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Stability and reliability analysis of metal plate connected wood truss assemblies Song, Xiaobin 2009

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STABILITY AND RELIABILITY ANALYSIS OF METAL PLATE CONNECTED WOOD TRUSS ASSEMBLIES by Xiaobin Song B.Sc., Tongji University, 2000 M.Sc., Tongji University, 2003 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Forestry) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) March 2009 © Xiaobin Song, 2009 Abstract This thesis describes a study on the stability capacity and lateral bracing force ofwood beam-columns and metal plate connected (MPC) wood truss assemblies. A user-friendly computer program, SATA, was developed based on the finite element method (FEM). The program can be used to perform three-dimensional nonlinear structural analyses by using the Newton-Raphson and arc-length methods. The Monte Carlo simulation and response surface methods have also been incorporated into the program for the purpose of reliability analyses. Experimental studies were conducted to provide input parameters and verification for the developed software. Material property tests were performed to consider a variety of materials. Biaxial eccentric compression tests of wood beam-columns and full-scale tests of MPC wood truss assemblies were also carried out to study the critical buckling load and lateral bracing force. The program predictions were in good agreement with the test results. A reliability analysis was conducted for a simplified MPC wood truss assembly using the developed program. The effect of the variation of the structural behaviour and external loads on the critical buckling load of the truss assembly was studied. The adequacy of the 2% rule-of-thumb was also studied. This research bridges the knowledge gap that currently exists in the understanding and design of MPC wood truss assemblies and their lateral bracing systems. The test database and the output of the developed program contributes to the development of more efficient design methods for MPC wood truss assemblies and other structures where buckling failure is of concern. — 11 — Table of content Abstract.ii Table of content iii List of tables xi List of figures xii Notations xviii Acknowledgements xix Chaper 1. Introduction 1 1.1. MPC Wood Truss System in Canada 1 1.2. MPC Wood Truss System Design and Regulation 2 1.3. Research Motivation 3 1.4. Main Objectives and Research Work 5 1.5. Thesis Organization 6 1.6. Conclusion 7 Chaper 2. Literature Review 8 2.1. MPC Connections 8 2.1.1. Numerical models of MPC connections 8 2.1.1.1. Models based on the wood foundation theory 9 2.1.1.2. Models based on the load-slip relationship of individual tooth 10 2.1.1.3. Models based on equivalent springs or elements 12 — 111 — 2.1.2. Experimental study of MPC connection behaviour 12 2.1.3. Probabilistic characteristics of MPC connection behaviour 13 2.1.4. Conclusions on the modeling of the MPC connections 14 2.2. Stability Analysis of Wood Beam-Columns 14 2.2.1. Stability analysis of unbraced wood beam-columns 14 2.2.2. Stability analysis of laterally braced wood beam-columns 16 2.2.3. Experimental studies ofwood beam-columns 19 2.2.4. Conclusions on the stability analysis of wood beam-columns 20 2.3. MPC Wood Truss Assembly Behaviour 20 2.3.1. Numerical models of MPC wood truss assemblies 21 2.3.2. Experimental studies of MPC wood truss assemblies 23 2.3.3. Conclusions on MPC wood truss assembly behaviour 25 2.4. Reliability Analysis of Wood Truss Assemblies 26 2.5. Conclusion 29 Chaper 3. Finite Element Method Based Model Formulation 30 3.1. Formulation of FEM-based elements 30 3.1.1. Three-dimensional beam element 30 3.1.1.1. Local and natural coordinate systems 31 3.1.1.2. Nodal displacement and element deformation 32 3.1.1.3. Element internal force vector and stiffness matrices 33 3.1.1.4. Coordinate transformation matrix 34 3.1.2. Thinplateelement 36 - iv - 3.1.2.1. Local and natural coordinate systems 37 3.1.2.2. Nodal displacement and element deformation 38 3.1.2.3. Element internal force vector and stiffness matrices 39 3.1.3. Spring element 41 3.1.3.1. Nodal displacement 41 3.1.3.2. Force displacement relationship 42 3.1.3.3. Internal force vector and stiffhess matrices 43 3.1.4. Metal plate connection element 44 3.1.4.1. Local and natural coordinate system 45 3.1.4.2. Nodal displacement and tooth slip 46 3.1.4.3. Internal force vector and stiffness matrices 47 3.2. Newton-Raphson and Arc-Length Methods 48 3.3. Convergence Criteria 50 3.4. Failure Criteria 51 3.5. Parallel-to-Wood-Grain Stress-Strain Relationship 51 3.6. Size and Stress Distribution Effects 54 3.6.1. Size effect 55 3.6.2. Stress distribution effect 56 3.7. Conclusion 57 Chaper 4. Development of the Computer Program SATA 59 4.1. Programming Logic 59 4.2. Data Structure of SATA 61 4.2.1. Storage of the data.61 4.2.2. Major content of the data 63 4.2.3. Organization of the data and objects 64 4.3. An Overview of SATA 65 4.3.1. Configuration of the GUI 66 4.3.2. Pre-processor 67 4.3.3. FEM solver 69 4.3.4. Post-processor 71 4.4. Availability 72 4.5. Conclusion 72 Chaper 5. Experimental Study and Model Verification I: Biaxial Eccentric Compression Test of Wood Beam-Columns 73 5.1. Research Method 73 5.1.1. Material 73 5.1.2. Material property test 74 5.1.2.1. Parallel-to-wood-grain compression and tension tests 74 5.1.2.2. Nail connection tests 77 5.1.3. Biaxial eccentric compression tests of wood beam-columns 80 5.1.3.1. Specimen design 80 5.1.3.2. Support conditions 81 5.1.3.3. Loading system 83 5.1.3.4. Data acquisition 83 - vi - 5.2. Material Property Tests Results 84 5.2.1. Parallel-to-wood-grain compression and tension test results 85 5.2.2. Nail connection test results 86 5.3. Beam-Column Test Results and Model Verification 87 5.3.1. Failure modes of the wood beam-column tests 88 5.3.2. Model development 90 5.3.3. Model input parameters and size and stress distribution effects 91 5.3.4. Unbraced wood beam-column test results and model verification 92 5.3.5. Laterally braced wood beam-column test results and model verification 99 5.4. Conclusion 105 Chaper 6. Experimental Study and Model Verification II: Stiffness and Stability Capacity of MPC Wood Truss Assemblies 107 6.1. Research Method 107 6.1.1. Material 108 6.1.2. Individual truss and truss assembly designs 109 6.1.3. Material property tests 113 6.1.3.1. Dimension lumber tests 114 6.1.3.2. Plywood sheathing tests 115 6.1.3.3. Nailconnectiontests 117 6.1.4. Individual truss and truss assembly tests 121 6.1.4.1. Support conditions 122 6.1.4.2. Loading system 125 - vii - 6.1.4.3. Data acquisition .127 6.1.5. Test procedures for individual trusses and truss assemblies 129 6.1.5.1. Stiffness of individual trusses 129 6.1.5.2. Critical buckling load of individual trusses 129 6.1.5.3. Critical buckling load of the first truss assembly 130 6.1.5.4. Critical buckling load of the first truss assembly reinforced by a CLB system 130 6.1.5.5. Load distribution behaviour of the second truss assembly 131 6.1.5.6. Critical buckling load of the second truss assembly 131 6.2. Results of the Material Property Tests 132 6.2.1. MOE of dimension lumber 132 6.2.2. Flexural stiffness of plywood sheathing 134 6.2.3. Load displacement relationship of the nail connection 135 6.3. FEM-Based Models of Individual Trusses and Truss Assemblies 137 6.4. Test Results and Model Verification 139 6.4.1. Stiffness of individual trusses 139 6.4.2. Critical buckling load of individual trusses 142 6.4.3. Critical buckling load of the first truss assembly 146 6.4.4. Critical buckling load of the first truss assembly reinforced by a CLB system 150 6.4.5. Load distribution behaviour of the second truss assembly 151 6.4.6. Critical buckling load of the second truss assembly 153 - viii - 6.5. Discussion .158 6.5.1. Effect of plywood sheathing 159 6.5.2. Effect of a CLB system 160 6.6. Conclusion 160 Chaper 7. Reliability Analysis of Critical Buckling Load of MPC Wood Truss Assemblies 162 7.1. The Response Surface Method 162 7.2. Reliability Analysis of the Critical Buckling Load of MPC Wood Truss Assemblies.. 164 7.2.1 Truss configuration and FEM model 164 7.2.2 Fixity factors of the CLB bracing members 167 7.2.3 Random variables of the reliability analysis 171 7.2.4 Response surface function 171 7.2.5 Sampling scheme of the response surface method 173 7.3. Evaluation of the Response Surface Function Coefficients 175 7.4. Lateral Bracing Force 179 7.5. Reliability Analysis 180 7.6. Conclusion 182 Chaper 8. Conclusion and Suggestions 184 8.1. Summary of the Research Work 184 8.2. Significance of the Research Work 186 - ix - 8.3. Limitations and Suggestions .187 Bibliography 189 Appendices 197 Appendix A: Formulation of Finite Element Method Based Elements 197 Appendix B: MOE Profile of the MPC Wood Trusses 205 Appendix C: Truss Plate Placement, Gap Width and Initial Midspan Lateral Deflection 213 Appendix D: ANOVA analysis and Sampling Results 217 -x List of tables Table 5-1. Specimen design for the parallel-to-wood-grain compression and tension tests... 75 Table 5-2. Beam-column specimen design for the biaxial eccentric compression test 81 Table 5-3. Parallel-to-wood-grain compression tests results 85 Table 5-4. Parallel-to-wood-grain tension tests results 86 Table 5-5. Nail connection tests results 86 Table 5-6. Length effect factor, k, of wood compression and tensile strengths 91 Table 5-7. Maximum compression load and midspan biaxial moments of wood beam-columns 98 Table 5-8 Critical buckling load and lateral bracing force of Specimen Group 8 to 11 104 Table 6-1. Specimen configurations of plywood sheathing tests 116 Table 6-2. Material property tests results of dimension lumber, plywood sheathing and nail connections 137 Table 6-3. Calibrated MPC connection properties based on an individual tooth 140 Table 7-1. Simulation results of the fixity factors of CLB bracing members 170 Table 7-2. Probabilistic distribution parameters of the input random variables 173 Table 7-3. Coefficients of the response surface functions 177 Table 7-4. Probabilistic distribution parameters of the error term, 178 Table 7-5. Extreme value type I distribution parameters of the external load, qext 181 Table 7-6. Reliability indices and probabilities of buckling failure of the truss assembly.... 181 - xi - List of figures Figure 2-1. MPC connection model based on the wood foundation theory 9 Figure 2-2. MPC connection model based on the load-slip relationship of individual tooth (Foschi 1977) 11 Figure 2-3. Simply supported beam-column with elastic midspan support 16 Figure 2-4. Compression beam-columns with “full bracing” 17 Figure 3-1. Local coordinate system of the three-dimensional beam element 31 Figure 3-2. Natural coordinate system of the three-dimensional beam element 32 Figure 3-3. Direction cosines of the local and global coordinate systems 35 Figure 3-4. Local and natural coordinate systems of the plate element 37 Figure 3-5. Nodal displacements and force displacement relationship of a spring element... 42 Figure 3-6. Local and natural coordinate systems of the metal plate connection element 45 Figure 3-7. Solution scheme of the Newton-Raphson and arc-length methods 49 Figure 3-8. Parallel-to-wood-grain stress-strain relationship models 52 Figure 3-9. Polynomial model of the parallel-to-wood-grain stress-strain relationship 54 Figure 3-10. Non-uniform and uniform stress distributions of wood cross sections 57 Figure 4-1. Storage of SATA data 62 Figure 4-2. Organization of SATA’s data and objects 65 Figure 4-3. Layout of the GUI components of SATA 66 Figure 4-4. Input dialogs for material property and truss configuration 68 Figure 4-5. Graphic rendering modes and element information inquiry dialog of SATA 69 Figure 4-6. FEM-based solution configurations of SATA 70 - xli - Figure 4-7. Reliability analysis configurations and solution progress of SATA 71 Figure 4-8. Options of SATA’s post-processor 72 Figure 5-1. Parallel-to-wood-grain compression test setup 76 Figure 5-2. Parallel-to-wood-grain tension test setup 76 Figure 5-3. Nail connection behaviour in six degrees of freedom 77 Figure 5-4. Specimen configurations and test setup of the nail connection tests 79 Figure 5-5. Connection between the steel boot and the hydraulic actuator 82 Figure 5-6. The nail connection and load cell under the lateral bracing member 82 Figure 5-7. Loading system of unbraced wood beam-columns 83 Figure 5-8. Layout of the string pots used to measure the midspan lateral deflections 84 Figure 5-9. Load-displacement relationship of the nail connections 87 Figure 5-10. Failure modes of unbraced wood beam-columns 88 Figure 5-11. Failure modes of the laterally braced wood beam-columns 89 Figure 5-12. FEM-based model of an eccentrically loaded wood beam-column 90 Figure 5-13. Test results and model predictions of Specimen Group 1 93 Figure 5-14. Test results and model predictions of Specimen Group 2 94 Figure 5-15. Test results and model predictions of Specimen Group 3 95 Figure 5-16. Test results and model predictions of Specimen Group 4 95 Figure 5-17. Test results and model predictions of Specimen Group 5 96 Figure 5-18. Test results and model predictions of Specimen Group 6 97 Figure 5-19. Test results and model predictions of Specimen Group 7 97 Figure 5-20. Test results and model predictions of Specimen Group 8 100 - xlii - Figure 5-21. Test results and model predictions of Specimen Group 9 101 Figure 5-22. Test results and model predictions of Specimen Group 10 102 Figure 5-23. Test results and model predictions of Specimen Group 11 102 Figure 5-24. Relationship between the axial compression load and lateral bracing force ....104 Figure 6-1. Double Howe truss design 111 Figure 6-2. A auxiliary bracing system of a five-bay truss assembly 112 Figure 6-3. A CLB system of a five-bay truss assembly 113 Figure 6-4. E-grading of dimension lumber by using a Cook-Bolinders machine 114 Figure 6-5. Vibration test of dimension lumber 115 Figure 6-6. Test setup for the plywood sheathing Specimen Configuration Al 116 Figure 6-7. Test setup for the plywood sheathing Specimen Configuration A2 117 Figure 6-8. Test setup for the nail connection Specimen Configuration B1 119 Figure 6-9. Test setup for the nail connection Specimen Configuration B2 119 Figure 6-10. Test setup for the nail connection Specimen Configuration B3 120 Figure 6-11. Test setup for the nail connection Specimen Configuration B4 120 Figure 6-12. Test setup for individual truss testing 121 Figure 6-13. Test setup for truss assembly testing 122 Figure 6-14. Details of the roller support for individual truss testing 123 Figure 6-15. Details of the lateral support for individual truss testing 124 Figure 6-16. Load cell for the measurement of the lateral bracing force 125 Figure 6-17. Actuator- and wire- pulley system for loading purpose 126 Figure 6-18. Load cell underneath the wire rope of the loading system 127 - xiv - Figure 6-19. Measurements of the force and deflection responses 128 Figure 6-20. E-grading outputs of dimension lumber MOE 133 Figure 6-21. Vibration test results of dimension lumber MOE 134 Figure 6-22. Flexural stiffhess of plywood sheathing tested in Configuration Al 135 Figure 6-23. Flexural stiffness of plywood sheathing tested in Configuration A2 135 Figure 6-24. Load displacement relationship of the nail connection 136 Figure 6-25. FEM model of an individual truss 138 Figure 6-26. FEM model of a five-bay truss assembly 139 Figure 6-27. The reaction force and bottom chord deflection of individually tested trusses. 142 Figure 6-28. Web buckling failures of the individually tested trusses 143 Figure 6-29. Out-of-plane rotational stiffness of a MPC connection 144 Figure 6-30. Reaction force and midspan lateral deflection of W2 webs of individual trusses 145 Figure 6-31. Reaction force and bottom chord midspan deflection of individually tested trusses 146 Figure 6-32. Buckling failure of the first truss assembly 147 Figure 6-3 3. Metal truss plate bulging and teeth withdrawn 148 Figure 6-34. Applied load and midspan lateral deflection of W2 webs of the first truss assembly 149 Figure 6-3 5. Reaction force and bottom chord midspan deflection of the first truss assembly 149 - xv - Figure 6-3 6. Buckling failure of the first truss assembly reinforced by using a CLB system 150 Figure 6-3 7. Test setup for the load distribution behaviour test of the second truss assembly 151 Figure 6-38. Distribution of the reaction force and bottom chord midspan deflection of the second truss assembly 152 Figure 6-39. Buckling failures of the laterally braced W2 webs of the second truss assembly 154 Figure 6-40. Midspan tension failure and nail connection failure of the second truss assembly 155 Figure 6-41. Applied load and midspan lateral deflection of the W2 webs of the second truss assembly 156 Figure 6-42. Applied load and lateral bracing force of the W2webs of the second truss assembly 157 Figure 6-43. Applied load and bottom chord midspan deflection of the second truss assembly 157 Figure 6-44. Load distribution via plywood sheathing 159 Figure 7-1. Truss configuration (plate size shown in inches with 1 inch = 25.4 mm) 165 Figure 7-2. Three-bay MPC truss assembly based on a half truss configuration 166 Figure 7-3. FEM model of the three-bay MPC truss assembly 167 Figure 7-4. Fixity factor model for one CLB bracing member per W2 web 168 Figure 7-5. Fixity factor model for two CLB bracing members per W2 web 169 - xvi - Figure 7-6. Cumulative distribution of the fixity factors of CLB bracing members 170 Figure 7-7. Complete and partial factorial sampling schemes 174 Figure 7-8. Reliability analysis using response surface method based on a partial factorial sampling scheme 175 Figure 7-9. Sampled and predicted critical buckling loads of the truss assembly (zero CLB) 176 Figure 7-10. Sampled and predicted critical buckling loads of the truss assembly (one CLB) 176 Figure 7-11. Sampled and predicted critical buckling loads of the truss assembly (two CLB5) 177 Figure 7-12. Lateral bracing force ratio of W2 webs with one CLB bracing member 179 Figure 7-13. Lateral bracing force ratio of W2 webs with two CLB bracing members 179 Figure 7-14. Reliability indices of the truss assembly with different CLB systems 181 - xvii - Notations CLB: continuous lateral bracing CSA: Canadian Standards Association CSI: combined stress index DOF: degree of freedom FEM: finite element method GUI: graphic user interface KD: kiln dried LSM: least square method LVDT: linear variable differential transducer MOE: modulus of elasticity MPC: metal plate connected MSR: machine stress rated NBCC: National Building Code of Canada OOP: objective oriented programming SPF: Spruce Fine Fir TEAM: Timber Engineering and Applied Mechanics TPIC: Truss Plate Institute of Canada UBC: University of British Columbia W2: The webs with the highest slenderness ratio in a truss - xviii - Acknowledgements I like to thank my research supervisor, Prof. Frank Lam, for his direction and encouragement on my research work and personal life, without which I would not have been able to overcome the difficulties in the past years. I like to thank Prof. Ricardo Foschi for being available all the time and offering me help on every research topic addressed in this thesis. I also like to thank Prof. J. D. Barrett for his advice on my study of the size effect of wood strength properties. Special thanks are given to Prof. Gu and Prof. He from Tongj i University for leading me into this research area and encouraging me to continue the study. I want to thank Minghao Li, Thomas Tannert, Jungpio Hong, Hiba Anastas and others in our research group for inspiring me. I also want to extent the thanks to Hao Huang for helping me with the truss assembly tests, Xiaoqin Liu for providing me with the truss plate test data, Feng-Cheng Chang for helping me with the ANOVA analysis, and George Lee, Bob Myronuk, Larry Tong and others in our Lab for helping me setup the tests. This research work is sponsored by Natural Sciences and Engineering Research Council of Canada (grants CRDP J 315282), Canadian Wood Council, Western Wood Truss Association, Jager Building Systems, Alpine Systems Corp, and Miteck. Thanks are owed to them for providing financial support and truss design and fabrication. Finally, I want to give my deepest thanks to my parents, my elder brother and a very special friend, Liwei Zhang, for their continuous and selfless support on my life and research. - xix - Chaper 1. Introduction 1.1. MPC Wood Truss System in Canada Metal plate connectors were invented around the 1950’s to replace nailed plywood gussets in wood truss assemblies. Pre-engineered metal plate connected (MPC) wood trusses rapidly penetrated the North American residential market in the 1960’s and reached near saturation by the early 1980’s. In Canada, current sales of MPC wood trusses total up to $450 million annually. The Wood Truss Council ofAmerica estimates that MPC wood trusses are now used in over 75% of all new residential roofs in U.S. single family constructions. The percentage is probably closer to 90% in Canada. MPC wood trusses also dominate the agricultural building market, with some competition from steel trusses. Commercial buildings, where steel is the norm, represent the greatest potential for MPC wood trusses. Generally, MPC wood trusses are fabricated with visually graded or machine stress rated (MSR) dimension lumber with the cross-sectional size varying from 38 x 64 mm2 to 38 x 184 mm2.The metal truss plates are manufactured by high-speed stamping machines that punch out the teeth and shear the plates to desired sizes. In Canada, the metal truss plates are mainly stamped from 16, 18 or 20 gauge (US Standard gauge) sheet steel meeting the minimum requirements of the Canadian Standards Association (CSA) Standard 086-0 1. The strength of the metal truss plate connection is dependent on the shear and tensile capacities of the metal plate and the gripping of the teeth. 1.2. MPC Wood Truss System Design and Regulation In Canada, the design of MPC wood trusses is initiated by the building owner(s) or a designate. Typically, the building designer contacts a truss fabricator, who will supply a truss layout and structural design for a roof or floor system. The building designer takes responsibility for design and detailing of the truss supports and anchorages; however, the truss fabricator provides design of individual truss components. The building designer is also responsible for detailing the permanent truss bracings to resist lateral forces. The design process is regulated by provincial or territorial building codes. Wood trusses for houses and small buildings, regulated under Part 9 of the National Building Code of Canada (NBCC), are designed in accordance with the “truss design procedures and specification of light metal plate connected wood trusses” of the Truss Plate Institute ofCanada (TPIC), as per the NBCC Clause 9.23.12.11. Wood trusses for commercial or industrial buildings, on the other hand, are governed by the requirements of Part 4 of the NBCC and are designed in accordance with the wood design standard CSA 086-0 1, which refers to the truss plate testing standard CSA S347 “method of test for evaluation of truss plates used in lumber joints”. Trusses for agricultural building are also designed in accordance with Part 4 of the NBCC and CSA 086-0 1. Modifications for agricultural type structures in some provinces are based on the Canadian Farm Building Code where required. According to the TPIC truss design procedures, trusses are designed as a collection of single components by using a two-dimensional analogue. All chord members are considered as rigidly connected through web joints and lapped joints. Splices are considered as pinned unless -2- designed for moments; and all web joints and pitch break joints are also considered as pinned. Heel joints are represented by three fictitious joints and three fictitious members that are pinned to each other and rigidly connected to the top and bottom chords. This compound joint model allows the heel joint to be designed for moment with a moment factor ranging from 0.65 to 0.8. The joints are mainly designed for the lateral resistance of the teeth and the tensile and shear resistances of the plates. The truss members are designed for the member forces and moments obtained from static analysis of the structural analogue. A combined stress index is calculated for the panels on the basis of average member force combined with the greater ofthe maximum panel moment and the maximum panel point moment. For light frame truss systems consisting of three or more essentially parallel members spaced not more than 610 mm (2 feet) apart and so arranged that they mutually support the applied load, certain strength properties of the truss members made of sawn lumber are increased by 10% to account for the system and load sharing effects. These include the bending strength, longitudinal shear strength, and parallel-to-grain compression and tensile strengths. 1.3. Research Motivation MPC wood truss assemblies are highly repetitive and statically indeterminate. Due to the system and load sharing effects, the external loads are redistributed between and within trusses. The possibility of weak members within highly stressed sections is reduced; however, truss members can experience complicated stress strain situations and behave differently from the specimens in standard lab tests. MPC connections are semi-rigid rather than pinned or -3- completely rigid, as is assumed in the current design procedures. The critical buckling load and lateral bracing requirements of the truss webs under compression load effects have not been fully studied. The currently used 2% rule-of-thumb considers the lateral bracing force to be 2% of the compression force in the web (Throop 1947). This estimation is based on the assumption that the member is simply supported and out of plumb by 1% of the length. However, the MPC connections at the ends of the compression webs are not completely rotational free. The rotational stiffnesses of the IVIPC connections in the buckling plane can notably increase the critical buckling load. The stiffness of the nail connections between the webs and the lateral bracing members and the initial deflections of the webs in the buckling plane can also affect the critical buckling load and lateral bracing force. Performance-based design is more rational when compared to the conventional working stress design or limit state design. It can achieve an optimized design with respect to certain performance criteria by taking into account the variation of the structural behaviour and the external loads. In two international workshops on the performance-based building structural design (1998 and 2000), it was confirmed that efforts were needed in the following areas to develop the steps for reliability-based assessment of structural performance: • Robust structural analysis software; • Comprehensive test data for model input and verification; • Information on the random characteristics of the loads; • Reliability analysis software and procedures; and • Code implementation. -4- This thesis presents the development of a structural analysis computer program and the construction of a test database for model input and verification. Reliability analysis procedures are also discussed. 1.4. Main Objectives and Research Work The overall objective of this research work was to study the stability capacity and lateral bracing force of wood beam-columns and MPC wood truss assemblies. This was divided into several sub-objectives as follows: Objective 1: to develop a user-friendly computer program capable of three-dimensional buckling analysis of wood beam-columns, MPC trusses and truss assemblies with consideration of material nonlinearity and P-Delta effect of compression loads. Objective 2: to construct a database of basic material properties. The material of concern included full-size dimension lumber, nail connections fabricated with 6d and 1 Od common nails and plywood panels. The database would be used to establish the parallel-to-wood-grain stress-strain relationship of wood, load and displacement relationship of nail connections and flexure stiffness of plywood panels. Objective 3: to conduct biaxial eccentric compression tests of simply supported wood beam-columns with and without lateral bracing. The critical buckling load, midspan biaxial column deflection and lateral bracing force of the wood beam-columns subjected to different load eccentricities were to be studied. The test results were also to be used to verify the developed program. -5- Objective 4: to test full-scale MPC trusses and truss assemblies for the stifihess and load carrying capacity of individual trusses and truss assemblies, and the load distribution behavior of the truss assemblies. The test results were also to be used to calibrate and verif’ the developed program. Objective 5: to conduct a reliability analysis by using the verified program. The probability of failure of the truss assembly concerning buckling failure was to be evaluated. The randomness in the material properties, structural behavior and external load would be considered. Response surface method would be used to facilitate the reliability analysis. Objective 6: to study the adequacy of the 2% rule-of-thumb based on the results of the wood beam-column and MPC truss assembly tests and the reliability analysis. 1.5. Thesis Organization The thesis consists of eight chapters. The second chapter describes previous research work on the metal plate connections, stability capacity of wood beam-columns, structural behavior of MPC truss assemblies, and the reliability analysis methods used in wood truss assemblies. Chapter 3 describes the formulation of four types of FEM based elements, the incorporation of the Newton-Raphson and arc-length methods, the convergence and failure criteria, the parallel-to-wood-grain stress-strain relationship, and the size and stress distribution effects of wood strength properties. Chapter 4 describes the development of a computer program, SATA, as a user-friendly package, to perform three-dimensional nonlinear structural and stability analyses. The -6- Newton-Raphson and arc-length methods were incorporated for nonlinear buckling analysis, and the response surface method and Monte Carlo simulate were also incorporated for reliability analysis. Chapter 5 and 6 present the results of the experimental studies ofwood beam-columns and full-scale MPC trusses and truss assemblies. The calibration and verification of the developed program are also described. The adequacy of the 2% rule-of-thumb and the other two methods for evaluation of the lateral bracing force was also studied. Chapter 7 presents a reliability analysis of a simplified truss assembly subjected to uniformly distributed roof load. The probability of failure of the system concerning buckling failure of the compression webs was evaluated using the response surface method in conjunction with a partial factorial sampling scheme. The adequacy of the 2% rule-of-thumb was further studied. Chapter 8 summarizes the results, significance and limitation of the research work, and the suggestions of future research. 1.6. Conclusion This thesis presents the results of a study on the stability capacity and lateral bracing force ofwood beam-columns and MPC wood truss assemblies. The generated database and the output of the developed program contribute to the development of more efficient design methods for MPC wood truss assemblies where buckling failure is of concern. -7- Chaper 2. Literature Review Introduction This chapter describes the previous research work on the modeling of metal plate connected (MPC) joints, the stability capacity ofwood beam-columns, the system behaviour of MPC wood truss assemblies, and reliability analysis of wood truss assemblies. Many of the findings referred to in this chapter were considered during the development of the finite element method (FEM) based structural analysis computer program presented in this thesis. 2.1. MPC Connections MPC connections are widely used in the fabrication of light frame truss assemblies due to their low cost, ease of installation and strength efficiency. However, the mechanical properties ofMPC connections have not been fully understood because of the diverse geometry of the metal truss plates and the interaction between the truss members and the teeth of the metal truss plates. Much effort has been made to develop numerical models and construct test databases, which are normally used as input parameters and verification for the numerical models. 2.1.1. Numerical models of MPC connections Numerous models of MPC connections have been developed for computer program implementation. The models can be divided into three categories: -8- [1] Models that consider the tooth as a steel beam resting on a wood foundation; [2] Models that consider the tooth as a set of springs or similar elements, of which the load displacement behaviour is of concern; and, [3] Models that consider the whole connection as a set of springs or similar elements with equivalent mechanical properties, such as the flexural stiffness. The major progresses on these models are described in Sections 2.1.1.1 to 2.1.1.3. 2.1.1.1. Models based on the wood foundation theory The common assumption of this type of model is that an individual tooth can be represented by a steel beam resting on a wood foundation. The steel beam is assumed to be subjected to flexural moments, and the foundation is assumed to be under compression. Both the lateral deflection of the steel beam and the foundation bearing contribute to the overall deformation of the tooth. A free body diagram of such a tooth is shown in Figure 2-1. Figure 2-1. MPC connection model based on the wood foundation theory Foschi (1974) studied the load-slip characteristics of a nail. It was assumed that the L x I — N + dM k -9- nail yielded in bending and the wood underneath failed in bearing. The model was able to deal with the elasto-plastic problem of nail slip and the estimate ultimate loads. The idea was also used to model the teeth of metal truss plates. Crovella et al. (1990) studied the tension splice joint by using a FEM based model and an elastic foundation model. It was found that the FEM based model over-predicted the stiffness, while the elastic foundation model agreed quite well with the experiment results. Groom et a!. (1992) developed a model based on the elastic foundation theory. The governing differential equations were solved by the Runge-Kutta numerical procedures. Riley et al. (1999) quantified the wood foundation modulus and predicted the axial and rotational stiffnesses. 