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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 of wood 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 Chaper 1.  xix  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. 2.1.  8  Literature Review  MPC Connections  2.1.1.  8  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 of wood 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. 3.1.  Finite Element Method Based Model Formulation  30  Formulation of FEM-based elements  3.1.1.  30  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.  36  Thinplateelement -  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.  41  Spring element  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.  44  Metal plate connection element  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  Chaper 4.  Development of the Computer Program SATA  57  59  4.1.  Programming Logic  59  4.2.  Data Structure of SATA  61  4.2.1.  Storage of the data  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  61  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  Research Method  73  5.1.1.  Material  73  5.1.2.  Material property test  74  5.1.  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  Chaper 6.  105  Experimental Study and Model Verification II: Stiffness and Stability  Capacity of MPC Wood Truss Assemblies 6.1.  107  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.  121  Individual truss and truss assembly tests  6.1.4.1. Support conditions  122  6.1.4.2. Loading system  125 -  vii  -  6.1.4.3. Data acquisition 6.1.5.  .127  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  Chaper 7.  160  Reliability Analysis of Critical Buckling Load of MPC Wood Truss  Assemblies  162  7.1.  The Response Surface Method  7.2.  Reliability Analysis of the Critical Buckling Load of MPC Wood Truss Assemblies..  162  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,  181  qext  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 149  assembly  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 2 to rated (MSR) dimension lumber with the cross-sectional size varying from 38 x 64 mm 38 x 184 mm . The metal truss plates are manufactured by high-speed stamping machines that 2 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 of Canada (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 of wood 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 of wood 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 of wood 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 of MPC 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.  L x I  —  +  N  dM  k  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  -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-  2  x  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-  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 mm 2 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 mm 2 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  (a)  ‘2  2 M  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: 1 sin 2u srn2u 2  =  1 +u 2(u 1 ) 2 sin2(u +u ) 2  1112 —  (l +12) 2  1 ( 2 a 1 +12)  (2-1)  -  16-  where u 1  =  0.5l JP/EI, u 2  =  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  F 2  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= 1 4 +-  (2-2) Fbr  4P ) 0 KA_(A+A  0 and A are the initial deflection and the deflection caused by external loading, K is the where A  -  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)  Fbr  (2-3)  ) 0 KA_(A+1.5A  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 of multiple 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 as 2 N(e+u)/1 where N, e and u are 7r , 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 mm 2 (nominally 2 by 4 in ) in cross-sectional size was considered. The metal plates were oversized 2 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 mm 2 (nominally 2 by 4 in ) in cross-sectional size with a combination of 16 gauge and 2 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 of the trusses and the strength and stiffness properties of No. 