06 3.82 1.69 0.878 0.915 12-4.70-D5.48 Type I-R1.96 5.48 4.70 1.96 0.882 0.967 20-1.72-D4.48 Type I-R1.82 4.48 1.72 1.82 0.224 0.224 20-3.06-D7.44 Type I-R1.12 7.44 3.06 1.12 0.224 0.223 20-1.77-D5.85 Type I-R1.49 5.85 1.77 1.49 0.224 0.223 20-2.10-D5.23 Type I-R1.05 5.23 2.10 1.05 0.223 0.222 20-3.17-D6.40 Type I-R2.51 6.40 3.17 2.51 0.227 0.226 137 Table 7-20. Verification models for joists with a web hole and retrofit Type II Model A: Opening Diameter B: Opening Location D: Retrofit Extension Ratio Stiffness (Actual) Stiffness (Predicted) 12-2.11-D4.47 Type II-R1.50 4.47 2.11 1.50 0.873 0.872 12-1.65-D5.81 Type II-R1.34 5.81 1.65 1.34 0.872 0.872 12-1.45-D4.11 Type II-R2.37 4.11 1.45 2.37 0.874 0.881 12-3.27-D5.70 Type II-R2.06 5.70 3.27 2.06 0.909 0.898 12-2.56-D6.50 Type II-R1.46 6.50 2.56 1.46 0.886 0.882 20-5.16-D6.65 Type II-R1.66 6.65 5.16 1.66 0.227 0.227 20-6.84-D7.54 Type II-R1.11 7.54 6.84 1.11 0.225 0.224 20-4.76-D5.89 Type II-R2.49 5.89 4.76 2.49 0.230 0.229 20-5.54-D4.92 Type II-R1.15 4.92 5.54 1.15 0.225 0.224 20-1.01-D6.97 Type II-R1.55 6.97 1.01 1.55 0.228 0.228 138 Table 7-21. Verification models for joists with a flange notch and retrofit Type I Model A: Notch Depth B: Notch Length C: Notch Location D: Retrofit Extension Ratio Stiffness (Actual) Stiffness (Predicted) 12-3.98-5.52x4.93 Type I-R1.30 4.93 5.52 3.98 1.30 0.735 0.710 12-2.32-7.02x5.33 Type I-R1.76 5.33 7.02 2.32 1.76 0.477 0.541 12-3.52-5.87x6.50 Type I-R1.21 6.50 5.87 3.52 1.21 0.551 0.554 12-3.06-5.89x4.50 Type I-R2.20 4.50 5.89 3.06 2.20 0.654 0.685 12-2.68-4.08x7.19 Type I-R2.87 7.19 4.08 2.68 2.87 0.377 0.442 20-7.52-5.02x7.72 Type I-R1.80 7.72 5.02 7.52 1.80 0.194 0.169 20-3.39-7.01x5.11 Type I-R1.76 5.11 7.01 3.39 1.76 0.159 0.138 20-4.61-6.84x4.30 Type I-R2.11 4.30 6.84 4.61 2.11 0.177 0.171 20-5.26-5.77x4.07 Type I-R1.38 4.07 5.77 5.26 1.38 0.189 0.177 20-1.56-5.08x5.54 Type I-R1.56 5.54 5.08 1.56 1.56 0.110 0.109 139 Table 7-22. Verification models for joists with a flange notch and retrofit Type II Model A: Notch Depth B: Notch Length C: Notch Location D: Retrofit Extension Ratio Stiffness (Actual) Stiffness (Predicted) 12-2.76-6.14x5.08 Type II-R1.68 5.08 6.14 2.76 1.68 0.596 0.629 12-1.82-5.28x5.24 Type II-R1.34 5.24 5.28 1.82 1.34 0.475 0.523 12-4.76-6.19x4.16 Type II-R1.29 4.16 6.19 4.76 1.29 0.845 0.846 12-3.55-6.44x6.59 Type II-R2.20 6.59 6.44 3.55 2.20 0.612 0.603 12-4.04-4.04x7.57 Type II-R2.64 7.57 4.04 4.04 2.64 0.641 0.585 20-2.96-5.50x5.05 Type II-R2.92 5.05 5.50 2.96 2.92 0.150 0.159 20-5.69-4.03x7.10 Type II-R1.52 7.10 4.03 5.69 1.52 0.155 0.148 20-6.93-5.94x4.29 Type II-R2.21 4.29 5.94 6.93 2.21 0.219 0.196 20-4.67-6.41x6.25 Type II-R2.62 6.25 6.41 4.67 2.62 0.149 0.146 20-7.65-7.26x5.22 Type II-R1.55 5.22 7.26 7.65 1.55 0.220 0.197 140 Figure 7-17. Actual vs predicted response for joists with a web hole and retrofit Type I: 12-ft (left) and 20-ft (right) Figure 7-18. Actual vs predicted response for joists with a web hole and retrofit Type II: 12-ft (left) and 20-ft (right) R² = 0.82360.850.860.870.880.890.900.910.85 0.90 0.95 1.00 1.05Actual (kN/mm)Predicted (kN/mm)R² = 0.99610.2220.2230.2240.2250.2260.2270.2280.222 0.223 0.224 0.225 0.226 0.227Actual (kN/mm)Predicted (kN/mm)R² = 0.90140.870.880.890.890.900.910.87 0.88 0.89 0.89 0.90Actual (kN/mm)Predicted (kN/mm)R² = 0.97670.2240.2250.2270.2280.2300.2310.224 0.225 0.227 0.228 0.230Actual (kN/mm)Predicted (kN/mm)141 Figure 7-19. Actual vs predicted response for joists with a flange notch and retrofit Type I: 12-ft (left) and 20-ft (right) Figure 7-20. Actual vs predicted response for joists with a flange notch and retrofit Type II: 12-ft (left) and 20-ft (right) 7.8. Summary The objective of this chapter has been to assess and quantify the effectiveness of I-joists with openings, retrofitted using two or three layers of OSB boards. Two retrofit types, referred to as Type I and Type II have been studied. For each type, different lengths of retrofit have been examined, which depend on the size of the opening. In addition to the retrofit type and size, the location and size of the openings are also considered in the study: opening diameter for web-holed joists, and length and depth of the notch for joists with a flange notch. A DoE set is designed in which a total of 270 joists with different configurations of opening and retrofit details have been generated, modeled and analyzed. The load-deflection responses of the models have been obtained R² = 0.96390.300.400.500.600.700.800.30 0.40 0.50 0.60 0.70 0.80Actual (kN/mm)Predicted (kN/mm)R² = 0.92710.1000.1200.1400.1600.1800.2000.100 0.120 0.140 0.160 0.180 0.200Actual (kN/mm)Predicted (kN/mm)R² = 0.90970.400.500.600.700.800.900.40 0.50 0.60 0.70 0.80 0.90Actual (kN/mm)Predicted (kN/mm)R² = 0.95180.1400.1600.1800.2000.2200.140 0.160 0.180 0.200 0.220Actual (kN/mm)Predicted (kN/mm)142 using numerical simulation. It is found that the retrofits can have significant contributions in the load-carrying behavior of the beam, which depend on the type of the opening, its size, its location along the length of the joist and finally the length of the span length of the joist. It was also found that, for joists with a web hole, the retrofit techniques can be highly effective, particularly when the opening is close to the supports, where the joist requires shear-bearing capacity. For joists with a flange notch, it was found that even though the retrofit can increase the capacity of the beam significantly, it is still not enough to compensate for the loss due to the notching of the flange. After obtaining the results from the analyses, response surface methodology has been employed to plot 3-D response surfaces and develop regression models to predict the response of new observations that fall within the range of the input characteristics. The derived equations are then evaluated by looking at their fit statistics and diagnostic measures to ensure the adequacy of their fitted models. To verify the obtained predictive models, forty additional joists with randomly-generated opening and retrofit configurations were simulated and analyzed, whose results were compared with the predicted responses. The comparison shows that the actual and predicted responses are in good agreement, which confirms the validity of the proposed predictive models. The results of this chapter, along with the findings of the previous sections will be used in the following chapter to suggest a limit state design approach for I-joists with web openings or flange notches. 143 Chapter 8: LIMIT STATE DESIGN OF TIMBER I-JOISTS WITH OPENINGS IN FLOOR SYSTEMS 8.1. General In Canada, timber members shall be designed following the Canadian wood design code, called Engineering Design in Wood (CSA O86), whose most recent version has been published in 20194. This standard presents wood design using limit state design (LSD) approach for both strength and serviceability aspects. According to this standard, the limit state is defined as “a condition of a structure in which the structure ceases to fulfill the design function.” Using the limit state design approach, the strength of the structure or the structural component is first determined, taking into account the basic specified strengths of the member and the conditions it is subjected to. This strength is then factored by a resistance factor, which is meant for adjusting the calculated strength for the variability of material properties, manufacturing and treatment procedure, the dimension of the member, the failure type that the member is expected to experience and, generally, the uncertainty that is involved in predicting the capacity. The adjusted strength is known as the factored resistance of the member under consideration. In this approach, it has to be made certain that, in the proposed design, the effects of applied loads, individually or combined, do not exceed the factored resistance of the member. In this chapter, we focus on proposing a limit state design approach for the strength of timber I-joists with openings, to be used as floor and roof beams. This is done using the strengths found in previous chapters for the joists with and without openings, and applying the effects of the opening properties, including the opening shape, size, and location. Design procedures will be suggested for joists with a web hole and those with a flange notch. As well, design approaches will be suggested for I-joists with openings retrofitted using the studied retrofit types. Finally, design examples will be given to illustrate the proposed design procedures. 4 Canadian Standards Associations: CSA O86-19 144 8.1.1. Scope New modification factors are introduced to reflect the effects of the existence of an opening on the resistance of timber I-joists in an individual setting or as a member in a group of members known as a timber system. The value of these modification factors is then proposed for the calculation of the factored resistance of such members. Furthermore, a limit state design procedure is suggested for these structural components based on the findings of this study. This procedure is then followed by examples to clarify and demonstrate the design workflow. 