2.1.1.2. Models based on the load-slip relationship of individual tooth Foschi (1977) constructed a nonlinear load-slip model of an individual tooth embedded in wood. The model parameters depended on the applied load orientation, grain direction and major axis of the metal truss plates. Four standard test configurations, as outlined in CSA S347-99 “method of test for evaluation of truss plate used in lumberjoints”, were used to evaluate the parameters. Hankinson’s formula was used to extrapolate the model parameters to intermediate grain directions (Hankinson 1921). The connection behaviour was then determined by considering all the teeth within the plate-wood contact area. The model is shown in Figure 2-2. - 10- Figure 2-2. MPC connection model based on the load-slip relationship of individual tooth (Foschi 1977) Gebremedhin et al. (1992) fitted the test results of the load-slip curves to Foschi’s model. For simplicity, the model was assumed to level out at large slip. Triche and Suddarth (1988) employed Foschi’s model to develop a FEM based model of MPC connections. Ellegaard (2002) modified Foschi’s model by using two types of steel beams that were perpendicular and parallel to the major axis of the metal truss plates. Vatovec et al. (1996) developed a three-dimensional model of MPC tension splice joints by simulating the tooth-wood interface by three uniaxial springs acting along the principal axes of the joint. The load-slip relationships of the springs were calibrated from test data. Cramer et al. (1990) constructed a model to study the tension and moment resistances of the MPC wood splice joints. The tooth-wood interface was simulated by nonlinear springs. The characteristics of the springs were obtained by back-calculation from test data. The model -11- 2 x distinguished deformation due to wood crushing, plate yielding and wood-tooth interaction. The modified Newton-Raphson iteration procedure was used to solve the nonlinear problem introduced by the nonlinearity of the plate yielding and tooth-wood interface springs. 2.1.1.3. Models based on equivalent springs or elements Khalil et a!. (1984) studied plane frame structures by using the matrix method. Each joint of the structures was modeled by three linear springs. The mechanical properties of the springs were derived from test results. Sasaki et al. (1988) analyzed semi-rigid wood truss joints by using a set of spring groups, each with three linear elastic springs, to model the axial, tangent and rotational stiffnesses of the joints. Riley et a!. (1993) proposed a procedure for truss analysis in which the semi-rigid MPC joints were modeled by fictitious members. The lengths of the fictitious members were equal to the lengths of the metal truss plates. The other dimensions were derived based on the equivalence of the axial and rotational stiffnesses of the joints. 2.1.2. Experimental study of MPC connection behaviour McCarthy et al. (1988) tested MPC truss joints to derive the parameters based on Foschi’s model. The joints were fabricated with Southern Pine lumber and 20 gauge metal plates. Factorial analysis was performed to assess the sensitivity of the parameter to the resultant forces. - 12 - Gupta et al. (1990, 1992) tested three types of MPC joints, including the web-to-bottom chord, tension splice and heel joints, to determine the strength and stiffness properties. The joints were fabricated with Southern Pine No. 2 KD lumber of 38 x 89 mm2 in cross-sectional size and 20 gauge metal truss plates. The joint failure was found to be a combination of wood and teeth failure. More tests have been conducted on MPC joints which were fabricated with Southern Yellow Pine, Douglas Fir and Spruce Pine Fir lumber and loaded by tension, shear and moments to determine the stiffness and strength properties (Crovella and Gebremedhin 1990, Wolfe 1990, Gebremedhin et al. 1992, Groom and Polensek 1992, Vatovec and Gupta 1995 and Riley and Gebremedhin 1999). 2.1.3. Probabilistic characteristics of MPC connection behaviour Results of the probabilistic characteristics of MPC connection behaviour are scarce. Gupta et al. (1990, 1992) characterized three types of MPC joints with probability density functions. The tested joints were fabricated with 38 x 89 mm2 Southern Pine No. 2 KD lumber and 20 gauge metal truss plates. The probability plot technique, in conjunction with the Kolmogorov-Smirnov and Chi-square statistics theories, was used to determine the best distribution. It was found the strength data of the heel and web-to-bottom chord joints were best described by the normal distribution. The strength data of the tension splice joints, however, fitted none of the distributions considered in the study. - 13 - 2.1.4. Conclusions on the modeling of the MPC connections Most models have focused on the in-plane behaviour of the MPC connections. The out-of-plane behaviour has not been studied. The model parameters were evaluated based on test results that were sensitive to the wood species, lumber grade and thickness of the metal truss plates. Probabilistic characteristics of the MPC connections have not been fully studied. Further research is needed for the purpose of reliability analyses. 2.2. Stability Analysis of Wood Beam-Columns Light frame timber structures, such as MPC truss assemblies, consist of members that are 38 mm in thickness. These members are susceptible to buckling failure when subjected to compression load effects. Lateral bracing is commonly used to increase the critical buckling load. Previous research studies on the stability capacity and lateral bracing requirements of wood beam-columns are reviewed in this section. 2.2.1. Stability analysis of unbraced wood beam-columns The stability analysis of beam-columns originally focused on those made of steel and concrete. The well known Euler’s column formula (Timoshenko 1953) was used to estimate the maximum attainable compression load for a slender column. The formula was based on the assumption that the column behaved linearly when buckling occurred. Later, Engesser (1889, 1895) proposed the tangent modulus approach to take into account the material nonlinearity. - 14 - The method was further modified and called the reduced modulus approach or the double modulus approach, when it considered the strain reversal. However, it was found that the reduced modulus approach overestimated the maximum load compared to test results. This discrepancy was explained eventually by Shanley’s concept about the true column behaviour (1947). Current formulae for wood beam-columns appear to date from the work by Newlin and Trayer in 1925 when they suggested a transition formula of the column capacity between the crushing type failure and the long column linear buckling failure (Buchanan 1984). Larsen and Thielgaard (1979) developed a general theory for laterally loaded timber columns with consideration of the biaxial effect and lateral torsional buckling based on linear elastic material assumptions. For intermediate beam-columns, which can experience nonlinear stress-strain situations and large lateral deflections, closed form solutions are only available for simple cases. Numerical methods are more feasible in most real situations. Buchanan (1984, 1986) and Buchanan et al. (1985) proposed a two-dimensional strength model for wood beam-columns based on the column deflection curve method, which also considered combined axial load and bending moment, the variation of wood mechanical properties and the size effect. Zohn (1982) introduced a nonlinear FEM based program with experimental verification in terms of the interaction between the axial compression load and the bending moment. Koka (1987) investigated the laterally loaded wood beam-columns by using a - 15 - second-order FEM model. Lau (2000) studied the strength of wood beam-columns by a two-dimensional FEM model with consideration of the lengthwise variation of wood mechanical properties. Song and Lam (2006) extended Buchanan’s strength model to consider three-dimensional behaviour of wood beam-columns with consideration of the biaxial bending moments, the nonlinear wood stress-strain relationship and the initial lateral deflections. 2.2.2. Stability analysis of laterally braced wood beam-columns Timoshenko (1961) constructed a general model to study the beam-column subjected to concentric compression load and lateral load simultaneously. This model for a simply supported beam-column with elastic midspan support is shown in Figure 2-3. — 11 ‘2(a) M2 Figure 2-3. Simply supported beam-column with elastic midspan support The deflection and moment of the column were obtained by using the principle of superposition. The characteristic equation of the moment and deflection at midspan was constructed based on the condition of continuity: 1112 _______ srn2u1 sin 2u = 2(u1+u2)sin2(u1+u2) 2 — (2-1)(l +12) a2(11 +12) - 16- where u1 = 0.5l JP/EI, u2 = 0.5/2 JP/EI , Elis the bending stiffness, and a is the stiffness of the elastic support. The critical axial loads were solved as the roots of the characteristic equation. Winter (1958) developed a simplified method to compute the lower limits of the strength and stiffness requirements of a laterally braced beam-column. The beam-column was assumed to be braced by a “full bracing”, with which the inflection point of the column’s deflection curve was supposed to exist at the midspan. As the moment zeroed out at the inflection point, a fictitious hinge was placed at the midspan as shown in Figure 2-4. L/4 F Figure 2-4. Compression beam-columns with “full bracing” The minimum stiffness of the lateral bracing to achieve “full bracing” was calculated by static analysis. For example, the lateral bracing stiffness and force of a simply supported beam-column can be calculated by K=41+- (2-2) 4P Fbr KA_(A+A0) where A0 and A are the initial deflection and the deflection caused by external loading, K is the F 2 - 17 - lateral bracing stiffness, Fbr is the lateral bracing force, P is the concentric compression load and L is the length of the beam-column. Plaut and Yang (1993) and Plaut (1993) studied the stability capacity of an elastic beam-column with two spans of arbitrary ratio. The critical load of the beam-column was solved from the characteristic equations which were derived from equilibrium differential equations and continuity conditions. A formula was proposed based on Equation (2-2) for design implementation: K =1’i+i.5--” L A) (2-3) Fbr KA_(A+1.5A0) Tsien (1942) investigated the buckling of linear-elastic imperfect beam-columns supported by nonlinear elastic lateral bracing at midspan. The relationship amongst the external load, the initial deflection, the midspan deflection due to the external load and the lateral bracing force were derived. Medland (1977) studied the critical loads of inter-braced parallel compression beam-columns and the strength requirements of the lateral bracings. A normal stiffness matrix was formulated using stability functions to incorporate the weakening effect of the compression load. The critical load of the beam-columns was defined as when the stiffness matrix turned singular. Later, Segedin and Medland (1978) simplified the problem down to the solution of a pair of simultaneous differential equations and the continuity criteria. Zhang et al. (1993) developed a criterion that was based on potential energy, in order to study the minimum lateral bracing stiffness. A global stiffness matrix was constructed with consideration of the material and the geometric stiffness matrices. The stability problem was - 18 - then converted into an eigenvalue problem. Underwood et al. (2001) investigated the net lateral bracing forces ofmultiple webs or chords in a row braced by one or more continuous lateral braces. The ratio between the net lateral bracing force per web and the axial compression load in the web was found to range from 2.3% to 3.1%. Munch-Andersen (2004) studied the strength requirements of the lateral bracings of wood truss systems. The lateral bracings were assumed to be connected to an external support frame. The maximum lateral bracing force was determined as7r2N(e+u)/1,where N, e and u are the axial load, the maximum value of the initial deflection and the maximum allowable deflection of the bracing structure, respectively. 2.2.3. Experimental studies of wood beam-columns Few test data have been published on the stability capacity of wood beam-columns. The first scientific test on wood beam-columns can be dated back to 1729 by Musschenbrock (Timoshenko 1953). Booth (1964) described compression test of large members performed by Girard in 1798. Bryson (1866) tested 40 small specimens of various lengths made of dry white pine lumber, Buchanan (1984) tested wood beam-column made of visually graded lumber, and Lau (2000) tested wood beam-columns made of MSR lumber. Even fewer data have been reported on laterally braced wood beam-columns. Pienaar (1986) conducted tests on pitched roof trusses and estimated the lateral load at each top chord brace to be approximately 10% of the axial force in the chord. Hoyle (1984) measured the lateral bracing forces of discretely braced chords of parallel-chord wood trusses. It was found - 19- that the lateral bracing forces varied between 0.003 and 0.05 of the axial force in the chords depending on the type of lateral restraint provided. Waltz (1998) and Waltz et al. (2000) conducted tests on the discrete compression web bracing. Three theoretical models — Winter’s model, Plaut’s model, and the 2% rule-of-thumb — were discussed based on the test results. It was found that Plaut’s and Winter’s models provided a more rational basis for the discrete compression web bracing design than the 2% rule-of-thumb. 2.2.4. Conclusions on the stability analysis of wood beam-columns The theoretical and numerical models of the stability capacity and lateral bracing requirements of wood beam-columns have experienced tremendous improvement as the models were refined from linear elastic to nonlinear, from two-dimensional to three-dimensional and from individual member analysis to system analysis. Further study needs to be focused on the wood compression members within a system with consideration of the system effect, the support conditions and the lateral bracing stiffness. Reliability analysis is also needed to study the effect of variation of these factors. 2.3. MPC Wood Truss Assembly Behaviour MPC wood truss assemblies are highly repetitive and exhibit strong system effects. Structural analyses of these systems need to consider the interaction amongst the roof sheathing, the individual trusses, the lateral bracing members, and the semi-rigid connections - 20 - made of metal truss plates or nails. Many numerical models have been developed for software implementation. Full-scale tests of individual trusses and truss assemblies have also been conducted to provide input parameters and verification for the numerical models. A brief review is presented in this section. A detailed summary can also be found in Gupta’s work (2005). 2.3.1. Numerical models of MPC wood truss assemblies Varoglu and Barrett (1984) developed a structural analysis program for roof systems (SAR). The model considered the contribution of the roof sheathing to the stiffness of the systems and the load sharing between trusses. The formulation was based on minimization of the total potential energy of the systems with respect to the generalized displacements. The program was verified with full-scale truss assembly test results (Varoglu 1986, Lam and Varoglu 1988). Cramer et al. (1988) proposed two approaches to model the load distribution of light-frame pitched-chord truss systems. The first approach considered linear elastic truss members, pinned connections and crossing beam analogues for the roof sheathing. The second approach was developed based on the first approach to improve the stiffness of the sheathing model and to consider the nonlinearity of the IVIPC connections. The first approach was embodied into a matrix frame analysis program ROOFSYS (Cramer et al. 1989). LaFave et al. (1992) developed a three-dimensional frame element model to study the distribution of the loads in wood truss roof systems. The semi-rigidity of the joints was taken into account by incorporating a joint fixity factor into the stiffness matrix associated with a -21 - three-dimensional, rigidly connected frame element. The roof sheathing was simulated by two different models. In the first model, the roof sheathing stiffness was lumped into five rows of elements, which connected the top chord panel points of the trusses. In the second model, the sheathing stiffness was distributed within the roof plane and represented by elements that had stiffness properties of a sheet of plywood with standard width. The second frame model was found to perform better than the first one. Mtenga et al. (1995) investigated the load sharing effect and the partial composite action on the strength and reliability of MPC roof truss systems by using two-dimensional linear elastic frame elements. Crossing beams were used to model the roof sheathing for the load distribution effect; and, numerical simulation was carried out for the strength of the pitched Fink truss roof systems. The results showed that the repetitive factor of 1.15 used in the design codes was conservative. Li et al. (1998) constructed a three-dimensional model to investigate the system performance of ]VJPC wood roof systems. The trusses and plywood sheathing members were modeled by beam elements. The heel and bottom chord tension splice joints were modeled by spring elements. A truss system model was constructed with nine trusses connected by sheathing beams. The plywood sheathing was modeled by three sheathing beam elements on each side of the system based on the tributary areas. The bending stiffness of the top chord members were increased to consider the partial composite action. Cramer et al. (2000) quantified the load-sharing effect of MPC trusses using a detailed structural analysis model with statistical characterizations of the stiffness and strength properties of the lumber. The influence of the partial composite action was ignored. The load - 22 - sharing effect was quantified by the ratios of the combined stress indices of otherwise identical unsheathed and sheathed assemblies at the design load and twice the design load. Monte Carlo simulation was conducted for six common truss configurations and one dimension lumber joist floor. The computed mean value of the load sharing factors ranged from 1.06 to 1.24 in the truss assemblies and from 1.17 to 1.19 for the joist assembly. 2.3.2. Experimental studies of MPC wood truss assemblies Mayo (1982) summarized the load sharing capacity of a laboratory tested roof system fabricated with Fink truss rafters. The load sharing effect was assessed by applying a range of load combinations to the rafters and ceiling members and measuring the resulted deflections. Uniformly distributed loads over the complete roof and a single concentrated load placed at different positions on the rafters and ceiling members were considered. The test results showed that the uniformly distributed loads were redistributed within a range of 20% to -30%. For the concentrated loads, it was found that the load actually carried by the loaded rafter varied between 50% and 75% of the applied load. Wolfe et al. (1986) conducted tests on MPC roof trusses in two configurations to characterize strength and stiffness performance. Forty-two full size trusses were tested: twenty-four were tested to failure, and the other eighteen were tested to 1.25 times design load. Both truss configurations were Fink trusses of an 8.5 m (28 feet) span. Two top chord slopes (3:12 and 6:12) were considered. Southern Pine lumber of No. 2 grade and 38x 89 mm2 (nominally 2 by 4 in2) in cross-sectional size was considered. The metal plates were oversized to entice failure in wood. - 23 - Wolfe and McCarthy (1989) tested two full-scale light frame IVIPC wood truss assemblies to study the assembly interactions and to model the load distribution and the capacity of the assemblies. The trusses were fabricated with No. 2 Southern Pine lumber of 38 x 89 mm2 (nominally 2 by 4 in2) in cross-sectional size with a combination of 16 gauge and 20 gauge metal truss plates. The heavier truss plates used at the critical joints were intended to induce failure in wood members. The trusses were fabricated with two configurations and of three stiffness groups. The tests included applying loads to individual trusses in the assembly, to the defined sections of the roof and to the full assemblies up to failure. The results showed that the variation of truss stiffness, roof pitch and assembly configuration had significant effects on the ratio of the load capacity of the assembly to the load capacity of the weakest individual truss in the assembly. Wolfe and LaBissoniere (1991) tested conventional truss assemblies to study the assembly interaction effects on the stiffness and strength properties of individual trusses. Fink trusses with a top chord pitch of 6:12 and 3:12 and scissor trusses were tested. The trusses were fabricated with No. 2 Dense Southern Pine lumber and 20 gauge metal plates. The test procedures included testing individual trusses outside the assembly, testing each truss in the assembly and fully loading the roof assembly to measure the stiffness and strength. It was found that the composite action between the roof sheathing and the top chords contributed 24% and 7% to the stiffness increase of the Fink and scissor trusses, respectively. It was also found that the first failure observed usually defined the assembly failure mode. The interaction effect on truss strength was found to be dependent on the truss configurations and the assumptions made on the strength of the weakest member. - 24 - Wolfe et a!. (1988) presented a summary on the truss assembly interaction and failure mechanism observed from five tested truss assemblies. Two of the five truss assemblies were constructed using trusses with overdesigned metal plate connections to force failure in wood members. The testing procedures included material properties tests, individual trusses tested outside the assembly for both strength and stiffhess, and roof assembly tests consisting of a series of loadings within the design load range. The gable end effect was simulated by sheathing an additional truss of the assembly with a plywood diaphragm and supporting it at midspan. The characteristics of the roof assembly performance within the design load range comprised the linear relationship of the load and the deflection. The load displacement characteristics beyond the design load range turned out to be linear up to the point of first failure. Karacabeyli et al. (1993) tested twelve 3:12 pitched chord and three parallel chord MPC glue-laminated (glulam) trusses, all of which had spans of 13.7 m. The trusses were composed of 79 mm wide and 152 to 2299 mm deep glulam members connected by 16 gauge metal truss plates. All trusses were tested to failure. It was found that the ratio of the ultimate load to the design load was 3.5 on average and varied between 2.7 and 4.4. Both wood and metal truss plate failure modes were observed. 2.3.3. Conclusions on MPC wood truss assembly behaviour Previous research activities have been mainly focused on evaluating the load distribution and partial composite effects by comparing the test results of roof assemblies and individual trusses. The plywood sheathing was normally modeled by beam elements with - 25 - equivalent mechanical properties. The effect of the system behaviour on the stability capacity and the lateral bracing requirements of the individual trusses and the truss assemblies had not yet been studied. 2.4. Reliability Analysis of Wood Truss Assemblies The structural behaviour of wood truss assemblies exhibits notable variation, due to the system effect and the inherent uncertainties ofthe material properties. Reliability analysis is a natural choice to account for these uncertainties. Much effort has been made to quantify the variation of individual member behaviour (Ellingwood 1981, Foschi et al. 1993, Drummond et a!. 2001) and external loads on wood structures (Drummond et al. 2001, Bulleit et al. 1995, Foschi 1984). Varoglu (1986) conducted a study of the short-term reliability of truss roof assemblies by computer simulation. Different slopes and spans ofthe trusses and the strength and stiffness properties ofNo. 2 grade Spruce Pine Fir (SPF) lumber and MSR lumber were considered. The snow loads of two Canadian cities, Vancouver and Quebec City, were considered. The failure probability was evaluated using the Rackwitz-Fiessler algorithm (Rackwitz and Fiessler 1978). It was found that the 6:12 sloped roof trusses consistently exhibited a larger allowable span than the 4:12 sloped roof trusses at a common target reliability index. The trusses fabricated with MSR lumber had a significantly larger allowable truss span compared to the trusses fabricated with visually graded lumber. Lam and Varoglu (1988) calculated the baseline reliability indices for four conventional truss designs. The strength and stiffness properties of the truss members were - 26 - based on the Canadian Lumber Properties Program in-grade data. The snow load parameters used for the computer simulation considered rain load on the snow pack. The reliability indices were calculated using the Rachwitz-Fiessler algorithm (Raekwitz and Fiessler 1978). It was found the baseline reliability indices ranged from 1.96 to 3.40. Lam (1989) established the short-term baseline reliability indices for twenty-four conventional truss designs for six Canadian cities. The reliability indices were found to range from 1.63 to 4.02. The average reliability index of the trusses was found to be 2.86 if only the wood failure mode was considered in linear analyses. Lam (1990) also evaluated the short-term baseline reliability indices for eighteen conventional truss designs using MSR lumber for six Canadian cities. The reliability indices were found to range from 2.30 1 to 4.207. Folz and Foschi (1989) used the first-order reliability method, as implemented in the Rackwitz-Fiessler algorithm and solution procedures proposed by Der Kiureghian and Liu (1986), to study the reliability of light frame wood structural systems. The reliability of a series system was obtained by using the Ditlevsen bound procedures (Madsen et al. 1986). Hammon et al (1985) studied the effect of the correlations of the strength properties on the reliability of the roof trusses. Computer models were developed to simulate the structural behaviour of five common roof truss patterns. Monte Carlo simulation was conducted to calculate the probabilities of failure for each truss pattern and the level of correlation of the strength properties. The results indicated that the ratio of the snow load to the dead load had an effect on wood truss reliability. Rojiani and Tarbell (1985) investigated the reliability of wood roof trusses with consideration of various failure modes, variation of external loads and strength parameters, and - 27 - correlations between the loads and the resistances. The truss configurations studied included Fink, double-W, triple-W, and flat Howe. It was found that the risk levels for the wood truss members were comparable to those for steel and concrete members. It was also found that computed risk levels were influenced by the choice of the distribution for the random variables and the correlation between the load and the resistance parameters. Bulleit (1991) used stochastic finite element analysis procedures to calculate the first four moments of the nodal deflections and member forces to estimate the reliability of each structural element. In another work, Bulleit (1995) used the first-order reliability analysis method combined with the order statistics of the modulus of rupture of the wood members to evaluate the system reliability. Lam (1999, 2000) studied the influence of the length effect and multiple members loaded with non-uniform loads on the reliability and performance of MPC truss chords under tension. It was found that the reliability could be significantly lower than the target reliability level for single members adopted during the code development process. Foschi et al. (2000) developed a user-friendly reliability analysis software, RELAN, which software can be used to evaluate the performance of a given structure or to determine the key parameters of a structural system to achieve a target reliability level. More recently, Hansson and Ellegaard (2006) studied the system effect of a roof truss subjected to snow load by using Monte Carlo simulation. The MPC connections were assumed nonlinear and the wood members were assumed linear. No significant difference was found between the linear and nonlinear analyses. The system effect was found to be 5% and 9% for the wood and the metal plate failures, respectively. It was also determined that the system effect - 28 - was dependent on the assumed distribution of the external loads. In conclusion, previous research activities have been focused on wood member and the MPC connection failures. The buckling failure of the compression truss members and the influencing factors, such as the initial lateral deflection, had not been studied. 2.5. Conclusion This chapter described the previous research work on the modeling and testing of the structural behaviour of MPC truss assemblies. The limitations and research needs were discussed. - 29 - Chaper 3. Finite Element Method Based Model Formulation Introduction This chapter describes the formulation of four types of finite element method (FEM) based elements, the Newton-Raphson and arc-length methods, and the criteria of convergence and failure. The formulation is based on other researchers’ work and the methodologies in textbooks and is presented for the completeness of the thesis. The parallel-to-wood grain stress-strain relationship and the size and stress distribution effects of wood strengths are also described. 3.1. Formulation of FEM-based elements The formulation of the FEM-based elements is described in the sequence of the coordinate systems, the nodal displacements, the internal force vectors, and the stiffness matrices. Several common assumptions were made during the formulation: • Plane sections remain plane after deformation; • The deformation is small compared to the element dimensions; • The stress-strain relationship of wood is independent of the loading rate; and, • Torsion and shear failure can be ignored. 3.1.1. Three-dimensional beam element The beam element was used to model the wood beam-columns and the metal plate -30- connected (MPC) wood truss members. The element nodal displacements, internal forces and stiffness matrices were formulated within a local coordinate system. A transformation matrix was developed to convert them to the global (structural) coordinate system. 3.1.1.1. Local and natural coordinate systems The local coordinate system of the beam element was defined by element geometry. The x axis of the local coordinate system coincided with the longitudinal axis of the element. They and z axes corresponded with the weak and strong axes of the element’s cross section, respectively. The local coordinate system of the beam element is shown in Figure 3-1, where the nodal displacements are explained in Equation (3-2). y(v1,v) The natural coordinate system consisted of three dimensionless axes: , and K which were parallel to the x, y and z axes of the local coordinate system, respectively. The use of the natural coordinate system can facilitate the evaluation of the element force and stiffness matrices. The natural coordinate system is shown in Figure 3-2. y(v,v) (w , w) Figure 3-1. Local coordinate system of the three-dimensional beam element -31 - zQc) z=h/2 K1 y(77) _____________ y(i) y=b/2 Figure 3-2. Natural coordinate system of the three-dimensional beam element The two coordinate systems can be transformed to each other by using the Jacobian matrix, J, which can be expressed as: ãx 8); ôz ôx -- --=J.-- (3-1) 8i 877 th7 th7 8y 8y 8 8 8K8K8K 3.1.1.2. Nodal displacement and element deformation A beam element was defined by two nodes, each with six degrees of freedom (DOFs). The nodal displacement vector, a, consisted of twelve components with one in each DOF as: aT = [“i v1 w1 é wi V1 U2 V2 W2 82 w v j (3-2) where u, v and w and w’ , v and 8 are the translational and the rotational displacements defined in the x, y and z axes, respectively. The operator, ‘,represents the partial differentiation with respect to x. The displacement vector, u = {u, v, w,8}T, which consisted of three translational displacements and one torsional displacement, was derived from the nodal displacement vector, x x=O = —1 - 32 - a, based on the shape functions. The calculation can be expressed as u = Ba, where B is the kinematic matrix and can be assembled as: N10 000 0N2 0000 B1 O 0 M1 0 M2 0 0 0 Al3 0 M4 0 B2Br: (3-3)0L1 00 0L3 00 0L4 B3 00 ON1 0000 ON2 00 B4 The row vectors, B1, where i1 to 4, consisted of the shape functions for the specific components of the element displacement vector, u. For brevity in the main text of this thesis, the shape functions, N, L1 and M, are presented in Appendix A. For beam-like problems, the normal strain, s, and the torsional strain, y, are of maj or concern. In this study, 6 and y were calculated from the element displacements as: 1 2 ii 1 ‘2 ii 1 ‘26—U +—(w) —zw +—(v) —yv +—(pO) y=p& where w” is the second derivative of w with respect to x and p is the radial distance to the centroid of the cross section. The element stresses, including the normal stress, u, and the shear stress, r, were calculated from the strains as: U = and r = Gy (3-5) where Esec is the secant modulus of the stress-strain relationship and G is the modulus of rigidity. 3.1.1.3. Element internal force vector and stiffness matrices The internal force vector and the stifihess matrices were derived based on the - 33 - principle of virtual work. The results are described in this section. The details of the derivation are presented in Appendix A. The internal force vector, was calculated by ‘int =aTK+Pt (3-6) where K is the element stiffness matrix and P is the nonlinear internal force. In this study, K and P were calculated by: P T I 2 2 T TK=IEsec(Bi B1 +z B2 B2 +y B3 )+Gp24B4dV (3-7) 1 2 2 1 2 2 T 1 = f[Esec(Bi B1 +h K B2 B2 27 B3 B3)+Gp B4 JDet(J)dd’7dK ‘t Esec[aCaB +BaaTC]Det(J)ddildK (3-8) where C=(B2 B2+B3B3+p24B4). The tangent stiffness matrix, K, was obtained by differentiating the internal force vector, with respect to the nodal displacement vector, a, as: Kt= Io(B12T +BTB +p2BiTBi}Iv+ IGP2B14TB1dV (3-9) = S [EtanCTC+c{B2TB1 + BTB +p2B’ T + Gp2BTB]Det)dd1dK where is the tangent modulus of the stress-strain curve and Det(J) is the determinant of the Jacobian matrix. 