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(v,v)  (w w) ,  ,v 1 y(v )  Figure 3-1. Local coordinate system of the three-dimensional beam element  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.  -31  -  _____________  z=h/2  zQc)  K1  y(77)  x  y(i) y=b/2  x=O = —1  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.-877 th 7 th 7 8y 8y 8 8K8K8K  --  8i 8  (3-1)  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  1 v  1 w  é  wi  1 V  2 U  2 V  2 W  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  =  }T, {u, v, w, 8 which consisted of three translational  displacements and one torsional displacement, was derived from the nodal displacement vector,  -  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: N 000 0 1 0 2 0N 0000 O 0 1 M 0 M 2 0 Al 0 M 0 0 3 4 0 Br: 1 00 0L 0L 00 2 0L 3 4 0L 00 ON 1 0000 2 00 ON  1 B 2 B  (3-3)  3 B 4 B  The row vectors, B , where i1 to 4, consisted of the shape functions for the specific 1 components of the element displacement vector, u. For brevity in the main text of this thesis, the shape functions, N, L 1 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: 6—U  1  2  ii  1  1  ‘2 ‘2 +—(w) —zw +—(v) —yv +—(pO) ii  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, ‘int  was calculated by  =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  1 +z 2 B 2 B 2 +y 2 B 3 K=IEsec(Bi B 1  1 +h f[Esec(Bi B  =  2 2 K  2 2 B B  T  T  2 ) 3 4 B +Gp 4 B d V  1  2 2  T  1  )+Gp B 3 27 B 3 B 4 B JDet(J)dd’7dK 4  Esec[aCaB +BaaTC]Det(J)ddildK  ‘t  (3-7)  (3-8)  where C=(B +B B 2 B 2 3 2 + 3 4 p 4 B ) .  The tangent stiffness matrix, K, was obtained by differentiating the internal force vector,  with respect to the nodal displacement vector, a, as:  TB +BTB +p2BiTBi}Iv+ 4 2 Io(B1 TB1 IGP2B1 d V  Kt=  (3-9) =  where  TB1 + BTB + p c{B B’ 2 S [EtanCTC+ 2  T  + Gp2BTB]Det)dd1dK  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)  3 T  =  cos(x, y) cos(y,x) cos(y,y) cos(z, x) cos(z, y)  cos(x, z) cos(y,z) cos(z, z)  (3-10) 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, a , in the local coordinate system can be transformed from the nodal 1 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  cos(z, x)  cos(z, y)  cos(z, z)  0  0  0  w  0  0  cos(x,x)  W  —  a  —  T6 r1  —  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  (3-11)  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 a=  6 rT  LajJ L°  O1Ir1 I =Tr  J 1 ] 6 T  (3-12)  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, , 12 as: T K  2 TKtanTi  (3-13)  TP  where the superscript, superscript,  T  g  is for the items defined in the global coordinate system and the  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, T , which is equal 1 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  =J.  a  a —  h 2  0  =  0  —  ay  a  ooLa  az  ôz  2  —  aIC  a —  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: aT  =  aT 2  e’i  where af  aT 3 v  =  aT 4  (315)  w  , 1 w  ,,), i 1 w  =  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,  =  1 lOy and wj, Ow  =  1 lOxOy are the partial derivatives of the w 2 8  displacement, w. The element displacement vector, u  =  {u  v  w}T was established from the nodal  displacement, a, and the kinematic matrix, B as:  u=  0 1 0L 0  0  1 a 2 0 0 N 0 0 0 1 B a 2 0000 =B a 2 2 0... 0 0L 0 3 a 2 M 3 M 4 0 M 1 M M 3 B 0 5 M 1 5 1 M 6 4 a  0  0  0  ...  ...  where N 1 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,  u, + v + ww  —  (3-16)  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:  {:}D{z;}  :: ::::::  0  (3-17) G{1}  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  -  =aTK+Pint*  “mt  (3-18)  where K is the stiffhess matrix and can be calculated by: K = i(klBlTBl  +  TB + 3 B k z 2 k y y y,bet(J)dd,ldK yTB 2B T +2 3 B 2 kiz  YTB + BiTB 2 [B X 2 +z YYTB (B X X +3 TB )]Det(J)dd7 B 23 dic 7 1k 1 3  +  TB + 2 B yTB + B 1 B x TBi )Det(J)dd7 2 + fG(BlTBl + 2 dic 7  The nonlinear internal force vector, 1  [ k y 2 y y aaTB TB B + aTByTByaBy]DetJ)dd7lJK xTB + aTBTBa] + 3 Ikl[Bl.xaaTB x 3  =  +  aTBxTBy]Detcgdl?iK [ 1k y 2 x x 3 aaTB TB B +Bi,aa 3 +. aBBaBi + B T  1 T 3 )+aTB Bya(B taT 3 TB +B yTB y (B +i)]Det@cIüiIic  +1 iG[(B +B where k 1 =  was calculated by:  =  E 1(1  E 1(1  2 k  —  1(1 —  =  and  was calculated by differentiating the internal force  with respect to the nodal displacement vector, a.  =  1(1 —  are the stiffness constants.  —  The tangent stiffness matrix, vector  =  =  ô(aTK+ P*)  can be expressed as:  K+  (3-19)  where 1flt 81  =  +  XTaTB 1 [B X X TB 13 Jk  xaBlx B x TB jDetT)dd7iK + BiaB B +3 T 3  T3 2 yTB + 3 aTB y yTB 2 B y y aB + 2 yTB B y 3 y aB ]Det1)dd7dic 1k [B 2  yTaTB 2 [B x x TB + 3 xTaTB 1 B y y TB + BbaB + fk 33 yaB + 3 2 B T x B }DetJ)d4i7dK T y 3  yaBix (B y TB + B 2 XTB X 3 Y aB + 3 fk 3 + fG[(B 1 +B 2  XTB + 3 (B Y TB )DetJ)d4d1?dK B )+ G(B 1 +B 2 )TaT 3  TB B +3  XTB + 3 (B TB (B B )+ 3 1 +B 2 )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: 6 T  1 a  aj a=  r• r 3  6 T =  6 T  ak 1 a  6 T  rk 1 r  r 24 =T  (3-20)  where T 6 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  -  !  1 (v,v  y ) 1 (u,O  Deformation  A  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  1 w  1 V  =  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: 100000—100000 0100000—10000 v 00100000—1000 w a =Ba= 000100000—100 6 0000100000—10 w’ 0—1 0000010000 V u  (3-21)  3.1.3.2. Force displacement relationship  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:  ktan  =  ksec  =  clF dA  —  =  =  1 +(k+ m  \ km z 1 0 m  1 A)[1 0 +m (m  —  0  e —m ) 1  (3-23) e1m0  ]  /A  The spring force vector, f, was calculated based on the element displacement vector, u, as:  =  F F F6 F,  u  ksec,u  F  ksec,v = =  Dsecu  V  ksec  W  0  ksec, ksec,w’  F,  W  ksec,v  where Dsec is the secant modulus matrix, and ksec,u, ksec,v  (3-24)  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 =  1 can be calculated by: , 1 from which the internal force vector, P nt i1  =  fTB  =  B T (Dsecu)  =  aTBTDsecB  =  aTK  where K is the stiffness matrix and can be calculated by K  (3-25) =  BTDsecB.  The tangent stiffness matrix, Ktan, was calculated as the first derivative of the internal force vector, =  3Pjt  =  6(f TB)  with respect to the nodal displacement, a, as: =  (f TB)  =  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 of the 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)  Grain direction  ,  ‘iO  = =  ‘  4  2  1)  2(-1, -1)  3(1, -1)  Plate principle axis  Wood truss  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  x  =  3  (,i)y 1 1 and y = N N(,i)x 1=0  (3-27)  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  -  0 (m  F(Ax)  F(Ay)  =  0 (m  +mi)[1_e_u1m0  +m 1  I  (3-29)  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 =  }lxdY = 1(ktan,xB’Bx + ktan,yBBy )Jxdy (3-31)  AÔ&  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)ddi 7  (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  -  uII  1  A  -:; 1X2z\u 1 m+1  2m  Ui  U  Urn  Figure 3-7. Solution scheme of the Newton-Raphson and arc-length methods  Consider the equilibrium equation at step Ktan Au =AP+P’ —F’ where K  m,  which can be expressed as: (3-33)  A2Pref +R’  is the tangent stiffhess matrix evaluated at displacement Urn, Au is the current  th displacement increment, P and F’ are the external and internal force vectors evaluated at  sub-step, respectively, and R’ = P’  —  F’ is the out-of-balance load vector.  1 and Au”)as: Au can be expressed in two parts (Au Au=A2•Au’  (3-34)  11 +Au  Substitution in equation (3-33) yields: (335)  •Au 2Pref +R’ Ktan A2Au’ +K 11  1 and The two parts of the displacement increment, Au 1 Au  Ktan’Prej  Au”  Ktan’R’  , 11 Au  can be solved as: (3-36)  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: , fl2j). (Au, /JA%) = 0 1 (u  (3-37)  -  49  -  where ,B is a scaling factor. Substituting the expressions in Equation (34) into Equation (3-37) yields: 1 (u,,,8 ) (A2Au ’ 2 2 2 +Au”,f + 1 A 2)=A2 /3 2= Au’u+ Au”u O  (3-38)  The coefficient, A1, of the load increment can be solved as: u 11 —Au  A2=  Au’u,  (339)  +,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  s  II  or  Au!  (3-40)  uI  where the operator,  ,  calculates the Euclidean norm of a vector; P is the external load;  the displacement solution achieved at the previous step; and,  8  ,is  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  U  U  E  Compression  <‘Eo S  Tension ---ft  (a)  (b)  (c)  (d)  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  -  wheref and 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: 0 E  f/Eo>e>0  u=  O66  p 6  (343)  p 8 6u<6<8p  where a and s are the stress and strain, respectively; J andf are the tensile and compressive strengths, respectively; E 0 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.  0•  u 6  8  Ed  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  =  6  (3-44)  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 of wood 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  -  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 of unit 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  ±1V (U)kdv m  Pf =1e  (3-46)  -  55  -  where V 0 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  1  4 1 0 1 k,  1m“ Jv 1—e v  =1—e  JV2  m  As the stresses are uniformly distributed, crl and  (347) 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:  =  2  3.6.2.  J 1 1V  (3-48)  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  Maximum tensile stress  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:  1—e  _Sv(L)k dV m  =1—e  Jkdv Iv(”Y 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: 1  V  (3-50)  U  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  -  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 of a series of dialogs, 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.  ,,__..  ‘Content of cell i—i  ‘Content of cell z  •I  cEçJ  I  ext-Pointer  •  Content of’ cell z+1  Browsing operation  of’ •:‘Content cell i—I :.  of’ •:‘Content cell i ,  ‘Content of’ cell i+1  •,  Deleting operation  of .:‘Content cell i—i  ,  ‘Content of ‘ cell i ,  .,  ‘Content of 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  -  Probabilistic Stress strain  Problem definition  Elemiit configuration J Attributes • Material parameter • Calculate transfer matrix Calculate strain and stress Calculate tangent stiffness matrix Methods  • Single step/ Ultimate analysis Load/displacement control Arc-length analysis Reliability analysis  -  and internal force vector  FEM solver  Nodal solution Element solution  Calculate consistent load, etc.  results Nodal displacement Element force and moments ‘Step by step deformation plotting Animation of deformation history  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 -  65  -  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  mmIon  Element sisbIe T Roof element  r Top chord r Goltom chord  1J  I  Zoom  Web  r Bracing r Spring element I  Boorrdary  [s’] [  J LJ l Pan  rinad Rendenag oplrons  t1 LJ 1J  2lJhne 3Olrne  I  dy  Flat shade  Agrimahon  Smooth shade Resutis from step 4th calculationt P0 FX FY Plate element 4 node 185 node 188  —140.452 —95.994  791.954 —1098.37  Teoture render —366.147 565.045  OK  Cancel  .3  jJ  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  -  _  Ciia,olbarnereto Roof ccoli9u&on  fRool  Later hac.g oopety Roof pand popeIty  Grorniofion  1  Told nu,nbea  Totaf noo,be  CUcoof fl  Cooonk nxther  Baneate.  Trusolayout.  Jood paopetV  ecmetjic  .  -:  H.  Loodjon  hean  —  [  [I’e  Vdoo  i°  octon of loan piano Ctoescton ri  [  Tionocort.nalion hnçft  [  .  .  I  Joec4pi  I  Op0000  _J  ..iJ  s1 OK  Cancol  J  .zJ  Appj  _LiJ 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  -  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  I Fl.  iuo V4  aooaIyoo ooI Po.ç,00..,  Dn,  EI.,n.nt yp.:  8EM,1  EI.m.nI nuni NO3oI rl.no  Nod, I  12S  Nod•j. Nod.I.  Nodek. Nodal arc, drop plc 000c y  flodO. dl  .  .0<  r  Food,  Foo  aOP4OONOg  Figure 4-5. Graphic rendering modes and element information inquiry dialog of SATA  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  -  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.  dotinition of solver options Ionc, oothod opO j SpOio Loooono.o  SoI’ofr Sbonbdoná  roblem define  000rgopoom j Loodsoo*odomjos, Oo nvth odepOn  So  optno Lo.dsnpooio Oo.o,neIhodoponno  Solvrg pomoo  Gaussian Inteoration nodes Nnodo.sOoooss,Oos,on  Obolomnoono no,*d noo4oio Sçpomoseoe,o  Arc-length method parameters S&oononoo  Ino  oodbooK  Isi o4 Uhn.te.n..,j.  