8.2. Design of I-joists as per CSA O86 Clause 15 of the Canadian wood design manual provides the design provisions for proprietary lumber products used in structural applications, including prefabricated wood I-joists. Characteristic design values used in this approach are to be provided by the manufacturer of these products as advised in this standard. In the derivation of these properties, it must be made sure that these values comply with the intent of the National Building Code of Canada (Canadian Commission on Building and Fire Codes, 2010). Strength, serviceability, and reliability should be in compliance with the wood design standard. For wood I-joists, in particular, the requirements of the ASTM D5055 standard should also be fulfilled (ASTM International, 2019). This document provides the standard practice for obtaining, monitoring and examining the capacities of these structural elements. 8.2.1. Introduction to specified strengths and design modification factors Specified strengths of a member are known as basic strengths of the material that get assigned to the member, following the application in which the resistance of the member is to be calculated. These strengths, along with the appropriate moduli of elasticity, are provided by CSA for a given set of timber materials based on relevant experiments. Some of the applications that these specified strengths are provided for include: member in bending 𝑓𝑏, shear 𝑓𝑣, fracture shear 𝑓𝑓, compression parallel to the grain 𝑓𝑐, bearing (compression perpendicular to the grain) 𝑓𝑐𝑝 and tension 𝑓𝑡. Modification factors that are used in the calculation of the resistance of timber members are used for adjusting the basic strength of the member for the conditions that the member is subjected to. The very first modification factor that is used in almost every factored resistance calculation is the Load Duration Factor, which is shown by 𝐾𝐷. In CSA, the specified strengths and capacities are 145 provided assuming the members being subjected to a standard-term duration load. A standard-term duration load is defined as a load whose duration is more than a total of seven days during the expected service life of the structure and cannot be categorized as a continuous load. The Load Duration Factor is introduced to increase or decrease the basic strength of the member according to the duration category of the applied loads. A very widely-used modification factor is the Service Condition Factor. The specified strengths are usually given considering the member servicing in a dry condition. The Service Condition Factor, denoted by 𝐾𝑆, reduces the strength of the member when subjected to a wet service condition. This factor reduces the capacity of the member by 4% to 33% for sawn lumber in different applications in a wet condition, compared to that of the member in a dry condition. A dry service condition, as per CSA O86, is known as a condition in which the moisture content of the servicing timber member never exceeds 19% and its average moisture content remains below 15% over one year. Timber members that are intended to serve in a permanent structure should be protected against decay or any form of deterioration using approved treatment procedures. This is usually done by pressure-treating the timber with preservative chemicals. The capacity of such members can potentially be reduced as a result of the treatment procedure. Another modification factor is introduced to take this fact into account, which is called Treatment Factor and is denoted by 𝐾𝑇. This factor decreases the basic strength of a sawn lumber member to 75% to 95% of the original strength, with respect to its intended application. Members that are treated using fire-retardants or any other strength-reducing chemicals also fall in this category and their capacity needs to be reduced using appropriate factors. In certain conditions, specified strengths of sawn lumber in different applications are influenced by the size of the member. This mostly applies to visually stress graded lumber, where the suggested specified strengths are determined by evaluating a piece of the lumber on the basis of visual appearance, presence of defects such as knots, wane, and slope of grain. Due to this fact, another modification factor named Size Factor, denoted by 𝐾𝑍, is introduced to adjust the strength of the member, if applicable. Depending on the size of the cross-section of the member, this factor can reduce the strength of the member by near 30% or even increase it by 70%. 146 The Lateral Stability Factor 𝐾𝐿 is also a size-dependent modification factor and is intended to take into consideration the depth-to-width ratio of the member, along with its level of lateral support. This modification factor is used in the calculation of the bending moment resistance of joist and beam members. For those that are provided with enough lateral support at points of bearing, or at the minimum required spacing on the compression side, this factor is allowed to be taken as unity. The supports are meant to prevent lateral displacements and rotation of the beam. For I-joists, and for conditions in which the lateral support is deemed not enough, the specified strengths should be obtained and supported by documented tests and engineering principles. In applications where multiple timber members are meant to carry the imposed load mutually, a concept of designing the members as a system is introduced by introducing another modification factor. This factor is referred to as System Factor in the Canadian wood design code and is commonly shown by 𝐾𝐻. It is also often known as a load-sharing factor, group factor, repetitive member factor or parallel support factor in some other design codes (American Wood Council, 2015; Canadian Standards Association, 2019; European Committee for Standardization (CEN), 2004; Ministry of Business, 2014). It is introduced to increase the strength of the individual members when they are a part of a system that constraints all its members to the same deformation. Figure 8-1 conceptually illustrates this fact. This increase in the capacity is based on the natural variability of specified strengths and moduli of elasticity that exist in timber members of a certain species, grade, and size. Figure 8-1. Schematic plot of a parallel support system for System Factor This modification factor is meant to account for the assembly (system) configuration, the natural variability of the stiffnesses of framing members, and the extent of the interaction between the strength and stiffness of the structural elements. An investigation led by ASTM International showed that the system factor conventionally introduced for sawn lumber members does not fully 147 reflect the effects of load sharing and composite action effects in wood I-joists (ASTM International, 2019). As well, this factor was shown to be lacking the correlation between the proposed modification factors and the specified strengths provided by the manufacturers of prefabricated I-joists. This study also proved that the repetitive member factor for I-joists is a function of the depth of the joist. Considering the fact that the variability of the stiffness of the framing members is not as extensive as conventional sawn lumber, this investigation concluded that the suggested system factors sawn lumber are not safe to apply to I-joists. Thus, it is instructed to use a system factor of unity (KS = 1.0) until further study advises factors taking into account the joist depth and manufacturer design procedures for suggesting the flexural stiffness and moment capacities (ASTM International, 2019; Canadian Wood Council, 2017). 