3.1.1.4. Coordinate transformation matrix In this study, the nodal displacement, internal force and stiffness matrices were -34- originally formulated in the local coordinate system and needed to be transformed into the global coordinate system for consistency. A coordinate transformation matrix, T, was assembled for this purpose. Assuming the two coordinate systems share the same origin point, the coordinate transformation matrix, T, can be established based on the direction cosines between the x, y and z axes of the global coordinate system and the x, y and z axes of the local coordinate system. The definition of the direction cosines is shown in Figure 3-3. J(; y Figure 3-3. Direction cosines of the local and global coordinate systems A 3 x 3 transformation matrix can be assembled as: cos(x, x) cos(x, y) cos(x, z) T3 = cos(y,x) cos(y,y) cos(y,z) (3-10) cos(z, x) cos(z, y) cos(z, z) 3x3 Based on the 3 x 3 transformation matrix, a 6 x 6 transformation matrix can be assembled to transform the nodal displacements with six components. For example, the nodal displacements, a1, in the local coordinate system can be transformed from the nodal displacements, r•, in the global coordinate system as: -35- U cos(x,x) cos(x,y) cos(x,z) 0 0 0 u v cos(y,x) cos(y,y) cos(y,z) 0 0 0 v a — W — T r — cos(z, x) cos(z, y) cos(z, z) 0 0 0 w (3-11)6 1 0 0 cos(x,x) cos(x,y) cos(x,z) 8 311 0 0 0 cos(y,x) cos(y,y) cos(y,z) w 11 0 0 0 cos(z, x) cos(z, y) cos(z, z) 11 Similarly, the nodal displacement vector, a, of a beam element, which consisted of 12 components, was transformed by a 12 x 12 matrix as: 1a1 rT6 O1Ir1 a= I =Tr (3-12)LajJ L° T6]1J where i andj are the node numbers. The tangent stiffness matrix and internal force vectors in the local coordinate system of a beam element were transformed by the same 12 x 12 transformation matrix, T12, as: K TKtanTi2 (3-13) TP where the superscript, g is for the items defined in the global coordinate system and the superscript, T is for the transpose of the matrix. The transformation from the global to local coordinate systems can be achieved simply by replacing the transformation matrix, T, by its inverse matrix, T1,which is equal to the transpose matrix, TT. 3.1.2. Thin plate element The thin plate element was used to model the plywood sheathing, which can be subjected to both in-plane and out-of-plane load effects. The element formulation was based on the small deflection plate theory. Some additional assumptions were made: - 36 - • The plate is elastic and orthotropic; • The element is initially flat and rectangle in its plane; • The plate thickness is constant and small compared to other dimensions; • The out-of-plane deformation is such that a straight line, initially normal to the middle surface, remains straight and normal to the middle surface after deformation; and, • The stresses normal to the middle surface are negligible. 3.1.2.1. Local and natural coordinate systems The origin of the local coordinate system coincided with the geometrical centroid of the element. The x andy axes of the local coordinate system were parallel to the edges of the plate element, and the z axis was perpendicular to the plane of the plate element. The natural coordinate system was defined at the same origin point with its axes parallel to the axes of the local coordinate system. The two coordinate systems are shown in Figure 3-4. y ‘7 b Top view Side view Figure 3-4. Local and natural coordinate systems of the plate element - 37 - Similar to the beam element, the two coordinate systems of the plate element were related by the Jacobian matrix, which can be calculated based on the element dimensions as: a b o0 a a h a =J. — = 0 — 0 — 2 ay a a ooLa — 2 aIC az ôz where b, h and t are the width, length and thickness of the element, respectively. 3.1.2.2. Nodal displacement and element deformation A plate element was defined by four nodes, each with six DOFs. The nodal displacement vector, a, consisted of 24 components as: T T T Ta = e’i a2 a3 a4 (315) where af = v w w1, w1,,), i = ito 4, and u, v and w are the translational displacements defined in the x, y and z axes of the local coordinate system, respectively. = 6 lox, = Ow1 lOy and wj, =82w1 lOxOy are the partial derivatives of the displacement, w. The element displacement vector, u = {u v w}T was established from the nodal displacement, a, and the kinematic matrix, B as: a1 0 0 0 0 0 N2 0 0 ... 0 0 B1 a2 u= 0L1 0000 0L2 0... 0 0 =B2a a3 0 0 M1 M2 M3 M4 0 0 M5 ... M15 M16 B3 a4 where N1 and L (i = ito 4) are the shape functions of the displacements, u and v; - 38 - M (i = ito 16) are the shape functions of the displacement, w. The shape functions are presented in Appendix A. The normal strains, e. and and the shear strain, Y.xji, of the plate element were derived from the element displacements, u, v and w, as: gx =u =v —zw, (3-16) = u, + v + ww — 2zw The detailed derivation of the shear strain, y, can be found in He’s work (2002). The stress components of the plate element were calculated from the strains and the orthotropic material property matrix, D. The calculation can be expressed as: _________ 0 {:}D{z;} :: :::::: G{1} (3-17) where E and E are the moduli of elasticity defined in the x andy axes of the local coordinate system, v and v,, are the Poisson’s ratios, and G is the modulus of rigidity. 3.1.2.3. Element internal force vector and stiffness matrices The internal force and the stiffness matrices of the plate element were also derived based on the principle of virtual work. The results are presented here for the completeness of the thesis. The details of the derivation are presented in Appendix A. The internal force vector, was calculated by: - 39 - “mt =aTK+Pint* (3-18) where K is the stiffhess matrix and can be calculated by: K = i(klBlTBl +kiz2B3T+k2BTB +k2z2B3yyTBy,bet(J)dd,ldK + 1k3[B2YTB1X+ BiTB2+ z2(B3YYTBXX +B3TB)]Det(J)dd77dic + fG(BlTBl +B2TB +B1yTB2x+B2TBi)Det(J)dd77dic The nonlinear internal force vector, was calculated by: 1 = Ikl[Bl.xaaTB3xTB+aTBTBa] +k2[ByaaTB3T +aTByTByaBy]DetJ)dd7lJK +1k3[B2yaaTBxTB+Bi,aaT3+. aBBaBi+aTBxTBy]Detcgdl?iK + iG[(B1+B taT(B3T +B yTB )+aTB3TBya(1+i)]Det@cIüiIic where k1 = E 1(1 — k2 = 1(1 — = 1(1 — and = E 1(1 — are the stiffness constants. The tangent stiffness matrix, was calculated by differentiating the internal force vector with respect to the nodal displacement vector, a. can be expressed as: = = ô(aTK+ P*) K + (3-19) where 811flt = Jk1[BXTaTB3TB +BiaB3T+B3xTBaBlxjDetT)dd7iK + 1k2[B2TaTB3y +B2yaB3T +B3yTBa2]Det1)dd7dic + fk3[B2yTaTBx B +B2yaB3xT+B1xTaTB3y B + BbaB3yT}DetJ)d4i7dK + fk3(ByTBa ix + B3XTBa2Y)+ G(B1 + B2)TaT(B3XTBY+B3TB)DetJ)d4d1?dK + fG[(B1+ B2 +B3TB)+(B3XTB +B3TB(B1+B2)JDetJ)cid7iic - 40 - The coordinate transformation matrix between the local and global coordinate systems of the plate element was derived in the same way as for the beam element. For example, the nodal displacement vector, a, in the local coordinate system can be transformed from the nodal displacement vector, r, in the global coordinate system as: a1 T6 r• aj T6 r3 a= = =T24r (3-20) ak T6 rk a1 T6 r1 where T6 can be calculated by using Equation (3-11). Transformation of the internal force vector and stiffness matrices can be done similarly, as shown in Equation (3-13). 3.1.3. Spring element The nonlinear spring element was used to model the nail connections. The element formulation was conducted in the local coordinate system of the spring element, which was defined by two additional reference nodes. 3.1.3.1. Nodal displacement The spring element was defined by two nodes, each with six DOFs. The nodal displacements and local coordinate system are shown in Figure 3-5. -41 - Figure 3-5. Nodal displacements and force displacement relationship of a spring element The nodal displacement vector, a, of a spring element can be expressed as: aT = V1 w1 6 Wj v u v w 6 Wj where i andj are the node numbers, u, v and w and 6, w. and v. are the translational and rotational displacements, as shown in Figure 3-5. The element displacement vector, u, was calculated as the difference between the nodal displacements of the two spring nodes in all six DOFs. The kinematic matrix, B, for the calculation of u can be expressed as: u 100000—100000 v 0100000—10000 w 00100000—1000 =Ba= a (3-21) 6 000100000—100 w’ 0000100000—10 V 0000010000 0—1 3.1.3.2. Force displacement relationship (v,v1 y (u,O1) ! Deformation A The relationship between the spring forces and the spring displacements in individual DOFs was defined separately. The load and displacement relationship in an individual DOF of the spring element was characterized by an exponential model based on Foschi’s work (1977): F(A) (mo + mi jA)[1 - em0] (3-22) - 42 - where A is the spring displacement, F(A) is the spring force, and mU, m and k are the parameters representing the intercept and slope of the asymptote and the initial stiffness of the force and displacement curve in Figure 3-5. The secant and tangent moduli of the force displacement curve were calculated from the quotient and the first derivative of the spring force, F(AI), with respect to the displacement, A, as: clF km1z\ ktan = — = m1 +(k+ —m1)e 0dA m0 (3-23) ksec = = (m0 + m1 A)[1 — e1m0 ] /A The spring force vector, f, was calculated based on the element displacement vector, u, as: F ksec,u u F ksec,v V F ksec W = F = Dsecu = (3-24) 6 ksec, 0 F, ksec,w’ W F, ksec,v V where Dsec is the secant modulus matrix, and ksec,u, ksec,v and ksec,v are the secant moduli of the force and displacement curves in all six DOFs of the spring element. 3.1.3.3. Internal force vector and stiffness matrices The internal force vector and stiffness matrices were derived based on the principle of virtual work. Considering the virtual work done by the spring force, f, at a virtual displacement, - 43 - = = fT = from which the internal force vector, P11, can be calculated by: 1int = fTB = (Dsecu)TB= aTBTDsecB = aTK (3-25) where K is the stiffness matrix and can be calculated by K = BTDsecB. The tangent stiffness matrix, Ktan, was calculated as the first derivative of the internal force vector, with respect to the nodal displacement, a, as: = 3Pjt = 6(f TB) = (fTB) = BTDB (3-26) where Dt is the tangent modulus matrix and can be calculated similarly to Equation (3-24) with the secant moduli replaced by the tangent moduli. The coordinate transformation matrix of the spring element was the same as that of the beam element. 3.1.4. Metal plate connection element The metal plate connection element was developed to model the metal truss plate connections of MPC wood truss assemblies. The element was formulated to consider both the in-plane and out-of-plane behaviour of the connections. The formulation for the in-plane behaviour of the connections was based on Foschi’s work (1977). The out-of-plane behaviour was modeled by using nonlinear springs, as formulated in last section. The following descriptions in Sections 3.1.4.1 to 3.1.4.3 are focused on the in-plane behaviour of the connections. - 44 - 3.1.4.1. Local and natural coordinate system The origin of the local coordinate system of the element coincided with the geometric centroid ofthe metal truss plate. The x-y plane of the local coordinate system was coplanar with the metal truss plate. The x andy axes were parallel and perpendicular to the major axis of the metal truss plate, and, the axis z was normal to the plane of the metal truss plate. The origin of the natural coordinate system coincided with the geometric centroid of the contact area between the metal truss plate and the truss member. The axes of the natural coordinate system were parallel to those of the local coordinate system. The two coordinate systems are shown in Figure 3-6. ________________________ 1(-i, 1) 1) Grain direction , ‘iO ‘ = = 2 4 2(-1, -1) 3(1, -1) Wood truss Plate principle axis Figure 3-6. Local and natural coordinate systems of the metal plate connection element The contact area was defined by four nodes, the coordinates of which were measured in the local coordinate system. The contact area was isoparametrically transformed into a rectangle in the natural coordinate system to facilitate numerical integration. By definition, the coordinates of a point within the contact area were transformed between the two coordinate systems as: 3 3 x = N(,i)x1 and y =N1(,i)y (3-27) 1=0 1=0 - 45 - where and , are the natural coordinates, x andy are the local coordinates, and i is the number of the four nodes. The shape functions, N, are presented in Appendix A. Based on these shape functions, the Jacobian matrix can be evaluated using Equation (3-1) with x andy substituted with and ii, as in Equation (3-27). 3.1.4.2. Nodal displacement and tooth slip The metal plate connection element was defined by two nodes, with one node on the metal truss plate and the other on the truss member. Each node was considered with six DOFs. The nodal displacement vector, a, can be expressed as: aT = v w u v w o, where u and v are the two in-plane translation displacements and O is the rotation angle around the z axis. The tooth slip was defined as the difference between the displacements of the wood and the metal truss plate at the position of the individual tooth. For an individual tooth within the contact area, the components of the tooth slip, Ax and Ay, were calculated based on the kinematic matrix, B, and the nodal displacement vector, a, as: Ax={i 0 0 0 0 —y —1 0 0 0 0 y}a=Ba Ay={01O00x0—1O00—x}a=Ba (3-28) where B and B are the row vectors for the tooth slip in x andy axes, respectively. The tooth force corresponding to such a slip was calculated by substituting the components, Ax and Ay, in the same exponential function (Foschi 1977) as for the spring elements: - 46 - F(Ax) (m0 +mi)[1_e_u1m0 I (3-29) F(Ay) = (m0 + m1 AY)[1 — e The parameters were evaluated based on the four standard tests defined in CSA S347-99 “method of test for evaluation of truss plate used in lumber joints”. Hankinson’s formula (1921) was used for the tooth loaded at intermediate grain directions of the truss members. 3.1.4.3. Internal force vector and stiffness matrices The internal force vector and stiffness matrices were also derived based on the principle of virtual work. The virtual work of the element was calculated by the summation of the virtual work done by the tooth force, F, at a virtual slip, A, of the individual teeth. Replacing the summation by integration over the contact area, the virtual work can be expressed as: = f(F(Ax)&x + F(Ay)SAy)Jxdy = S(F(/xx)BXT + F(Ay)B )lxdybà = Pbli (3-30) where A is the contact area and P is the internal force vector, which can be expressed by: P = f(F(Ax)BT + F(Ay)B )ixdy Similarly, the tangent stiffness matrix was evaluated by differentiating P with respect to the nodal displacement, a, as: kt = AÔ& + }lxdY = 1(ktan,xB’Bx + ktan,yBBy )Jxdy (3-31) where ktan,x and are the tangent stiffnesses of the load-slip curve of a individual tooth, as in Equation (3-29). - 47 - The integration was conducted in the natural coordinate system by replacing dxdy with Det(J)dd i . For example, the integration of the internal force vector can be carried out as: = 1(bx(,hl)BxT +Fy(4,ii)B)Jet(J)ddi7 (3-32) where Det(J) is the determinant of the Jacobian matrix, J. The coordinate transformation matrix of the metal plate connection element was the same as that of the beam element. 3.2. Newton-Raphson and Arc-Length Methods The Newton-Raphson and arc-length methods are commonly used in nonlinear structural analyses. The arc-length method uses a scalar, 2, called load coefficient and a reference load vector, Pref, to represent the external loads. The arc-length method can be used to model post-buckling behaviour (Ramm 1980, Crisfield 1982, Forde and Stiemer 1987). In this study, the Newton-Raphson and arc-length methods were incorporated into the FEM-based models to study the critical buckling load and the post-buckling behaviour of the structures. The algorithm of the arc-length method is briefly described here for better understanding. The solution scheme of a nonlinear analysis using the updated Newton-Raphson and the arc-length methods is shown in Figure 3-7. - 48 - Figure 3-7. Solution scheme of the Newton-Raphson and arc-length methods Consider the equilibrium equation at step m, which can be expressed as: Ktan Au =AP+P’ —F’ A2Pref +R’ (3-33) where K is the tangent stiffhess matrix evaluated at displacement Urn, Au is the current displacement increment, P and F’ are the external and internal force vectors evaluated at th sub-step, respectively, and R’ = P’ — F’ is the out-of-balance load vector. Au can be expressed in two parts (Au1 and Au”)as: Au=A2•Au’ +Au11 (3-34) Substitution in equation (3-33) yields: Ktan A2Au’ +K •Au11 2Pref +R’ (335) The two parts of the displacement increment, Au1 and Au11, can be solved as: Au1 Ktan’Prej (3-36) Au” Ktan’R’ The increment vector (Au, A2), which consists of the load and the displacement increments, is assumed to be perpendicular to the previous solution , 2) at th sub-step to entice convergence. Therefore, the scalar product of the two vectors should approach zero: (u1,fl2j). (Au, /JA%) = 0 (3-37) 1 A 2m uII -:; 1X2z\u m+1 Ui Urn U - 49 - where ,B is a scaling factor. Substituting the expressions in Equation (34) into Equation (3-37) yields: (u,,,821)(A2Au’ +Au”,f2)=A2Au’u+Au”u1/32A2=O (3-38) The coefficient, A1, of the load increment can be solved as: A2= —Au11u (339)Au’u, +,822 The displacement increment, Au, can then be obtained by substituting A2 into Equation (3-34). A negative A2 implies that the external load needs to be reduced in order to achieve convergence. 3.3. Convergence Criteria The solutions obtained from the FEM models were checked at the end of each load step. The out-of-balance load vector, R, or the increment displacement vector, Au, was compared to a prescribed tolerance. The convergence was considered to be met when the Euclidean norm of these vectors was less than the tolerance. The convergence criteria can be expressed as: RI Au! s or (3-40)II uI where the operator, , calculates the Euclidean norm of a vector; P is the external load; ,is the displacement solution achieved at the previous step; and, 8 is the prescribed tolerance. The magnitude of the tolerance is essentially arbitrary. Normally, it is determined based on the balance between the computational cost and the precision of the solution. A tolerance level of 1 .OE-3 was used in this study. - 50 - 3.4.Failure Criteria Different failure modes were considered in this study; and failure criteria were used to determine whether the structures or individual members had failed. In general, two types of failure modes and the corresponding failure criteria were considered in this study. The first type of failure mode was material failure, which was characterized by the exceedance of the material strength, due to the stresses induced by the external load effects. Normally, the material strength was evaluated based on test results. For wood, additional reductions were considered to account for the size and stress distribution effects. The material failure criterion, based on the internal stresses or forces calculated at the integration points of the elements, was checked at the end of each load step. The second type of failure mode was instability failure, also called buckling failure. Normally, compression beam-columns with high slenderness ratio are more prone to buckling failure than material failure. Unbraeed compression webs of MPC truss assemblies are also susceptible to buckling failure due to insufficient lateral stiffness. The buckling failure criterion for the load- and displacement-controlled analysis was the singularity of the tangent stiffness matrix; and, if the arc-length method was used, a negative load increment, &, would imply buckling failure. 3.5.ParalleI-to-Wood-Grain Stress-Strain Relationship Wood truss members are generally designed to be stressed in the wood grain direction; therefore, the parallel-to-wood-grain stress-strain relationship is one of the most important -51 - input parameters for the truss analysis models. Many models have been proposed to quantify’ the stress-strain relationship. Most of the efforts have been made to model the stress-strain relationship of wood under compression. (Normally wood is assumed to be nonlinear under compressive stresses and linear under tensile stresses). Comprehensive literature reviews have been made by Buchanan (1984), Koka (1987), and Lau (2000). Some well-known models are shown in Figure 3-8. 0• U f E Compression <‘Eo Tension S ---ft (a) Figure 3-8. Parallel-to-wood-grain stress-strain relationship models Model (a) was proposed by Ylinen (1956), who employed a logarithmic function: e = -—[c.f —(1 — c).f ln(1 — i)J (3-4 1) where e andfare the strain and stress, respectively; E is the modulus of elasticity; J is the maximum compressive stress; and c is a parameter defining the shape of the curve, which can be determined based on test data. O’Halloran (1973) proposed an exponential model (Figure 3-8 b) for clear dry wood under compressive stress at various angles to the grain orientation. The model can be expressed as: f=Ee—Ae (3-42) - 52 - U U (b) (c) (d) wherefand E are the same as in Equation (3-42), and A and n are constants which can be determined by test data. A simple bilinear model (Figure 3-8 c) was put forward by Bazan (1980) in which the slope of the stress-strain curve beyond the maximum stress was assumed to be a fraction of the modulus of elasticity, E. The mathematical expression is a simple step function. Glos (1978) proposed a stress-strain relationship model of timber with defects (Figure 3-8 d). The model utilized four parameters that were determined from the density, moisture content, knot ratio and percentage compression wood. Logarithmic and exponential functions are less desirable for computer program implementation. O’Halloran’s model cannot be used for the stress-strain curve beyond the maximum stress, as the stress drops rapidly to negative stresses (Buchanan 1984). Bazan’s model may not be able to capture the nonlinear stress-strain behaviour. Glos’s model requires information that may not be available in some testing projects. A polynomial model, which is easy to implement into computer programs, was developed in this study for the parallel-to-wood-grain stress-strain relationship. The model can be expressed as: E0 f/Eo>e>0 u= O66 (343) 6p 8p 6u<6<8p where a and s are the stress and strain, respectively; J andf are the tensile and compressive strengths, respectively; E0 and Ed are the initial modulus of elasticity and the slope of the falling branch of the stress strain curve, respectively; s, is the strain corresponding to the - 53 - compression strength, f; r= e,Eo/f is a measure of the nonlinearity of the model; and s, and ft/Eo are the maximum compressive and tensile strains, respectively. The model is shown schematically in Figure 3-9. Figure 3-9. Polynomial model of the parallel-to-wood-grain stress-strain relationship The secant modulus, Esec, and the tangent modulus, Etan, can be calculated from the quotient and first derivative of the stress, a, to the strain, e, as: Esec = and (3-44) 6 3.6. Size and Stress Distribution Effects The strength properties of wood can be affected by the specimen sizes and the distributions of the stresses. The phenomena are normally referred to as the size and stress distribution effects, which are mostly caused by the inhomogeneity of wood material, attributable to the natural characteristic ofwood and the circumstances in which the trees grew. For visually graded lumber, the grading rule also brings up certain size effects (Buchanan 1984). Wood failures mostly occur in the vicinity of local defects, such as knots, cracks and - 54 - 0• 6u 8 Ed the slope of grain. Normally failures, especially those caused by tensile stresses, are abrupt and brittle. The brittle fracture theory, also known as the weakest link theory, has been used extensively to quantify the size and stress distribution effects. It was originated by Pierce (1926), who applied it to cotton yam. Major contributions were made by Weibull (1939). By using an exponential distribution function, Weibull showed that the strength depended on the stressed volume of a test specimen, assuming all specimens consisted of statistically independent but similar elements. The classic weakest link theory was first applied to wood products under flexural loading by Bohannan (1966). Its application in wood and timber structure strengths has been summarized by Barrett (1974), Buchanan (1984), Madsen (1992), Lam (1999), and Lau (2000). The formulation of the size effect is briefly described here, in order to lead to the formulation of the stress distribution effect for biaxially stressed wood members. 3.6.1. Size effect Assume the strength of a wood specimen ofunit volume is fitted to the two-parameter Weibull distribution as: F(x) =1— m)k where F(x) is the cumulative distribution function, x is the strength, and k and m are the shape and the scale parameters of the Weibull distribution, respectively. The failure probability, Pf, of a specimen subjected to a uniformly distributed stress, u, can be calculated by ±1 (U)kdv Pf =1e V m (3-46) - 55 - where V0 is a reference volume, and V is the stressed volume. Consider the strengths of two groups of wood specimens evaluated under different stressed volumes at a common probability of failure, Pj, their stresses at failure (o- and U2) can be related by: 1 10k,4 1 v Jv1 “ JV21—e m =1—e m (347) As the stresses are uniformly distributed, crl and 2 are constant over the stressed volumes and can be removed from the integrations. Striking out the common factors and rearranging Equation (3-47) yields: = (3-48) 2 1VJ 3.6.2. Stress distribution effect The stress distribution effect was also evaluated based on the weakest link theory. Consider two groups of wood specimens, of which the first group of wood specimen is subjected to uniformly distributed stresses with the maximum stress off The second group is subjected to non-uniformly distributed stresses with the maximum stress of o-. An example of such two stress distributions is shown in Figure 3-10, where the non-uniformly distributed stresses are the tensile stresses caused by the combined biaxial bending moments and axial compression load. - 56 - Stressed area Uniform stress f Figure 3-10. Non-uniform and uniform stress distributions of wood cross sections Assuming the two maximum stresses, a, and, j, were evaluated at a common probability of failure, Pfi they can be related by: _Sv(L)k dV Iv(”YJkdv 1—e m =1—e ° • m (349) where u(x,y,z) is the biaxially distributed stress, and the normalized stress, u(x,y,z)/a, is only a function of the external loadings and independent of the maximum stress, a. Note that bothfand ci are constant over the stressed volumes and can be removed from the integrations. Striking out the common factors and rearranging Equation (3-49) yields: (3-50) Equation (3-50) is similar to the calculation of the size effect in Equation (3-48). 3.7. Conclusion This chapter described the formulation of the FEM-based elements and the Newton-Raphson and arc-length methods for the completeness of the thesis. The criteria of - 57 - Maximum tensile stress 1 V U convergence and failure, the polynomial model of the parallel-to-wood-grain stress-strain relationship, and the size and stress distribution effects of wood strengths were also described for the purpose of computer program implementation. - 58 - Chaper 4. Development of the Computer Program SATA Introduction In this thesis, a finite element method (FEM) based computer program, SATA, was developed based on the C++ programming language and OpenGL graphic functions. This software can be used to perform three-dimensional nonlinear structural analysis and reliability analysis. The software development, including the programming logic, the data structure, the graphic user interface and the major functions of the program, is briefly described in this chapter for a better understanding of the program. 4.1. Programming Logic The program was codified following the objective oriented programming (OOP) logic, which was chosen because it is favoured by many program developers for its advantages in program debugging, maintenance and extensions (Haukaas 2004). Typically, an OOP based program consists mainly of objects (also called classes) and messages. It may be seen as a collection of cooperating objects, as opposed to the traditional view in which a program is seen as a group oftasks (subroutines). Each object ofthe program is capable of receiving messages, processing data, and passing messages to other objects. The objects are standardized and self-contained, so that the modifications made to one object will not affect other objects of the program. Each object is essentially an abstract presentation of a subject in the real world. Four - 59 - common concepts — abstraction, encapsulation, inheritance and polymorphism — are followed, although not all are necessary used in the same program, to organize the objects in a standardized and interchangeable manner. Abstraction is to the simplification of complex subjects, focusing only on the most pertinent properties and functions. For example, a metal plate connected (IVIPC) truss member exhibits many properties, such as texture, dimensions, and weight, etc; however, only those related to its structural behaviour were considered in SATA. Encapsulation is the construction of self-contained objects in the program. Each object is encapsulated with all attributes that are sufficient to carry out its functions. Using the concept of encapsulation also helps to protect the attributes of the individual objects from being mistakenly altered by other objects or functions. In SATA, the access to the attributes was classified by defining them as “protected”, “private” or “public”. Inheritance is the development of a new object (a child object) based on one or several existing objects (parent objects). By default, the new object inherits all attributes and functions from the parent objects. This concept is particularly useful when extensions to an existing program are of interest, since as the additions can be achieved by simply generating child objects. The inherited attributes and functions in the child objects can then be modified to have extended functions. For example, in SATA, all FEM elements were generated from a common parent object called an “element”. This parent object was used as a prototype and consisted of the most general attributes and functions for structural analysis purposes. This concept notably increased the code efficiency of SATA. Polymorphism allows the child objects to have extended attributes and functions and - 60 - yet be treated consistently with the parent objects. This consistent treatment is achieved by using virtual functions. The child objects inherit the virtual functions from the parent objects and overload them. With polymorphis, a newly added object or function can fit into the program with no modification needed. For example, in order to assemble the system stiffness matrix for a FEM-based analysis, SATA simply called a member function “CalStiffMatrix”, which was defined in the parent object element and inherited by all the FEM elements as child objects. 4.2. Data Structure of SATA SATA relies on the input parameters to perform a structural analysis. The input parameters can be either imported from an existing data file or defined within the program with the aids ofa series ofdialogs, which are described in Section 2.4.1. The generated data from the input parameters need to be stored in the program before they can be used or disposed of. The objective of designing a data structure is to have a memory efficient storage plan and fast access to these data. 4.2.1. Storage of the data A structural analysis may comprise a large amount of data. Much of the data are the same types. For example, there can be thousands of nodal information data, each consisting of the same kind of information, such as the nodal coordinates, the nodal displacements, the nodal forces, etc. SATA stores these data by using a structure called an “array”, which is comprised of - 61 - many cells. Each cell is used to store the information of the individual data, i.e. the information of a FEM-based node. The array provides convenient operations, such as browsing, adding and deleting, to its cells. In SATA, the browsing operation is achieved by links between adjacent cells. The cells of the array can be considered as the many links of a chain. Each cell has the connections to the previous and next cells. The connections are realized by using the C++ “pointers”, which are defined to be of the same type as the cells. Figure 4-1 shows the organization of the array and the realization of the browsing, adding and deleting operations. ,,__.. cEçJ ext-Pointer ‘Content of ‘Content of I Content of’ •I •cell i—i cell z cell z+1 Browsing operation ‘Content of’ ‘Content of’ ‘Content of’•: :. •: •,cell i—I cell i , cell i+1 Deleting operation ‘Content of ‘Content of ‘ ‘Content of .: . •,cell i—i , , cell i , cell i+1 Adding operation Figure 4-1. Storage of SATA data - 62 - 4.2.2. Major content of the data The cells of an array are used to store various types of data that are pertinent to the operation of the structural analysis or the interaction between the computer and the user. The cells are also encapsulated with the functions for the standard input and output operations and the random generation of the data for reliability analysis. This kind of organization of the data and functions not only improves the security of the data, but also facilitates the extensions to an existing cell. The major content of the data includes the material property, geometry and topology, boundary condition and load, and environmental parameters and user control message. Material properties are used as the input parameters for the FEM-based elements. One type of material property was designed for one type of FEM-based element. For example, the material properties of dimension lumber are used for beam elements, and the material properties of plywood sheathing are used for the plate elements, etc. Sub-types are also generated when more than one kind of material are considered in the structural analysis. For example, sub-types of the metal truss plate properties were generated separately for the metal truss plates of model MT18HS and MT2O plates. The geometry and topology information mainly comprises the nodal coordinates, the FEM element configurations and the placement of the nodes and the elements. Each node is assigned with one unique identification number. The FEM-based elements use the identification numbers of the nodes to specify their geometry and the reference nodes to construct the local coordinate systems. The boundary conditions specify the fixity of the boundary nodes in six degrees of - 63 - freedom (DOFs). For the nodes with restrained DOFs, such as a pinned or fixed support, a displacement called “support settlement” is initialized and ready to be used in a displacement control analysis. Two types of loads are incorporated in SATA: the concentrated loads at specified nodes and the distributed loads on specified surfaces. For each type of load, a special attribute is used to distinguish between the dead and live loads. The environmental parameters are mainly used for graphic functions. The parameters include the origin and scaling ratio of the graphic window, the view angle for three-dimensional rendering, the colours for individual lines and areas and the rendering modes supported by the OpenGL graphic library, and the show and hidden options for individual members. The changes in these parameters lead to the “pan” movements, zooming, three-dimensional rotations and color changing, etc., of the structural model. The control messages are used to facilitate the interaction between the computer and the user. This feature allows the user to interrupt or redirect the operating process of the program by clicking a button on the tool bar or a command in the menu. 4.2.3. Organization of the data and objects Most of SATA’s data can be accessed by the objects defined in the program via the pointers, which contain the addresses of the data in the computer memory. The pointers, in conjunction with the objects that are generated following the four OOP concepts, contribute to an efficient solution of the data storage plan. The organization of SATA in terms of the objects and data is shown in Figure 4-2 by a flow chart. - 64 - Figure 4-2. Organization of SATA’s data and objects 4.3. An Overview of SATA SATA consists of a user friendly graphic interface and several functional moduli that cover the major aspects of a structural analysis. Commonly used dialogs were developed to receive user input and display program status or analysis results. A brief introduction to the configuration of the graphic user interface (GUI) and the program moduli is presented in Probabilistic Stress strain Problem definition • Single step/ Ultimate analysis Load/displacement control Arc-length analysis Reliability analysis Elemiit configuration - • Material parameter J Attributes • Calculate transfer matrix Calculate strain and stress Calculate tangent stiffness matrix Methods and internal force vector Calculate consistent load, etc. FEM solver Nodal solution Element solution results Nodal displacement Element force and moments ‘Step by step deformation plotting Animation of deformation history - 65 - mmIon Element sisbIe T Roof element r Top chord r Goltom chord 1J [s’] [ I Web Zoom r Bracing J LJ l r Spring element __________ I Boorrdary Pan rinad Rendenag oplrons t1 LJ 1J 2lJhne 3Olrne Flat shade Agrimahon Smooth shade Teoture render .3 OK Cancel jJ Sections 4.3.1 to 4.3.4. 