Soivernethods  S 0  r r  Oe*odoo  .,omnoo Tdesom, O  Dntn*on4oio  oeoo  ConvoooenveDlomo noOdOnI  r AeOooOo the ná,thoo, no.  Oth.sopkn,  Cdeooedfeot  inn  Ns,ntho.donode [s  r i&  oorcobood*og,th.ne  I  1  Mnoâ,o.ove,e  D.O.E  Pboononi  For  Ktvop  -  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.  -  70  -  ___________________ ________  ___________________ _______________  S ttlCtt 41 yttt ii 41141y!zlt OPoPeY  I  Wood type p000ody  -  MoOeodiy Rodpoeoe.e.odpoioob.d  Rod p.eoooe ond3Rd  . .  Olo boi.ioo type [t odott  —  .-.———————  Sod.oelood  URwntot ooot1oiø i  zi  Aootpoeype  1.  oi.doii,.otype  PCnootrdlaiAIjRtc AoONtyPiN epPooto ThSIOt.P[T  j,<  ooi,oot  Oddodioo dpl.t. mdeod RoPeotY•]  Dijiontyp.  .  [.  vthe  t  Peoneeteo  ..  pt  PWetypertypeO.o  .  j  cao.  too.yoáodlo.d  .  P&Nio.doooo  loodO  NO  f  SRPOPodY  0  NO  FJ  DbtW.i  .  V.  lJ  &.  I  °  I  Cet*rotpd by moot om of fiti  4 Coo  F J  pooo.oeN  tcpctoe tNUtttnco0oRIIy  Mao ooutobet  r--i  vod.  °‘  o.%oiot  Solution progress  Dioondmodolpoop.oty Sroeogtype typoO  r.  ttuled b  pyy  Co.to.t  [oj  %  Appty  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.  -  71  -  .I7Ifl0d41  Oupulanatyi8resuhs Ol4UIead8F20ces Ebrmlcces Node av1e  Node tooolped  El20ceo  Ndsnee  0047 I.pIo4om 7778  :  /7  -  ETdIOce4d  Erevth  4  -  node 3 -1033  OOFloooe  ‘1  .......J  DOFtoipoi 1F  -  V.-.  -0433  Coordinate system. -470 -374 -202 -101  Cancel  Ap  [  OK  j  Canc  44  0.0  4.4  00.8  202 374  Apçj  474  OK  Figure 4-8. Options of SATA’s post-processor  4.4. Availability  The source codes of SATA were compiled using a C++ compiler. An executable file is T04 window 98 to XP available with some conditions applied. The program works with Microsoft 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.  -  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 1650 f-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 mm , nominally 2 by 4 in. 2 ; 2 38x 139 mm , nominally 2 by 6 in. 2 ) and one length (4267.2 mm or 14 feet). The lumber was 2 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 design for the parallel-to-wood-grain compression and tension tests Test  Lumber grade  Compression  Tension  MSR165Of-1.5E MSR2400f-2.OE MSR165Of-1.5E MSR2400f-2.OE  Specimen size* (mm ) 3  Replication  38x89x304.8  30  38 x 89 x 457.2  30  38x139x457.2  30  38x89x2133  30  38x89x3048  30  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.  direction  Lateral  3  Degree of freedom  Grain direction 2  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.  -  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  x axis  y axis  bracing  0  20  NO  30  2134  20  20  NO  30  2134  20  50  NO  30  38x89  3048  20  20  NO  30  38x 139  2134  20  20  NO  30  Group  Lumber grade  1  MSR1650f-1.5E  38x89  2134  2  MSR165Of-l.5E  38x89  3  MSR165Of-1.5E  38x89  4  MSR165Of-1.5E  2 (mm)  Length (mm)  Replication  5  MSR165Of-l.5E  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  -  Figure 5-5. Connection between the steel boot and the hydraulic actuator  For the laterally braced specimens, a lateral bracing member was placed vertically and connected to the edge side of the 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  -  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, E , and of the slope of the falling branch, Ed, 0 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 Parameters  MSR165Of-1.5E shape’  scale  mean  MSR2400f-2.OE St.d.  shape  scale  mean  St.d.  38x 89x 304.8 mm 3 specimens j (MPa) e, (10j  7.18  24.47  23.15  4.27  8.82  32.44  30.83  4.34  6.87  3.91  3.7  0.73  5.66  4.0  3.74  0.89  3 specimens 38 x 89 x 457.2 mm 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 mm 3 specimens  J (MPa) e, (10)  5.41 5.73  25.04  23.18  4.48  10.6  29.66  28.36  3.26  3.50  3.27  0.73  4.967  3.579  3.33  0.87  All three specimen sizes (MPa) 0 E  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 Grade  2-P Weibull  3-P Weibull  2 shape  scale  shape  scale  location  38x89x2133  2.52  32.76  2.47  32.37  38x89x3048  4.2  