8.2.2. Design considerations for I-joists with openings The Canadian wood design manual limits the use of openings in I-joists. It is known that the existence of openings in the web directly reduces the shear strength of the beam, as the web is primarily meant to take the shear forces in joist and beam applications. ASTM D5055 requires the reductions in shear capacity of I-joists with web openings to be obtained using valid tests. The web holes shall be located in segments of the joist with proper shear levels, depending on the size and shape. Special considerations should be given to openings near supports, where shear forces are commonly the highest. The manufacturer of the joist is allowed to specify a maximum web hole size that is allowed to be located anywhere along the joist, supported by valid test data. Multiple holes can be allowed, provided that the minimum spacing between them is suggested by the manufacturer. Flange notches are more rigorously proscribed in CSA O86, as the existence of discontinuities in the flanges has a significant impact on the flexural capacity of beams and joists, as well as their lateral stability as bending members. Nonetheless, if verified by the manufacturer through test results, they are permitted to be used. 8.2.3. Design considerations for serviceability In addition to strength design considerations, I-joists need to meet serviceability design criteria as per CSA O86. To this end, the effect of all loads applied to the joist should be examined, which includes loads such as dead, live, snow, wind and earthquake loads. The effect of the loads should 148 be assessed both individually and combined as instructed in this standard and the National Building Code of Canada. Serviceability criteria include elastic deflection of the joist, permanent deformations caused by long-term loads, ponding effects for roof members, vibration performance of the members particularly those deemed slender, and structural movements attributed to shrinkage and swelling due to moisture content change of the wood. As per this standard, the maximum allowable deflection for I-joists is not allowed to exceed 1/180 of the length of the joist span. In this research, it was found that the existence of an opening in joists causes stiffness decrease, which significantly increases the deflection of these bending members. This causes the beams to exceed the serviceability deflection limits before exceeding the strength criteria. As a result, the maximum allowable deflection for the span of the joist was used to assess the performance of the beam. 8.2.4. Design considerations for strength Like other bending members, I-joists are required to have sufficient bending moment resistance and shear resistance, as per CSA O86 design code. The bending moment resistance Mr and shear resistance Vr are calculated as in equations (8.1) and (8.2), respectively. 𝑀𝑟 = 𝜙𝑀𝑟′(𝐾𝐷𝐾𝑆𝐾𝐻𝐾𝑇)𝐾𝐿 (8.1) 𝑉𝑟 = 𝜙𝑉𝑟′(𝐾𝐷𝐾𝑆𝐾𝑇) (8.2) In these equations, ϕ is considered as the resistance factor for which a value of 0.9 has been advised for both bending moment and shear resistances. KD, KS, KH, and KT are the design modification factors: Load Duration Factor, Service Condition Factor, System Factor, and Treatment Factor, respectively, as described in section 8.2.1. KL, used in bending moment resistance calculation, is the Lateral Stability Factor. For prefabricated I-joists, compound ϕM'r and ϕV'r values are known as ultimate resistance values, should be obtained from the CCMC Registry of Product Evaluations (National Research Council of Canada, 2019) and stated in manufacturer’s product evaluation report. Besides, I-joists should be checked for their resistance at their end and intermediate bearing locations, known as ERr and IRr, respectively. The product evaluation report of the I-joist should document the ultimate resistances ER'r and IR'r, which will then be adjusted using the modification 149 factors, KD, KS and KT, and a resistance reduction factor of ϕ = 0.9. Consideration of these resistances for I-joists with openings is beyond the scope of this research. If required, the joists should also meet bearing length and web stiffener requirements set out by the manufacturer. 8.3. Design modification factors for I-joists with openings This research proposes new design modification factors for considering the effects of openings in prefabricated I-joists. These factors are meant to adjust the nominal capacity of the I-joist with respect to reduced capacities of those with openings found in previous chapters of this research. Eq. (8.1) is considered as the basis for moment capacity calculation, and a design modification factor called Opening Factor, KO has been suggested to be used as shown in Eq. (8.3). The introduction of this modification factor is meant to pave the way for extending this design procedure to other pre-fabricated and custom joists with openings. 𝑀𝑟 = 𝜙𝑀𝑟′𝐾𝑂(𝐾𝐷𝐾𝑆𝐾𝐻𝐾𝑇)𝐾𝐿 (8.3) The Opening Factor KO has been obtained by comparing joists with an opening with control I-joists. It is calculated as the ratio of the stiffness of the joists with a web hole or a flange notch to that of those without one. The predictive equations presented earlier in this research are used to obtain the capacity of joists with openings. Table 8-1 and Table 8-2 contain the calculated Opening Factors for different possible configurations of 12- and 20-ft I-joists with a web hole, respectively. For I-joists with web holes, the opening size refers to the diameter of the web hole. Table 8-3 and Table 8-4, in turn, list these factors for joists with a flange notch. For I-joists with flange notches, the depth and length of the notch are specified separately, denoted by D and L, respectively. In all cases, opening location is measured from the mid-span point of the joist and denotes the centerline location of the opening. 150 Table 8-1. Opening Factors for 12-ft I-joists with a web hole Opening Location ft (mm) Opening Size in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 1.000 0.997 0.978 1 (304.8) 0.988 0.974 0.953 2 (609.6) 0.980 0.965 0.942 3 (914.4) 0.981 0.964 0.939 4 (1219.2) 0.985 0.965 0.939 5 (1524.0) 0.986 0.965 0.936 Table 8-2. Opening Factors for 20-ft I-joists with a web hole Opening Location ft (mm) Opening Size in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.995 0.989 0.982 1 (304.8) 0.992 0.987 0.979 2 (609.6) 0.991 0.985 0.978 3 (914.4) 0.991 0.985 0.977 4 (1219.2) 0.991 0.985 0.978 5 (1524.0) 0.992 0.986 0.978 6 (1828.8) 0.992 0.986 0.979 7 (2133.6) 0.993 0.987 0.980 8 (2438.4) 0.993 0.987 0.980 9 (2743.2) 0.993 0.987 0.980 Table 8-3. Opening Factors for 12-ft I-joists with a flange notch Notch Size, in. (mm) Notch Location ft (mm) D → 4(101.6) 6 (152.4) 8 (203.2) L → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.301 0.273 0.301 0.138 0.063 0.082 0.082 0.000 0.000 1 (304.8) 0.364 0.346 0.358 0.210 0.155 0.167 0.138 0.045 0.057 2 (609.6) 0.459 0.451 0.447 0.315 0.278 0.283 0.227 0.161 0.175 3 (914.4) 0.586 0.588 0.568 0.452 0.434 0.432 0.347 0.309 0.325 4 (1219.2) 0.745 0.757 0.721 0.621 0.622 0.612 0.499 0.489 0.507 5 (1524.0) 0.937 0.958 0.906 0.822 0.841 0.825 0.684 0.701 0.721 151 Table 8-4. Opening Factors for 20-ft I-joists with a flange notch Notch Size, in. (mm) Notch Location ft (mm) D → 4(101.6) 6 (152.4) 8 (203.2) L → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.326 0.294 0.324 0.154 0.072 0.087 0.075 0.000 0.000 1 (304.8) 0.369 0.344 0.365 0.207 0.138 0.150 0.124 0.012 0.014 2 (609.6) 0.420 0.402 0.415 0.268 0.213 0.222 0.181 0.089 0.095 3 (914.4) 0.480 0.469 0.473 0.338 0.296 0.303 0.247 0.174 0.184 4 (1219.2) 0.549 0.545 0.540 0.416 0.388 0.392 0.321 0.268 0.282 5 (1524.0) 0.627 0.630 0.615 0.503 0.488 0.490 0.404 0.371 0.389 6 (1828.8) 0.713 0.723 0.699 0.599 0.597 0.596 0.496 0.483 0.504 7 (2133.6) 0.807 0.825 0.792 0.703 0.715 0.711 0.596 0.603 0.628 8 (2438.4) 0.911 0.936 0.894 0.816 0.842 0.835 0.705 0.732 0.761 9 (2743.2) 1.000 1.000 1.000 0.938 0.977 0.968 0.823 0.870 0.902 In earlier sections, two retrofit types were also studied to find out their effectiveness in strengthening the joist with an opening. This study was conducted for both web-holed and flange-notched I-joists. The Opening Factors for these retrofitted joists were also calculated, using the metamodels brought in the previous chapter. Table 8-5 to Table 8-28 report the obtained modification factors for retrofitted joists. The unavailable cells in these tables indicate the impractical retrofit conditions – i.e., where the opening is so close to the supports so that the retrofit with the given size will not fit the joist at the opening location. A clearance of ½ feet between the retrofit and joist end is considered to determine this. In these tables, R denotes the extension ratio of the retrofits, defined in section 7.2. 152 Table 8-5. Opening Factors for 12-ft I-joists with a web hole, Type I retrofitted, R=1 Opening Location ft (mm) Opening Size in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.994 1.000 1.000 1 (304.8) 1.000 1.000 1.000 2 (609.6) 1.000 1.000 1.000 3 (914.4) 1.000 1.000 1.000 4 (1219.2) 1.000 1.000 1.000 5 (1524.0) 1.000 - - Table 8-6. Opening Factors for 20-ft I-joists with a web hole, Type I retrofitted, R=1 Opening Location ft (mm) Opening Size in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.884 0.887 0.887 1 (304.8) 0.881 0.883 0.883 2 (609.6) 0.883 0.883 0.884 3 (914.4) 0.886 0.885 0.886 4 (1219.2) 0.887 0.886 0.887 5 (1524.0) 0.887 0.885 0.885 6 (1828.8) 0.884 0.881 0.882 7 (2133.6) 0.881 0.878 0.878 8 (2438.4) 0.880 0.876 0.877 9 (2743.2) 0.887 - - 153 Table 8-7. Opening Factors for 12-ft I-joists with a web hole, Type I retrofitted, R=2 Opening Location ft (mm) Opening Size in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.999 1.000 1.000 1 (304.8) 1.000 1.000 1.000 2 (609.6) 1.000 1.000 1.000 3 (914.4) 1.000 1.000 1.000 4 (1219.2) 1.000 1.000 - 5 (1524.0) - - - Table 8-8. Opening Factors for 20-ft I-joists with a web hole, Type I retrofitted, R=2 Opening Location ft (mm) Opening Size in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.890 0.894 0.896 1 (304.8) 0.887 0.891 0.893 2 (609.6) 0.889 0.892 0.894 3 (914.4) 0.893 0.895 0.897 4 (1219.2) 0.895 0.896 0.899 5 (1524.0) 0.895 0.895 0.898 6 (1828.8) 0.893 0.892 0.895 7 (2133.6) 0.890 0.889 0.892 8 (2438.4) 0.890 0.888 - 9 (2743.2) - - - 154 Table 8-9. Opening Factors for 12-ft I-joists with a web hole, Type I retrofitted, R=3 Opening Location ft (mm) Opening Size in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 1.000 1.000 1.000 1 (304.8) 1.000 1.000 1.000 2 (609.6) 1.000 1.000 1.000 3 (914.4) 1.000 1.000 1.000 4 (1219.2) 1.000 - - 5 (1524.0) - - - Table 8-10. Opening Factors for 20-ft I-joists with a web hole, Type I retrofitted, R=3 Opening Location ft (mm) Opening Size in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.896 0.903 0.908 1 (304.8) 0.893 0.899 0.903 2 (609.6) 0.895 0.900 0.904 3 (914.4) 0.897 0.902 0.906 4 (1219.2) 0.899 0.902 0.907 5 (1524.0) 0.898 0.901 0.906 6 (1828.8) 0.896 0.897 0.902 7 (2133.6) 0.893 0.894 0.899 8 (2438.4) 0.892 - - 9 (2743.2) - - - 155 Table 8-11. Opening Factors for 12-ft I-joists with a web hole, Type II retrofitted, R=1 Opening Location ft (mm) Opening Size in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.988 0.992 0.996 1 (304.8) 0.992 0.996 1.000 2 (609.6) 0.995 0.999 1.000 3 (914.4) 0.998 1.000 1.000 4 (1219.2) 1.000 1.000 1.000 5 (1524.0) 1.000 - - Table 8-12. Opening Factors for 20-ft I-joists with a web hole, Type II retrofitted, R=1 Opening Location ft (mm) Opening Size in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.889 0.893 0.897 1 (304.8) 0.889 0.893 0.896 2 (609.6) 0.889 0.892 0.895 3 (914.4) 0.889 0.892 0.895 4 (1219.2) 0.889 0.891 0.893 5 (1524.0) 0.889 0.890 0.892 6 (1828.8) 0.888 0.889 0.890 7 (2133.6) 0.887 0.888 0.889 8 (2438.4) 0.886 0.886 0.886 9 (2743.2) 0.885 - - 156 Table 8-13. Opening Factors for 12-ft I-joists with a web hole, Type II retrofitted, R=2 Opening Location ft (mm) Opening Size in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 1.000 1.000 1.000 1 (304.8) 1.000 1.000 1.000 2 (609.6) 1.000 1.000 1.000 3 (914.4) 1.000 1.000 1.000 4 (1219.2) 1.000 1.000 - 5 (1524.0) - - - Table 8-14. Opening Factors for 20-ft I-joists with a web hole, Type II retrofitted, R=2 Opening Location ft (mm) Opening Size in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.899 0.908 0.917 1 (304.8) 0.899 0.908 0.916 2 (609.6) 0.899 0.907 0.915 3 (914.4) 0.898 0.906 0.913 4 (1219.2) 0.897 0.904 0.911 5 (1524.0) 0.896 0.903 0.909 6 (1828.8) 0.895 0.901 0.907 7 (2133.6) 0.893 0.899 0.905 8 (2438.4) 0.891 0.897 - 9 (2743.2) - - - 157 Table 8-15. Opening Factors for 12-ft I-joists with a web hole, Type II retrofitted, R=3 Opening Location ft (mm) Opening Size in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 1.000 1.000 1.000 1 (304.8) 1.000 1.000 1.000 2 (609.6) 1.000 1.000 1.000 3 (914.4) 1.000 1.000 1.000 4 (1219.2) 1.000 - - 5 (1524.0) - - - Table 8-16. Opening Factors for 20-ft I-joists with a web hole, Type II retrofitted, R=3 Opening Location ft (mm) Opening Size in. (mm) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.910 0.924 0.938 1 (304.8) 0.909 0.923 0.936 2 (609.6) 0.908 0.921 0.934 3 (914.4) 0.907 0.919 0.932 4 (1219.2) 0.905 0.917 0.929 5 (1524.0) 0.904 0.915 0.927 6 (1828.8) 0.901 0.913 0.924 7 (2133.6) 0.899 0.910 0.921 8 (2438.4) 0.897 - - 9 (2743.2) - - - 158 Table 8-17. Opening Factors for 12-ft I-joists with a flange notch, Type I retrofitted, R=1 Notch Size, in. (mm) Notch Location ft (mm) D → 4(101.6) 6 (152.4) 8 (203.2) L → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.483 0.483 0.483 0.300 0.300 0.300 0.118 0.118 0.118 1 (304.8) 0.583 0.583 0.583 0.407 0.407 0.407 0.231 0.231 0.231 2 (609.6) 0.684 0.684 0.684 0.514 0.514 0.514 0.344 0.344 0.344 3 (914.4) 0.785 0.785 0.785 0.621 0.621 0.621 0.457 0.457 0.457 4 (1219.2) 0.886 0.886 0.886 0.728 0.728 0.728 0.570 0.570 0.570 5 (1524.0) 0.986 - - 0.835 - - 0.683 - - Table 8-18. Opening Factors for 20-ft I-joists with a flange notch, Type I retrofitted, R=1 Notch Size, in. (mm) Notch Location ft (mm) D → 4(101.6) 6 (152.4) 8 (203.2) L → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.526 0.526 0.526 0.333 0.333 0.333 0.140 0.140 0.140 1 (304.8) 0.542 0.542 0.542 0.363 0.363 0.363 0.183 0.183 0.183 2 (609.6) 0.566 0.566 0.566 0.400 0.400 0.400 0.234 0.234 0.234 3 (914.4) 0.597 0.597 0.597 0.444 0.444 0.444 0.292 0.292 0.292 4 (1219.2) 0.635 0.635 0.635 0.497 0.497 0.497 0.358 0.358 0.358 5 (1524.0) 0.682 0.682 0.682 0.556 0.556 0.556 0.431 0.431 0.431 6 (1828.8) 0.735 0.735 0.735 0.624 0.624 0.624 0.512 0.512 0.512 7 (2133.6) 0.797 0.797 0.797 0.698 0.698 0.698 0.600 0.600 0.600 8 (2438.4) 0.865 0.865 0.865 0.781 0.781 0.781 0.696 0.696 0.696 9 (2743.2) 0.942 - - 0.870 - - 0.799 - - 159 Table 8-19. Opening Factors for 12-ft I-joists with a flange notch, Type I retrofitted, R=2 Notch Size, in. (mm) Notch Location ft (mm) D → 4(101.6) 6 (152.4) 8 (203.2) L → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.519 0.519 0.519 0.320 0.320 0.320 0.122 0.122 0.122 1 (304.8) 0.620 0.620 0.620 0.428 0.428 0.428 0.235 0.235 0.235 2 (609.6) 0.722 0.722 0.722 0.535 0.535 0.535 0.349 0.349 0.349 3 (914.4) 0.823 0.823 0.823 0.643 0.643 0.643 0.463 0.463 0.463 4 (1219.2) 0.924 0.924 - 0.750 0.750 - 0.576 0.576 - 5 (1524.0) - - - - - - - - - Table 8-20. Opening Factors for 20-ft I-joists with a flange notch, Type I retrofitted, R=2 Notch Size, in. (mm) Notch Location ft (mm) D → 4(101.6) 6 (152.4) 8 (203.2) L → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.557 0.557 0.557 0.351 0.351 0.351 0.145 0.145 0.145 1 (304.8) 0.573 0.573 0.573 0.381 0.381 0.381 0.189 0.189 0.189 2 (609.6) 0.598 0.598 0.598 0.419 0.419 0.419 0.240 0.240 0.240 3 (914.4) 0.630 0.630 0.630 0.464 0.464 0.464 0.299 0.299 0.299 4 (1219.2) 0.669 0.669 0.669 0.517 0.517 0.517 0.365 0.365 0.365 5 (1524.0) 0.716 0.716 0.716 0.577 0.577 0.577 0.439 0.439 0.439 6 (1828.8) 0.770 0.770 0.770 0.645 0.645 0.645 0.521 0.521 0.521 7 (2133.6) 0.832 0.832 0.832 0.721 0.721 0.721 0.609 0.609 0.609 8 (2438.4) 0.902 0.902 - 0.804 0.804 - 0.706 0.706 - 9 (2743.2) - - - - - - - - - 160 Table 8-21. Opening Factors for 12-ft I-joists with a flange notch, Type I retrofitted, R=3 Notch Size, in. (mm) Notch Location ft (mm) D → 4(101.6) 6 (152.4) 8 (203.2) L → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.555 0.555 0.555 0.340 0.340 0.340 0.125 0.125 0.125 1 (304.8) 0.657 0.657 0.657 0.448 0.448 0.448 0.240 0.240 0.240 2 (609.6) 0.759 0.759 0.759 0.557 0.557 0.557 0.354 0.354 0.354 3 (914.4) 0.861 0.861 0.861 0.665 0.665 0.665 0.468 0.468 0.468 4 (1219.2) 0.963 - - 0.773 - - 0.583 - - 5 (1524.0) - - - - - - - - - Table 8-22. Opening Factors for 20-ft I-joists with a flange notch, Type I retrofitted, R=3 Notch Size, in. (mm) Notch Location ft (mm) D → 4(101.6) 6 (152.4) 8 (203.2) L → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.585 0.585 0.585 0.366 0.366 0.366 0.147 0.147 0.147 1 (304.8) 0.602 0.602 0.602 0.397 0.397 0.397 0.191 0.191 0.191 2 (609.6) 0.627 0.627 0.627 0.435 0.435 0.435 0.243 0.243 0.243 3 (914.4) 0.660 0.660 0.660 0.481 0.481 0.481 0.303 0.303 0.303 4 (1219.2) 0.700 0.700 0.700 0.535 0.535 0.535 0.370 0.370 0.370 5 (1524.0) 0.747 0.747 0.747 0.596 0.596 0.596 0.444 0.444 0.444 6 (1828.8) 0.802 0.802 0.802 0.664 0.664 0.664 0.527 0.527 0.527 7 (2133.6) 0.865 0.865 0.865 0.741 0.741 0.741 0.616 0.616 0.616 8 (2438.4) 0.935 - - 0.824 - - 0.713 - - 9 (2743.2) - - - - - - - - - 161 Table 8-23. Opening Factors for 12-ft I-joists with a flange notch, Type II retrofitted, R=1 Notch Size, in. (mm) Notch Location ft (mm) D → 4(101.6) 6 (152.4) 8 (203.2) L → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.525 0.525 0.525 0.333 0.333 0.333 0.141 0.141 0.141 1 (304.8) 0.619 0.619 0.619 0.440 0.440 0.440 0.260 0.260 0.260 2 (609.6) 0.714 0.714 0.714 0.547 0.547 0.547 0.380 0.380 0.380 3 (914.4) 0.808 0.808 0.808 0.653 0.653 0.653 0.499 0.499 0.499 4 (1219.2) 0.902 0.902 0.902 0.760 0.760 0.760 0.619 0.619 0.619 5 (1524.0) 0.996 - - 0.867 - - 0.738 - - Table 8-24. Opening Factors for 20-ft I-joists with a flange notch, Type II retrofitted, R=1 Notch Size, in. (mm) Notch Location ft (mm) D → 4(101.6) 6 (152.4) 8 (203.2) L → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.528 0.528 0.528 0.385 0.385 0.385 0.242 0.242 0.242 1 (304.8) 0.583 0.583 0.583 0.421 0.421 0.421 0.258 0.258 0.258 2 (609.6) 0.633 0.633 0.633 0.462 0.462 0.462 0.291 0.291 0.291 3 (914.4) 0.677 0.677 0.677 0.508 0.508 0.508 0.339 0.339 0.339 4 (1219.2) 0.716 0.716 0.716 0.560 0.560 0.560 0.405 0.405 0.405 5 (1524.0) 0.750 0.750 0.750 0.618 0.618 0.618 0.486 0.486 0.486 6 (1828.8) 0.778 0.778 0.778 0.681 0.681 0.681 0.584 0.584 0.584 7 (2133.6) 0.801 0.801 0.801 0.750 0.750 0.750 0.699 0.699 0.699 8 (2438.4) 0.819 0.819 0.819 0.825 0.825 0.825 0.830 0.830 0.830 9 (2743.2) 0.832 - - 0.904 - - 0.977 - - 162 Table 8-25. Opening Factors for 12-ft I-joists with a flange notch, Type II retrofitted, R=2 Notch Size, in. (mm) Notch Location ft (mm) D → 4(101.6) 6 (152.4) 8 (203.2) L → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.573 0.573 0.573 0.361 0.361 0.361 0.150 0.150 0.150 1 (304.8) 0.667 0.667 0.667 0.468 0.468 0.468 0.269 0.269 0.269 2 (609.6) 