4.3.1. Configuration of the GUI The GUI consists of five major components, including the utility menu, the standard tool bar, the graphic area, the status and prompt area and the output window. A typical layout of the GUI components is shown in Figure 4-3. Graphic control panels dy I Resutis from step 4th calculationt FX FY P0 Plate element 4 node 185 —140.452 791.954 —366.147 node 188 —95.994 —1098.37 565.045 Figure 4-3. Layout of the GUI components of SATA The utility menu (©) contains the utility functions, such as the file controls, the view options, and the graphics controls. The functions can be reached by clicking the mouse keys or using the mnemonic characters (short-cut keys), which are indicated by underscoring of the captions of the menu. - 66 - The standard tool bar (©) provides fast access to the most frequently used functions. By default, the toolbar is loaded and initialized at the startup of the program and is located underneath the utility menu. The “hide and show” attribute of the toolbar can be changed at anytime during the running of the program, controlled by the “view” menu of the utility menu. The toolbar can also be dragged to any other positions within the GUI. The graphic area (®) is a window where the graphical results are displayed. Two graphic control panels were developed to provide fast access to the graphic functions. A floating menu is also available and can be activated by clicking the right key of the mouse. The status and prompt area (®) is located at the bottom of the GUI. This area is normally used to display the prompts and status of the program. It is also used to display a short description of a menu or a command button on the tool bar, when the curser is located near a menu or a button. The output window (©) is behind the GUI and is designed to display the intermediate information and final results of the analysis. The output window has most of the text editing functions, such as copy, cut and paste. The results in the output window can also be transplanted to other programs for more advanced data processing. 4.3.2. Pre-processor The pre-processor was designed to facilitate the process of constructing a structural analysis model. A series of dialogs are designed for this purpose. Definition of the material properties, the model geometry, the boundary conditions, and the external loads were facilitated by using the dialog components. Typical dialogs are shown in Figure 4-4. - 67 - fRool Ciia,olbarnereto Roof ccoli9u&on Grorniofion Totaf noo,be Baneate. Cooonk nxther Trusolayout. . ecmetjic OK Cancol J Appj OK j C&ef, Figure 4-4. Input dialogs for material property and truss configuration The dialogs are developed based on a standard windows property sheet, which consists of several property pages. Each property page is designed for one particular type of material. The property pages consist of various types of dialog components for data input and display functions: • The edit boxes are mainly used for text or number input; • The drag-down combo-boxes are for the selection of a list of options; • The bitmaps are used to provide graphic explanations of the data; and, The command buttons are linked to the built-in functions for data processing, such as adding, deleting and browsing the defined data. The pre-processor is also used to generate the digitized structural models based on the input data. The models are displayed in the GUI for the purpose of inspection of the input data. Various rendering options are available to improve the quality of the visualization. The Later hac.g oopety Roof pand popeIty Told nu,nbea 1 CUcoof fl Jood paopetV -: — hean Vdoo [I’e Ctoescton ri [ hnçft [ Op0000 _J s1 _ Loodjon H. [ i° octon of loan piano Tionocort.nalion . . I Joec4pi I ..iJ .zJ ______ _LiJ - 68 - structural members of the model can also be picked up by double clicking the mouse key close to a specific member. A member inquiry dialog was designed to display the basic information of the selected member. Figure 4-5 shows a three-bay truss assembly displayed by using the line mode and the smooth shading mode, respectively. A 33) MPC iuo aooaIyoo ooI 4.3.3. FEM solver The FEM solver is mainly used to set up the solution options and carry out the solution process of the REM-based structural analysis. Five property pages are designed for various solution options, as shown in Figure 4-6. The “solve options” property page is used to specify the type of analysis, including single step analysis and ultimate analysis. For an ultimate analysis, a secondary option is provided to specify the control mode, including displacement, load and arc-length controls. Detailed configurations of the control modes are included in three property page based dialogs. - 69 - I Fl. V4 Po.ç,00.., Dn, EI.,n.nt yp.: 8EM,1 EI.m.nI nuni NO3oI rl.no Nod, I 12S Nod•j. Nodek. Nod.I. Nodal arc, drop plc 000c flodO. . r Food, y dl .0< Foo aOP4OONOg Figure 4-5. Graphic rendering modes and element information inquiry dialog of SATA The “solving parameters” property page includes advanced configurations of the structural analysis, including numerical integration options, convergence criteria, the P-Delta effect and the size effect. Ionc, oothod opO j SoI’ofr SpOio Loooono.o Sbonbdoná roblem define Uhn.te.n..,j. Soivernethods r r Dntn*on4oio oorcobood*og,th.ne - OK j Apps Ot Figure 4-6. FEM-based solution configurations of SATA The FEM solver also provides the means to define the random variables for reliability analysis purposes, as shown in Figure 4-7. Two sampling methods, including Monte Carlo simulation and response surface method, are provided to approximate the performance functions of reliability analyses. During the solution process, a modeless dialog is used to display the progress and percentage of the finished analysis. A progress bar is used to visualize the solution process of the analysis. dotinition of solver options 000rgopoom j Loodsoo*odomjos, Oo nvth odepOn Solvrg pomoo Gaussian Inteoration nodes Nnodo.sOoooss,Oos,on ConvoooenveDlomo 0 .,omnoo Tdesom, O Oth.sopkn, r i& I Cdeooedfeot Ktvop So optno Lo.dsnpooio Obolomnoono no,*d noo4oio Oo.o,neIhodoponno Sçpomoseoe,o Arc-length method parameters S&oononoo Ino oodbooK Isi o4 S oeoo Oe*odoo 1 noOdOnI r AeOooOo the ná,thoo, no. inn Ns,ntho.donode [s D.O.E Mnoâ,o.ove,e Pboononi For - 70 - OPoPeY Rod p.eoooe ond3Rd Wood type I p000ody Olo boi.ioo type [t odott Oddodioo dpl.t. mdeod RoPeotY•] PWetypertypeO.o _______________ Dijiontyp. [. Peoneeteo t vthe Dioondmodolpoop.oty Sroeogtype typoO NO 0 SRPOPodY FJ DbtW.i V. ________ vod. °‘ lJ &. ________ ______ _ _ _ ______ Figure 4-7. Reliability analysis configurations and solution progress of SATA 4.3.4. Post-processor The post-processor is designed to process the analysis results by displaying the structural deformation in the GUI and exporting the results. Examples are shown in Figure 4-8. Basically, the post-processor shares all the graphic functions of the pre-processor. In addition, a curve plotting dialog is enclosed in the post-processor for displaying the interaction curves between the forces and the displacements of specified nodes or elements. The post-processor uses several property page type dialogs to facilitate the presentation of the structural analysis results. The results are extracted for individual nodes and individual elements in both the global and element local coordinate systems. The results can be either listed in a built-in text editing window or saved to an external data file. S ttlCtt 41 yttt ii 41141y!zlt - MoOeodiy Rodpoeoe.e.odpoioob.d .-.——————— — URwntot ooot1oiø . Sod.oelood . Aootpoeype zi i PCnootrdlaiAIjRtc oi.doii,.otype 1. AoONtyPiN epPooto j,< ThSIOt.P[T ooi,oot pt .. j cao. . too.yoáodlo.d . r. o.%oiot tcpctoe tNUtttnco0oRIIy . P&Nio.doooo loodO NO Solution progress f Cet*rotpd by moot om offiti . r--i Mao ooutobet 4 Coo ttuled b pooo.oeN F J pyy [oj % I ° I Co.to.t Appty - 71 - Cancel Ap Ndsnee El20ceo Erevth ETdIOce4d DOFtoipoi 1F Coordinate system. [ OK j Canc Apçj : /7 - 4 - -1033 - V.-. -0433 -470 -374 -202 -101 44 0.0 4.4 00.8 202 374 474 OK 4.4. Availability Figure 4-8. Options of SATA’s post-processor The source codes of SATA were compiled using a C++ compiler. An executable file is available with some conditions applied. The program works with MicrosoftT04 window 98 to XP operation systems. For more information please contact the Timber Engineering and Applied Mechanics (TEAM) research group of the Department of Wood Science in the Faculty of Forestry at the University of British Columbia, Vancouver, Canada. 4.5. Conclusion This chapter described the development of the computer program SATA, in terms of the programming logic, data structure, GUI and major functions of the program moduli. The program was used as the platform for the numerical modeling studies presented in the remainder of this thesis. Oupulanatyi8resuhs Ol4UIead8F20ces Ebrmlcces Node av1e Node tooolped node 3 OOFloooe ‘1 .......J .I7Ifl0d41 0047 I.pIo4om 7778 - 72 - Chaper 5. Experimental Study and Model Verification I: Biaxial Eccentric Compression Test of Wood Beam-Columns Introduction This chapter presents an experimental study to evaluate the basic material properties of dimension lumber and nail connections and to investigate the stability capacity and lateral bracing force of eccentrically loaded wood beam-columns. The test results were used to provide input parameters and verification for the computer program, SATA. Good agreement was achieved. 5.1. Research Method Basic material property tests, including parallel-to-wood-strain compression and tension tests and nail connection tests, were conducted to provide input parameters for the computer program, SATA. Biaxial eccentric compression tests of wood beam-columns, with and without a lateral bracing member, were performed to provide verification for the SATA program, in terms of the maximum compression load, midspan lateral deflections and lateral bracing forces of the laterally braced beam-columns. 5.1.1. Material Spruce Pine Fir (SPF) dimension lumber of two machine stress rated (MSR) grades (MSR 1650f-i. 5E and MSR2400f-2.OE, graded in conformance with the National Lumber - 73 - Grades Authority Special Product Standard 2 of Canada, SPS 2) were considered. The dimension lumber was of two cross-sectional sizes (38x 89 mm2, nominally 2 by 4 in.2; 38x 139 mm2, nominally 2 by 6 in.2) and one length (4267.2 mm or 14 feet). The lumber was used to fabricate the specimens for the material property and beam-column tests. Common nails (lOd —76.2 mm or 3 in. in length and 3.76 mm or 0.148 in. in diameter) were used to assemble the nail connection specimens and connect the beam-column specimens and the lateral bracing members. 5.1.2. Material property test The material property tests included parallel-to-wood-grain compression and tension tests and nail connection tests; and the test results were used to establish the parallel-to-wood-grain stress-strain relationship and the load displacement relationships of a two- lOd common nail connection in six degrees of freedom (DOFs). The size and stress distribution effects of wood strength properties were also studied based on the tests results. 5.1.2.1. Parallel-to-wood-grain compression and tension tests The tests were conducted in conformance with the ASTM standard D 1 98-05a “Methods of Static Tests of Timber in Structural Sizes”. Three specimen sizes, each with 30 replicates, were used for the compression and tension tests. The specimens were kept in a room conditioned at 65% relative humidity and 20°C, until the equilibrium moisture content was reached. Before loading, each specimen was measured for the net cross-sectional dimensions - 74 - (at three positions), specimen length and moisture content. Details of the specimens are listed in Table 5-1. Table 5-1. Specimen designfor the parallel-to-wood-grain compression and tension tests Test Lumber grade Specimen size* (mm3) Replication 38x89x304.8 30 MSR165Of-1.5E Compression 38 x 89 x 457.2 30 MSR2400f-2.OE 38x139x457.2 30 38x89x2133 30 MSR165Of-1.5E Tension 38x89x3048 30 MSR2400f-2.OE 38x139x3048 30 Note: * gross sizes are shown in the table The parallel-to-wood-grain compression tests were conducted on a Sintech machine with a capacity of 245 kN (55 kips) under displacement control. The loading rate was constant at an equivalent strain rate of 0.001 mm/mm per minute. The compression displacements were measured by two linear variable differential transducers (LVDTs) with a gauge length of 254 mm (10 in.), the readings from which were averaged to eliminate the influence of flexural deformation. The test setup is shown in Figure 5-1. - 75 - Figure 5-1. Parallel-to-wood-grain compression test setup The parallel-to-wood-grain tension tests were conducted on a Metriguard tension machine. All specimens were pulled to failure within 5 to 10 minutes. Each specimen was tightly fixed by a pair of tension grips of 1219 mm (4 feet) in length. The maximum tension loads were recorded and used to calculate the tensile strengths based on the net cross-sectional area of the specimens. The test setup is shown in Figure 5-2. Figure 5-2. Parallel-to-wood-grain tension test setup - 76 - 5.1.2.2. Nail connection tests The nail connection tests were conducted to establish the load displacement relationships of the connection between the wood beam-columns and the lateral bracing members that were used in the beam-column tests. The load displacement behaviour in the six DOFs, as illustrated in Figure 5-3, of the nail connections were studied separately. Lateral direction Grain direction Degree of freedom Main compression column Figure 5-3. Nail connection behaviour in six degrees of freedom The connection behaviour corresponding to each DOF was studied by testing specimens that were fabricated in one specific configuration. The dimension lumber of the specimens was aligned in conformance with the model shown in Figure 5-3. Two 1 Od nails were used to connect every two pieces of the dimension lumber. The specimen configurations and test setup are shown in Figure 5-4, in which the number of the types is in conformance with the number of the DOFs. 3 2 - 77 - 00 Figure 5-4. Specimen configurations and test setup of the nail connection tests The specimens were tested on a Sintech machine with a capacity of 245 kN (55 kips). - 79 - Compression loads were applied to the specimens that were configured per the translational DOFs; whereas moments, which were achieved by offsetting the compression loads by the prescribed distance of the lever arm, were applied to the other specimens. Displacement control was considered with the loading rate constant at 2.54 mm!min. Two LVDTs were used to measure the displacements or the angles of rotation. 5.1.3. Biaxial eccentric compression tests of wood beam-columns The wood beam-column specimens were tested by a compression load with a biaxial load eccentricity. Various load eccentricities were considered to simulate the biaxial end moments. 5.1.3.1. Specimen design The wood beam-column specimens were made of SPF dimension lumber of MSR165Of-1.5E and MSR2400f-2.OE. Two cross-sectional sizes, two lengths and three combinations of biaxial load eccentricities were considered. The biaxial load eccentricities were defined by their components in the weak axis (x) and the strong axis (y) of the specimen’s cross section. In total, 11 groups of specimens were tested. The specimen design is listed in Table 5-2. - 80 - Table 5-2. Beam-column specimen design for the biaxial eccentric compression test Cross section size Eccentricities (mm) Lateral Group Lumber grade 2 Length (mm) Replication(mm) x axis y axis bracing 1 MSR1650f-1.5E 38x89 2134 0 20 NO 30 2 MSR165Of-l.5E 38x89 2134 20 20 NO 30 3 MSR165Of-1.5E 38x89 2134 20 50 NO 30 4 MSR165Of-1.5E 38x89 3048 20 20 NO 30 5 MSR165Of-l.5E 38x 139 2134 20 20 NO 30 6 MSR2400f-2.OE 38x89 3048 20 20 NO 30 7 MSR2400f-2.OE 38x 139 3048 20 20 NO 30 8 MSR165Of-1.5E 38x89 3048 0 20 YES 15 9 MSR165Of-1.5E 38x89 3048 20 20 YES 15 10 MSR2400f-2.OE 38x89 3048 20 20 YES 15 11 MSR2400f-2.OE 38x 139 3048 20 50 YES 15 5.1.3.2. Support conditions The beam-column tests used two steel columns as the main supporting frame. A hydraulic actuator, with a capacity of 890 kN (200 kips), was placed horizontally and fixed on the first steel column. The specimens were placed horizontally between the actuator and the second steel column. The specimens were tightly fitted into a pair of steel boots. On the bottom plates of the steel boots, there were nine holes, the positions of which were determined by the prescribed load eccentricities. The two steel boots were connected to the actuator and the second steel column via one of the holes as shown in Figure 5-5. A steel bar was welded to each of the two steel boots, in order to restrain the torsional displacements of the specimens. - 81 - For the laterally braced specimens, a lateral bracing member was placed vertically and connected to the edge side ofthe midspan cross section of each beam-column specimen by two lOd nails. The bottom end of the lateral bracing member was rested on a load cell with a capacity of 22 kN (5 kips), which was used to measure the lateral bracing force. The details are shown in Figure 5-6. Figure 5-6. The nail connection and load cell under the lateral bracing member Figure 5-5. Connection between the steel boot and the hydraulic actuator - 82 - 5.1.3.3. Loading system The hydraulic actuator was controlled by a testing program in the application of the compression load to the specimens under displacement control. The loading rate was constant at 0.001 mm/mm per minute. The loading system for an unbraced wood beam-column is shown in Figure 5-7, where the wood construction underneath the specimen was installed to protect the testing equipment and sensors in case of abrupt failure. Figure 5-7. Loading system of unbraced wood beam-columns 5.1.3.4. Data acquisition For each beam-column specimen, the test results of concern included the axial compression load, midspan lateral deflections in the axes x andy and lateral bracing force. - 83 - The axial compression load was measured and recorded by a load cell built into the actuator. The midspan lateral deflections were measured and recorded by three string pots, each with a range of 635 mm (25 in.). The string pots were fixed onto the testing frame and orientated as either parallel or perpendicular to the specimen’s longitudinal axis, as shown in Figure 5-8. The lateral bracing force was measured and recorded by the load cell underneath the lateral bracing member. Figure 5-8. Layout of the string pots used to measure the midspan lateral deflections 5.2. Material Property Tests Results The test results were used to evaluate the parallel-to-wood-grain stress-strain relationship, as expressed in Equation (3-43), and the load displacement relationship of a nail connection in individual DOFs, as expressed in Equation (3-22), respectively. The parameters - 84 - of the relationships were evaluated by using the least square method (LSM). The parameters were firstly evaluated based on the test results of individual specimens and then compiled and fitted to the Weibull distribution (Weibull 1939) to quantify the variation within the replicates of individual specimen groups. 5.2.1. Parallel-to-wood-grain compression and tension test results The compression strength, and corresponding strain, er,, of the parallel-to-wood-grain stress-strain relationship are listed in Table 5-3 for specimens that were tested in different sizes. The moduli of elasticity, E0, and of the slope of the falling branch, Ed, of the stress-strain curve were evaluated based on the test results of specimens in all three sizes. The results are listed at the bottom of Table 5-3. The results of the tensile strength,J,, are listed in Table 5-4. Table 5-3. Parallel-to-wood-grain compression tests results MSR165Of-1.5E MSR2400f-2.OE Parameters shape’ scale mean St.d. shape scale mean St.d. 38x 89x 304.8 mm3 specimens ______________ j (MPa) 7.18 24.47 23.15 4.27 8.82 32.44 30.83 4.34 e, (10j 6.87 3.91 3.7 0.73 5.66 4.0 3.74 0.89 38 x 89 x 457.2 mm3 specimens J(MPa) 6.83 20.18 19.0 3.49 7.61 31.61 29.76 4.26 e, (10) 5.93 3.43 3.21 0.71 6.19 2.94 2.766 0.60 38x 139x 457.2 mm3 specimens J (MPa) 5.41 25.04 23.18 4.48 10.6 29.66 28.36 3.26 e, (10) 5.73 3.50 3.27 0.73 4.967 3.579 3.33 0.87 All three specimen sizes E0(MPa) 5.95 10562 9868 1970.6 7.95 12042 11359 1593 Ea(-MPa) 1.12 854.0 830.8 685.5 1.22 738.2 820.9 860.9 Note: 1 shape parameters are dimensionless - 85 - Table 5-4. Parallel-to-wood-grain tension tests results 2-P Weibull 3-P Weibull Grade Specimen (mni3)’ Mean St.d. shape2 scale shape scale location 38x89x2133 2.52 32.76 2.47 32.37 0.37 30.0 14.6 MSRI650f-1.5E 38x89x3048 4.2 32.1 2.16 19.41 12.22 29.41 7.95 38x139x3048 5.28 29.62 1.43 11.14 17.66 27.66 6.73 38x89x2133 6.01 50.98 3.08 30.16 20.51 47.54 9.33 MSR2400f-2.OE 38x89x3048 5.81 46.0 3.49 30.73 15.06 42.82 8.6 38x 139x 3048 4.14 42.86 3.3 36.03 6.67 39.05 10.29 Note: ‘the specimen’s length includes the gripping length of 1219.2 mm (4 feet) 2 shape parameters are dimensionless 5.2.2. Nail connection test results The results of the initial slope, k, intercept of the asymptote, m0, and slope of the asymptote, ml, of the load displacement curve of individual nail connections in six DOFs are listed in Table 5-5. For the purpose of simplicity, ml and mo were set to zero and the maximum load of the test results, respectively. Table 5-5. Nail connection tests results Maximum capacity m0 Slope of asymptote m1 Initial slope k Type Mean St.d. Mean St.d. Mean St.d. Unit N N N/mm N/mm N/mm N/mm 1 3141.03 557.60 0 0 824.71 122.04 2 3219.67 168.68 0 0 1370.99 297.09 3 862.42 130.79 0 0 2006.74 1166.62 Unit kN.m kN.m kN.m kN.m kN.m kN.m 4 0.039 0.005 0 0 1.813 0.484 5 0.018 0.005 0 0 0.441 0.203 6 0.025 0.009 0 0 3.33 0.939 The test results and model fittings of the load displacement relationship of the specimens in configurations 1, 2 and 4 are demonstrated in Figure 5-9. - 86 - 4000 3000 2000 1000 0 4000 g 3000 w 3 2000 1000 0 50000 40000 E 30000 20000 10000 0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Angle of rotation (radian) Figure 5-9. Load-displacement relationship of the nail connections 5.3. Beam-Column Test Results and Model Verification The failure modes of the beam-column tests were studied first. A finite element method (FEM) based model was developed based on the understanding of the failure modes. The Newton-Raphson and arc-length methods were used to evaluate the axial compression - 87 - 0 5 10 15 20 25 Displacement (mm) 0 3 6 9 12 15 Displacement (mm) load, lateral deflections and lateral bracing force. The model was verified based on the test results. The 2% rule-of-thumb, Winter’s and Plaut’s methods were also studied by comparing their predictions of the lateral bracing force to the test results. 5.3.1. Failure modes of the wood beam-column tests Major lateral deflections of the unbraced wood beam-columns occurred in the weak axis (x). Failures also took place in the weak axis and at positions near the midspan of the specimens, where the maximum moment was reached. The failure surfaces indicated an inclined neutral axis, which was caused by the biaxial end-moments. Figure 5-10 shows typical failure modes of the unbraced wood beam-columns. The laterally braced wood beam-columns behaved differently from the unbraced ones. The major deflections were observed in the strong axis (y). Failures also took place in the strong axis and at positions near the quarter nodes of the specimens, as shown in Figure 5-11. This difference was caused by the lateral bracing member, since it restrained the deflections in - 88 - Figure 5-10. Failure modes of unbraced wood beam-columns the weak axis and changed the distribution of the bending moments. Nail connection failure was observed once. The nails were partially pulled out, and the beam-column specimen rotated considerably around its longitudinal axis. The axial compression load at failure was much smaller than those measured with the other specimens. This failure was caused by the lateral bracing force, which was passed to the edge side of the specimen via the nail connection. The lateral bracing force was distanced from the specimen’s longitudinal axis by half of the width of the specimen; therefore, a torque equaling the product of the lateral bracing force and the half width was exerted on the specimen and caused the torsional displacement. Figure 5-11. Failure modes of the laterally braced wood beam-columns - 89 - 5.3.2. Model development A FEM-based model was constructed for the beam-column study. The model consisted of the FEM beam elements and spring elements, as formulated in Chapter 3. The beam-column was modeled by twenty beam elements; and, the lateral bracing member was modeled by three beam elements. The nail connection was modeled by a nonlinear spring element. The lateral bracing elements were distanced from the beam-column’s longitudinal axis by half of the column width, as shown in Figure 5-12. One unidirectional spring element was used at the bottom end of the lateral bracing elements, for the purpose of parametric study. The beam-column was subjected to an axial compression load and biaxial bending moments in constant ratio to the compression load, as specified by the load eccentricities. The support condition was assumed to be a pin-roller with the rotational displacement restrained at the end nodes of the beam-column member. Figure 5-12. FEM-based model of an eccentrically loaded wood beam-column P M Nail connection column Lateral bracing x M - 90 - 5.3.3. Model input parameters and size and stress distribution effects For verification purposes, the input parameters of the model were based on the mean values of the parallel-to-wood-grain stress-strain relationship and the load displacement relationship of the nail connection. The model geometry and the load eccentricities were based on the information in Table 5-2. The size and stress distribution effects were determined based on the parallel-to-wood-grain compression and tension test results. As the wood material property tests were conducted using specimens of the same width as specimens of the beam-column tests, only the length effects needed to be evaluated. The results are listed in Table 5-6. The k factors were evaluated based on the strengths of the specimens of 38 x 89 mm2 in cross-sectional size. Table 5-6. Length effectfactor, Ic, ofwood compression and tensile strengths k factor Strength Grade ki k2 k3 mean adjusted MSR165Of-1.5E 7.18 6.83 2.05 5.35 12.0 Compression MSR2400f-2.0E 8.82 7.61 11.48 9.30 12.0 MSR165Of-1.5E 2.52 4.2 35.0 13.91 14.0 Tension MSR2400f-2.0E 6.01 5.81 6.62 6.15 7.0 The results in the first two columns were calculated from the shape parameters of the two-parameter Weibull distribution, to which the strengths were fitted. The results in the third column were calculated using Equation (3-48), based on the mean values of the strengths. The results were averaged in the fourth column. As the results exhibited variation to a certain extent, due to the quality of material in the specimens (Barrett 1974), the factors were adjusted with - 91 - consideration of the beam-column test results. The final results are listed in the fifth column of the table. The stress distribution effect of the wood tension strength was considered in this study. As no test data on the wood bending strengths were available, the factor k was considered to be 14.0 according to Buchanan’s work (1984). 5.3.4. Unbraced wood beam-column test results and model verification The test results of the unbraced wood beam-columns are shown in Figures 5-13 to 5-19 for Specimen Groups ito 7, respectively. For each specimen group, the results consisted of the relationships between the axial compression loads and the midspan lateral deflections in the x andy axes. For the purpose of clarity, the results have been represented by curves of the mean values and the upper and lower bounds. For each specimen group, the mean values, and upper and lower bounds were evaluated by: • The mean values were evaluated based on the test results of all the specimens in the group. Each data point on the curve represented a mean axial compression load, corresponding to a specific lateral deflection; • The upper bound curve was based on the test results of the specimen that exhibited the stiffest load and lateral deflection relationship; and, • The lower bound curve was based on the test results of the specimen that exhibited the weakest load and lateral deflection relationship. - 92 - The model predictions are presented in comparison with the test results for the purpose of verification. The predictions were based on the FEM model in Figure 5-12 with the lateral bracing member and unidirectional spring ignored. The wood compression and tension strengths were adjusted using the factors in Table 5-8 and the lengths of the beam-column specimens. 14000 12000 10000 0 8000 E 6000 0 0 4000 x < 2000 0 14000 12000 10000 0 0 E 6000 0 0 4000 x < 2000 0 0 20 40 60 80 100 Midspan lateral deflection mx axis (mm) 0 2 4 6 8 10 Midspan lateral deflection in y axis (mm) 120 Figure 5-13. Test results and model predictions of Specimen Group 1 12 - 93 - gI0000 8000 Average max load .2 6000 4000 C) 2000 —model 0 I I 0 30 60 90 120 Midspan lateral deflection in x axis (mm) g 10000 8000 Upperbound Average maxload .2 6000 4000 7owerbou C) 2000 —model 0 I I 0 2 4 6 8 10 12 14 Midspan lateral deflection in y axis (mm) Figure 5-14. Test results and model predictions of Specimen Group 2 8000 7000 — Upperbound 6000 Aver9e max _____________ • 5000 Lowerbound 4000 3000 3 2000 —— mean 1000 —model 0 I I I I 0 20 40 60 80 100 120 Midspan lateral deflection in x axis (mm) - 94 - 8000 ‘ 7000 Upper bound 6000 • 5000 Co 4000 - 3000 8 2000 —‘-—mea 1000 — mod 0 0 3 6 9 12 15 Midspan lateral deflection in y axis (mm) Figure 5-15. Test results and model predictions of Specimen Group 3 5000 Upperbound • 4000 .2 .9 3000 U) Co 2000 0 1000 —*-—mean . —model 0 I 0 20 40 60 80 100 120 140 Midspan lateral deflection in x axis (mm) 5000 4000 .2 C .2 3000 Average max loadCo Co mean 2000 0 0 • 1000 —model 0 . I I I 0 2 4 6 8 10 12 Midspan lateral deflection in y axis (mm) Figure 5-16. Test results and model predictions of Specimen Group 4 - 95 - 14000 Upper bound12000 (0 .2 10000 A C .2 8000 (0 6000 E o 4000C) —4--- mean .2 2000 —model 0 0 30 60 90 120 Midspan lateral deflection in x axis (mm) 14000 12000 Upperbound (0 Average max load.2 10000 C .2 8000 (0 60000. E o 4000C.) an .2 2000 del 0 0 2 4 6 8 10 12 Midspan lateral deflection in y axis (mm) Figure 5-17. Test results and model predictions of Specimen Group 5 8000 Upperbound .2 5000 Averema .2 U) ba U) 4flflfl owerboun3000E ° 2000C) x —model1000 0 0 50 100 150 200 Midspan lateral deflection in x axis (mm) - 96 - 8000 - 7000 6000 • 5000 0 4000 - 3000 2000 10000 10 Figure 5-18. Test results and model predictions of Specimen Group 6 10000 z 8000 0 .2 6000 0(I) 4000 2000 x 0 10000 z 8000 0 C .2 6000 0 0 4000 2000 x 0 2 4 Midspan lateral deflection in y axis (mm) 200 6 Figure 5-19. Test results and model predictions of Specimen Group 7 The test results of the maximum compression loads and the corresponding midspan lateral deflections were used to calculate the maximum midspan bending moments. To include - 97 - 0 2 4 6 8 Midspan lateral deflection in y axis (mm) 0 50 100 150 Midspan lateral deflection in x axis (mm) 0 the P-Delta effect of compression load, the load eccentricities were added to the midspan lateral deflections for the calculation of the midspan moments. Table 5-7 lists the mean values of the test results and the model predictions of Specimen Groups ito 7. Table 5-7. Maximum compression load and midspan biaxial moments ofwood beam-columns Midspan bending moments (kN.m) Maximum load (kN) Group Test Model Error (%) Test Model Error (%) Test Model Error (%) 1 9.0 8.862 1.533 0.24 0.234 2.500 0.354 0.390 10.169 2 6.4 6.500 1.562 0.163 0.162 0.613 0.687 0.565 17.758 3 6.2 5.977 3.597 0.39 0.362 7.179 0.532 0.527 0.940 4 3.2 3.660 14.375 -- 0.093 -- -- 0.452 -- 5 9.5 9.707 2.179 0.203 0.211 3.941 1.01 0.825 18.317 6 4.8 4.930 2.708 0.122 0.130 6.557 0.793 0.688 13.241 7 6.3 6.623 5.127 0.133 0.147 10.526 1.081 1.011 6.475 Note: *M and M are defined in the axes x andy of the beam-column’s cross section. It can be seen from the figures and Table 5-7 that the model predictions agreed well with the test results. The predicted relationships of the axial compression load and midspan lateral deflections were very close to the mean values of the test results. For all seven specimen groups, the model predictions were within the upper and lower bounds. It was also found that the agreement ofthe maximum compression load was better than that of the midspan moments. This occurred for two reasons: • Firstly, the measuring errors of the midspan lateral deflections in the x axis became significant when the axial compression loads approached the maxima, where the load deflection curves were very flat, as can be seen from the figures. The agreement of the midspan moment, M, was better than that of the moment, - 98 - M, because the measurements of the midspan lateral deflections in they axis were more stable than those in the x axis. • Secondly, the parallel-to-wood-grain stress-strain relationship used in the model was established based on the test results with 30 replicates. The parameters of the relationship consisted of considerable variation, due to the quality of the material in the specimens. These variations can affect the agreement of the modeling results. For example, the slope of the falling branch of the stress-strain curve (Ed) can affect the maximum midspan moment (Buchanan 1984). Other reasons may include the alignment of the specimens, the compliance in the fixing of the specimens, and the connection between the steel boot and the actuator, which might not be perfectly rotational free. These issues need to be addressed with more advanced testing facilities and more test replicates. 5.3.5. Laterally braced wood beam-column test results and model verification The test results of concern for the laterally braced wood beam-columns consisted of the relationships between the axial compression loads and the midspan lateral deflections or the lateral bracing forces. The results were also represented by the curves of the mean values and the upper and lower bounds, which were established in the same way as those of the unbraced beam-columns. The model input parameters were also the same as for the unbraced wood beam-columns; however, the mean properties of the nail connection and lateral bracing - 99 - member were added to the model to study the influence of the lateral bracing member. The test results of the axial compression loads and midspan lateral deflections of Specimen Groups 8 to 11 are shown in Figures 5-20 to 5-23, in comparison with the model predictions. 20000 z 16000 0 .2 12000 U, U, 8000 4000 x 0 20000 z 16000 0 .2 12000 U, U, 8000 4000 0 0 2 4 6 8 10 Midspan lateral deflection in x axis (mm) 20 40 60 Midspan lateral deflection in y axis (mm) 80 Figure 5-20. Test results and model predictions of Specimen Group 8 12 0 - 100 - 20000 Upperbound 16000 Average max loadC .2 12000 8000 0 —*-— mean4000 model 0 o 2 4 6 8 10 Midspan lateral deflection in x axis (mm) 20000 U perbound 16000 .2 C .2 12000 Cd) Co 8000 0 C) - 4000 model 0 0 20 40 60 80 Midspan lateral deflection in y axis (mm) Figure 5-21. Test results and model predictions of Specimen Group 9 35000 30000 Upperbound .2 25000 C 20000 ________ Cl) 15000 E 3 10000 —UE-— mean • 5000 —model 0 0 2 4 6 8 10 Midspan lateral deflection in x axis (mm) - 101 - 35000 30000 .2 25000 20000 15000 E 3 10000 5000 0 0 20 40 60 80 Midspan lateral deflection in y axis (mm) 100 Figure 5-22. Test results and model predictions of Specimen Group 10 35000 30000 25000 .2 20000 15000 3 10000 5000 35000 30000 2 25000 20000 15000 3 10000 5000 0 5 10 15 20 Midspan lateral deflection in y axis (mm) 25 Figure 5-23. Test results and model predictions of Specimen Group 11 The test results and model predictions of the relationships between the axial compression loads and the lateral bracing forces are shown in Figure 5-24. For each specimen 0 5 10 15 20 Midspan lateral deflection in x axis (mm) -102- group, the curves of the mean values were established by calculating the mean values of the axial compression load corresponding to a specific lateral bracing force. The lateral bracing forces obtained from the 2% rule-of-thumb were based on the axial compression loads of the test results. Winter’s and Plaut’s methods were based on the load eccentricity and midspan lateral deflections in the axis x (the braced direction) and the axial compression loads of the test results. For Specimen Group 8, the two methods’ predictions were the same since the load eccentricity was zero. 20000 Upperbound Averagemaxload ::::: 8000 /x’Lowerbound —model C-) /ff”’ 2%rule 400: __—Wnterarid 0 200 400 600 800 1000 Lateral bracing force of specimen group 8(N) 20000 Upper bound 16000 - —— Averagemaxload 12000 ,/ owerbound mean 8000 — — 400: —Winter 0 500 1000 1500 2000 Lateral bracing force of specimen group 9(N) - 103 - z V Co 0 0 0 0 a! E 0 0 Co Critical buckling load (kN) Corresponding lateral bracing force (N) Group Test Model Error (%) Test Model Error (%) 8 16.5 15.2 7.9 318 193 39.3 9 15.7 14.3 8.9 1065 909 14.6 10 21.4 19.0 11.2 1365 1126 17.5 11 26.6 27.4 3.0 1842 1682 8.7 It can be seen from the figures and the table that the model predictions agreed well 0 500 1000 1500 Lateral bracing force of specimen group 10(N) 2000 35000 30000 25000 20000 15000 10000 5000 0 g 35000 30000 0 25000 20000 0. E 15000 0 0 10000 5000 0 1500 2000 2500 Lateral bracing force of specimen group 11(N) Figure 5-24. Relationship between the axial compression load and lateral bracing force The test results and model predictions of the critical buckling load and the corresponding lateral bracing force of Specimen Groups 8 to 11 are listed in Table 5-8. For simplicity purpose, only mean values were presented. Table 5-8 Critical buckling load and lateral bracingforce ofSpecimen Group 8 to 11 0 500 1000 -104- with the test results in terms of the nonlinear relationships of the axial compression forces, the midspan lateral deflections and the lateral bracing forces. The critical buckling load and the ratio between it and the lateral bracing force were also predicted well. The maximum modeling error took place in Specimen Group 8 on the lateral bracing force. This was most possibly due to the imperfection of the specimens, such as the initial lateral deflection, which can affect the lateral stiffness ofthe beam-columns and thus the lateral bracing force. In the other three specimen groups, the influence of the load eccentricities prevailed over the initial lateral deflection; therefore, the model predictions were in better agreement with the test results. The initial lateral deflection and its influence on the critical buckling load and lateral bracing force will be addressed in a follow-up study. The FEM model was found superior to the 2% rule-of-thumb, Winter’s and Plaut’s methods, in terms of the predictions of the lateral bracing forces. Plaut’s method was the second most accurate for most of the specimen groups, except for Specimen Group 8, of which the 2% rule-of- thumb provided the best predictions. However, the 2% rule of thumb significantly underestimated the lateral bracing forces of the other three specimen groups. This implies a limitation on use of the 2% rule-of-thumb for simply supported wood beam-columns, beyond which it may result in a significant underestimation of the lateral bracing force. 5.4.Conclusion This chapter presented the results of the first part of the experimental study and computer program verification. The study was focused on the stability capacity and lateral bracing force of wood beam-columns subjected to biaxial eccentric compression load. The - 105 - program was verified in terms of the predictions of the maximum compression load, midspan lateral deflections and lateral bracing force. Good agreement was achieved. Both the test results and the model predictions indicated that the maximum load carrying capacity of the wood beam-columns was affected by the member geometry, the load eccentricity and the lateral bracing member. The 2% rule-of-thumb, Winter’s and Plaut’s methods were found to underestimate the lateral bracing forces in the tested cases. The variation of the lateral bracing stiffness and the imperfection (initial lateral deflection) of the beam-column members were not considered in this study and will be addressed in future research. - 106 - Chaper 6. Experimental Study and Model Verification II: Stiffness and Stability Capacity of MPC Wood Truss Assemblies Introduction This chapter presents the results of an experimental study on the stiffness, stability capacity and lateral bracing force of metal plate connected (MPC) wood trusses and truss assembles subjected to concentrated loading. Material property and full-scale tests of MPC wood trusses loaded individually and in truss assemblies were conducted to provide input parameters and verification for the computer program, SATA. Good agreement was achieved. 6.1. Research Method Material property tests were conducted to study the modulus of elasticity (MOE) of dimension lumber, the flexural stiffness of plywood sheathing and the load displacement relationship of the nail connection. The properties of MPC connections were obtained from a companion research project (Liu 2008). Full-scale tests were conducted to study the stiffness, critical buckling load and lateral bracing force of MPC wood trusses and truss assemblies. Fifteen trusses were tested individually, and two five-bay truss assemblies were tested with various provisions for bracing systems. Concentrated loads were applied to entice buckling failures of the compression webs of the trusses. Finite element method (FEM) based models of the individual trusses and five-bay - 107 - truss assemblies were constructed, based on the configurations of the individual trusses and bracing systems. The input parameters of the models were derived from the material property test results and the MPC connection properties from Liu’s work (2008). The model of an individual truss was calibrated for the in-plane and out-of-plane behaviour of MPC connections. Based on the calibration results, the models of the individual trusses and truss assemblies were verified by using the tests results of the critical buckling load, the lateral deflections ofthe buckled webs, the bottom chord deflections, and the lateral bracing force. The influence of the plywood sheathing and continuous lateral bracing (CLB) system on the critical buckling load of the truss assemblies was also studied, based on the test results. 6.1.1. Material The raw materials used in this study included dimension lumber, metal truss plates, plywood sheathing and common nails: • Spruce Pine Fir (SPF) dimension lumber graded as MSR165Of-1.5E was provided by a truss fabricator for the truss fabrication. The lumber was of 38x 89mm2 (nominally 2 by 4 in.2) and 38x 139 mm2 (nominally 2 by 6 in.2) in cross-section size and of a variety of lengths. • Metal truss plates of two models (MT18HSTM and MT2OTM) were provided by Mitek Canada, Inc. The MT18HSTM plates were manufactured from minimum 18 gauge steel (1.18 mm or 0.0466 in. in thickness) and the MT2OTM - 108 - plates were manufactured from minimum 20 gauge steel (0.9 mm or 0.0356 in. in thickness). Both models had eight teeth per 645 mm2 (1 in.2) with each tooth 9.5 mm (3/8 in.) in length. • Twenty-six sheets of four-ply plywood sheathing of 1219 x 2438 mm2 (4 by 8 ft2) in size and 11.9mm (15/32 in.) in thickness were obtained from a local retailer and used as roof sheathing material for the truss assemblies. • lOd common nails (76.2mm or 3 in. in length and 3.76 mm or 0.148 in. in diameter) and 6d common nails (50.8mm or 2 in. in length and 3.05 mm or 0.12 in. in diameter) were used for the wood-wood connections (for bracing) and wood-to-plywood connections (for the sheathing), respectively. 6.1.2. Individual truss and truss assembly designs Double Howe trusses were used in this study. The trusses were designed for a (top chord) slope of 6:12 and a span of 12.19 m (40 feet); and the truss members were connected by metal truss plates. All connections were fabricated with the MT2OTM plates except for the tension splice joints of the bottom chords, which were fabricated with the MT1 8HSTM plates to prevent joint failure. All truss plates were slightly oversized to entice the wood failure. The trusses were designed by the truss fabricator for a uniformly distributed live load of 3.571 kN/m2 (74.6 lbf/&) and a dead load of 0.144 kN/m2 (3.0 lbf/&) on the top chords and 0.335 kN/m2 (7.0 lbf/ft2)on the bottom chords, which were in compliance with Part 9 of Ontario Building Code (OBC) 2006, Canada Standards Association CSA 086-0 1 and the Truss Plate Institute of Canada 1996. According to the truss design, a maximum combined stress - 109 - index (CSI) of 0.93 was reached in the bottom chords. The truss design was reevaluated by the truss fabricator for the concentrated loading to be used in this experimental study. The objective was to prevent the MPC connections from failing before web buckling failure. The individual truss design is shown in Figure 6-1, where the dimension of the truss is in the format of foot-inch-sixteenth inch, and the dimension of the metal truss plates is in the format of inch-inch. For example, 20-0-0 means 20 feet, 0 inch and 0 sixteenth-inch; and 7 x 8 means the size of the plate is seven inches by eight inches. -110- 9-9-0 I’0 a a en cd 0 0) C, ‘a 0 v, C 0, en Co en Co CD 0 9 0 en en en Ii en ‘-I 0 C.’ I CC x en en F C.’ en >< ‘a F CC -1 ‘C, F C.’ ‘a 0 C.’ F 0a a 0 th CL C 0, C.’ en en a CD eo en U, en 0 C 0 C.’ op a C I en CC. Co en en a en 0 F— C CL z \\ C 0 F en 0 C-) ‘LI \\ 0 C-I F to L 0 CC I— 0 en en to 0 CL F- C CL CL CC F to 9-9-Ot Figure 6-1. Double Howe truss design Two truss assemblies were constructed by using the double Howe trusses. Each truss assembly consisted of five trusses spaced 610 mm (2 feet) on centre and fastened to bearing — 111— I—i 9-9-0 plates by wood screws of 76.2 mm (3 in.) in length. The truss assemblies were sheathed by plywood sheathing, which was connected to the top chords of the trusses by 6d common nails spaced 152.4 mm (6 in.) on centre at the edges and 304.8 mm (12 in.) on centre in the interior area of the plywood sheathing. The truss assemblies were reinforced by an auxiliary bracing system to improve the system stability and prevent bottom chord sway. The auxiliary bracing system was comprised of five diagonal bracing members for the vertical webs (webs Wi, W3 and W5 in Figure 6-1) and five horizontal bracing members at the panel joints (nodes J, K, M, 0 and P in Figure 6-1) of the bottom chords. The auxiliary bracing system of a five-bay truss assembly is shown in Figure 6-2, where the plywood sheathing and inclined webs are not shown for the purpose of clarity. Figure 6-2. A auxiliary bracing system of a five-bay truss assembly - 112- A CLB system was installed to the truss assemblies at specific stages of the tests. The CLB system consisted of two CLB bracing members, one for each of the two groups of the inclined webs at the centre of the trusses (W2 webs in Figure 6-1). Both CLB bracing members were installed at the midspan of the W2 webs. The CLB bracing system of a five-bay truss assembly is shown in Figure 6-3, where the plywood sheathing is not shown for the purpose of clarity. Figure 6-3. A CLB system of a five-bay truss assembly Both the auxiliary bracing and CLB systems were installed by using two lOd common nails for each connection between the wood members. 6.1.3. Material property tests Dimension lumber, plywood sheathing and 6d common nails were tested to evaluate - 113 - the basic material properties. 1 Od common nails had been studied in the beam-column tests and were not considered in these material property tests for brevity. 6.1.3.1. Dimension lumber tests One hundred and twenty-four pieces of 38x 89 mm2 (nominally 2 by 4 in.2) dimension lumber and one hundred and forty-two pieces of38x 139 mm2 (nominally 2 by 6 in.2) dimension lumber were provided and sampled by a local truss fabricator. The lumber was in a variety of lengths for the truss fabrication. All the lumber was graded as MSR1 65 Of-i. 5E and tested at the University of British Columbia (UBC) Timber Engineering and Applied Mechanics (TEAM) lab for MOE property. The tests included grading by a Cook-Bolinders machine (E-grading) and vibration tests. Each piece of lumber was labeled with a unique number for the purpose of identification. Figures 6-4 and 6-5 show the test setup of the E-grading and the vibration test, respectively. Figure 6-4. E-grading of dimension lumber by using a Cook-Bolinders machine -114- Figure 6-5. Vibration test of dimension lumber 6.1.3.2. Plywood sheathing tests The plywood sheathing was tested by a centre-point loading to evaluate the flexural stiffness. The tests were conducted in conformance with the ASTM standard D3043-00 (2006) “Standard Methods of Testing Structural Panels in Flexure”. The tests were conducted on a Sintech machine with a capacity of 245 kN (55 kips). Two specimen configurations were considered, each with 30 replicates: the principal direction of the face ply of the specimens was the parallel-to-the-span direction in the first configuration (Al) and the perpendicular-to-the-span direction (A2) in the second configuration. The specimen configurations and loading rates are listed in Table 6-1. - 115 - Table 6-1. Specimen configurations ofplvood sheathing tests Specimen Length between Loading rate Width (mm) Length (mm) Replication configuration supports (mm) (mm/mm) Al 50 400 300 7.2 30 A2 50 700 600 1.8 30 Note: the specimens were of the same thickness as the plywood sheathing Each specimen was measured before testing for the thickness and width at three positions. The results were used to determine the net cross-section area in order to calculate the flexural stiffness. The test setups are shown in Figures 6-6 and 6-7 for the first and second specimen configurations, respectively. Figure 6-6. Test setup for the plywood sheathing Specimen Configuration Al -116- Figure 6-7. Test setup for the plywood sheathing Specimen Configuration A2 6.1.3.3. Nail connection tests Nail connections fabricated with 6d common nails were tested to evaluate the load displacement relationship of the nail connections between the plywood sheathing and the top chords. Each nail specimen consisted oftwo sheets of plywood, one piece of dimension lumber of 38 x 139 mm2 (nominally 2 by 6 in.2) in cross-sectional size and eight nails. The specimens were fabricated in different configurations to study the influence of the load orientation, grain direction and principle direction of the face ply of the plywood sheathing. In total, four specimen configurations were used, each with 10 replicates. The specimen configurations were defined as: Configuration B 1: The applied load was parallel to the wood grain direction, but perpendicular to the principle direction of the face ply of the plywood sheathing. -117- Configuration B2: The applied load was parallel to both the wood grain direction and the principle direction of the face ply of the plywood sheathing. Configuration B3: The applied load was perpendicular to both the wood grain direction and the principle direction of the face ply of the plywood sheathing. Configuration B4: The applied load was perpendicular to the wood grain direction, but parallel to the principle direction of the face ply of the plywood sheathing. The specimens were tested under a tension load, which was applied by a Sintech machine with a capacity of 55 kips (245 kN). The loading rate was maintained constant at 2.5 mm/mm. The displacements of the nails were measured by two linear variable differential transducers (LVDTs), the readings from which were averaged to eliminate the influence of the incidental flexural deformation. The test setups for the specimens in the four configurations are shown in Figures 6-8 to 6-11, respectively. -118- Figure 6-8. Test setup for the nail connection Specimen Configuration B 1 Figure 6-9. Test setup for the nail connection Specimen Configuration B2 -119- Figure 6-10. Test setup for the nail connection Specimen Configuration B3 Figure 6-1 1. Test setup for the nail connection Specimen Configuration B4 -120- 6.1.4. Individual truss and truss assembly tests The individual truss and truss assembly tests were conducted in the UBC TEAM Lab. A steel frame, which was bolted to the concrete ground, was used as the main supporting frame. The steel frame consisted of a steel I beam, two steel tubes and six steel columns. The trusses were placed on the steel I beam at one end and the steel tubes at the other end. Consequently, the bottom chords of the trusses had a clearance of 471 mm, which was enough for the bottom chord deflections. The steel columns were mainly used to restrain the lateral deflections of the top and bottom chords. The test setups for individual truss and truss assembly testing are shown in Figures 6-12 and 6-13, respectively. Figure 6-12. Test setup for individual truss testing - 121 - The details of the test setups, including the support conditions, loading system and data acquisition, are described in Sections 6.1.4.1 to 6.1.4.3. 6.1.4.1. Support conditions The trusses were considered as being a pin-roller supported in the truss plane. At one end of the trusses, a pin support was achieved by using wood screws or steel clamps. At the other end, the trusses were placed on load cells. A bearing board, in conjunction with two steel plates, was used between the trusses and the load cells to achieve a roller support. The load cells were used to measure the reaction forces. The details of the roller support for an individually tested truss are shown in Figure 6-14, where two load cells were used for the - 122 - Figure 6-13. Test setup for truss assembly testing purpose of stability. Figure 6-14. Details of the roller support for individual truss testing The six steel columns were used to support the individually tested trusses and the trusses at the edge of the truss assemblies. The objective was to restrain the lateral deflections of the top and bottom chords at selected positions. At each of these positions, two sets of steel rods, C channels, bearing boards and steel plates were used to restrain the lateral deflection without affecting the in-plane displacement of the trusses. Details of the lateral support are shown in Figure 6-15. - 123 - Figure 6-15. Details of the lateral support for individual truss testing For the CLB system shown in Figure 6-3, one of the two bracing members was fixed at one end for displacement in the horizontal direction. A load cell with a capacity of 22.24 kN (5 kips) was used at the fixed end to measure the lateral bracing force. Details of the connection between the CLB bracing member and the load cell are shown in Figure 6-16. - 124 - Figure 6-16. Load cell for the measurement of the lateral bracing force 6.1.4.2. Loading system Four sets of actuator- and wire- pulley systems were used for the purpose of loading. Each set consisted of a hydraulic actuator with a capacity of 89 kN (20 kips), a galvanized wire rope of 7-19 type and 9.53 mm (3/8 in.) in diameter, and two pulleys. The actuator was controlled by a testing program to pull the wire rope at a constant rate of displacement. The wire rope was fixed at both ends — to the loading head of the actuator and to the ground. During the application of pulling, the wire rope slid around the two pulleys and applied a compression force to the top chords of the trusses. Details of the loading system are shown in Figure 6-17 and can also be found in Figure 6-12 (Test setups of individual truss testing). - 125 - Pulley 2 Figure 6-. 17. Actuator- and wire- pulley system for loading purpose Two load cells, each with a capacity of 22.24 kN (5 kips), were used in two of the four loading systems to measure the load applied on the truss top chords. The load cells were fixed with the wire ropes. Assuming the second pulley in Figure 6-17 was concentric with the top chords, the force measured in the wire rope can be doubled to approximate the load applied on the top chords. The readings from the load cells were also used to verif’ the readings from the actuators of the forces in the wire ropes. The details of the connection between the load cell and the wire rope are shown in Figure 6-18. Loading clamps Top chord Hydraulic Actuator Wire rope Wire rope fixed to the actuator Wire rope fixed - 126 - 6.1.4.3. Data acquisition The force and displacement responses of the individual trusses and truss assemblies were recorded at selected positions. The recorded responses included the reaction forces at the roller supports, the applied loads at the loading points, the in-plane deflections of the top and bottom chords, the lateral deflections of the buckled webs, and the lateral bracing forces of the truss assemblies reinforced by CLB systems. The reaction forces and applied loads were measured by the load cells and actuators. The in-plane and lateral deflections were measured by using string pots, each with a range of 508 mm (20 in.) and 0.5% accuracy, and LVDTs, each with a range of 127 mm (5 in.). The positions of the measurements are shown schematically in Figure 6-19 for individual trusses - 127 - Figure 6-18. Load cell underneath the wire rope of the loading system (top) and truss assemblies (bottom). Figure 6-19. Measurements of the force and deflection responses D F --Deflection measurement 0 --Force measurement --Deflection measurement --Force measurement - 128 - 6.1.5. Test procedures for individual trusses and truss assemblies The full-scale individual truss and truss assembly tests were carried out in six steps, including two steps for the individual trusses and four steps for the truss assemblies. Details are described in Sections 6.1.5.1 to 6.1.5.6. 6.1.5.1. Stiffness of individual trusses Fifteen trusses were tested nondestructively, in order to evaluate the stiffness of the trusses. Concentrated loads were applied on nodes B, D, F and H (illustrated in Figure 6-1) under displacement control. The loading rates were 0.9 mm/mm for nodes B and H and 1.33 mm/mm for nodes D and F. The loading was continued up to roughly 25% of the critical buckling load of the trusses, which was obtained in a dummy test. Each truss was measured before loading for the metal truss plate placement, width of the wood-to-wood gaps at the MPC connection areas and initial lateral deflections of the W2 webs (illustrated in Figure 6-1). The truss plate placement was measured from the centre of the joints to one of the corner nodes of the plates as specified in the truss design (Figure 6-1). The gap width was measured at the heel joints, web-to-bottom chord joints and splice joints of the chords. For each joint, the average gap width along the member thickness dimension was recorded. The initial lateral deflections were measured at the midspan of the W2 webs. 6.1.5.2. Critical buckling load of individual trusses From the individual truss stiffness tests, it was determined that the variation of the - 129 - truss stiffness was small; therefore, five trusses were randomly selected to study the critical buckling load. The loads were applied at the same four nodes (B, D, F and H) with same loading rates (0.9 mm/mm for nodes B and H and 1.33 mm/mm for nodes D and F). Two string pots were used to measure the midspan lateral deflections of the W2 webs where the buckling failures took place. The string pots were initially stretched to half ranges to measure the lateral deflections in either direction (toward and away from the string pots). One issue arose from the fact that the trusses were fabricated with a few more keeper nails, which were used for positioning of the metal truss plates during truss fabrication. One truss was tested with the extra keeper nails at the W2 webs-to-chords connections pulled out to study their influence on the critical buckling load of the trusses. 6.1.5.3. Critical buckling load of the first truss assembly The first truss assembly was tested to buckling failure with no CLB system installed. Four concentrated loads were applied under displacement control at nodes D and F of the second and fourth trusses. All the loading rates were 1.33 mm/mm. The loading was stopped after the W2 webs reached a notable lateral deflection (80 mm) due to buckling failure. The midspan lateral deflections of the W2 webs of the second and fourth trusses were measured by using two string pots and two LVDTs. 6.1.5.4. Critical buckling load of the first truss assembly reinforced by a CLB system The first truss assembly was reinforced by a CLB system and retested with the same -130- loads. The results were used to study the influence of the residual deformation of the MPC connections, which was due to the buckling failures in the last step of tests, on the critical buckling load of the truss assembly. One of the two CLB bracing members was fixed in the horizontal direction, as shown in Figure 6-16, in order to measure the lateral bracing force. The W2 webs braced by the other CLB bracing member were measured at midspan for the lateral deflections. 6.1.5.5. Load distribution behaviour of the second truss assembly The second truss assembly, which was reinforced by a CLB system, was tested nondestructively to study the load distribution behaviour. Both CLB bracing members were free in the horizontal direction during this step of the tests. The load distribution behaviour was studied by loading one truss of the assembly at a time. All five trusses were loaded once to study the influence of the loading positions on the load distribution behaviour. For each loading time, two concentrated loads up to 8000 N were applied at nodes D and F of the truss that was under loading. The loading rates were all the same at 1.33 mm/mm. The bottom chord midspan deflections and reaction forces measured at the roller supports of all five trusses were measured. 6.1.5.6. Critical buckling load of the second truss assembly The second truss assembly was tested with four concentrated loads at nodes D and F of - 131 - the second and the fourth trusses up to buckling failure. One of the two CLB bracing members was fixed in the horizontal direction to measure the lateral bracing force. The loading rates were all the same at 1.33 mm/mm. The loading was continued up to the material failure of the W2 webs, which was caused by the excessive lateral deflections due to buckling failure. The lateral bracing force and bottom chord midspan deflections were measured. The midspan lateral deflections of the W2 webs braced by a unfixed CLB bracing member were also measured. 6.2. Results of the Material Property Tests 6.2.1. MOE of dimension lumber The output of the dimension lumber E-grading tests was in the form of a series of forces required to create a prescribed deflection at discrete positions of the individual lumber. At each of these positions, a 900 mm segment (in length) of the dimension lumber was deformed flatwise at the middle to a deflection of 4.5 mm. The average MOE of this segment was then calculated based on the bending theory. Each piece of lumber was tested twice by switching the compression and tension sides. A typical set of MOE results from the E-grading tests is shown in Figure 6-20. -132- 14000 120OO 0 .10000 0 0 0) 6000 4000 2000 0 400 600 800 1000 1200 1400 1600 1800 2000 Longitudinal coordinate (mm) Figure 6-20. E-grading outputs of dimension lumber MOE The vibration tests were also conducted with the dimension lumber placed flatwise. The MOE results of the 38 x 89 mm2 and 38 x 139 mm2 dimension lumber were compiled and fitted to the three-parameter Weibull distribution separately. The cumulative distributions and Weibull distribution fitting results of the MOE are shown in Figure 6-21. The distribution parameters are listed in Table 6-2. The original MOE results are presented in Appendix B. C 1• —p 0 3PWeibulldistribution: 0.8 Location :8833.65 MPa C Scale :1665.54 MPa 2 0.6 Shape:1.56 • 0.4 0.2 0 Testresults —3PWeibull E 0 D o 7000 8500 10000 11500 13000 14500 16000 17500 MOE of 38x89 mm2 lumber (MPa) Side 1 —Side 2 Mean - 133 - ci 0 0 0.8 2 0.6 D 0.4 . 0.2 o 7000 Figure 6-21. Vibration test results of dimension lumber MOE It was found that the E-grading and vibration test results were very close. Therefore, for convenience, the vibration test results of the MOE of dimension lumber were used in the remainder of this study. 6.2.2. Flexural stiffness of plywood sheathing The flexural stiffness of the plywood sheathing was calculated based on the net cross-sectional area and the load deflection relationship of individual specimens. The results of the specimens tested in both configurations (Configurations Al and A2) were compiled and fitted to the three-parameter Weibull distribution. The cumulative distributions and Weibull distribution fitting results of the flexural stiffness are shown in Figures 6-22 and 6-23 for the specimens in Configurations Al and A2, respectively. The distribution parameters are also listed in Table 6-2. 8500 10000 11500 13000 14500 16000 17500 MOE of 38x139 mm2 umber (MPa) - 134 - 1C 0 C.) C C 0 .0 C,) •0 ci) > C 2 D C-) 0 C 0.8 4- C 0 0.6 .0 . 04 I 0.8 0.6 0.4 0.2 3P Weibull distribution: Location :241.2 MPa Scale :8553.5 MPa Shape: 9.18 o Test results —3PWeibuIl 0 0 3000 6000 9000 12000 15000 Flexure stiffness (MPa) Figure 6-22. Flexural stiffness of plywood sheathing tested in Configuration Al 3PWeibull distribution: Location: 799.4 MPa Scale: 851.0 MPa Shape :2.41 o Test results —3PWeibull 0 500 1000 1500 2000 2500 3000 3500 Flexure stiffness (MPa) Figure 6-23. Flexural stiffhess of plywood sheathing tested in Configuration A2 6.2.3. Load displacement relationship of the nail connection The test results of the load displacement relationship of 6d common nail connections tested in four configurations (Configurations Bi, B2, B3 and B4) were compiled and fitted to Foschi’s exponential model (1977). The original model can be expressed by: F(A) = (mO + ml AI)[1 - em0] (6-1) where A is the lateral displacement, F(A) is the lateral load, mo, ml and k are the parameters representing the intercept and slope of the asymptote and initial stiffness of the load displacement curve, respectively. - 135 - In this study, the model was simplified by assuming the load displacement curve levels out at large nail displacement. By definition, mo was evaluated as the maximum load of the individual specimens; ml was set to zero; and, k was evaluated based on the test results by using the least square method (LSM). The test results and model fittings of the average load displacement relationship of the specimens in Configurations B 1 and B4 are shown in Figures 6-24. 1200 1000 800 (U .2 600 400 200 0 1200 1000 800 •0(U 2 600 a 400 c. 200 0 0 1 2 3 4 5 Nail displacement of specimen configuration BI (mm) 6 0 1 2 3 4 5 6 7 8 9 10 11 Nail displacement of specimen configuration B4 (mm) Figure 6-24. Load displacement relationship of the nail connection The variation of the parameters was evaluated based on the results of all 10 replicates. The mean value and standard deviation (St.d.) of the parameters are listed in Table 6-2. Model - 136 - Table 6-2. Materialproperty tests results ofdimension lumber, plywood sheathing and nail connections Weibull distribution parameters Material Configuration Property Unit * Mean St.d. location scale shape ________ Dimension 38x89mm2 8833.65 1665.54 1.56 10327.7 977.3 2 MOE MPalumber 38x 139 mm 8811.21 1982.2 1.5 10595.6 1200.2 Plywood Al 241.2 8553.5 9.18 8382.8 1165.3 MOE MPa sheathing A2 799.4 851.0 2.41 1558.1 324.2 m0 N -- -- -- 905.2 141.1 B1 m1 N/mm -- -- -- 0.0 0.0 k N/mm -- -- -- 2720.3 2128.9 m0 N -- -- -- 885.2 169.4 B2 m1 N/mm -- -- -- 0.0 0.0 Nail k N/mm -- -- -- 2510.6 1509.3 connection m0 N -- -- -- 1014.3 105.1 83 m1 N/mm -- -- -- 0.0 0.0 k N/mm -- -- -- 1645.0 577.5 m0 N -- -- -- 1010.3 46.6 B4 m1 N/mm -- -- -- 0.0 0.0 k N/mm -- -- -- 1142.7 295.2 Note:* shape parameters are dimensionless 6.3. FEM-Based Models of Individual Trusses and Truss Assemblies FEM-based models were constructed based on the configurations of the individual trusses and truss assemblies. The plywood sheathing, truss members, MPC connections, and nail connections were modeled by the thin plate elements, beam elements, metal plate connection elements, and nonlinear spring elements, respectively. The input parameters were based on the mean values of the material property test results in Tables 5-3, 5-4, 5-5 and 6-2, the size effect factors in Table 5-6, and the MPC connection properties from Liu’s work (2008). The models considered the individual trusses to be simply supported. The initial lateral deflections of the W2 webs were assumed to be of half sinusoidal shape with the -137- maximum values at the midspan. The gaps of the plywood sheathing were ignored. The models were assumed to be loaded by prescribed deflections at nodes B, D, F and H, in conformance with the loading positions and loading rates used in the individual truss tests and truss assembly tests. The models for an individual truss and a truss assembly are shown in Figures 6-25 and 6-26, respectively. Figure 6-25. FEM model of an individual truss - 138- Figure 6-26. FEM model of a five-bay truss assembly 6.4.Test Results and Model Verification 6.4.1. Stiffness of individual trusses The measurements of the truss plate placement, gap width and initial midspan lateral deflection of the W2 webs are presented in Appendix C. The stiffness of the individually tested trusses was quantified based on the relationship between the bottom chord midspan deflection and the reaction force measured at the roller support. The results were used to calibrate the individual truss model in terms of the in-plane behaviour of the MPC connections. The calibrated MPC connection properties based on an individual tooth are listed in Table 6-3, where the parameters k, mo and mi were defined in Equation (6-1) based on Foschi’s work (1977). During the calibration, m1 was assumed to be zero for simplicity. - 139 - Table 6-3. Calibrated MPC connection properties based on an individual tooth Parameter Direction* Mean k(N/mm) AA 3514.14 AE 1312.36 EA 3562.45 EE 1477.22 m0 (N) AA 487.46 AE 336.49 EA 435.24 EE 323.81 Note: the directions were defined in Foschi’s work (1977), where the first and second letters of the “A” and “E” combinations defined whether the load was parallel (A) or perpendicular (E) to the plate major axis and the grain direction of the truss member, respectively. The test results (at node M) and the calibrated model predictions are shown in Figure 6-27. The calibrated model was also used to predict the bottom chord deflections measured at nodes P, 0, K, and I. The model predictions were compared to the test results as verification. The results are shown in Figure 6-27 where, in order to distinguish the model predictions and the rest results, the external deflections in the FEM model were up to 125% ofthe loading point deflections applied in the tests. Good agreement was achieved. - 140 - 5000 4000 3000 2000 1000 0 5000 ‘ 4000 3000 . 