0.762 0.762 0.762 0.575 0.575 0.575 0.389 0.389 0.389 3 (914.4) 0.856 0.856 0.856 0.682 0.682 0.682 0.508 0.508 0.508 4 (1219.2) 0.950 0.950 - 0.789 0.789 - 0.628 0.628 - 5 (1524.0) - - - - - - - - - Table 8-26. Opening Factors for 20-ft I-joists with a flange notch, Type II retrofitted, R=2 Notch Size, in. (mm) Notch Location ft (mm) D → 4(101.6) 6 (152.4) 8 (203.2) L → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.543 0.543 0.543 0.372 0.372 0.372 0.201 0.201 0.201 1 (304.8) 0.593 0.593 0.593 0.407 0.407 0.407 0.221 0.221 0.221 2 (609.6) 0.637 0.637 0.637 0.447 0.447 0.447 0.258 0.258 0.258 3 (914.4) 0.676 0.676 0.676 0.494 0.494 0.494 0.311 0.311 0.311 4 (1219.2) 0.710 0.710 0.710 0.545 0.545 0.545 0.380 0.380 0.380 5 (1524.0) 0.739 0.739 0.739 0.602 0.602 0.602 0.466 0.466 0.466 6 (1828.8) 0.762 0.762 0.762 0.665 0.665 0.665 0.569 0.569 0.569 7 (2133.6) 0.779 0.779 0.779 0.733 0.733 0.733 0.687 0.687 0.687 8 (2438.4) 0.792 0.792 - 0.807 0.807 - 0.823 0.823 - 9 (2743.2) - - - - - - - - - 163 Table 8-27. Opening Factors for 12-ft I-joists with a flange notch, Type II retrofitted, R=3 Notch Size, in. (mm) Notch Location ft (mm) D → 4(101.6) 6 (152.4) 8 (203.2) L → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.621 0.621 0.621 0.390 0.390 0.390 0.159 0.159 0.159 1 (304.8) 0.715 0.715 0.715 0.497 0.497 0.497 0.278 0.278 0.278 2 (609.6) 0.809 0.809 0.809 0.604 0.604 0.604 0.398 0.398 0.398 3 (914.4) 0.904 0.904 0.904 0.710 0.710 0.710 0.517 0.517 0.517 4 (1219.2) 0.998 - - 0.817 - - 0.637 - - 5 (1524.0) - - - - - - - - - Table 8-28. Opening Factors for 20-ft I-joists with a flange notch, Type II retrofitted, R=3 Notch Size, in. (mm) Notch Location ft (mm) D → 4(101.6) 6 (152.4) 8 (203.2) L → 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 4 (101.6) 6 (152.4) 8 (203.2) 0 (0.0) 0.646 0.646 0.646 0.414 0.414 0.414 0.183 0.183 0.183 1 (304.8) 0.690 0.690 0.690 0.449 0.449 0.449 0.208 0.208 0.208 2 (609.6) 0.729 0.729 0.729 0.489 0.489 0.489 0.248 0.248 0.248 3 (914.4) 0.763 0.763 0.763 0.534 0.534 0.534 0.306 0.306 0.306 4 (1219.2) 0.791 0.791 0.791 0.585 0.585 0.585 0.380 0.380 0.380 5 (1524.0) 0.814 0.814 0.814 0.642 0.642 0.642 0.470 0.470 0.470 6 (1828.8) 0.832 0.832 0.832 0.704 0.704 0.704 0.577 0.577 0.577 7 (2133.6) 0.844 0.844 0.844 0.772 0.772 0.772 0.700 0.700 0.700 8 (2438.4) 0.851 - - 0.845 - - 0.839 - - 9 (2743.2) - - - - - - - - - 164 These tables provide the opening modification factors for a wide range of opening scenarios, including a wide range of sizes and locations of the opening. When using these tables, for sizes or locations not specifically listed, it is suggested that a more conservative Opening Factor be used; that is, the smaller KO from the nearest smaller and larger opening sizes available be used. 8.4. Design examples of a floor joist with opening In this section, examples of the design of I-joists with openings using the proposed design procedure will be demonstrated. The bending moment resistance of I-joists, as per CSA O86-19, is calculated using Eq. (8.3). The CCMC product evaluation report of the reference I-joist reports the descriptions, technical properties and design requirements of these I-joists (National Research Council of Canada, 2019). Per this report, the factored ultimate bending resistance ϕM'r is given as 7.310 kN.m. Assuming a standard-duration load, dry service condition and an untreated joist, modification factors KD, KS, and KT shall be taken as 1.0. Considering a system of I-joists meant to jointly carry the loads, supported at bearing points and along the compression edge, KH and KL factors can also be taken as 1.0. Based on these factors, the bending moment resistance of such joist will be calculated as follows: 𝑀𝑟 = 𝜙𝑀𝑟′(𝐾𝐷𝐾𝑆𝐾𝐻𝐾𝑇)𝐾𝐿𝐾𝐷=𝐾𝑆=𝐾𝐻=𝐾𝑇=𝐾𝐿=1.0⇒ 𝑀𝑟 = 7.310 𝑘𝑁.𝑚 (8.4) Let us assume a 12-ft I-joist of this series, with a 6-in. web hole at three feet from the mid-span. Table 8-1 provides the suggested opening factors for this joist. Referring to this table, the corresponding opening factor will be KO = 0.964. The bending moment resistance of this joist according to the proposed method in this chapter will then be calculated as follows: 𝑀𝑟 = 𝜙𝑀𝑟′𝐾𝑂(𝐾𝐷𝐾𝑆𝐾𝐻𝐾𝑇)𝐾𝐿𝐾𝑂=0.964⇒ 𝑀𝑟 = 7.310 × 0.964 = 7.047 𝑘𝑁.𝑚 (8.5) Referring back to results presented in Chapter 5, the total maximum load P for this joist is 16.993 kN given a deflection limit ratio of 1/180 of the joist span. Considering this load being applied as a distributed load, the maximum moment induced in the joist will be: 𝑀 =𝑤𝐿28=(16.99312 ) × 1228× 0.3048 = 7.769 𝑘𝑁.𝑚 (8.6) 165 It is seen that the moment capacity suggested by using the opening modification factor is in a good agreement with the capacity suggested in previous chapters. Referring to opening factor tables for joists with retrofits, it is found that for such opening, Type I retrofit with and extension ratio of R=1 would be enough to compensate for the reduced capacity of the joist due to the existence of the opening. Let us now consider an example of a configuration of this joist with a flange notch. Considering a 6-in.x6-in. flange notch three feet away from the mid-span of the joist, Table 8-3 suggests an opening factor of KO = 0.434. In previous chapters in this study, the maximum load for a joist with this configuration was obtained as 7.079 kN. This corresponds to a maximum moment of 3.237 kN.m in the joist. Following the design procedure proposed in this chapter, the capacity of this joist can be calculated as: 𝑀𝑟 = 𝜙𝑀𝑟′𝐾𝑂(𝐾𝐷𝐾𝑆𝐾𝐻𝐾𝑇)𝐾𝐿𝐾𝑂=0.434⇒ 𝑀𝑟 = 7.310 × 0.434 = 3.173 𝑘𝑁.𝑚 (8.7) It is observed that the suggested moment capacity using the opening modification factor agrees well with the capacity found from the simulations. 8.5. Summary The focus of this chapter has been to propose a limit state design procedure for prefabricated timber I-joists with a web hole or a flange notch. This proposed design procedure is meant to align with the current design procedure given in the CSA O86-19 code. As per CSA O86-19, the reference design capacity of prefabricated joists reported by the manufacturer is adjusted using modification factors accounting for service condition, treatment type, the system of beams the joist is used in, and the duration of loads acting on it. In this chapter, the CSA design procedure was reviewed, the conventional modification factors were briefly discussed, and new modification factors were introduced to account for the existence of openings in the joist. The new modification factors were named opening factors. The simulation data obtained in previous chapters for different configurations of web holes and flange notches were used to obtain these so-called opening factors. These factors are presented for 12-ft and 20-ft joists and account for the location of the opening, shape, and size of it. This 166 modification factor will be used in the same way as the other conventional modification factors to adjust the reported capacity of the joist. Besides, opening factors are also provided for joists with openings, retrofitted using the two retrofit types studied in previous sections of this study. For each retrofit type, the modification factors are given for three different lengths of the retrofit, defined by their extension ratio. The proposed opening factors are presented in form of tables based on the opening type (web hole or flange notch), joist length, retrofit type, and retrofit length. Finally, design examples of I-joists with web and flange openings are demonstrated to illustrate the proposed design procedure. 167 Chapter 9: SUMMARY, CONCLUSIONS AND RECOMMENDED FUTURE STUDY 9.1. Summary I-joists are engineered timber beams, mainly purposed to carry loads in roof and floor systems. With the increasing interest in timber construction, these beams gave gained popularity, thanks to their advantages over conventional sawn lumber joists. During the construction phase, these joists are often seen to be drilled into their web or notched into the flange for the convenience of the passage of electrical and mechanical systems. The structural capacity of the joists can significantly be affected by introducing such openings to these beams. The research in this dissertation was aimed at investigating the structural response of timber I-joists with a web hole or a flange notch. The overall objective of this research has been to gain a better understanding of the performance of I-joists with openings in their web and flanges. This is used to recommend procedures to measure the extent of deficiency caused to the joist, design the beam accordingly and propose remedies, if possible. One primary contribution of this research was developing finite element models for timber I-joists, with and without openings. The techniques introduced can be used to capture the structural response of the joists and estimate their capacity with good precision. Parts of this research were purposed to characterize timber material for the modeling of parts of the beams. The finite element models were used to conduct a sensitivity analysis on the response of joists with different opening configurations. These findings were then used to develop predictive equations to estimate the response of the joists as a function of opening characteristics. Furthermore, the modeling techniques were employed to also assess the effectiveness of retrofits on stiffening joists with an opening. Several retrofit characteristics were studied, and equations were derived to predict the effectiveness of the retrofits on achieving the goal of elevating the performance of the joists. The findings of the research were finally used to propose design procedures for I-joists with openings. 168 As well, procedures were recommended to introduce retrofits for bringing the joists back to desired serviceability levels. In this dissertation, the following objectives were accomplished: • Characterizing timber material behavior and proposing a robust finite element model for timber I-joists, • Identifying the effects of opening characteristics, including its shape, size, and location, on limit state behavior of I-joists, • Evaluating the effectiveness of retrofits on the performance of I-joists with openings, • Developing predictive equations for the response characteristics of I-joists with an opening, with and without retrofits, and • Proposing a guideline for designing I-joists with a web hole or a flange notch, and recommending a retrofit for the joist to regain its performance. 9.2. Conclusions 9.2.1. Characterization of timber material The timber material used in I-joist flanges was experimentally tested, to find its material properties. A total of 92 specimens were fabricated and tested in the facilities of the University of British Columbia. The fabrication and testing of the specimens complied with the ASTM standards. The timber material was tested in three directions, namely longitudinal, transverse and radial. The material behavior was characterized by tension and compression, using uni-axial testing machines. Tests were replicated to take into account the effects of non-homogeneity of wood and potential variations in specimens. 9.2.2. Development of finite element models Efforts were made to develop finite element models for I-joists, with and without openings. These models paved the way for expanding the research by conducting several analyses on these joists. The results of the material characterization in the previous chapter were used to model the timber material of the flanges. 169 Full-scale 3D models of the I-joists were developed. The results of the simulations were validated against experimental tests, and a close match was observed. It was concluded that the developed finite element models can consistently and reliably be used to simulate the structural response of the joists. A sensitivity analysis was conducted to study the sensitivity of the developed models to the number of elements and mesh size. It was observed that a relatively coarse mesh could be used, without compromising the accuracy of the results. This led to a substantial reduction in the computational efforts and analysis time, given a large number of models under study. Solid elements were used in the developed models. Boundary conditions were defined to exactly match those of the experimental tests. These boundary conditions included the supports at beam ends, as well as lateral supports constraining the compression flanges. A higher mesh density was introduced near the openings to more accurately capture the behavior and consider the effects of stress concentrations in this area. Besides, the symmetry of the models about their longitudinal axis was taken as an advantage to further reduce the computational efforts. Given the reliable results of the models, they were found to be reliable in predicting the load-deflection response of the joists. This facilitated the study of a wide range of I-joist configurations, presented in the rest of the research. 9.2.3. Sensitivity analysis of the performance of I-joists with openings The effects of the existence of a web hole or a flange notch in an I-joist were evaluated in a section of this research. A full-factorial experiment was designed to assess the effects of the opening configuration on the load-deflection response of the joists. The input characteristics included the shape of the opening, the location of the opening measured from the mid-span of the beam, the diameter of the opening in case of web holes, and length and depth of the opening in case of flange notches. Besides, the mentioned factors were examined on I-joists in two span lengths. In the case of I-joists with web hole, a total of 48 configurations were studied. Web holes were varied in size from 2 to 8 inches, and at locations with 1 ft increments measured from the mid-span of the beam. The load-deflection response of the joists was recorded, and the response was evaluated. It was observed that the existence of a web hole can significantly affect the structural capacity of the joists, depending on the size and location of the hole. Smaller holes (4-in. diameter) 170 had negligible effects, regardless of their location. Larger openings, however, had more considerable impacts on the stiffness of the beam. Unlike smaller holes, the effect of the location of these holes was seen to be more imperative. Holes closer to beams supports, meaning high shear locations, were seen to more significantly decrease the capacity of the beam. For all cases, 3D graphs of the responses were graphed to better study the effects of the input parameters simultaneously. 144 cases of joists with a flange notch with different notch configurations were examined. The effect of flange notch openings was found to be considerably more devastating for the beams. A decrease of up to 92% and 85% of the original capacity was observed for 12-ft and 20-ft joists, respectively. In contrary to those with a web hole, joists with a flange notch closer to the mid-span were seen to have the highest reduction in capacity. It was shown by the results that the depth of the joist had more contribution to the decrease in the capacity, compared to the length of the notch. Despite the significance of the effects of the size of the notch, the location was seen to be a more important factor in impacting the beam’s capacity. The high level of impact of the flange notches observed supported the idea of them not being permitted by many manufacturers. 9.2.4. Predictive equations for I-joists with openings The overall structural stiffness of the I-joists was observed as the main response characteristics of the joists. The study resulted in deriving the predictive equations to predict the response of any given configuration of the openings falling in the applicable range of the parameters of the study. The response surface method was used for this purpose. The variables in the predictive equations included the size and location of the opening, as well as interactions between them, considering their significance. The predictive equations paved the way for developing a suggested procedure for designing I-joists with a web hole or a flange notch. 9.2.5. Evaluation of retrofitting of I-joists with openings In order to fulfill another objective of this research, the effectiveness of retrofitting I-joists with an opening using OSB boards was investigated. The finite element method was used to conduct this investigation. Two types of retrofits suggested in previous research were considered, namely Type I and Type II retrofits. Type I retrofit represented OSB panels reinforcing the web, filling the space between the two flanges of the joist. Type II retrofit included an additional panel effectively 171 connecting the top and bottom flanges. For each type, various lengths of the retrofit were examined. The length of the retrofit was defined as a ratio of the extension of the retrofit on each side of the opening to the opening’s length. The effect of the retrofits was examined for openings with different sizes and locations along the length of the beam. A design of experiment was employed, studying a total of 270 joists with different configurations of openings. It was observed that retrofits can have a significant contribution in reinforcing joists with openings. Factors affecting this contribution include the shape, size, and location of the opening, in addition to the length of the retrofit. It was found that for joists with a web hole, the proposed retrofit types can be highly effective, particularly for openings closer to the supports, where the joist requires shear-bearing capacity. In most cases, retrofit configurations could be found to bring the joist’s capacity back to that of the control joists. For joists with a flange notch, despite the significant contribution of the retrofits in increasing the capacity, in most cases, the retrofits were not enough to compensate for the lost stiffness. Furthermore, three-dimensional effect plots were plotted using response surface methodology. Regression models were also developed to predict the response of new observations falling within an applicable range of the study parameters. Studying the fit statistics of the predictive equations showed that these equations are capable of predicting the responses with a good accuracy. These equations, along with those of un-retrofitted joists, were used for proposing a procedure to design these beams. 