2000 C.) 1000 0 5000 ‘ 4000 3000 2000 0 1000 0 5000 4000 3000 C . 2000 C) ‘1) ° 1000 0 — — — e . —0—FEM model 40 1 2 3 Bottom chord deflection at node P (mm) 0 — FEM model 0 1 2 3 4 5 Bottom chord deflection at node 0 (mm) 6 0 — model 0 1 2 3 4 5 Bottom chord deflection at node M (mm) 6 —€-• FEM model 0 1 2 3 4 5 Bottom chord deflection at node K (mm) 6 - 141 - 5000 4000 3000 . 2000 C.) 1000 0 Bottom chord deflection at node J (mm) Figure 6-27. The reaction force and bottom chord deflection of individually tested trusses 6.4.2. Critical buckling load of individual trusses As previously mentioned, five trusses were randomly selected and tested to study the critical buckling load. All buckling failures took place at the W2 webs. It was found that the directions of the buckling failures of the individual trusses differed from each other. However, the critical buckling loads were very close. Therefore, the FEM model assumed the two W2 webs with same initial lateral deflections with the maximum values at midspan equaled -0.93 mm, which was based on the mean value of the initial lateral deflection measurements made in the first step of the tests. Figure 6-28 shows the buckling failures of the two W2 webs of an individually tested truss. 0 1 2 3 4 - 142 - Figure 6-28. Web buckling failures of the individually tested trusses The critical buckling loads were used to calibrate the out-of-plane rotational stiffness of the MPC connections at the W2 webs-to-chords joints of the individual trusses. The out-of-plane rotational stiffness of the MPC connections at other joints was ignored, as its effect on the critical buckling load was negligible. No models have been developed to evaluate the out-of-plane rotational stiffness of MPC connections. Instead, the webs are normally assumed pinned and the contribution of the out-of-plane rotational stiffness on the critical buckling load is ignored. In this study, a simplified model was developed to obtain a rough estimation of this stiffness, allowing for an improved estimation via calibration. The simplified model of the out-of-plane rotational stiffness of a MPC connection is shown in Figure 6-29. - 43 - Top chord Figure 6-29. Out-of-plane rotational stiffness of a MPC connection Based on the geometry of the connection, a rough estimation of the stiffness, k, can be made based the gap width, Al, and bending stiffliess, El, of the metal truss plates as: k_M_M 1 El 62 _o_ø ( where M, 0 and are the bending moment, angle of rotation and curvature, respectively. Considering a web of 38 x 89 mm2 in cross-section size, a metal truss plate of 0.9 mm in thickness and MOE of2.Ox io N/mm2,k can be calculated for a gap width of 2.0 mm as 5.78x i0 N.mm/rad. This estimation did not consider the truss plate buckling, teeth withdrawn and gap opening under large connection deformation. The reduction of the cross-sectional area of the truss plates due to the tooth slots was not taken into account either. However, these effects were accounted for by using a reduction coefficient, which was determined as 0.006 from the truss model calibration. This resulted in an out-of-plane rotational stiffness of 3.5 x i0 N.mm/rad. The model predictions and test results of the relationship between the reaction force at the roller support and the midspan lateral deflection of the W2 webs of the individually tested Steel plate -144- trusses are shown in Figure 6-30. The model predictions corresponded to the out-of-plane rotational stiffnesses of a pin connection (0 N.mmlrad), the rough estimation (5.78 x 1 0 N.mm/rad) and the calibrated estimation (3.5 x 1 N.mm!rad), respectively. The test results of the truss with the extra keeper nails removed are also displayed in Figure 6-30 (broken lines). Based on these results, it is believed that the keeper nails had no noticeable influence on the critical buckling load of individual trusses. 20000 16000 12000 . 8000 C.) 4000 0 0 20 40 60 80 100 Midspan lateral deflection of W2 webs (mm) Figure 6-3 0. Reaction force and midspan lateral deflection of W2 webs of individual trusses The test results and calibrated model predictions of the relationship between the reaction force at the roller support and the bottom chord midspan deflection are shown in Figure 6-31. Good agreement was achieved. 120 - 145 - 20000 16000 12000 . 8000 () 4000 0 Figure 6-31. Reaction force and bottom chord midspan deflection of individually tested trusses 6.4.3. Critical buckling load of the first truss assembly The first truss assembly was tested without a CLB system. The buckling failures took place at the W2 webs of the second and fourth trusses, which were directly loaded. Without a CLB system, the W2 webs of the truss assembly deflected separately. Consequently, noticeable lateral deflections were not observed in the W2 webs of the first, third and fifth trusses, as they were not directly loaded. Figure 6-32 shows the lateral buckling of a W2 web of the second truss of the truss assembly. It can be seen that the W2 webs of the adjacent trusses remained straight in the truss plane. 0 5 10 15 20 25 Bottom chord midspan deflection (mm) -146- Figure 6-32. Buckling failure of the first truss assembly The metal truss plates of the MPC connections between the buckled W2 webs and the top and bottom chords were subjected to irrecoverable deformations, due to buckling failure and the resulting lateral deflections of the W2 webs. The residual deformations observed after the loads were removed are shown in Figure 6-33, including bulging of the metal truss plates and teeth withdrawn at the edges of the truss plates. - 147 - Figure 6-33. Metal truss plate bulging and teeth withdrawn The test results and model predictions of the relationship between the applied loads on the individual loading points and the midspan lateral deflections of the W2 webs of the second and fourth trusses are shown in Figure 6-34. The model predictions were based on the calibrated out-of-plane rotational stiffness and an initial midspan lateral deflection of -0.93 mm that was considered for all the W2 webs. -148- 30000 25000 z 20000 0 15000 a) 10000 5000 0 Figure 6-34. Applied load and midspan lateral deflection of W2 webs of the first truss assembly The test results and model predictions of the relationship between the reaction forces of the truss assembly (the sum of the reaction forces at the five roller supports) and the bottom chord midspan deflections of the individual trusses (numbered from BV1 to BV5) are shown in Figure 6-3 5. 60000 50000 z 40000 0 30000 0 20000 a) 10000 0 3 6 9 12 15 Bottom chord midspan deflection (mm) Figure 6-35. Reaction force and bottom chord midspan deflection of the first truss assembly It can be seen from Figures 6-34 and 6-35 that the model provided very good predictions of the critical buckling load and midspan lateral deflections of the W2 webs. For the bottom chord midspan deflections, the model predictions for the second and fourth trusses -149- 0 25 50 75 100 125 Midspan lateral deflection of W2 web (mm) 0 were also very good. The predictions of the other three trusses were affected by the nonlinear load redistribution behaviour, due to the web buckling failures. However, the predictions were still satisfactory during the linear stage of the truss assembly. 6.4.4. Critical buckling load of the first truss assembly reinforced by a CLB system The first truss assembly was reinforced using a CLB system and retested. The same loading was used as in the last test. One of the two CLB bracing members was fixed in the horizontal direction to measure the lateral bracing force. The midspan lateral deflections of the W2 webs braced by the other bracing member were recorded. Figure 6-36 shows the buckling failure of the W2 webs with the fixed CLB bracing member. Figure 6-3 6. Buckling failure of the first truss assembly reinforced by using a CLB system - 150- The test results are presented together with the test results of the second truss assembly for comparison. The influence of the residual deformations of the MPC connections (as shown in Figure 6-33) on the critical buckling load is discussed in Section 6.4.6. 6.4.5. Load distribution behaviour of the second truss assembly The second truss assembly was tested nondestructively to study the load distribution behaviour. The truss assembly was reinforced by a CLB system. Both CLB bracing members were free to displace in the horizontal direction. Figure 6-37 shows the test setups of the truss assembly, in which, the third truss was under load. Figure 6-37. Test setup for the load distribution behaviour test of the second truss assembly The test results consisted of the reaction forces measured at the roller supports of the - 151 - E E 0 C.) ci) ci 0 C) E 0 0 individual trusses and the bottom chord midspan defleetions, when one of the five trusses was loaded at a time by two concentrated loads at nodes D and F. The test results and model predictions are shown in Figures 6-38, in which each curve represents the distribution of the reaction forces or the bottom chord deflections amongst the five trusses when one specific truss, which is distinguished by the Figure’s legend, is under load. The model predictions are represented by the broken lines for clarity. 5000 4000 0 3000 2000 1000 0 -1000 6 5 4 3 2 0 Figure 6-38. Distribution of the reaction force and bottom chord midspan deflection of the second truss assembly As can be seen from Figure 6-3 8, the model predictions agreed very well with the test results. The amount of load distributed away from the directly loaded truss, which can be calculated as twice of the sum ofthe roller support reaction forces measured from the other four - 152 - 1 2 3 4 5 Truss position trusses, was found to be dependent on the position of the truss within the truss assembly. The ratio of this amount of load to the total applied load was the highest when the middle truss was loaded, and the lowest when the edge trusses were loaded. In this study, this ratio was found to be between 43% and 76%. 6.4.6. Critical buckling load of the second truss assembly The second truss assembly was tested to buckling failure with one of the two CLB bracing members fixed in the horizontal direction in order to measure the lateral bracing force. The W2 webs braced by the unfixed CLB bracing member buckled in a half sinusoidal shape (first mode). In the end, material failures took place at the midspan of the webs, due to excessive lateral deflections and the resulting tensile stresses. On the other hand, the W2 webs braced by the fixed CLB bracing member were forced into second mode buckling. The lateral deflections were in a complete sinusoidal shape. The first material failure took place at the quarter node of the W2 web of the fourth truss. Nail connection failure took place at the W2 webs with the fixed CLB bracing member. The connection failure occurred right after the material failure of the adjacent webs. The wood at the connection area split due to the lateral bracing force passed from the nails. The first and second buckling modes and the failure at the quarter node of a W2 web are shown in Figure 6-39. The midspan tension and nail connection failures due to wood splitting are shown in Figure 6-40. - 153 - Figure 6-39. Buckling failures of the laterally braced W2 webs of the second truss assembly Tension failure at the quarter node -154- Figure 6-40. Midspan tension failure and nail connection failure of the second truss assembly The test results and model predictions of the midspan lateral deflections of the W2 webs, lateral bracing force and bottom chord midspan deflections are shown in Figures 6-41 to 6-43, respectively. The model predictions were based on an initial midspan lateral deflection of -0.93 mm for all the W2 webs. - 155 - In Figures 6-41 to 6-43, the applied load was calculated as the average of the two concentrated loads on each group of the W2 webs, or, in other words, the same side ofthe truss assembly. The midspan lateral deflections were measured from the W2 webs braced by the unfixed CLB bracing member. The lateral bracing force was measured by the load cell at the end of the fixed CLB bracing member. The results of the first truss assembly, which was retested with a CLB system, are shown in Figures 6-41 and 6-42. The predictions of the lateral bracing force by using the 2% rule-of-thumb, as discussed in Chapter 5, are also shown in Figure 6-42 for comparison. 0 20 40 60 80 100 120 140 Midspan lateral deflection of the W2 webs (mm) I, I- / 40000 z 30000 0 0 . 20000 0. 10000 0 First truss assembly Second truss assem bly — Model Figure 6-41. Applied load and midspan lateral deflection of the W2 webs of the second truss assembly - 156 - 40000 — z 30000 - - - - .2 20000 - - - a. .-‘ ———- Firsttruss assembly Second truss assembly 10000 2% rule predictions — Model 0 I I -200 100 400 700 1000 1300 1600 Lateral bracing force of theW2 webs (N) Figure 6-42. Applied load and lateral bracing force of the W2webs of the second truss assembly 40000 30000 -a 20000 10000 0 5 10 15 20 Midspan vertical deflection of the bottom chord (mm) Figure 6-43. Applied load and bottom chord midspan deflection of the second truss assembly The model predictions agreed very well with the test results of the two truss assemblies. The relationship between the applied load and the midspan lateral deflection of the W2 webs was predicted well, up to the critical buckling load. The model predictions of the lateral bracing force were between the test results of the first and the second truss assemblies. The bottom chord deflections of the second and fourth trusses were predicted well. The predictions for the other three trusses were less accurate, due to the nonlinear load redistribution and the interaction amongst the W2 webs braced by the 0 25 - 157 - same bracing member. Comparison of the test results of the two truss assemblies, as shown in Figures 6-41 and 6-42, indicated that the influence of the residual deformation of the MPC connections on the critical buckling load and lateral deflection was not significant. This finding can be used as reference for the retrofitting and reinforcing of the MPC wood truss assemblies after buckling failures. The predictions of the lateral bracing force from the 2% rule-of-thumb were evaluated based on the total compression load in the five W2 webs. Since no test results of the compression load were available, the total compression load was determined based on the FEM model predictions. Figure 6-42 indicates that the 2% rule-of-thumb overestimated the lateral bracing force of the two truss assemblies tested. This was caused by three reasons: 1) three of the five W2 webs were not directly loaded due to the concentrated loading pattern; 2) the initial lateral deflections of the W2 webs were in different directions; and, 3) the out-of-plane rotational stiffness of MPC connections increased the flexural stiffness of the W2 webs. The loading pattern, initial lateral deflection and out-of-plane rotational stiffness ofMPC connections were further studied in a reliability-based analysis, which is described in Chapter 7. 6.5. Discussion The test results of the individually tested trusses and the two truss assemblies were used to investigate the influence of plywood sheathing and a CLB system on the critical buckling loads of IvIPC truss assemblies. - 158 - 6.5.1.Effect of plywood sheathing Plywood sheathing can increase the critical buckling load of a MPC truss assembly in two ways. Firstly, the plywood sheathing can distribute the load away from the directly loaded truss to the adjacent trusses. Secondly, the plywood sheathing can increase the stiffness of the top chords via the composite effect, which reduces the load passed to the webs where buckling failure may take place. Figure 6-44 shows the deflections of the plywood sheathing amongst the trusses, which indicates the load distribution. The influence of the plywood sheathing on the critical buckling load of a truss assembly was studied by comparing the test results of the individual trusses and the first truss assembly, which was tested without a CLB system. The results can be found in Figures 6-30 and 6-34, respectively, where it can be seen that the critical buckling load of the tested trusses was increased by about 50% due to the plywood sheathing. Figure 6-44. Load distribution via plywood sheathing -159- 6.5.2. Effect of a CLB system A CLB system is able to increase the critical buckling load of a truss assembly by laterally supporting the compression webs. The lateral support is attributed to the end fixity and axial stifihess of the bracing member itself, the stiffliess of the nail connections and the flexural stiffnesses of the other compression webs that are braced by the same bracing member. The contribution of the lateral bracing member and the nail connection to the lateral support has been studied in the beam-column tests in Chapter 5. In this study, the influence of the lateral support with a CLB system was studied by comparing the test results of the first and second truss assemblies, which represented a truss assembly without and with a CLB system, respectively. The test results are shown in Figures 6-34 and 6-41, where it can be seen that the critical buckling load of the tested truss assembly was increased by about 60%. 6.6.Conclusion This chapter described an experimental study on the stability capacity and lateral bracing force of individual MPC trusses and MPC truss assemblies. The influence of plywood sheathing and a CLB system on the maximum load carrying capacity of the truss assemblies was also studied. The test results were used to provide input parameters and verification for the computer program, SATA. Good agreement was observed. The 2% rule-of-thumb was also studied based on the lateral bracing force of the truss assemblies. ft was found that the stiffness of the fifteen tested trusses was similar. The critical - 160 - buckling load of the five tested trusses exhibited variability due to the variation of the initial out-of-plane deformation of the W2 webs. It was also found that the out-of-plane rotational stiffness of the MPC connections had a significant effect on the maximum load carrying capacity of the trusses, whereas the extra keeper nails did not. The plywood sheathing and CLB system were found to be able to increase the critical buckling load of the individual trusses and truss assemblies tested, by approximately 50% and 60%, respectively. The 2% rule-of-thumb was found to overestimate the lateral bracing force of the truss assemblies tested in this study. - 161 - Chaper 7. Reliability Analysis of Critical Buckling Load of MPC Wood Truss Assemblies Introduction This chapter presents the results of a reliability analysis, using the verified truss assembly model, on the critical buckling load of a metal plate connected (MPC) wood truss assembly, which is subjected to a uniformly distributed roof load. The response surface method was used to approximate the performance function based on a partial composite sampling scheme. The probability of buckling failure of the truss assembly was evaluated with consideration ofthe influence ofa continuous lateral bracing (CLB) system and variation ofthe external load. 7.1. The Response Surface Method The response surface method, originally proposed by Box and Wilson (1954), is a well developed statistical technique by which a simplified functional relationship is established between a scalar variable (response or output variable) and a number of variables (input variables) that are believed to have an influence on the response (Pinto et al. 2004). Without the response surface method, a dilemma exists in the selection of reliability analysis methods. On the one hand, the performance function and its gradients required by the first- and second- order reliability methods cannot be easily obtained in structural reliability analyses (Yu et al. 2002). On the other hand, simulation methods, such as Monte Carlo simulation, involve a large number of structural analyses for different realizations of the - 162 - random variables (Gupta and Manohar 2004). The response surface method establishes an approximate performance function of simple polynomials based on a limited number of samplings of the structural response, with which the evaluation of the probability of failure can be greatly facilitated. A full second-order response surface model can be expressed as: Y=1 +/3x+/3,1x+6 (71) i=1 1=1 ji where y is the response of interest with I ranging between 1 to k; k is the number of input variables, x; /30, /3, and / are unknown coefficients; and e is a random variable that accounts for the fitting error, which usually arises from the lack of fit of the model and the statistical incompleteness (Pinto et al. 2004). The polynomial response model does not generally include terms of a higher order than quadratic, as it can lead to a large number of coefficients and sometimes irregular shapes of the response surface (Gomes and Awruch 2004). Even the quadratic polynomial does not need to be complete. Some of the linear, quadratic or mixed terms can be omitted if it is supported by either physical reasons or preliminary analysis results (Wong 1985, Bucher and Bourgund 1990, Kim and Na 1997). Experimental design and determination of the response surface function (coefficients) are the central issues in the response surface method. The purpose of an experimental design is to prepare a proper plan in order to sample the response levels at different values of the random variables, which are usually initiated around the mean values. Several sampling techniques have been proposed, including the central composite design, the fractional factorial design, the random design and the partially balanced incomplete box design (Gomes and Awruch 2004). - 163 - Some sampling techniques result in an exponential increase ofthe total number of experiments, due to the number of random variables, and lead to unacceptably high computational costs (Wong 2005). To overcome this difficulty, alternative sampling techniques are proposed, such as Bucher and Bourgund’s design (1990), Rajashekhar and Ellingwood’s adaptive sampling plan (1993) and Kin and Na’s vector projection sampling technique (1997). Obtaining the results from a sampling scheme, the coefficients of the response surface function can be determined by using regression analysis methods. The error term, s, is ignored during this process and can be evaluated afterwards as the difference between the sample responses and the responses predicted by the response surface function using the same or a new set of sample values of the input random variables. 7.2. Reliability Analysis of the Critical Buckling Load of MPC Wood Truss Assemblies The critical buckling load of a MPC wood truss assembly can be affected by the variation of both the structural behaviour and the external loads. Reliability analysis is a natural choice to evaluate the probability of buckling failure of a MPC truss assembly for certain distributions ofthe external loads. A case study considering a simplified MPC truss assembly is described in following sections to illustrate the approach of using the response surface method to evaluate the probability of buckling failure of truss assembly. 7.2.1. Truss configuration and FEM model The MPC wood truss assembly was assumed, for convenience, to be consisted of the - 164 - identical double Howe trusses as those used in Chapter 6. The material properties of the individual trusses, plywood sheathing and nail connections were also assumed to be identical with those used in Chapters 5 and 6. Since the double Howe trusses were symmetric about the centre plane, for simplicity only the left half of the trusses was considered in this study. The webs at the centre plane were halved, in order to make the structural behaviour of the half trusses comparable to the complete trusses. This resulted in a cross-sectional size of 38 x 70 mm2.The configuration of a half truss is shown in Figure 7-1. The notations of the dimensions were organized in the same way as in Figure 6-1. Q 7x6 MT2O 6x10 MT20 Figure 7-1. Truss configuration (plate size shown in inches with 1 inch 25.4 mm) The MPC truss assembly was assumed to consist of three trusses spaced 610 mm (2 feet) on centre. It was also assumed that an auxiliary bracing system and a CLB system, as - 165 - E 7x8 MT2O D 4x8MT20 C 5x6 MT2O B B A 0 N M 24 MT2O 35x6 MT2O 5x12 MTI8HS 81 I I T 4. 2210 mm 1943 mm 1943 mm 6096 mm MT2O illustrated in Chapter 6, were installed in the truss assembly. The CLB system was considered with up to two bracing members per compression web to study its influence on the critical buckling load. A finite element method (FEM) based model was constructed based on the geometry of the truss assembly using the same FEM elements as in Chapter 6. The model evaluated the critical buckling load of the truss assembly, in which the W2 webs were the most susceptible to buckling failure. The truss assembly and FEM model are shown in Figures 7-2 and 7-3, respectively. In Figure 7-2, the bracing members are shown in colours for clarity. Figure 7-2. Three-bay MPC truss assembly based on a half truss configuration - 166 - Figure 7-3. FEM model of the three-bay MPC truss assembly 7.2.2. Fixity factors of the CLB bracing members The fixity factor defines the stiffness of a lateral support asserted at the end of a bracing member. Considering the three-bay truss assembly as a part of a larger truss assembly, this lateral support is mainly attributed to the flexural stiffness of the other webs that are braced by the same bracing member(s). Diagonal bracing members, although not considered in the three-bay truss assembly, are usually required to be installed in the larger truss assembly and can also contribute to the lateral support. In this study, the three-bay truss assembly was assumed to be at the center of a fifteen-bay truss assembly with the trusses spaced 610 mm (2 feet) on centre. It was also assumed that the diagonal bracing members were used at both ends of the truss assembly and - 167 - Rdy :j /1 aligned at 45 degrees to the CLB bracing members, as in conformance with the document “Handling, Erection and Bracing of Wood Trusses” that is prepared by the Truss Plate Institute of Canada (TPIC). The fixity factor was mainly dependent on the structural behaviour of the W2 webs and the bracing members in the same plane; therefore, two-dimensional models were constructed to study the fixity factors of two different provisions of the CLB system. The models consisted of the W2 webs (15 pieces), CLB and diagonal bracing members, and nail connections between them. Each web was assumed to be simply supported with a rotational spring of 3.5 x 1 O N.mm/rad, which was based on the calibration results of the out-of-plane rotational stiffness of MPC connections in Chapter 6. The models corresponding to a truss assembly with one and two CLB bracing members per W2 web are shown in Figures 7-4 and 7-5, respectively. W2 webs of the three-bay truss assembly E S 00 Figure 7-4. Fixity factor model for one CLB bracing member per W2 web S 1 S -+- c SI c” El c’I 610 mm k3.5 x N.mm/rad 8540 mm - 168 - 7 F —‘-———_.f- — j 6 DLI / Compression web / 7 Diago 610 mm k=3.5 x i07 N.mm/rad 8540 mm Figure 7-5. Fixity factor model for two CLB bracing members per W2 web The fixity factor was calculated based on the load and displacement responses of the intersection nodes between the CLB bracing member and the first W2 web of the three-bay truss assembly, as shown in Figures 7-4 and 7-5, where the W2 webs of the three-bay truss assembly are highlighted. The nodal displacement, ô, caused by a concentrated lateral load, F, was used to calculate the fixity factor as FIö. In the second model, the fixity factors of the two CLB bracing members equaled each other, due to the assembly’s symmetry, and were calculated in the same way. The variation of the fixity factor(s) was quantified by using Monte Carlo simulation. The variation of the modulus of elasticity (MOE) of the W2 webs and the translational stiffness of the nail connections were considered in the simulations. The results of 100 simulations were fitted to the three-parameter Weibull distribution. The input parameters and the simulation results are listed in Table 7-1. —1-- c L W2 webs of the three-bay truss assembly T Ig icc I Ice —4- nal bracing -169- Table 7-1. Simulation results ofthe fixityfactors ofCLB bracing members Three-parameter Weibull distribution Item Sub-item Unit location scale. shape2 Mean St.d MOE of dimension lumber -- GPa 8.834 1.666 1.56 10.328 0.977 Type 11 N/mm -. -- -- 824.7 122.0Nail connection stiffness Type2i N/mm -- -- -- 1371.0 297.1 One CLB N/mm 354.8 70.0 3.22 417.6 21.3 Fixity factors two CLBs N/mm 391.5 90.0 3.81 472.9 23.6 Note: nail connection types are described in section 5.1.2.2 of Chapter 5. 2 shape parameters are dimensionless. C 0 0.8 C o 0. 0 0.4 0.2 ( D 350 The cumulative distributions and the Weibull distribution fitting results of the fixity factors are shown in Figure 7-6 for comparison. 400 450 500 550 Fixity factor of one CLB per web (N/mm) C 40.8 . 0.6 D 0 0.4 . 0.2 350 400 450 500 550 Fixity factor of two CLBs per web (N/mm) Figure 7-6. Cumulative distribution of the fixity factors of CLB bracing members - 170 - 7.2.3. Random variables of the reliability analysis Only the most relevant factors to study the buckling failure mode ofthe W2 webs were considered as random variables in the reliability analysis in order to reduce the number of samplings required to establish the response surface function. In total, up to eight random variables were considered. These included: • The MOE of the W2 webs; • The initial midspan lateral deflection of the W2 webs; • The translational stiffness of the nail connections; and • The fixity factors of the CLB bracing members of the W2 webs. The MOE values of the three W2 webs were assumed to be of the same distribution, as were the initial midspan lateral deflections of the W2 webs. They were sampled separately in the reliability analysis in order to distinguish their influence on the critical buckling load. The variation of the fixity factors of the CLB bracing members was based on the simulation results in Table 7-1. Since the variation of the nail connection stiffness was already considered in the evaluation of the fixity factors, constant nail connection properties were used in the analysis of the truss assembly. 7.2.4. Response surface function The response surface function of the critical buckling load of the truss assembly consisted of all the linear quadratic terms of the input random variables, and with selected - 171 - mixed terms for the interaction between the random variables. The selection of the mixed terms was based on the results of an analysis of variance (ANOVA) with a confidence level of 95%. The results of the ANOVA are presented in Appendix D for brevity. Eventually, the response functions for the truss system with zero, one and two CLB members per W2 web can be expressed as: i i C0X45+C1X46+C2X56+6 y=a0+ax +b1x? + c0x37+c1x45+c2x46+c3x56+6 (7-2) 1=1 ii c0x14+c1x45+c2x46+c3x48+x4x56+C5X68+6 where y is the critical buckling load, expressed as a coefficient of a reference load; x, is the input random variable; 1 is the total number of random variables; ao, a, b1 and c are unknown coefficients; and, s is the error term. The statistical information of the input random variables were obtained from the test results in Chapter 6 and the Monte Carlo simulation results using the FEM models in Figures 7-4 and 7-5. The detailed information of the input parameters is listed in Table 7-2. -172- Table 7-2. Probabilistic distribution parameters ofthe input random variables 3-parameter Weibull distribution Variable Description Unit Mean St.d. location scale shape’ Basic random variables x1 MOE GPa 8.834 1.666 1.56 10.328 0.977 x2 MOE GPa 8.834 1.666 1.56 10.328 0.977 x3 MOE GPa 8.834 1.666 1.56 10.328 0.977 x4 Midspan deflection2 mm -- -- -- -0.93 2.42 x5 Midspan deflection mm -- -- -- -0.93 2.42 x6 Midspan deflection mm -- -- -- -0.93 2.42 Additional random variables for one CLB per W2 web x7 Fixity factor N/mm 354.8 70.0 3.22 417.6 21.3 Additional random variables for two CLBs per W2 web x7 Fixity factor N/mm 391.5 90.0 3.81 472.9 23.6 x8 Fixity factor N/mm 391.5 90.0 3.81 472.9 23.6 Note: ‘shape parameters are dimensionless 2 the midspan deflection refers to the initial out-of-plane deformation of the braced webs 7.2.5. Sampling scheme of the response surface method A complete factorial sampling scheme is desirable, as it leads to independent sampling results and a normally distributed error term, e (Pinto et al. 2004). In such a sampling scheme, each random variable is sampled at the mean value, u, and two other points at, p± ha, where a is the standard deviation and h is random number generated between 1.0 and 3.0. The number of samplings for a response surface function with seven random variables goes up to 37 = 2187. Alternatively, a partial factorial sampling scheme was adopted in this study. One hundred sample values of the random variables were randomly selected from the complete factorial sampling scheme. The coefficients of the response surface function were then evaluated based on the responses of the 100 samples of the random variables. The error term, e, - 173 - was calculated afterwards, as the difference between the sample response, y, and the predicted response, y, from the response surface function at the same sample values of the random variables, x1. The difference between a complete and a partial factorial sampling scheme for a response surface function with three random variables is shown in Figure 7-7. Figure 7-7. Complete and partial factorial sampling schemes With the explicitly postulated response surface function, the probability of buckling failure and the design point of the truss assembly subjected to specific external loads can be calculated by using first- and second-order reliability methods. The obtained design point was checked; and a re-sampling would be conducted around the design point, if it was significantly different from the mean values ofthe random valuables. The procedures ofa reliability analysis using the response surface method are explained with the flowchart in Figure 7-8. h3o- p3 Th h33 p3 x2 x1 x2 x1 -174- [start 1 Conduct a complete factorial sampling 4, Randomly select 100 samples I Evaluate the response surface function I Use first and second order reliability methods Calculate the failure probability and the design point NO The design point is close to the mean values ( Done I Figure 7-8. Reliability analysis using response surface method based on a partial factorial sampling scheme 7.3. Evaluation of the Response Surface Function Coefficients The critical buckling load of the truss assemblies with zero, one and two CLB bracing members per W2 web was evaluated based on the 100 sample values of the random variables. The reference load used to evaluate the critical buckling load was assumed to be a uniformly distributed roof load with an intensity of 1.0 x 1 0 N/mm2. The obtained critical buckling loads were used to calculate the coefficients of the response surface function, which was used to predict the critical buckling load at the same sample values ofthe random variables. The predictions were compared to the sample responses to study the error term, 8. The results of all the three CLB provisions (zero, one and two CLB - 175 - bracing members per W2 web) are shown in Figures 7-9 to 7-11, respectively. 11.0 R2= 0.77 110.5 7 1:.: Ezzz qrcf=-0.001 NImm2 9.0 9.0 9.5 10.0 10.5 11.0 Sampled critical buckling load y Figure 7-9. Sampled and predicted critical buckling loads of the truss assembly (zero CLB) 18 R2= 0.84,, B i: 16.5 --‘ a) 0 0 qref=-O.OO1 N/mm2 16 16 16.5 17 17.5 18 Sampled critical buckling load Ys Figure 7-10. Sampled and predicted critical buckling loads of the truss assembly (one CLB) - 176 - 30 V(U 0 C 0 D .0 CU 0 0 V a) C) V ci) 0 Zero CLB One CLBs Two CLBs Index a c a1 b c a1 0 8.143 -- -0.006 11.471 -- 0.000 34.290 -- 0.004 1 -0.052 0.005 -0.002 -0.072 0.003 0.001 -0.355 0.019 -0.001 2 0.426 -0.018 -0.006 0.038 0.016 -0.005 -2.821 0.147 -0.006 3 -0.241 0.015 -- 0.621 -0.029 0.000 1.418 -0.064 0.000 4 -0.004 -0.002 -- -0.004 0.003 -- 0.091 -0.002 -0.007 5 0.035 0.007 -- -0.023 0.003 -- 0.128 0.002 -0.001 6 0.019 0.003 -- -0.020 -0.007 -- 0.261 0.002 -- 7 -- -- -- 0.00 1 0.000 -- -0.009 0.000 -- 8 -- -- -- -- -- -- 0.013 0.000 -- The coefficients in the table are the results of minimizing the fitting error of the 29 28 27 26 26 27 28 29 30 Sampled critical buckling load Ys Figure 7-11. Sampled and predicted critical buckling loads of the truss assembly (two CLBs) The coefficients of the response surface function in Equation (7-2) for the truss assembly with different number of CLB bracing members are listed in Table 7-3. The results were obtained by using least square method. The sample values of the critical buckling loads and the input random variables are presented in Appendix D for brevity. Table 7-3. Coefficients ofthe response surface functions - 177 - response surface function at the sampled points of the random variables. Direct physical meanings of some coefficients were weaken or lost due to the complexity of the problem; therefore, these coefficients should only be used in the vicinity of the mean values of the random variables, where the sampling was conducted. If the resulting design point is distanced from the mean values, the random variables should be re-sampled and the coefficients should be re-evaluated based on the new sample values. The error term, 6, was calculated as the difference between the sampled and predicted critical buckling loads as: = (Yp,i Ys,i)’Ys,i wherey,, andy31are the predicted and sampled critical buckling load coefficients, respectively, corresponding to the ith sample values, x1, of the input random variables, i = 1 to n, and n is the total number of samplings. The results of the error term, e, were fitted to the normal distribution. The mean values and standard deviations for the truss assembly with the different CLB systems are listed in Table 7-4. Table 7-4. Probabilistic distribution parameters ofthe error term, e Parameter No CLB One CLB Two CLBs Mean 1.3E-4 1.23E-4 1.78E-5 St.d. 1.15E-2 1.13E-2 4.3E-3 As can be seen from Table 7-4 and Figures 7-9 to 7-11, the error term, 6, of the response surface function is very small, which implies that both the response surface model and the sampling scheme used in this study are well defined. - 178 - 7.4. Lateral Bracing Force 14 I U- 6 4 2 0 o 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Lateral bracing force ratio % Figure 7-12. Lateral bracing force ratio of W2 webs with one CLB bracing member 20 18 16 14 C.) C 10 U- 6 4 2 0 It can be seen from Figures 7-12 and 7-13 that the lateral bracing force ratios, which were evaluated based on randomized MOE values, initial lateral defleetions of W2 webs and -179- The lateral bracing force corresponding to the critical buckling load of the truss assembly was studied based on the sample results. For each sample result, the ratio between the lateral bracing force and the total compression load in the W2 webs was calculated and compared to the 2% rule-of-thumb. The frequency distribution ofthe ratio are shown in Figures 7-12 and 7-13 for one and two CLB bracing members per W2 web, respectively. 18 16 2% rule Frequency — Cumulative % 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 I 2% rule Frequency Cumulative % 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 bracing members 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Lateral bracing force ratio % Figure 7-13. Lateral bracing force ratio of W2 webs with two CLB fixity factors ofthe CLB bracing members, range between 0.1% and 0.8%. The 95 percentile of this ratio of the truss assembly with one and two CLB bracing members per W2 web is 0.77% and 0.63%, respectively, and is significantly smaller than the 2% rule-of-thumb. 7.5. Reliability Analysis The performance function of the truss assembly was established as the difference between the critical buckling load and the external load qext: G = qrefyp (1—8) — Jext (74) where qref is the reference load used to evaluate the critical buckling load coefficient, yp; and s is the error term. In this study, both the reference load, qrej, and the external load, q1, were uniformly distributed roof loads. qext was further assumed to be of the extreme value type I distribution with the mean value at 3.571 x i0 N/mm2,which is the design value of the live load of the individual trusses. qext can then be expressed as: qexi =B—ln(—lnp)/A (7-5) where A and B are the distribution constants and can be calculated from the mean value, 1u, and the coefficient of variation (COy), v, as: A=1.282/(pv) B=p—0.577/A (7-6) Three coefficients of variation (COVs) of the external load were considered to study their effect on the reliability of the truss assembly. The values were selected with consideration of the statistic data of the snow load of six Canadian cities (Foschi 1989). The distribution parameters of the external, qext, are listed in Table 7-5. - 180- Table 7-5. Extreme value type I distribution parameters ofthe external load, qext Level Mean (10 N/mm2) COV A B 1 3.571 1.0 0.359 1.964 2 3.571 0.5 0.718 2.767 3 3.571 0.2 1.795 3.250 With the explicitly expressed performance function, the probability ofbuckling failure can be calculated by first- and second-order reliability methods. In this study, a computer program, RELAN, developed by Foschi (2000), was used for its convenience. The reliability indices and probabilities of buckling failure of the truss assembly with different CLB systems are listed in Table 7-6. The reliability indices are also plotted in Figure 7-14. Table 7-6. Reliability indices andprobabilities ofbucklingfailure ofthe truss assembly External load variation (COV) CLB provision 1.0 0.5 0.2 No CLB 1.548 (0.61E-1) 2.457 (0.7E-2) 4.261 (0.1E-4) One CLB 2.55 (0.54E-2) 3.878 (0.53E-4) 6.456 (0.54E-10) Two CLBs 3.814 (0.68E-4) 5.63 (0.9E-8) 8.649 (0.26E-17) Note: *enclosed in the parenthesis are the probabilities of buckling failure 9 8 7 6 5 4 3 2 0 10 cov=1.o a2. x COVO.5 0 .9 —e-- COV=O.2 No CLB One CLB Two CLBs CLB provisions Figure 7-14. Reliability indices of the truss assembly with different CLB systems - 181 - As can be seen from the figure, the reliability indices, which represent the chance of the critical buckling load not being exceeded by the external load, was significantly increased when more CLB bracing members were provided or smaller variation of the applied load was encountered. 7.6. Conclusion This chapter presents a reliability analysis of the critical buckling load of a three-bay MPC truss assembly subjected to a uniformly distributed roof load. The response surface method was used to approximate the performance function, in conjunction with a partial composite sampling scheme. The influence of the CLB system and the variation ofthe material properties and external load on the probability of buckling failure of the truss assembly were studied. The adequacy of the 2% rule-of-thumb was also studied. The results in this study indicated that the reliability of the truss assembly concerning buckling failure of compression webs can be significantly increased by using more CLB members or in cases with lower variation of the external load. It was also found that the variation of the modulus of elasticity and the initial out-of-plane deformation of the braced webs can also affect the system reliability, although not so much as the preceding two factors. The 2% rule-of-thumb was found to overestimate the lateral bracing force for the system with one and two lateral bracing members per braced web. The research work in this chapter provides a framework to study the reliability of the truss assemblies concerning buckling failure of the compression truss members. The fi values are for the specific truss configuration, material properties of the dimension lumber and truss -182- plates, and nail type/stiffliess. The results cannot be directly extrapolated to other cases. More comprehensive studies can be conducted under the developed framework to study the general problem of reliability of truss assembly concerning the buckling of compression truss members. - 183- Chaper 8. Conclusion and Suggestions 8.1. Summary of the Research Work This research work was focused on the stability capacity and lateral bracing force of wood beam-columns and metal plate connected (MPC) wood truss assemblies. This was achieved by both experimental and analytical research. A three-dimensional computer program, SATA, was developed with the capacity to perform three-dimensional nonlinear structural and stability analyses. The program was codified into a user-friendly package. Four types of FEM based elements and the Newton-Raphson and arc-length methods were incorporated into the program for the nonlinear buckling analysis. Response surface method and Monte Carlo simulation were also incorporated for reliability analysis purpose. Parallel-to-wood-grain compression and tension tests and the nail connections tests of 1 Od common nails were conducted to construct the material property database. Biaxial eccentric compression tests of wood beam-columns with and without lateral bracing member were conducted to verify the developed program. The program predictions were in good agreement with the test results. Both results indicated that the maximum load carrying capacity of the beam-columns was affected by the member geometry, load eccentricity and the provision of lateral bracing member. The 2% rule-of-thumb, Winter’s and Plaut’s methods were found to underestimate the lateral bracing force of the tested members. Additional material property tests were conducted with the dimension lumber, plywood sheathing and nail connections fabricated with 6d common nails. The variation of the - 184- MOE of the dimension lumber, the flexure stiffness ofthe plywood in both principle directions and the load and displacement relationship of the nail connections were quantified. Full-scale tests of MPC trusses and truss assemblies were also conducted to further verifi the program. It was found the program predictions of the maximum load carrying capacity and lateral bracing force were in very good agreement with the test results. It was also found that the out-of-plane rotational stiffness of the MPC connections had a significant effect on the maximum load carrying capacity of the trusses, whereas the extra keeper nails did not. The plywood sheathing and CLB system were found to be able to increase the critical buckling load of the individual trusses and truss assemblies, by approximately 50% and 60%, respectively. The 2% rule-of-thumb was found to overestimate the lateral bracing force of the truss assemblies tested. The results of the single member tests and the truss/truss assembly tests clearly indicate the importance of consideration of system behaviour and the out-of-plane rotational stiffness of metal truss plate for the consideration of lateral bracing forces in compression truss members. A reliability analysis was conducted by using the verified program for a three-bay truss assembly subjected to uniformly distributed roof load. It was found that the number of lateral bracing members used per braced web and the variation of the external load had the most significant effect on the probability of buckling failure of the truss assembly. The variation of the MOE and initial out-of-plane deformation of the braced webs also influenced truss performance. - 185 - 8.2. Significance of the Research Work This research work comprised a thorough study on the critical buckling load and lateral bracing force of wood beam-columns and MPC truss assemblies. The developed database and software was used to evaluate the adequacy ofthe 2% rule-of-thumb. Much of the work is original, and the study on the adequacy of the 2% rule-of-thumb leads to improvement of the design methods and more efficient and economical designs. The inclusion of the out-of-plane rotational stiffness of the MPC connections in the stability analysis of the MPC trusses is original. This is pertinent to the understanding of the conservativeness of the 2% rule-of-thumb when it is applied to the MPC truss assemblies. The implementation of this stiffness consideration into the design process may substantially reduce the requirements on the lateral bracing members and the connections between them and the neighboring truss members, which are used to pass the forces to the supporting frames underneath. More work is needed to expand the study to consider different truss configurations, truss spans, material and metal plate connector properties. The computer program exhibited good accuracy in predicting the critical buckling load and lateral bracing force for the wood beam-columns and MPC truss assemblies. Given appropriate input parameters, the program provides a cost efficient platform to the analysis of other structural systems of different materials, configurations and loading situations. The work in Chapter 7 provided a framework for the reliability analysis of MPC truss assemblies concerning buckling failure of compression webs. The work also demonstrated the procedures using the response surface method in conjunction of a partial factorial sampling. The database of the material properties and structural behavior of the wood - 186- beam-columns and MPC trusses and truss assemblies can be used as input parameters and for verification purpose for other numerical analysis models and programs, thus avoid unnecessary repeated testing costs. 8.3. Limitations and Suggestions The databases constructed in this research work were based on specific wood species, grades, specimen sizes and truss configurations and cannot be extrapolated to other situations. The computer program has been calibrated and verified with reference to tests of wood beam-columns and MPC truss assemblies. Application to other types of materials or structures may need further calibration and verification. The lateral bracing stiffness can affect both the load carrying capacity and the lateral bracing force of the wood beam-columns. Due to limited resources, a constant lateral bracing stiffness was considered in the experimental study. Using the verified program, it is convenient to consider different lateral bracing stiffness and study its influence on the wood beam-column’s stability behavior. Torsional buckling is a common issue for beam-columns with open cross sections or cross sections with high depth-to-width ratio. Wood members fabricated in composite cross sections, such as an I beam, can also be susceptible to torsional buckling. Advanced numerical models are needed to take into account large torsional displacement and the resulting geometric nonlinearity. Test results are also needed to calibrate and verifS’ the models. As indicated in this research work, the out-of-plane rotational stiffness of MPC connections has a significant effect on the critical buckling load of the individual wood trusses - 187- and truss assemblies. Although a simplified model was developed in this thesis to estimate this stiffness, more sophisticated models are needed to take into consideration the effect of wood-wood contact, truss plate buckling and teeth withdrawal, due to large deformation of the connections. The structural behaviour and critical buckling load of MPC wood truss assemblies under unbalanced wind load have not been fully studied. Various factors, including the wind speed, loading direction and truss assembly geometry, can influence the truss member forces and the resulting critical buckling load. - 188- Bibliography ASTM Standard D 198-05a “Standard Test Methods of Static Tests of Lumber in Structural Sizes”, ASTM International, West Conshohocken, PA. ASTM Standard D 3043-00 (2006) “Standard methods of testing structural panels in flexural”, ASTM International, West Conshohocken, PA. “Handling, Erection and Bracing of Wood Trusses” of Truss Plate Institute of Canada (TPIC), pp. 7. Barrett, J. D., 1974. “Effect of size on tension perpendicular-to-grain strength of Douglas-fir. Wood and Fiber.” Wood and Fiber Science, 6(2), PP. 126-143. Bazan, I.M.M, 1980. “Ultimate bending strength of timber beams”, Thesis presented to Nova Scotia Technological College, at Halifax, N.S., in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Bohanan, B., 1966. “Effect of size on bending strength of wood members.” Res. Paper FPL 56, USDA Forest Service, Forest Product Lab., Madison, Wisc. Booth, L.G., 1964. “The strength testing of timber during the 17th and 18th centuries.” Jo. Inst. Wood Science. No.13, pp. 5-30. Box GEP and Wilson K.B. (1954). “The exploration and exploitation of response surface: some considerations and examples.” Biometrics, 10, pp. 16-60. Bryson, W., 1866. “Strength of cast iron and timber pillars: a series of tables showing the breaking weight of cast iron, dantzic oak and red deal pillars.” J. Franklin Inst. 81, pp. 3 12-322. Buchanan, A. H., 1984. “Strength model and design methods for bending and axial load interaction in timber members.” Ph.D. thesis, UBC, Department of Civil Engineering. Buchanan, A.H., 1986. “Combined bending and axial loading in lumber”, Journal of Structural Engineering, Vol.112, No.12, December, pp. 2592-2609. Buchanan, A.H., Johns, K. C., and Madsen B., 1985. “Column design method for timber engineering”, Canadian journal of Civil Engineering, 12(4), pp. 731-744. Bucher C.G. and Bourgund U. (1990). “A fast and efficient response surface approach for structural reliability problems.” Structural Safety, 7, pp. 57-66. Bulleit, W. M. and Liu, W. F., 1995. “First-order reliability analysis of wood structural systems.” Journal of Structural Engineering, 12 1(3), pp. 5 17-529. -189- Bulleit, W. M. and Yate, J. L., 1991. “Probabilistic analysis of wood trusses.” Journal of Structural Engineering, 117(10), PP. 3008-3025. Bulleit, W. M., Rosowsky, D. V., Fridley, K. J., and Criswell, M. E., 1993. “Reliability of wood structural systems.” Journal of Structural Engineering, 119(9), pp. 2629-2641. Cramer, S. M. and Wolfe, R. W., 1989. “Load distribution model for light frame wood roof assemblies.” Journal of structural engineering, Vol. 115, No. 10, October, 1989, pp. 2603-2616. Cramer, S. M., Drozdek, J. M. and Wolfe, R. W., 2000. “Load Sharing Effects in Light-frame Wood-truss Assemblies.” Journal of Structural Engineering, 126(12), pp. 1388-1394. Cramer, S. M., Shrestha, D., and Fohrell, W. B., 1990. “Theoretical consideration of metal plate connected wood splice joints.” Journal of Structural Engineering, 116(12), pp. 3458-3475. Cramer, S. M., Wolfe, R. W. and Peyrot, A., 1988. “Modeling roof system for reliability analysis.” Vol.1, in R. Y. Itani, ed. Proc. of the 1988 International Conference on Timber Engineering, pp. 143-150; Crisfield, M.A., “A fast incremental/iterative solution procedure that handles “snap-through”. Computers & Structures, Vol. 13. 1981, pp.55-62. Crovella, P. L. and Gebremedhin, K. G, 1990. “Analysis of light frame wood truss tension joint stiffness.” Forest Products Journal, 40(4), pp. 41-47. Der Kiureghian, A. and Liu, P. L., 1986. “Structural reliability under incomplete probability information.” Journal of Engineering Mechanics, 112(1), pp. 85-103. Drummond, 1., Kermani, A. and Wamuziri, S.C. 2001. “Reliability of timber structural systems: a review.” Proceedings of the Institution of Civil Engineers: Structures and Buildings, 146(1), pp. 101-108, Ellegaard, P., 2002. “Analysis of timber joints with punched metal plate fasteners with focus on knee joints.” Ph.D. thesis defended publicly at Aalborg University on February 5th, 2002 Ellingwood, B., 1981. “Reliability of wood structural elements.” ASCE J Structural Division, 107(1), pp. 73-87. Engesser, F., 1889. “Ueber die Knickfestigkeit Gerader Stabe.” Z. Architekten und Ingenieur. 35, pp. 456-462. Engesser, F., 1895. “Ueber Knickfragen. “Schweiz. Bauzgh. 26, pp. 24-26. Folz, B. and Foschi, R. 0., 1989. “Reliability-based design of wood structural systems.” Journal of Structural Engineering, 115(7), pp. 1666-1680. Forde, B.W,R. and Stiemer, S.F., “Improved arc length orthogonality methods for nonlinear finite element analysis”, Computers & Structures, Vol. 27, No.5, 1987, pp. 625-630. Foschi, R. 0., 1974. “Load-slip characteristics of nails.” Wood Science, 7(1), pp. 69-76. - 190 - Foschi, R. 0., 1977. “Analysis of wood diaphragms and trusses, part II: Truss-plate connections.” Canadian Journal of Civil Engineering, Vol. 4, pp. 353-363. Foschi, R. 0., 1984. “Reliability of wood structural systems.” Journal of Structural Engineering, 110(12), pp. 2995-3013. Foschi, R. 0., Folz, B.R., and Yao, F. Z. (1989). “Reliability-based design ofwood structures.” Structural research series, Report No. 34, Department of Civil Engineering, University of British Columbia, Vancouver, Canada Foschi, R. 0, Folz, B. and Yao, Z. C., 1993. “Reliability-based design of wood structures: background to CSA-086. l-M89.” Canadian Journal of Civil Engineering, 20(3), pp. 349-357. Foschi, R. 0., Li, H., Folz, B., Yao, F. and Baldwin, J., 2000. “RELAN: A general software package for reliability analysis.” Department of Civil Engineering, University of British Columbia, Vancouver, Canada Gebremedhin, K. G, Jorgensen, M. C., and Woelfel, C. B., 1992. “Load-slip characteristics of metal plate connected wood joints tested in tension and shear.” Wood and Fiber Science, 24(2), pp. 118-132. Glos, P. 1978. “Reliability theory for timber structures: determination of compression strength behaviour of Glulam components for interaction of material properties.” Technische Universitat Munchen, pp. 335. Gomes H. M. and Awruch A. M. (2004). “Comparison of response surface and neural network with other methods for structural reliability analysis.” Structural Safety 26, pp. 49-67. Gromala, D. S., and Sharp, D. J., 1988. “Concepts of wood structural system performance.” Vol.1, in R. Y. Itani, ed. Proc. of the 1988 International Conference on Timber Engineering, pp. 136-142. Groom Leslie and Polensek Anton, 1992. ‘Nonlinear modeling of truss plate joints.” ASCE 118(9), pp. 25 14-253 1. Gupta Sayan and Manohar C.S. (2004). “An improved response surface method for the determination of failure probability and importance measures.” Structural Safety 26, pp. 123-139. Gupta, R. 2005. “System behaviour of wood truss assemblies.” Prog. Struct. Engng Mater. 7, pp. 183-193. Gupta, R., and Gebremedhin, K. G, 1990. “Destructive testing of metal-plate connected wood truss joints.” J. Structural Engineering. 116(7), pp. 1971-1982. Gupta, R., Gebremedhin, K. G. and Grigoriu, M. D., 1992. “Characterizing the strength of wood truss joints.” Transactions of American Society of Agricultural Engineers, 35 (4), pp. 1285-1290. Hammon, D. C., Woeste, F. E., and Green, D. W., 1985. “Influence of lumber property correlations on roof truss reliability.” Transactions oftheASAE, 28(5), pp. 1618-1625. - 191 - Hankinson, R. L., 1921. “Investigation of crushing strength of spruce at varying angles of grain.” United Sates Air Service Information Circular, 3, pp. 259. Hansson, M. and Ellegaard, P, 2006. “System reliability of timber trusses based on nonlinear structural modeling.” Materials and Structures (2006), 39, pp. 593-600. Haukaas, Terje and Der Kiureghian, Armen, 2004. “Finite Element Reliability and Sensitivity method for performance-based earthquake engineering.” Peer report 2003/14, Pacific Earthquake Engineering Research Center, College of Engineering, University of California, Berkeley. He, M., 2002. “Numerical modeling of three-dimensional light wood-framed buildings.” A thesis submitted in partial fulfillment of the requirement for the degree of Doctor of Philosophy in the faculty of forestry, department of wood science, University of British Columbia, April, 2002 Hoyle, R. J., Galligan, W. L. and Haskell, J. H., 1979. “Characterizing lumber properties for truss research. Proceedings, Metal Plate Wood Truss Conference.” Forest Products Research Laboratory, Madison, WI, pp. 32-5 1. Karacabeyli, E., Lum, C. and Olson, L., 1993. “Strength and stiffness of glulam trusses with punched metal plated joints.” Canadian Journal of Civil Engineering, 20(4), pp. 622-630. Khalil M. and Itani, R. Y., 1984. “Influence of truss plate connectors on the analysis of light frame structures.” Wood and Fiber Science, 16 (3), pp. 306-332. Kim S. and Na S. (1997). “Response surface method using vector projected sampling points.” Structural Safety, 19(1), pp. 3-19. Koka, E. N., 1987. “Laterally loaded wood compression members: Finite element and reliability analysis.” A thesis submitted in partial fulfillment of the requirements for the degree of mater of applied science, University of British Columbia LaFave, K. and Itani, R. Y., 1992. “Comprehensive Load Distribution Model for Wood Truss Roof Assemblies.” Wood and Fiber Science, 24 (1), pp. 79-88. Lam F., 1999. “Length effect on the tensile strength of truss chord members.” Can. J. Civil. Eng. 27, pp. 48 1-489. Lam, F. and Varoglu, E., 1988. “Reliability of trusses and roof systems.” Research report, Forintek Canada Corporation, Vancouver, B.C., Canada Lam, F., 1989. “Short-term baseline reliability of trusses.” Research report, Forintek Canada Corporation, Vancouver, B.C., Canada - 192 - Lam, F., 1990. “Short-term baseline reliability of trusses-MSR lumber.” Research report, Forintek Canada Corporation, Vancouver, B.C., Canada Lam, F. 2000. “Length Effect on the Tensile Strength of Truss Chord Members.” Canadian Journal of Civil Engineering, 27(3), pp. 481-489. Larsen, H. J. and Thielgaard, E., 1979. “Laterally loaded timber columns.” Proc. ASCE 105, pp. 1347-1363. Lau, W., W.S., “Strength model and FEM analysis of wood beam columns in truss application”, Ph.D. thesis, Department of Wood Science, University of British Columbia, Vancouver, B.C., 2000 Li, Z., Gupta, R. and Miller, T. H., 1998. “Practical Approach to Modeling of Wood Truss Roof Assemblies.” Practice Periodical on Structural Design and Construction, 3(3), pp. 119-124. Liu, X. Q. 2008. Ph.D thesis under preparation of the Department of Wood Science, University of British Columbia, Vancouver, Canada Madsen, B., 1992. “Structural behaviour of timber.” Timber Engineering Ltd., North Vancouver, B.C., Canada, pp. 237. Madsen, H. 0., Krenk, S., and Lind, N. C. 1986. “Methods of structural safety.” Prentice-Hall Publishing Co., pp. 94-112. Mayo, A. P., 1982. “Trussed rafter roofs—load distribution and lateral stability.” Building Research Establishment Information Paper McCarthy, M., and Little, J. K., 1988. “Sensitivity of truss plate model parameters to parameter determination methods.” Forest Products Journal, 38(5), pp. 63-67. Medland, I. C. 1977. “A basis for the design of column bracing.” Structural Engineer, 55(7), pp. 3 01-307. Mtenga, P. V., Cramer, S. M., Peyrot, A. H. and Wolfe, R. W., 1995. “System Factors for Light-frame Wood Truss Assemblies.” Journal of Structural Engineering, 121(2), pp. 290-300. Munch-Andersen, J., 2004. “Bracing of Timber Members in Compression.” International Council for Research And Innovation in Building and Construction, Working Commission W18-Timber Structures National Building Code of Canada 2005, Canadian Commission on Building and Fire Codes, Institute for Research in Construction. O’Halloran, M.R., 1973. “Curvilinear stress strain relationship for wood in compression”, Thesis presented to Colorado State University at Fort Collins, Cob., in partial fulfillment of the requirements for the degree of Doctor of Philosophy, 1973. - 193 - Pienaar, F. R. P. 1986. “The effective length and bracing requirements for out ofplane buckling of timber rafters in compression.” South African Forestry Journal, 137 (June), pp. 13-18. Pierce, F. T., 1926. “Tensile tests for cotton yarns.” Jo. Text. Inst. 17, pp. 355-368. Pinto P. E., Giannini Renato and Franchin Paolo (2004), “Seismic reliability analysis of structures.” IUSS Press, Pavia, Italy Plaut, R. H., 1993. “Requirements for lateral bracing of columns with two spans.” Journal of Structural Engineering, 119(10), pp. 2913-2931. Plaut, R. H., and Yang, J., 1993. “Lateral bracing forces in columns with two unequal spans.” Journal of Structural Engineering, 119(10), pp. 2896-29 12. Rackwitz, R., and Fiessler, B., 1978. “Structural reliability under combined random load sequences.” Computers and Structures, Vol. 9, pp. 489-494. Rajashekhar, M.R., and Ellingwood, B. R. (1993). “A new look at the response surface approach for reliability analysis.” Structural Safety, 12(3), pp. 205-220. Ramm, E., 1980. “Strategies for tracing the nonlinear response near limit points”, In Europe-U.S. Workshop, nonlinear finite element analysis in structural mechanics, Ruhr Universität Bochum, Germany, pp.63-89. Riley, G. J. and Gebremedhin, K. G., 1999. “An empirical model for predicting the foundation modulus of wood for metal plate connector teeth.” Transactions of the ASAE, 42(3), pp. 755-760. Riley, G. J., Gebremedhin, K. G., and White, R. N., 1993. “Semi-rigid analysis of metal plate-connected wood trusses using fictitious members.” ASAE, 36 (3), pp. 887-894. Rojiani, K. B. and Tarbell, K. A., 1985. “Analysis of the reliability of wood roof trusses.” Structural Safety and Reliability, Vol. 1: Proceedings of ICOSSAR, the 4th International Conference., Kobe, Japan, pp. 365-374. S347-99 “Method of test for evaluation of truss plate used in lumber joints” (reaffirmed 2004), ISSN 0317-5669, published in March 1999 by Canadian Standards Association, 178 Rexdale Boulevard, Etobicoke, Ontario, Canada. Sasaki, Y., and Takemura, T., 1988. “Non-linear analysis of semi-rigid jointed metal-plate wood truss.” J. Japanese Wood Res. Soc. 34 (2), pp. 120-125. Segedin, C. M. and Medland, I. C., 1978. “The buckling of inter-braced columns.” mt. J. Solids Structure. 1978, Vol. 14, pp. 375-384. Shanley, F.R., 1947. “Inelastic Column Theory,” J. Aero. Sci., 14 (5), pp. 26 1—267. - 194 - Song, X. and Lam, F., “Three dimensional stability analysis of wood beam-columns”, In Proceedings of the 9th WCTE. Portland USA CD-ROM Proceedings, 2006 Throop, C. M. 1947. “Suggestions for safe lateral bracing design.” Engineering News Record, pp. 90-91. Timoshenko, S. P. and Gere, J. M., 1961. “Theory of Elastic Stability.” New York: McGraw-Hill Book Company, pp. 70. Triche, M. h., and Suddarth, S. K., 1988. “Advanced design of metal plate connector joints.” Forest Products Journal, 38 (9), pp. 7-12. Truss design procedures and specifications for light metal plate connected wood trusses, Limit State Design, TPIC 2007. Truss Plate Institute of Canada. Tsien, H. S., 1942. “Buckling of a column with nonlinear lateral supports.” Journal of the Aeronautical Sciences 9(4),pp. 119-132. Underwood, C. R., Woeste, F. E., Dolan, J. D., and Hoizer, S. M., 2001. “Permanent Bracing Design for MPC Wood Roof Truss Webs and Chords.” Forest Products Journal, 51(7/8), pp. 73-81. Varoglu, E. 1986. “Reliability of trusses and roof systems.” Research report, Forintek Canada Corporation, Vancouver, B.C., Canada Vatovec, M., Miller, T. H., and Gupta, R., 1996. “Modeling of metal-plate-connected wood truss joints.” Transactions oftheAmerican Society ofAgricultural Engineers 39(3), pp. 1101-1111. Vaziri, R. 2004. Lecture notes for Computational Mechanics 1, a course offered by the Department of Civil Engineering, University of British Columbia, Vancouver, Canada Waltz, M. E. Jr., Thomas E. M., Thomas H. M., and Robert J. L., 2000. “Discrete Bracing Analysis for Light-frame Wood-truss Compression Webs.” Journal of Structural Engineering, 126(9), pp. 1086-1093. Waltz, M. E., 1998. “Discrete compression web bracing design for light frame wood trusses.” MS thesis, Oregon State University, Corvallis, Oregon Weibull, W., 1939. “A statistical theory of the strength of materials.” Proc. Royal Swed. Inst. Eng. Res. No. 151. Stockholm, pp. 45. Winter, G, 1958. Lateral bracing of columns and beams. American society of civil engineers, transaction, paper No. 3044, pp. 807-826. Wolfe, R. W. and LaBissoniere, T., 1991. “Structural Performance of Light-Frame Roof Assemblies II. Conventional Truss Assemblies.” Forest Products Laboratory Research Paper FPL-RP-499 - 195 - Wolfe, R. W. and McCarthy, M., 1989. “Structural Performance of Light-Frame Roof Assemblies I Truss Assemblies with High Truss Stiffness Variability.” Forest Product Laboratory, September, 1989, Rep. FPL-RP-492, pp. 1-4 1. Wolfe, R. W., 1990. “Metal plate connections loaded in combined bending and tension.” Forest Products Journal, 40(9), pp. 17-23. Wolfe, R. W., Labissoniere, T. and Cramer, S. M., 1988. “Performance Tests of Light-frame Roof Assemblies.” Proc., 1988 International Timber Engineering Conference, Forest Products Research Society, pp. 3-11 Wolfe, R. W., Percival, D. H. and Moody, R. C., 1986. “Strength and stiffness of Light-Frame Sloped Trusses.” Res. Paper FPL 471, Forest Products Laboratory, pp. 1-16. Wong, F.A. (1985). “Slope reliability and response surface method.” Eur. J. Gastroenterol. Hepatol, 111(1), pp. 32-53. Ylinen, A., 1956. “A method of determining the buckling stress and the required cross sectional area for centrally loaded straight columns in elastic and inelastic range.” IABSA Publications, 16, pp. 529-549. Yu, L., Das, P. K. and Zheng, Y., 2002. “Stepwise response surface method and its application in reliability analysis of ship hull structure.” Transactions of the ASME, 124, pp. 226-230. Zahn, J. J., 1982., “Strength of lumber under combined bending and compression”, Forest Products Laboratory, Research paper FPL391 Zhang, H. Y., Beliveau, J.-G, and Huston, D., 1993. “Minimum lateral stiffness for equally spaced braces in columns.” Journal of Engineering Mechanics, ASCE, 119(9), pp. 1888-1897. - 196 - Appendices Appendix A: Formulation of Finite Element Method Based Elements If not otherwise specified, the formulation ofthe elements in this section was based on Dr. Vaziri’s lecture notes on computational mechanics (2004). 