9.2.6. Design guideline development for I-joists with openings with and without retrofits The final objective of this research was to develop a guideline for the design of I-joists with an opening. This design procedure is intended to be a limit state design, resembling the procedure in the current Canadian wood design code for regular I-joists. To this end, the findings of the studies conducted on numerous I-joists, as well as the predictive equations were employed. New design modification factors, named Opening Factors, were introduced to adjust the reported capacity of the joists to account for the existence of the opening. The opening factors were obtained using the results of the numerical simulations and were presented in the form of tabulated data for ease of use. The Opening Factor tables take into account the size and location of the opening and 172 are generated for 12-ft and 20-ft joists, Besides, tables are provided to suggest opening factors, including the effects of the studied retrofit types. These tables can also be used to suggest a retrofit for a given opening configuration to bring the joist back to its expected serviceability level. Finally, design examples are provided to demonstrate the proposed design procedure. 9.3. Contributions of this research The research in this dissertation sheds light upon the structural response of timber I-joists with openings: a web hole or a flange notch. Major contributions of this research are summarized as follows: • Robust and accurate finite element models were developed to simulate the response of timber I-joists; experimental tests were conducted to characterize the timber material for this purpose. • A thorough sensitivity analysis was conducted on the impact of the opening characteristics on the structural performance of the joists, studying a large number of opening configurations. Important factors were recognized, and effect plots were produced. • The effectiveness of retrofits on stiffening the joists and improving their performance was evaluated. Two retrofit types and several retrofit properties were examined. • Predictive equations were derived for obtaining the stiffness and load at span/180 deflectopm of joists with and without retrofits. • Procedures were proposed for the design of I-joists with an opening: a web hole or a flange notch. Besides, procedures were recommended to suggest retrofits to bring the joists back to expected serviceability levels. 9.4. Limitations of the study and recommendations for future research The study of openings in timber I-joists in this research was limited to one joist section size and two distinct span lengths of joists. As a result, the resulting empirical equations apply only to 301mm deep wood joists constructed with 38mm x 63mm SPF No. 2 grade lumber flange and 9.5mm thickness OSB, with span of either 12ft or 20ft, and are based on the design assumption of span/180 deflection limit and a two-point bending loading type. Efforts can be made to expand the research to include the joist section size as an input factor in the analysis. Predictive equations can also be derived to take into account the size of the section. Furthermore, more studies can be 173 conducted to derive predictive equations and design modification factors for generic lengths of joists. This research focused on the effect of openings on prefabricated I-joists. For these joists, the nominal capacity is prescribed by their manufacturer, and design modification factors are suggested as per the wood design code. It is recommended to study the effect of web and flange openings for built-up panel-web beams, the design procedure for which is given in Clause 10.5 of CSA O86-19. Besides, the research can be expanded to include manufacturer variabilities in the limit state design procedure. In the finite element models developed in this research, the web-flange connection in the joists, as well as the connection between retrofits and the joist, were assumed as a perfect bond to simplify the analysis. Although this assumption led to results with the desired accuracy, efforts can be made to investigate the effects of slippage and breakage of the bond during the analysis and find the extent of these effects during the loading of the beam. This study has not taken into account strength considerations and material failure. The structural stiffness of the joist is taken as a response characteristic, and the experiments are continued up to a deflection of span/180, thanks to the linear nature of the load-deflection response. Even though this limit was found applicable to most of the cases studied in this research, it may not apply to all joist configurations, and material failure should be taken into consideration in every scenario. It is recommended for the finite element modeling to be improved to include and track the failure of the joist material. This can also be expanded to take into consideration the existence of defects such as knots, checks and splits in the joist components, particularly the flanges. The study on I-joists in this research has been based on joists used in typical floor and roof systems covered by sheathing, whose compression flanges are sufficiently supported against lateral movement. In the rare case of applications in which the joist is not laterally supported, the effects of torsion need to be taken into consideration. The following topics are also recommended to expand the scope of the research: • Development of design procedures for I-joists with openings, considering compliance with other serviceability limit state aspects such as vibration performance, ponding, shrinkage and swelling, and permanent deflections due to long-term loads 174 • Development of design modification factors for joists with more than one opening, including the effects of spacing between them • Study of end and intermediate reaction resistances of I-joists with openings • Study of possible interactions between service condition and treatment factors and opening factors • Evaluation of the effectiveness of the connection between retrofits and the joist and suggesting a proposed connection type such as adhesive type and nailing pattern. • Study of more retrofitting techniques, including more retrofit types and hybrid reinforcers and deriving design procedures and modification factors for these types. • Proposing retrofitting techniques that provide the joist with a better flange integrity • Investigation of the effect of the thickness of the retrofits, and consideration of damage in retrofits while loading 175 Bibliography Afzal, M. T., Lai, S. Y., Chui, Y. H., & Pirzada, G. (2006). Experimental evaluation of wood I-joists with web holes. Forest Products Journal, 56(10), 26–30. American Forest & Paper Association Inc. (2001). Allowable stress design (ASD) manual for engineered wood construction. AF&PA. American Wood Council. (2015). National Design Specification for Wood Construction (p. 90). ANSYS Inc. (2006). ANSYS Software, Documentation (11.0). ANSYS Ltd. ANSYS Inc. (2012). ANSYS Software, Mechanical APDL (14.0). ANSYS Ltd. ASTM International. (2014). ASTM D143-14, Standard Test Methods for Small Clear Specimens of Timber (p. 31). https://doi.org/10.1520/D0143 ASTM International. (2019). ASTM D5055-19, Standard Specification for Establishing and Monitoring Structural Capacities of Prefabricated Wood I-Joists. https://doi.org/10.1520/D5055-19 Pirzada, G. B., Chui, Y. H., & Lai, S. (2008). Predicting Strength of Wood I-Joist with a Circular Web Hole. Journal of Structural Engineering, 134(7), 1229–1234. https://doi.org/10.1061/(ASCE)0733-9445(2008)134:7(1229) Baylor, G., & Harte, A. M. (2013). Finite element modelling of castellated timber I-joists. Construction and Building Materials, 47, 680–688. https://doi.org/https://doi.org/10.1016/j.conbuildmat.2013.05.076 Building Safety Standards Branch. (2018). British Columbia Building code (p. 1586). Crown Publications. Canada Wood Group. (2017). SPF Products. http://canadawood.org/products/spf/ Canadian Commission on Building and Fire Codes. (2010). National Building Code of Canada (p. 1404). National Research Council Canada. Canadian Standards Association. (2012). Construction sheathing (CSA O325-07). Canadian Standards Association. (2013a). Douglas Fir Plywood (CSA O121-13). 176 Canadian Standards Association. (2013b). Poplar plywood (CSA O153-13). Canadian Standards Association. (2014a). Canadian softwood plywood (CSA O151-14). Canadian Standards Association. (2014b). Evaluation of adhesives for structural wood products (CSA O112-14). Canadian Standards Association. (2019). Engineering Design in Wood - CSAO86-19. In CSA O86-19 (p. 337). CSA Group. Canadian Wood Council. (2017). Wood Design Manual (8th ed.). Chen, G., Li, H., Zhou, T., Li, C., Song, Y., & Xu, R. (2015). Experimental evaluation on mechanical performance of OSB webbed parallel strand bamboo I-joist with holes in the web. Construction and Building Materials, 101, 91–98. https://doi.org/https://doi.org/10.1016/j.conbuildmat.2015.10.041 Chui, Y. H., Pirzada, G., & Lai, S. (2005). Enhancing Shear and Bearing Strength of Wood I-joists. Clinch, R. W. (1993). Effect of holes on static flexural stiffness of a wood composite I-beam. In Forest products journal (Vol. 43). Damery, D., Clouston, P., & Fisette, P. (2007). Wood science education in a changing world: A case study of the UMASS-Amherst building materials & wood technology program, 1965-2005. Forest Products Journal, 57. Davids, W., Rancourt, D., & Dagher, H. (2011). Bending Performance of Composite Wood I-Joist/Oriented Strand Board Panel Assemblies. Forest Products Journal, 61, 246–256. https://doi.org/10.13073/0015-7473-61.3.246 Dinehart, D. W., & Morrissey, G. C. (2006). Experimental Evaluation of Wood I-joists With Web Openings. European Committee for Standardization (CEN). (2004). Eurocode 5: Design of timber structures (EC 5 EN 1995) (p. 121). European Standards. Fergus, D. A. (1979). Effect of web voids and stiffeners on the structural performance of composite beams. Purdue University. 177 Grandmont, J.-F., Cloutier, A., Gendron, G., & Desjardins, R. (2010a). Effect of Density on the Properties of Oriented Strandboard Web Stock Used in Wood I-Joists. Forest Products Journal, 60, 592–598. https://doi.org/10.13073/0015-7473-60.7.592 Grandmont, J.-F., Cloutier, A., Gendron, G., & Desjardins, R. (2010b). Wood i-joist model sensitivity to oriented strandboard web mechanical properties. Wood and Fiber Science, 42, 352–361. Guan, Z. W., & Zhu, E. C. (2004). Non-linear FE modelling of crack behaviour of openings in OSB webbed I-beams. ASCE Journal of Structural Engineering, 130(10), 1562–1569. https://doi.org/10.1061/(ASCE)0733-9445(2004)130:10(1562) Harte, A. M., & Baylor, G. (2011). Structural evaluation of castellated timber I-joists. Engineering Structures, 33(12), 3748–3754. https://doi.org/10.1016/J.ENGSTRUCT.2011.08.011 Islam, M. S. (2017). Performance of un-retrofitted and retrofitted wood I-joists with flange notch or web hole. Islam, M. S., Shahnewaz, Md., & Alam, M. S. (2015). Structural capacity of timber I-joist with flange notch: Experimental evaluation. Construction and Building Materials, 79, 290–300. https://doi.org/http://dx.doi.org/10.1016/j.conbuildmat.2015.01.017 Kirkham, W. J. (2018). Reducing construction defects and design errors in timber buildings. ASCE Structure Congress. Kisternaya, M., Kozlov, V., Grishina, I., & Leri, M. (2016). Tree rings as criteria for selection of timber for building of chapels in the Republic of Karelia. Dendrochronologia, 40, 143–150. https://doi.org/10.1016/j.dendro.2016.10.002 Kouris, L. A. S., & Kappos, A. J. (2014). A practice-oriented model for pushover analysis of a class of timber-framed masonry buildings. Engineering Structures, 75, 489–506. https://doi.org/10.1016/j.engstruct.2014.06.012 Lam, F. (2009). Timber products and manufacturing processes. In ICE manual of Construction Materials (pp. 717–726). https://doi.org/10.1680/mocm.35973.0717 Leichti, R. J., & Tang, R. C. (1983). Analysis of Wood Composite I-Beams with Glued Flange-Web Joints. Society for Experimental Stress Analysis, 45–50. 178 Loferski, J. R., & Hindman, D. P. (2008). Cold-Formed Steel Reinforcement for Improperly Cut Wood Composite I-Joists. https://doi.org/10.1061/(ASCE)1084-0680(2008)13:4(198) Mark, A., María, G.-M. L., & Enrique, H.-M. (2010). Engineered Bamboo I-Joists. Journal of Structural Engineering, 136(12), 1619–1624. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000235 Metwood Building Solutions. (2019). Joist Reinforcers. https://www.metwood.com/joist-reinforcers/ Ministry of Business, I. & E. (2014). New Zealand Building Code Handbook (p. 208). Modirzadeh, M., Tesfamariam, S., & Milani, A. S. (2012). Performance based earthquake evaluation of reinforced concrete buildings using design of experiments. Expert Systems with Applications, 39(3), 2919–2926. https://doi.org/10.1016/J.ESWA.2011.08.153 Montgomery, D. C. (2012). Design and Analysis of Experiments, 8th Edition. John Wiley & Sons, Incorporated. Morrissey, G. C., Dinehart, D. W., & Dunn, W. G. (2009). Wood I-Joists with Excessive Web Openings: An Experimental and Analytical Investigation. Journal of Structural Engineering, 135(6), 655–665. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000013 Myers, R. H., Montgomery, D. C., & Anderson-Cook, C. M. (2009). Response surface methodology: process and product optimization using designed experiments. John Wiley & Sons, Incorporated. NASCOR. (2010). I-joist specifier guide (p. 9). National Institute of Standards and Technology. (2010). Performance standard for wood-based structural-use panels. Voluntary product standard (DOC PS2-10). Department of Commerce. National Instruments. (2016). LabVIEW. National Research Council of Canada. (2019). Evaluation Report CCMC 13535-R - Nascor NJ10-12, NJH10-16, and NJM10-18 Series I-Joists. Polocoser, T. (2012). Evaluation of Remediation Techniques for Circular Holes in the Webs of Wood I- Joists. Oregon State University, Corvallis, Oregon, USA. 179 Polocoser, T., Miller, T. H., & Gupta, R. (2013). Evaluation of remediation techniques for circular holes in the webs of wood I-Joists. Journal of Materials in Civil Engineering, 25(12), 1898–1909. https://doi.org/10.1061/(ASCE)MT.1943-5533.0000737 Reynolds, T., & Casagrande, D. (2016). Comparison of multi-storey cross-laminated timber and timber frame buildings by in situ modal analysis. Construction and Building Materials, 102, 1009–1017. https://doi.org/10.1016/j.conbuildmat.2015.09.056 Shahnewaz, M., Alam, S., & Tannert, T. (2018). In-Plane Strength and Stiffness of Cross-Laminated Timber Shear Walls. Buildings, 8(8). https://doi.org/10.3390/buildings8080100 Shahnewaz, M., Islam, M. S., Ahmadipour, M., Tannert, T., & Alam, M. S. (2017). Reinforced Wood I-Joists with Web Openings. Journal of Structural Engineering (United States), 143(6). https://doi.org/10.1061/(ASCE)ST.1943-541X.0001747 Shahnewaz, M., Islam, M. S., Tannert, T., & Alam, M. S. (2019). Flange-notched wood I-joists reinforced with OSB collars: Experimental investigation and sensitivity analysis. Structures, 19, 490–498. https://doi.org/https://doi.org/10.1016/j.istruc.2019.02.009 Simpson, T. W., Poplinski, J. D., Koch, P. N., & Allen, J. K. (2001). Metamodels for Computer-based Engineering Design: Survey and recommendations. Engineering with Computers, 17(2), 129–150. https://doi.org/10.1007/PL00007198 Stat-Ease Inc. (2018). Design-Expert Software (11). The MathWorks Inc. (2011). MATLAB (R2010a). Vining, G. (2010). Technical Advice: Residual Plots to Check Assumptions. Quality Engineering, 23(1), 105–110. https://doi.org/10.1080/08982112.2011.535696 Wang, A., & Cheng, J. J. R. (1995). Shear behaviour of OSB wood composite I-beams with web opening. Wisniewski, B., & Manbeck, H. (2003). Residential Floor Systems: Wood I-Joist Creep Behavior. Journal of Architectural Engineering, 9. https://doi.org/10.1061/(ASCE)1076-0431(2003)9:1(41) Wood I-Joist Manufacturers Association. (2008). Establishing Prefabricated Wood I-Joist Composite EI (Issue July). 180 Wu, W. (2014). Experimental Analysis of Bending Resistance of Bamboo Composite I-Shaped Beam. Journal of Bridge Engineering, 19(4), 4013014. https://doi.org/10.1061/(ASCE)BE.1943-5592.0000557 Zhang, X., Hu, Y., & Fang, Y. (2011). Study of the bonding property between flanges and web plates of wood I-joist. Advanced Materials Research, 150, 1092–1095. https://doi.org/10.4028/www.scientific.net/AMR.150-151.1092 Zhou, D. (1990). A study of oriented structural board made from hybrid poplar. Holz Als Roh- Und Werkstoff, 48(7), 293–296. https://doi.org/10.1007/BF02626522 Zhu, E. C., Guan, Z. W., Pope, D. J., & D, R. P. (2007). Effect of Openings on Oriented Strand Board Webbed Wood I-Joists. Journal of Structural Engineering, 133(1), 145–149. https://doi.org/10.1061/(ASCE)0733-9445(2007)133:1(145) Zhu, E. C., Guan, Z. W., Rodd, P. D., & Pope, D. J. (2005a). A constitutive model for OSB and its application in finite element analysis. Holz Als Roh- Und Werkstoff, 63(2), 87–93. https://doi.org/10.1007/s00107-004-0513-y Zhu, E. C., Guan, Z. W., Rodd, P. D., & Pope, D. J. (2005b). Finite element modelling of OSB webbed timber I-beams with interactions between openings. Advances in Engineering Software, 36(11–12), 797–805. https://doi.org/http://dx.doi.org/10.1016/j.advengsoft.2005.03.027 Zhu, E. C., Pope, D. J., Rodd, P. D., & Guan, Z. W. (2005). Buckling of Oriented Strand Board Webbed Wood I-Joists. https://doi.org/10.1061/(ASCE)0733-9445(2005)131:10(1629)
Thesis/Dissertation
eng
Civil Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Alam, M. Shahria
Limit state design of timber I-joists with web openings and flange notches
Text