1. Beam Element 1.1. Shape functions For a finite element method (FEM) based beam element, the shape functions were defined in its natural coordinate system and can be expressed as: N1 =--(1-),N2=--(1+) 1 1 21 3 (1)M2—L=l[-—(+1)——(+l) +—(+1)] M3 = L3 = O.75(+1)2 —O.25(+1) M4 =L4 1.2. Derivation of the internal force vector and stiffness matrices The internal force and stiffness matrices were derived based on the principle of virtual work. Consider the virtual work done by stresses, o and r, at virtual strains, 56 and Sy as: = USEdV f5rdV = - JEsec6SEdV Gy5ydV (2) - 197 - Substituting the expression of the strains in Equation (3-4) in Equation (2) and following the law of variation yields: = — fEsec(U’ +(w’)2 —zw +--(v’)2 l()2) (3) (Sit’ +w’Sw’ —zSw” +v’Sv’ —ySv’+p858’)dV—JGp8’p58’dV Ignore the terms of higher order and note that fzwdV = fyv”dV = 0 1Ese1’&t’ +z2w””+y2v”” +u’(w’’ +v’’ +p28’SO’) (4) + I[(w’)2 + (v’)2 + (p8)25u’}dV — JGp28’58’dV Replacing the element displacement, u, v, w and 8 in Equation (4) by the kinematic matrix, B, and nodal displacement vector, a, yields: = _aT + z2BTB + y2B0TB)dv — aT(B2T +B3TB + p2BTB)aB + BaaT(BT+ BTB3+ p2B T )]dVöa (5) — aT fGp2B4TBldVoa = Therefore, the internal force vector, P, can be calculated by: T * = a K + Pt (6) where the element stiffness matrix, K, and nonlinear internal force vector, P, can be calculated by: KSEsec(BBfz2B2TBy2TB+BiTB4dI7 E3e01 B+—h2,cB2÷—b B3 B3)+Gp B4 ]Det(J)dd77dK Pt = v aT2T+B3T +B3T +p2B4TB)]Det(J)dd1ldJc - 198 - where Det(J) is the detenuinant of the Jacobian matrix. The tangent stiffness matrix was calculated by differentiating the internal force vector, With respect to the nodal displacement, a. Note that the internal virtual work can also be expressed as: = _Irn5sdv = — Jo-osdV _fGYo?dV = + wow —zOw + vOv —yOv +p20’09)dV _jGp2eoOdV —zB2 +aTBTB —yB3 +p2aTBTB)dVOa_jGp2aTTdVOa Consequently, the internal element force P1 can be expressed as: irnt = +aTBTB —zB2÷aTB3TB —yB3+p2aTBTB)dV+fGp2aTBTBdV = 2TT where c=Bi+aTB22—zB+a B3 —yB+p a B4 B4. Partially differentiating with respect to the nodal displacement vector, a, yields: $dv+ IGP2aTB4TBdV] Ktan 8Pint V V = —---Cdv+ fJ[BTB2÷B3T +p2B4TBJdV+ fGP2B4TBdV a8aa V V V Notethat ão/ô6=Etan andã8/8a=C, = JEtCTCdV + JO.(BTB2+ BTB +p2B4TJdV+ fGp2B4TBdV V V (7) = V.[ + JBTB2+B3TB + p2B4TB4 +Gp2BTB]Det(J)dd?ldK where Etan is the tangent modulus ofelasticity and can be calculated based on the stress-strain relationship. - 199- 2. Plate Element 2.1. Shape functions The shape functions of a plate element were defined in the natural coordinate system and can be expressed as (He 2002): N1 (,i) L1(,i) = N2(, ij) = L2 (, i) 1(1 + )(1 - N3(,i) = = N4(,) = L (, i) =1(1 )(1 + ) M1 = 0.25 —0.375+ 0.125 —O.375i + 0.125 + O.562 —0.1875 —0.1875 ÷0.06257 = (0.125 —O.1875+O.0625 —0.125’ _0.125172 +0.125 +0.1875 + 0.1875?72 —0.0625 —0.1875 _0.06253172 +0.0625373)! M3(,i7)=(0.125—0.125—0.1252+0.125 —0.1875+0.0625i +0.187517+0.18752,7_0.187537 — 0.0625 — 0.06252173 + 0.06254) — M4()=0.0625(1—— __ _3 _23 _32 = 0.25+ 0.375—0.125 —0.375i +0.125 —0.5625 + 0.1875 + 0.18757 M6(,i) = (0.125 + 0.1875 — 0.0625 — 0.125i _0.125172 +0.125’ —0.1875 _0.1875172 +0.0625 +0.187577 +0.06253172 M7(,i) = (—0.125 —0.125+ 0.1252 + 0.125 +0.1875 —0.062577 + 0.1875 _0.187527 —0.1875 — 0.0625 + 0.06252773 + ) — M8()=0.0625(—1—+ _3 _3_22 _3 23 _32 = 0.25 + 0.375— 0.125 + 0.375i —0.125 + 0.562 —0.1875 — 0.1875 + 0.0625 = (—0.125 —0.1875+ 0.0625 —0.125i + 0.125,72 + 0.125 —0.1875,7 + 0.1875772 + 0.0625,7 +0.1875 0.06253i72_0.06253173)! - 200 - = (—0.125— 0.125 + 0.1252 +0.125 —0.1875i + 0.0625 —0.1875 + 0.187527 + 0.1875 + 0.0625 — 0.06252773 — 0.062577 ) - M12(,77)=0.06 1+— _+_ _3 _3772772 _3 23 3772 3773) M13(,77)= 0.25— 0.375 +0.125 +0.37577 —0.125 —0.5625ç + 0.18757 + 0.1875 — 0.062577 M14(,77) = (—0.125 + O.1875—0.06254 —0.125i + 0.125772 ÷0.12517 +0.187577 _0.1875772 — 0.0625 —0.187577 +0.06253?72 +0.062577)- = (0.125—0.125 _0.1252 + 0.125 + 0.187577 —0.062577 — 0.187577 _0.1875217 ÷0.187517 + 0.0625 +0.062542i73 M16(,77)=0.0 2—1++ — _77772 3772772 773 2773 3772 3773) 2.2. Derivation of the internal force vector and stiffness matrices Consider the virtual work done by a stress vector at a virtual strain vector & = _f(De)T 5dV = _1TDdV Substituting the expressions of the stresses in Equation (3-18) yields E vE vE c5Wjt j Cö6dV— j yx s68dV— J 6L58,dVv1_vxyvyx v1_vxyvyx v1_vxyyyx 8yôSydV_fGxyrxyôrxydV (8) =_1(klsxo6x +k2cyö6+k26xô8y +k3688 +Gxyyxyc5yxy)JV where k1 = E 1(1 — k2 = 1(1 — = 1(1 — and k3 = E 1(1 — are the material property constants. Substituting the expressions of the stains in Equation (3-17) into Equation (8) and carrying out the variation yields: -201 - = —fki [u&z + (zwxx )ô(zw )j+ k1 [(ut Xw )8(w )+ -- (w )2 o(u )]dv - fk2 [( )+( )]+ k[())+ ( o( )]dv — Jk3 [(v), )(u )+ (u )o(v )÷ (zw )(zw, )+ (zwy, )5(zw )J1v V (9) - fk3 [(vp Xw )o(w )+ (u Xwy )(wy )+ (w )2 )+ ! (wy o(u V _fG[(u + )o( +v)÷4z2(w)5(w)}jv _$G[(u + vxX(wx)3(wy)+(wy)5(wx))+(wxXwy)5(uy +v)Iiv Replacing the element displacements, u, v and w, by the nodal displacement vector, a, and the kinematic matrix, B, yields: = _aT jki [BIxTBlx + z2BTBJ+k2[BT + — aTJic3[B,7Bi + BlTB2+ z2(B3yyTxx+B3yJB,y)jiVöa — aT fo[BlTBl +B2xTB + BlTB2+B2XTBIYIjVöa — Ikl[B1XaaTB3T + aTB3xaBlx]dV —Jk2[ByaaTB3T +-aTB3ya2]dVöa —fk3[B2yaaTBxT +BiaaT3+-aTB3y Blx +-aTB3xa2y]dV3a — fG[(B1+2xaT(B3y+B3TB)+aTB3XTYa(1+B2)JiVea = — Therefore, the internal force vector, can be calculated by: Pint = aTK + (10) where the stiffness matrix, K, and the nonlinear internal force vector, can be expressed as: K = fk1[B1XTB +z2B3Tj+k2[BT +Z2B3TJ,Y1jV + fk3TB +B1xTB2y+ z2(B3yyTxx+B3xxTByy)11V +fG[B1YTBIY +B2T ÷BlTB2+B2TBl,jIV - 202 - = fkl[B1xaaT3XT + --aTB3xaBlx]dV+1k2[BYaaTB3T +aTB3ya2]dV +1k3[B2YaaTBxTBX+BixaaTy’ +.aTB3YaBIX +.aTB3xa2y]dV + fG[(B1+2x)IaT(B3XTBy+B3TB)+aTB3xya(Bly +B2)JiV The tangent stiffness matrix of the plate element was also obtained by differentiating the internal force, with respect to the nodal displacement, a as: Ktan = ãPjfl = (aK+Pint)K a11t (11) where 811flt = fk1 BlXTaTB3xTB + BiaB3T+B3xTBaBlx}IV +1k2[B2yTaTB3B+B2ya3TB+B3YTBa2IJV +1k3[BYTaTBXTB+B2aB3T+B1xTaTB3y B +B1xaB3yTIJV +1k3[ByTBaX+BxTBayljV+ jG[(B1+B2x)TaT(B3By+B3T)}IV + B2(B3XTBY+B3YTB)+(X +B3YTBX( l+B2)V The integration over the volume, d V, in the local coordinate system was replaced by the integration over the volume, Det(J)dd l7dK, in the natural coordinate system, to facilitate the numerical integration. 3. Shape functions of the MPC connection element for isoparametric transformation The shape functions for the isoparametric transformation of a MPC connection element were defined in the element’s natural coordinate system and can be expressed as: - 203 - N0(,i7)=(1+)(1+i)/4 (,i)=(1—)(1+i)/4 (12) N2(,i) = (1—)(1—i)I4 N3(,i) =(1+)(1—i)/4 where and are defined in Figure 3-6. - 204 - Appendix B: MOE Profile of the MPC Wood Trusses 1. MOE Values of the Dimension Lumber The dimension lumber used for the truss fabrication was tested flatwise by vibration tests to measure the MOE values. For clarity, the lumber was divided into six types based on the cross-sectional dimensions and lengths. The results are listed in Tables B-i and B-2. Table B-i. Definition of the six types ofdimension lumber No. TypeA Type B Type C TypeD TypeE Type F Width (mm) 38 38 38 38 38 38 Depth (mm) 89 89 139 139 139 139 Length (mm) 2450 3050 3674 2443 3054 4903 Table B-2. Vibration test results (MOE) No. Type A Type B Type C Type D Type E Type F 1 9190 10950 10470 11580 10220 11176 2 10265 9980 9340 11000 10695 11020 3 13200 10490 11260 11340 10775 11040 4 9895 10515 9570 10970 13195 9810 5 9575 10060 10355 10755 9495 10506 6 9750 9860 11210 10610 15425 10132 7 11825 10870 11795 10350 10400 10169 8 14135 10790 10495 10035 10515 11569 9 10975 10840 10545 11030 11080 10911 10 11145 10355 10845 10290 11880 12431 11 12110 9680 11620 10765 10340 11987 12 10780 9360 10165 10355 11205 10895 13 9730 10520 10480 10525 11355 13747 14 11580 10680 11275 10620 13005 10194 15 10630 10685 10650 9375 11875 11224 16 10170 10340 11205 10425 11115 10969 17 9355 11560 11280 10575 12690 11166 - 205 - No. Type A Type B Type C Type D Type E Type F 18 10670 10860 11115 9925 9450 11325 19 9755 9230 10665 (endofTypeD) (endofTypeE)10690 20 9635 10310 11135 11087 21 10570 10230 11645 10339 22 10885 9500 9180 11091 23 9755 9890 10480 9644 24 9775 13350 10495 10204 25 11625 10545 10785 9965 26 9395 10845 11795 10852 27 8715 11030 9830 10300 28 10180 9525 10745 10294 29 11495 10470 9215 10128 30 9895 11225 9545 10820 31 9730 8795 10685 10598 32 10345 12530 11730 10688 33 9555 10455 10460 10203 34 9375 11570 10945 8814 35 10085 9760 10485 9587 36 10745 10580 9130 11138 37 9935 (end of Type B) 11710 (end ofTypeF) 38 11145 9490 39 10275 11755 40 9750 10540 41 11715 10530 42 9670 10575 43 11185 9190 44 10835 8990 45 10195 10150 46 9180 9645 47 10345 10165 48 10000 10135 49 9185 10755 50 10755 13085 51 10060 11680 52 10125 9205 53 9605 11900 54 8980 12045 55 8890 11665 56 11690 11655 57 8945 9320 58 10170 10895 59 9315 10910 60 9570 11595 - 206 - No. 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 Type A 9340 9585 9035 10110 11205 9665 9145 10380 11380 10170 12850 9375 9005 12660 10305 10320 8795 10295 9225 9670 9975 10005 10385 9785 11055 9400 10965 10295 2. MOE Profile of the Trusses The MOE values of the individual truss members of the fifteen trusses were recorded. The results are shown in Figures B-i to B-15. For each truss member in the figures, its MOE value can be found in Table B-2 by the member’s number. For example, the centre web in Figure B-i was numbered E13; its MOE can be found at row 13 and column E of Table B-2, which is 11355 MPa. Type B Type C TypeD TypeE Type F 10195 11730 10915 10320 11150 10180 11680 10370 11875 10830 (end of Type C) - 207 - The number of the short vertical web of the twelfth truss, as shown in Figure B-12, is missing since it was trimmed away from the truss member during the fabrication process. Figure B-i. MOE profile of Truss No. 1 Figure B-2. MOE profile of Truss No. 2 Figure B-3. MOE profile of Truss No. 3 C67 C61 F18 D3 F5 C2 C12 8 Fli D12 F17 C28 C20 Fl D4 F20 -208- C59 Figure B-4. MOE profile of Truss No. 4 Figure B-5. MOE profile of Truss No. 5 Figure B-6. MOE profile of Truss No. 6 C66 F25 D7 F26 C49 C50 F4 D17 F12 C57 C53 F33 D8 F27 - 209 - C65 Figure B-7. MOE profile ofTruss No. 7 Figure B-8. MOE profile of Truss No. 8 Figure B-9. MOE profile of Truss No. 9 F34 D5 F2 C33 C29 F16 D14 F19 C41 C46 F24 D6 F6 -210- Cl Figure B-b. MOE profile of Truss No. 10 Figure B-il. MOE profile of Truss No. ii Figure B-12. MOE profile of Truss No. 12 C4 FlO D18 Fl Cs’ C22 F14 D13 F9 C58 C3 F13 D1O F8 - 211 - C9 Figure B-13. MOE profile of Truss No. 13 Figure B-14. MOE profile of Truss No. 14 Figure B-15. MOE profile of Truss No. 15 c55 F32 D2 F35 C17 C64 F3 Dl P36 C25 C70 19 F15 Dli F28 - 212 - Appendix C: Truss Plate Placement, Gap Width and Initial Midspan Lateral Deflection 1. Truss Plate Placement The placement of the metal truss plates was checked at the selected joints by measuring the distances between the reference nodes and edge nodes, as specified in the truss design. Since no noticeable misplacement of the truss plates was observed during the measurements of the first five trusses; therefore, no measurement was made for the other ten trusses. The selected joints with the specified reference nodes and edge nodes (represented by circles) are shown in Figure C-i. The results are shown in Table C-i. A Figure C-i. Metal truss plate placement measurements E D C B R Q P 0 N M -213- Table C-i Metal truss plate placement measurements Node Q Node M Node B Node DTruss No. x (mm) x (mm) y (mm) x (mm) y (mm) x (mm) y (mm) 1 63.5 82.6 101.6 63.5 57.2 76.2 82.6 2 63.5 82.6 101.6 63.5 57.2 76.2 82.6 3 63.5 82.6 101.6 63.5 57.2 76.2 82.6 4 63.5 82.6 95.3 63.5 57.2 76.2 82.6 5 63.5 82.6 101.6 63.5 57.2 76.2 76.2 Note: axes x andy are parallel and perpendicular to the major axis of the metal truss plates 2. Gap Width The width of the gaps between adjacent wood truss members of the MPC connections was measured at the selected joints. The measurements were based on the mean values of the gap width throughout the thickness of the truss members. The positions of the measurements are shown in Figure C-2; and the results are listed in Table C-2. Figure C-2. Positions of the gap width measurements -214- Table C-2. Gap width measurements (mm) Truss no. Gi G2 G3 G4 G5 G6 G7 G8 1 1.0 1.0 1.0 0.5 1.0 2 1.0 0.2 1.0 0.5 0.5 3 1.0 0.5 1.0 1.0 1.0 4 0.5 0.5 0.5 2.0 0.5 5 1.0 0.2 1.0 1.0 1.0 0.5 2.0 0.5 6 0.5 1.0 1.5 1.0 1.5 0.2 1.0 0.5 7 0.5 1.0 2.0 0.5 1.0 1.0 1.0 1.0 8 2.0 0.5 2.0 1.0 1.0 0 2.0 1.0 9 1.0 0.5 0.5 2.0 1.0 0.5 1.0 0.5 10 3.0 0.2 1.0 0.2 1.0 0.5 1.0 0.5 Ii 1.0 0.2 1.0 1.0 0.5 12 1.0 0.5 1.0 1.0 1.0 1.0 2.0 0.5 13 1.0 1.5 1.0 3.0 0.5 0.5 1.5 0.5 14 1.0 1.0 1.0 2.0 1.5 0.5 3.0 0.5 15 2.0 2.0 2.0 2.0 0.5 0.5 0.2 0.5 3. Initial 1’lidspan Lateral Dellections of W2 Webs G9 GlO Gil Gi2 G13 0.5 0.5 1.0 0.5 1.5 1.0 1.0 0.5 1.0 0.5 2.0 2.0 0.5 1.0 1.5 0.5 0.5 1.5 2.0 1.0 3.0 1.0 0.5 0.5 0.5 3.0 1.0 0.5 0.2 1.0 1.0 1.0 0.2 1.0 0 1.0 0.5 0.2 0.5 1.0 0.5 5.0 0.2 0.2 0.5 0.5 1.0 1.0 1.0 0.5 0.5 1.0 1.0 1.0 1.0 The initial midspan lateral deflections of the W2 webs of the fifteen trusses were measured. Each measurement was made by attaching a steel wire to the flat side of a W2 web and measuring the distance between the steel wire and the midspan of the web. The initial midspan lateral deflections were distinguished by the positive and negative signs, based on their directions to the webs. For each truss, the measurements of the two W2 webs were designated by LD 1 and LD 2, as shown in Figure C2. The results of the measurements are listed in Table C-3. -215 - Table C-3. Midspan lateral deflection ofcompression web W2 Truss No. LD 1 LD 2 Truss No. LD 1 LD 2 Truss No. LD 1 LD 2 1 1.5 -2.5 6 -2 -3 11 -1.4 -2.6 2 1 -2 7 0 -1.5 12 -1 -1.5 3 -7 2 8 -2 -1.8 13 3 -3 4 -2.5 -1.5 9 -1.9 -3 14 1 2.5 5 2.2 -3.5 10 2.5 -3 15 -1 4 The mean value and standard deviation of the initial midspan lateral deflections were -0.93 mm and 2.42 mm, respectively. -216- Appendix D: ANOVA analysis and Sampling Results This section presents the ANOVA analysis results of the mixed terms of the response surface function used in Chapter 7 for the reliability analysis of a three-bay truss assemblies. The sample values of the random variables and the responses of the truss assembly in terms of the critical buckling load are also presented. The ANOVA analysis was performed to find out the mixed terms that are significant to the system response. A commercially available program, SAS, was used based on the GLM procedures, in which least square method was used to fit general linear models. The analysis was performed based on a confidence level of 95% (CL equals to 0.05). Initially, the response surface function comprised of all the mixed terms, which were then discarded if not significant to the system response. The ANOVA analysis results of the truss assembly with zero, one and two CLB members per W2 web are listed in Tables D-1, D-2 and D-3. Table D-1 ANOVA analysis results ofthe truss assembly (no CLB) Source DF* Sum of Square Mean Square F value P>F Model 24 48.39 2.02 147.38 <.0001 Error 75 1.03 0.01 -- -- xl 2 0.182 0.091 6.66 0.0022 x2 2 11.037 5.518 403.42 <0.0001 x3 2 0.696 0.348 25.43 <0.0001 x4 2 3.272 1.636 119.6 <0.0001 x5 2 19.873 9.937 726.41 <0.0001 x6 2 0.727 0.364 26.58 <0.0001 x45 4 0.427 0.107 7.81 <0.0001 X46 4 0.728 0.182 13.3 <0.0001 X56 4 0.464 0.116 8.48 <0.0001 Note: * degree of freedom - 217 - Table D-2 ANOVA analysis results ofthe truss assembly (One CLB) Source DF Sum of Square Mean Square F value Pr>F Model 30 48.39 2.02 147.38 <.0001 Error 69 1.03 0.01 -- -- xl 2 0.177 0.088 7.34 0.0013 x2 2 10.671 5.336 443.39 <0.0001 x3 2 0.436 0.218 18.1 <0.0001 x4 2 2.477 1.239 102.93 <0.0001 x5 2 17.942 8.971 745.52 <0.0001 x6 2 0.650 0.325 27 <0.0001 x7 2 0.035 0.018 1.47 0.2380 X37 4 0.158 0.040 3.29 0.0158 X45 4 0.480 0.120 9.97 <0.0001 X46 4 0.721 0.180 14.98 <0.0001 X56 4 0.452 0.113 9.39 <0.0001 Note: * degree of freedom Table D-3 ANOVA analysis results ofthe truss assembly (Two CLBs) Source DF* Sum of Square Mean Square F value Pr>F Model 40 48.39 . 2.02 147.38 <.0001 Error 59 1.03 0.01 -- -- xl 2 0.081 0.040 3.54 0.0352 x2 2 8.029 4.015 351.22 <0.0001 x3 2 0.548 0.274 23.96 <0.0001 x4 2 2.695 1.347 117.86 <0.0001 x5 2 12.945 6.473 566.25 <0.0001 x6 2 0.444 0.222 19.44 <0.0001 x7 2 0.082 0.041 3.57 0.0345 x8 2 0.045 0.022 1.95 0.1517 x14 4 0.151 0.038 3.31 0.0163 x45 4 0.391 0.098 8.55 <0.0001 x46 4 0.853 0.213 18.65 <0.0001 x48 4 0.158 0.040 3.46 0.0133 x56 4 0.226 0.056 4.94 0.0017 X68 4 0.153 0.038 3.34 0.0156 Note: * degree of freedom The critical buckling load of the truss assembly was evaluated based on the FEM-based model in Chapter 7. The result was then represented by a load coefficient in -218 - conjunction with a reference load, which was assumed to be a uniformly distributed roof load of 0.001 N/mm2.The results for the truss assembly with zero, one and two CLB members per W2 webs are listed in Tables D-4, D-5 and D-6, respectively. For clarity, the random variables were designated by RV1 to RV8. Table D-4. Sampling results ofthe truss assembly with zero CLB bracing member RV1 RV2 RV3 RV4 RV5 RV6 Coeff. No. GPa GPa GPa mm mm mm -- 1 12.953 12.613 12.704 -0.930 2.951 5.240 10.480 2 8.190 12.613 8.439 -0.930 -0.930 -0.930 9.485 3 8.190 10.572 12.704 -4.751 -4.811 -0.930 9.648 4 8.190 12.613 8.439 2.891 2.951 -7.100 9.833 5 12.953 12.613 8.439 2.891 -0.930 -7.100 9.715 6 10.572 12.613 8.439 2.891 -0.930 5.240 10.019 7 8.190 10.572 10.572 2.891 -4.811 5.240 9.804 8 8.190 10.572 12.704 2.891 -4.811 -7.100 9.982 9 10.572 12.613 8.439 2.891 2.951 -0.930 9.742 10 8.190 8.530 8.439 -4.751 -4.811 -0.930 9.234 11 8.190 10.572 10.572 -0.930 -0.930 -0.930 9.579 12 12.953 12.613 8.439 -0.930 -0.930 5.240 10.067 13 8.190 8.530 8.439 -0.930 -4.811 -0.930 9.280 14 10.572 12.613 12.704 -4.751 -0.930 5.240 10.197 15 10.572 10.572 10.572 2.891 -0.930 5.240 9.989 16 8.190 12.613 8.439 -4.751 2.951 5.240 9.840 17 12.953 12.613 8.439 -0.930 2.951 -7.100 10.162 18 10.572 8.530 12.704 -4.751 -4.811 -0.930 9.613 19 10.572 10.572 12.704 -0.930 -4.811 -0.930 9.769 20 12.953 8.530 8.439 2.891 2.951 5.240 9.598 21 10.572 8.530 8.439 -4.751 2.951 -7.100 9.775 22 10.572 8.530 8.439 -0.930 -0.930 5.240 9.779 23 12.953 8.530 8.439 2.891 2.951 -7.100 10.051 24 12.953 8.530 8.439 -0.930 -0.930 -0.930 9.418 25 8.190 10.572 10.572 -0.930 2.951 5.240 9.901 26 8.190 10.572 10.572 2.891 -4.811 5.240 9.804 27 10.572 8.530 12.704 -4.751 -0.930 -7.100 9.584 28 8.190 8.530 8.439 -0.930 -4.811 5.240 9.657 29 8.190 8.530 12.704 -0.930 -4.811 5.240 9.922 30 12.953 8.530 12.704 2.891 2.951 -7.100 10.002 31 8.190 12.613 12.704 -4.751 2.951 -0.930 10.171 - 219 - RVI RV2 RV3 RV4 RV5 RV6 Coeff. No. OPa GPa OPa mm mm mm -- 32 8.190 8.530 8.439 -0.930 2.951 -7.100 9.695 33 12.953 8.530 12.704 -0.930 -0.930 -7.100 9.722 34 10.572 10.572 10.572 -0.930 2.951 -0.930 10.145 35 12.953 12.613 8.439 -4.751 -0.930 -0.930 9.571 36 8.190 8.530 8.439 -4.751 -4.811 -0.930 9.234 37 8.190 12.613 12.704 2.891 -0.930 -7.100 10.036 38 12.953 12.613 12.704 -0.930 2.951 -0.930 10.558 39 10.572 8.530 12.704 -0.930 -0.930 -0.930 9.682 40 12.953 12.613 8.439 -0.930 -4.811 -7.100 9.625 41 12.953 12.613 8.439 -4.751 2.951 -7.100 10.079 42 8.190 10.572 12.704 -0.930 -0.930 -7.100 9.669 43 12.953 8.530 8.439 -0.930 -4.811 -7.100 9.454 44 12.953 12.613 8.439 2.891 -0.930 -7.100 9.715 45 12.953 8.530 8.439 -4.751 -4.811 -0.930 9.362 46 10.572 12.613 8.439 2.891 -4.811 -0.930 9.610 47 12.953 8.530 8.439 2.891 2.951 -7.100 10.051 48 8.190 8.530 8.439 -0.930 -4.811 5.240 9.657 49 8.190 8.530 8.439 -0.930 2.951 -7.100 9.695 50 10.572 12.613 8.439 -4.751 -0.930 5.240 9.867 51 12.953 8.530 8.439 -0.930 -0.930 5.240 9.888 52 12.953 10.572 10.572 2.891 -4.811 -0.930 9.726 53 12.953 10.572 10.572 -0.930 2.951 5.240 10.281 54 10.572 8.530 8.439 -0.930 2.951 -7.100 9.842 55 10.572 12.613 8.439 2.891 2.951 5.240 9.663 56 10.572 12.613 8.439 2.891 2.951 -7.100 10.008 57 8.190 12.613 8.439 2.891 -0.930 -7.100 9.550 58 12.953 8.530 8.439 -4.751 2.951 -7.100 9.874 59 10.572 8.530 8.439 2.891 2.951 -0.930 9,532 60 12.953 10.572 10.572 -0.930 -4.811 5.240 10.161 61 12.953 10.572 10.572 -4.751 -4.811 -7.100 9.637 62 10.572 8.530 12.704 -4.751 2.951 -0.930 10.162 63 12.953 12.613 8.439 2.891 -0.930 5.240 10.181 64 10.572 10.572 12.704 2.891 -4.811 -7.100 9.811 65 8.190 8.530 12.704 -0.930 2.951 -7.100 9.934 66 12.953 8.530 8.439 2.891 2.951 5.240 9.598 67 8.190 10.572 12.704 2.891 -0.930 -0.930 10.008 68 10.572 12.613 8.439 2.891 -0.930 -7.100 9.646 69 12.953 8.530 8.439 -4.751 -0.930 5.240 9.809 70 8.190 8.530 12.704 -4.751 -0.930 5.240 9.882 71 10.572 8.530 8.439 2.891 2.951 5.240 9.457 72 10.572 8.530 8.439 -4.751 -4.811 5.240 9.725 73 10.572 12.613 8.439 -0.930 -0.930 -7,100 9.564 - 220 - RV1 RV2 RV3 RV4 RV5 RV6 Coeff. No. GPa GPa GPa mm mm mm -- 74 12.953 12.613 8.439 -0.930 -4.811 -0.930 9.594 75 10.572 8.530 12.704 2.891 2.951 -7.100 9.862 76 8.190 12.613 12.704 -4.751 2.951 -0.930 10.171 77 8.190 8.530 12.704 -0.930 -4.811 -7.100 9.559 78 8.190 10.572 10.572 -0.930 -0.930 -0.930 9.579 79 12.953 12.613 12.704 -0.930 -4.811 5.240 10.365 80 10.572 10.572 10.572 -0.930 -4.811 -7.100 9.629 81 8.190 8.530 12.704 -4.751 2.951 5.240 9.864 82 10.572 12.613 8.439 -0.930 -4.811 5.240 9.967 83 8.190 12.613 12.704 2.891 -4.811 -0.930 10.101 84 10.572 10.572 10.572 -0.930 -0.930 -7.100 9.638 85 10.572 10.572 10.572 -4.751 -0.930 -0.930 9.583 86 10.572 10.572 12.704 -0.930 -0.930 5.240 10.184 87 10.572 12.613 12.704 -0.930 -0.930 -0.930 9.884 88 10.572 12.613 8.439 -4.751 -4.811 5.240 9.893 89 12.953 8.530 8.439 2.891 -4.811 5.240 10.368 90 12.953 12.613 8.439 -0.930 -0.930 5.240 10.067 91 12.953 8.530 12.704 -0.930 -0.930 -7.100 9.722 92 8.190 8.530 8.439 -0.930 2.951 -0.930 9.752 93 12.953 12.613 8,439 -0.930 -4.811 -0.930 9.594 94 10.572 12.613 8.439 -0.930 -0.930 -0.930 9.570 95 10.572 12.613 12.704 -4.751 -0.930 -0.930 9.838 96 10.572 10.572 10.572 2.891 -0.930 -7.100 9.711 97 12.953 10.572 10.572 2.891 -0.930 -0.930 9.793 98 10.572 10.572 12.704 -4.751 2.951 -7.100 10.140 99 12.953 8.530 12.704 -4.751 -4.811 5.240 10.115 100 10.572 8.530 8.439 -4,751 -4.811 -0.930 9.298 Table D-5. Sampling results ofthe truss assembly with one CLB bracing member per W2 web RV1 RV2 RV3 RV4 RV5 RV6 RV7 Coeff. No. GPa GPa GPa mm mm mm N/mm -- 1 10.572 12.157 12.138 -4.078 -0.930 2.571 417.600 16.824 2 8.852 10.572 10.572 -0.930 -5.914 -4.431 463.801 16.962 3 10.572 10.572 12.138 -4,078 -5.914 2.571 463.801 16.962 4 12.291 10.572 10.572 2.218 -0.930 2.571 417.600 16.815 5 8.852 12.157 12.138 -0.930 -0.930 2.571 417.600 16.824 6 8.852 12.157 9.005 2.218 -0.930 -0.930 417.600 16.824 7 10.572 8.986 9.005 -0.930 -5.914 2.571 463.801 16.952 8 8.852 10.572 12.138 2.218 4.054 -4.431 371.399 16.708 9 12.291 10.572 10.572 -0.930 4.054 2.571 417.600 16.708 10 8.852 10.572 12.138 -4.078 -0.930 2.571 417.600 16.824 -221 - RV1 RV2 RV3 RV4 RV5 RV6 RV7 Coeff. No. GPa GPa GPa mm mm mm N/mm -- 11 12.291 10.572 10.572 -0.930 -0.930 2.571 417.600 16.824 12 8.852 8.986 9.005 -4.078 -0.930 -0.930 463.801 16.824 13 12.291 12.157 9.005 -4.078 4.054 -4.431 371.399 16.708 14 12.291 12.157 9.005 -0.930 -5.914 -4.431 417.600 16.968 15 8.852 10.572 12.138 -0.930 4.054 -0.930 371.399 16.708 16 10.572 12.157 9.005 2.218 -5.914 2.571 417.600 16.943 17 10.572 10.572 12.138 -0.930 -0.930 -0.930 417.600 16.824 18 12.291 10.572 12.138 2.218 -0.930 -4.431 371.399 16.824 19 12.291 10.572 12.138 2.218 4.054 -4.431 371.399 16.708 20 12.291 10.572 10.572 -4.078 4.054 2.571 417.600 16.708 21 12.291 12.157 9.005 2.218 -5.914 -0.930 371.399 17.792 22 8.852 12.157 9.005 2.218 -5.914 -0.930 371.399 17.782 23 10.572 10.572 10.572 -0.930 4.054 -0.930 463.801 16.824 24 10.572 8.986 12.138 2.218 -0.930 2.571 417.600 16.279 25 12.291 10.572 10.572 -4.078 -0.930 -0.930 417.600 16.956 26 10.572 10.572 12.138 -0.930 -5.914 -0.930 371.399 17.129 27 8.852 12.157 9.005 2.218 -0.930 2.571 371.399 17.614 28 8.852 12.157 9.005 2.218 -5.914 -4.431 463.801 17.782 29 10.572 12.157 9.005 -4.078 4.054 -4.431 417.600 17.494 30 8.852 8.986 12.138 -0.930 4.054 -0.930 463.801 16.187 31 8.852 12.157 9.005 2.218 -0.930 2.571 417.600 17.614 32 12.291 12.157 9.005 -0.930 4.054 -0.930 463.801 17.484 33 10.572 8.986 9.005 2.218 -0.930 2.571 463.801 16.193 34 8.852 12.157 9.005 2.218 -5.914 2.571 417.600 17.783 35 12.291 12.157 12.138 -0.930 4.054 -4.431 463.801 17.578 36 10.572 12.157 9.005 -0.930 -5.914 -0.930 371.399 17.795 37 10.572 12.157 9.005 -4.078 -0.930 -0.930 371.399 17.642 38 12.291 8.986 9.005 -0.930 4.054 -4.431 417.600 16.136 39 8.852 8.986 9.005 -0.930 -5.914 -0.930 463.801 16.268 40 8.852 10.572 12.138 -0.930 -5.914 -4.431 371.399 17.139 41 8.852 12.157 12.138 -4.078 -0.930 2.571 463.801 17.717 42 10.572 12.157 9.005 2.218 -5.914 -0.930 463.801 17.783 43 8.852 12.157 9.005 -4.078 -0.930 -0.930 463.801 17.633 44 12.291 8.986 9.005 -0.930 -0.930 2.571 371.399 16.237 45 10.572 12.157 12.138 -0.930 -5.914 -0.930 417.600 17.883 46 10.572 12.157 9.005 -4.078 4.054 2.571 417.600 17.484 47 8.852 8.986 9.005 -0.930 -0.930 -0.930 417.600 16.190 48 12.291 10.572 12.138 2.218 -5.914 -0.930 463.801 17.129 49 10.572 8.986 9.005 -0.930 4.054 -4.431 417.600 16.127 50 10.572 12.157 12.138 -0.930 4.054 -0.930 463.801 17.576 51 12.291 12.157 9.005 -4.078 -0.930 2.571 371.399 17.642 52 12.291 10.572 10.572 -4.078 -5.914 -4.431 371.399 17.099 - 222 - RV1 No. GPa 53 12.291 54 10.572 55 8.852 56 10.572 57 10.572 58 8.852 59 12.291 60 10.572 61 10.572 62 12.291 63 10.572 64 12.291 65 10.572 66 8.852 67 10.572 68 8.852 69 10.572 70 10.572 71 8.852 72 12.291 73 8.852 74 8.852 75 10.572 76 12.291 77 12.291 78 12.291 79 8.852 80 12.291 81 10.572 82 8.852 83 8.852 84 12.291 85 8.852 86 8.852 87 8.852 88 8.852 89 8.852 90 10.572 91 12.291 92 8.852 93 8.852 94 8.852 RV2 RV3 RV4 GPa GPa mm 12.157 9.005 2.218 12.157 9.005 -4.078 8.986 9.005 -4.078 10.572 10.572 -0.930 10.572 10.572 -0.930 8.986 9.005 2.218 10.572 10.572 -4.078 12.157 9.005 2.218 12.157 9.005 -4.078 8.986 9.005 -0.930 10.572 10.572 -0.930 10.572 12.138 -4.078 10.572 10.572 -4.078 8.986 9.005 2.218 8.986 12.138 2.218 8.986 12.138 -0.930 8.986 9.005 -4.078 12.157 9.005 -4.078 12.157 12.138 -4.078 12.157 9.005 -4.078 12.157 9.005 -0.930 12.157 12.138 -4.078 10.572 10.572 -0.930 10.572 10.572 2.218 8.986 12.138 2.218 8.986 9.005 -4.078 10.572 12.138 2.218 12.157 9.005 -0.930 10.572 10.572 -4.078 12.157 9.005 -0.930 10.572 12.138 -0.930 8.986 9.005 -4.078 8.986 9.005 2.2 18 10.572 12.138 -4.078 12.157 12.138 2.218 8.986 12.138 -0.930 10.572 10.572 -0.930 12.157 9.005 2.218 10.572 12.138 2.218 8.986 9.005 -4.078 10.572 12.138 -0.930 12.157 12.138 -4.078 RV6 RV7 Coeff. mm N/mm —— -4.431 463.801 17.484 2.571 463.801 17.484 -4.431 371.399 16.190 -0.930 37 1.399 17.086 2.571 463.801 16.824 2.571 371.399 16.279 -4.431 417.600 16.958 -4.43 1 371.399 17.633 2.571 463.801 17.484 2.571 371.399 16.324 2.571 417.600 17.075 -0.930 371.399 16.883 -0.930 371.399 17.096 -0.930 417.600 16.085e -0.930 463.801 16.190 -4.43 1 417.600 16.277 -0.930 371.399 16.127 2.571 463.801 17.783 2.571 371.399 17.552 -0.930 371.399 17.494 -0.930 371.399 17.471 2.571 417.600 17.886 -4.431 463.801 16.824 -0.930 371.399 16.956 2.571 371.399 16.193 2.571 417.600 16.247 -0.930 417.600 16.962 2.571 463.801 17.484 -4.431 463.801 17.096 -4.43 1 417.600 17.484 -0.930 417.600 17.123 -4.431 371.399 16.136 2.571 371.399 16.279 2.571 371.399 16.837 -4.431 417.600 17.717 -4.431 417.600 16.188 -4.431 463.801 17.364 2.571 417.600 17.774 -4.431 417.600 17.129 -4.431 463.801 16.127 -0.930 371.399 16.968 2.571 463.801 17.717 RV5 mm 4.054 4.054 -0.930 -5.914 4.054 -5.914 -0.930 -0.930 4.054 -5.914 -5.914 4.054 -5.914 4.054 4.054 -0.930 4.054 -5.9 14 4.054 4.054 4.054 -5.9 14 4.054 -0.930 4.054 -0.930 -0.930 4.054 -5.9 14 4.054 -5.9 14 4.054 -5.9 14 4.054 -0.930 4.054 -5.914 -5.9 14 -5.914 4.054 -0.930 -0.930 - 223 - RV1 RV2 RV3 RV4 RV5 RV6 RV7 Coeff. No. GPa GPa GPa mm mm mm N/mm -- 95 10.572 12.157 9.005 -4.078 -0.930 -0.930 417.600 17.633 96 12.291 12.157 12.138 -0.930 4.054 -4.431 417.600 17.578 97 12.291 10.572 10.572 -0.930 -0.930 -0.930 371.399 16.956 98 12.291 10.572 10.572 -0.930 4.054 -4.431 417.600 16.837 99 10.572 10.572 10.572 -4.078 4.054 -4.431 463.801 16.824 100 10.572 12.157 9.005 -4.078 -5.914 -0.930 371.399 17.802 Table D-6. Sampling results ofthe truss assembly with two CLB bracing members per W2 web RV1 RV2 RV3 RV4 RV5 No. GPa GPa OPa mm mm 1 11.769 9.038 8.314 -7.460 3.900 2 9.374 12.105 8.314 -7.460 -0.930 3 11.769 10.572 10.572 -7.460 -0.930 4 10.572 10.572 12.830 5.600 3.900 5 11.769 9.038 12.830 -0.930 3.900 6 9.374 10.572 10.572 -7.460 3.900 7 10.572 9.038 8.314 -0.930 -0.930 8 10.572 12.105 8.314 5.600 3.900 9 10.572 12.105 8.314 -0.930 -0.930 10 9.374 10.572 10.572 5.600 -5.760 11 10.572 9.038 12.830 -7.460 -0.930 12 9.374 9.038 8.314 5.600 -0.930 13 10.572 10.572 10.572 -7.460 -5.760 14 11.769 9.038 8.314 -7.460 -5.760 15 9.374 10.572 10.572 -0.930 3.900 16 11.769 9.038 12.830 -7.460 -5.760 17 11.769 9.038 8.314 5.600 3.900 18 11.769 10.572 10.572 -7.460 -0.930 19 10.572 12.105 12.830 -0.930 3.900 20 11.769 12.105 12.830 5.600 3.900 21 9.374 12.105 8.314 5.600 3.900 22 9.374 10.572 10.572 -7.460 3.900 23 11.769 9.038 8.314 -0.930 -0.930 24 9.374 12.105 12.830 -7.460 -5.760 25 10.572 12.105 8.314 -7.460 -5.760 26 9.374 10.572 12.830 -7.460 -5.760 27 9.374 12.105 8.314 5.600 3.900 28 10.572 12.105 8.314 -7.460 -0.930 29 11.769 12.105 8.314 5.600 3.900 30 10.572 12.105 8.314 5.600 3.900 31 10.572 12.105 12.830 5.600 -0.930 RV6 RV7 RV8 Coeff. mm N/mm N/mm —- -0.930 406.980 5 14.097 27.674 2.399 406.980 472.900 27.887 -4.259 406.980 472.900 27.311 -0.930 406.980 5 14.097 28.238 2.399 406.980 431.703 28.419 -0.930 538.820 431.703 28.091 2.399 406.980 431.703 27.3 15 -4.259 406.980 472.900 28.964 -0.930 406.980 472.900 28.049 2.399 538.820 43 1.703 27.405 -0.930 538.820 514.097 27.098 2.399 406.980 43 1.703 27.680 -0.930 406.980 43 1.703 26.837 2.399 406.980 514.097 26.568 -4.259 406.980 431.703 28.301 -4.259 406.980 5 14.097 26.507 2.399 538.820 5 14.097 27.854 -0.930 538.820 472.900 27.473 2.399 406.980 472.900 29.235 -0.930 406.980 5 14.097 28.964 -0.930 538.820 5 14.097 28.791 2.399 406.980 431.703 28.328 -0.930 406.980 431.703 27.185 -4.259 472.900 514.097 27.311 -0.930 538.820 514.097 27.317 2.399 472.900 472.900 27.037 2.399 472.900 431.703 28.385 2.399 406.980 431.703 27.886 -4.259 406.980 5 14.097 29.006 -4.259 538.820 43 1.703 29.030 -0.930 538.820 43 1.703 28.492 - 224 - RV1 RV2 RV3 RV4 RV5 RV6 RV7 RV8 Coeff. No. GPa GPa GPa mm mm mm N/mm N/mm -- 32 9.374 12.105 8.314 5.600 -0.930 -0.930 406.980 472.900 28.365 33 11.769 9.038 8.314 -0.930 3.900 -4.259 472.900 472.900 27.821 34 9.374 10.572 12.830 -0.930 -5.760 2.399 538.820 472.900 27.257 35 11.769 10.572 10.572 5.600 3.900 -4.259 472.900 472.900 28.916 36 10.572 9.038 8.314 5.600 3.900 2.399 538.820 472.900 27.720 37 9.374 9.038 8.314 -7.460 -0.930 2.399 406.980 431.703 26.990 38 9.374 12.105 8.314 5.600 3.900 -0.930 538.820 514.097 28.791 39 9.374 10.572 10.572 -7.460 3.900 2.399 406.980 431.703 28.328 40 11.769 9.038 8.314 -0.930 -0.930 -0.930 406.980 431.703 27.185 41 9.374 9.038 8.314 -0.930 -5.760 2.399 472.900 514.097 26.720 42 10.572 9.038 8.314 5.600 3.900 -4.259 538.820 431.703 28.298 43 10.572 9.038 8.314 5.600 -0.930 -0.930 472.900 472.900 27.478 44 9.374 9.038 12.830 -0.930 3.900 2.399 406.980 472.900 28.353 45 10.572 9.038 12.830 -0.930 -0.930 -0.930 406.980 431.703 27.299 46 11.769 9.038 12.830 -0.930 -5.760 -0.930 472.900 431.703 26.758 47 9.374 10.572 10.572 5.600 -0.930 -4.259 406.980 431.703 27.849 48 11.769 12.105 8.314 5.600 3.900 2.399 472.900 472.900 28.523 49 10.572 12.105 8.314 -7.460 -0.930 -0.930 406.980 472.900 27.770 50 9.374 12.105 8.314 -0.930 -0.930 -0.930 406.980 431.703 28.008 51 11.769 9.038 12.830 -0.930 -5.760 -0.930 406.980 431.703 26.730 52 10.572 12.105 8.314 5.600 -0.930 -4.259 406.980 431.703 28.199 53 10.572 12.105 8.314 -7.460 3.900 -0.930 472.900 472.900 28.529 54 10.572 10.572 10.572 5.600 -0.930 -4.259 406.980 514.097 27.808 55 11.769 9.038 8.314 5.600 -5.760 -0.930 406.980 431.703 26.842 56 10.572 9.038 12.830 -7.460 3.900 2.399 472.900 431.703 27.986 57 11.769 10.572 10.572 -0.930 -5.760 -4.259 406.980 514.097 27.059 58 11.769 12.105 8.314 5.600 -0.930 2.399 538.820 431.703 28.699 59 9.374 9.038 12.830 5.600 -0.930 -0.930 406.980 514.097 27.608 60 11.769 10.572 12.830 5.600 3.900 2.399 538.820 472.900 28.415 61 10.572 9.038 8.314 -0.930 -5.760 -0.930 472.900 472.900 26.637 62 9.374 9.038 12.830 5.600 -5.760 -4.259 538.820 431.703 26.828 63 11.769 9.038 12.830 5.600 -5.760 2.399 406.980 514.097 27.148 64 9.374 9.038 8.314 -0.930 -5.760 -4.259 472.900 514.097 26.536 65 9.374 10.572 10.572 -7.460 -0.930 -0.930 406.980 514.097 27.400 66 10.572 12.105 8.314 -7.460 -5.760 2.399 472.900 472.900 27.365 67 10.572 10.572 10.572 5.600 -5.760 -0.930 472.900 431.703 27.348 68 9.374 12.105 8.314 -7.460 3.900 -4.259 406.980 472.900 28.297 69 9.374 10.572 10.572 -0.930 -0.930 -4.259 472.900 472.900 27.578 70 10.572 12.105 8.314 5.600 -5.760 -4.259 406.980 472.900 27.552 71 11.769 10.572 10.572 5.600 -5.760 -0.930 472.900 514.097 27.368 72 11.769 9.038 8.314 -7.460 -0.930 2.399 472.900 431.703 27.063 73 10.572 12.105 8.314 5.600 -0.930 -0.930 538.820 472.900 28.372 - 225 - RV1 RV2 RV3 RV4 RV5 RV6 RV7 RV8 Coeff. No. GPa GPa GPa mm mm mm N/mm N/mm -- 74 11.769 10.572 10.572 5.600 -0.930 2.399 538.820 472.900 28.171 75 10.572 9.038 8.314 -7.460 3.900 -0.930 406.980 514.097 27.611 76 11.769 12.105 8.314 -7.460 3.900 2.399 472.900 514.097 28.781 77 11.769 10.572 10.572 -0.930 -0.930 -4.259 472.900 514.097 27.592 78 9.374 12.105 12.830 -7.460 3.900 2.399 406.980 514.097 28.920 79 10.572 10.572 10.572 -0.930 -5.760 -0.930 472.900 431.703 27.108 80 11.769 10.572 10.572 -7.460 -5.760 -0.930 472.900 514.097 26.996 81 9.374 9.038 8.314 -7.460 3.900 -4.259 472.900 431.703 27.382 82 10.572 10.572 12.830 -7.460 3.900 -0.930 472.900 472.900 28.247 83 11.769 12.105 12.830 5.600 3.900 -4.259 406.980 472.900 29.297 84 10.572 12.105 8.314 -0.930 -0.930 -0.930 538.820 431.703 28.049 85 11.769 9.038 8.314 -7.460 -0.930 2.399 472.900 514.097 27.136 86 9.374 12.105 8.314 5.600 -5.760 -4.259 472.900 472.900 27.538 87 9.374 9.038 12.830 5.600 3.900 -0.930 472.900 431.703 28.004 88 9.374 10.572 10.572 5.600 -0.930 -4.259 538.820 514.097 27.862 89 11.769 10.572 10.572 -7.460 3.900 -4.259 406.980 472.900 27.949 90 11.769 12.105 8.314 -7.460 -5.760 -0.930 538.820 472.900 27.317 91 11.769 12.105 12.830 5.600 -5.760 -0.930 472.900 514.097 27.849 92 11.769 10.572 10.572 5.600 3.900 -4.259 472.900 472.900 28.916 93 10.572 12.105 8.314 5.600 -0.930 -4.259 538.820 514.097 28.219 94 11.769 10.572 10.572 5.600 3.900 -4.259 538.820 514.097 29.052 95 9.374 9.038 8.314 -0.930 3.900 -4.259 406.980 431.703 27.776 96 9.374 10.572 10.572 -0.930 -0.930 -4.259 472.900 431.703 27.541 97 10.572 9.038 12.830 -7.460 3.900 2.399 406.980 472.900 27.998 98 9.374 12.105 8.314 -7.460 3.900 2.399 472.900 431.703 28.748 99 11.769 10.572 12.830 -7.460 -5.760 2.399 472.900 431.703 27.077 100 11.769 9.038 12.830 -7.460 -5.760 -0.930 406.980 472.900 26.579 - 226 -

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