32.1  2.16  38x139x3048  5.28  29.62  1.43  38x89x2133  6.01  50.98  38x89x3048  5.81  38x 139x 3048  4.14  )’ 3 Specimen (mni  MSRI650f-1.5E  MSR2400f-2.OE  Mean  St.d.  0.37  30.0  14.6  19.41  12.22  29.41  7.95  11.14  17.66  27.66  6.73  3.08  30.16  20.51  47.54  9.33  46.0  3.49  30.73  15.06  42.82  8.6  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, m , and slope of the 0 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 Type  Maximum capacity m 0  Slope of asymptote m 1  Initial slope k  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 0  5  10  15  20  25  9  12  15  Displacement (mm) 4000  g  3000  w  3 2000 1000 0 0  3  6  Displacement (mm)  50000 E  40000 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  -  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.  Figure 5-10. Failure modes of 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  -  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.  P  M  column Nail connection  Lateral bracing  x M  Figure 5-12. FEM-based model of an eccentrically loaded wood beam-column  -  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 mm 2 in cross-sectional size.  Table 5-6. Length effectfactor, Ic, ofwood compression and tensile strengths Strength Compression Tension  Grade  k factor ki  k2  k3  mean  adjusted  MSR165Of-1.5E  7.18  6.83  2.05  5.35  12.0  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  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 0 0  6000 4000  x  <  2000 0 0  20  40  60  80  100  120  Midspan lateral deflection mx axis (mm) 14000 12000 10000 0 0  E 6000 0 0  4000 x < 2000 0 0  2  4 6 8 10 Midspan lateral deflection in y axis (mm)  12  Figure 5-13. Test results and model predictions of Specimen Group 1  -  93  -  _____________  0000 gI 8000 .2  Average max load 6000 4000  C)  2000 —model 0  0  30  I  I  60  90  120  Midspan lateral deflection in x axis (mm)  g  10000 Upperbound  8000  Average maxload  .2 6000  7owerbou 4000 C)  2000 —model I  0  0  2  I  4  8  6  10  12  14  Midspan lateral deflection in y axis (mm)  Figure 5-14. Test results and model predictions of Specimen Group 2  8000 —  Upperbound  7000 Aver9e max  6000 •  5000 4000 3000  3  Lowerbound  2000  mean  1000  —model  ——  0  0  20  I  I  I  I  40  60  80  100  120  Midspan lateral deflection in x axis (mm)  -  94  -  8000 ‘  Upper bound  7000 6000  • 5000 Co  4000 -  8  3000 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 U) Co  3000 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 Co  Average max load  Co  2000 0 0  •  1000 0  mean —model .  0  2  I  I  4  6  I  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 bound  12000 (0  .2 C  .2 (0  A  10000 8000 6000  E o  C)  4000  .2  2000  —4---  mean  —model  0 0  30  60  90  120  Midspan lateral deflection in x axis (mm) 14000 Upperbound  12000 (0  .2 C  .2  Average max load  10000 8000  (0 0.  6000  C.)  E o  4000  .2  2000  an 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 .2 U) U)  E  5000 4flflfl  Averema ba owerboun  3000  ° C)  2000  x  1000  —model  0 0  50  100  150  200  Midspan lateral deflection in x axis (mm)  -  96  -  8000 -  7000  6000 • 5000 0  -  4000 3000 2000 10000 0  2  4  6  8  10  Midspan lateral deflection in y axis (mm)  Figure 5-18. Test results and model predictions of Specimen Group 6  10000  z 8000 0  .2 0 (I)  6000 4000 2000  x  0 0  100  50  150  200  Midspan lateral deflection in x axis (mm) 10000  z 8000 0 C  .2  0 0  6000 4000  x  2000 0 0  2  4  6  Midspan lateral deflection in y axis (mm)  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  -  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  Error  (%)  Test  Model  Test  Model  1  9.0  8.862  1.533  0.24  0.234  2  6.4  6.500  1.562  0.163  3  6.2  4  3.2  5.977  3.597  0.39  3.660  14.375  5  9.5  9.707  2.179  0.203  6  4.8  4.930  2.708  7  6.3  6.623  5.127  (%)  (%)  Test  Model  2.500  0.354  0.390  10.169  0.162  0.613  0.687  0.565  17.758  0.362  7.179  0.532  0.527  0.940  --  --  0.211  3.941  1.01  0.825  18.317  0.122  0.130  6.557  0.793  0.688  13.241  0.133  0.147  10.526  1.081  1.011  6.475  --  0.093  Error  0.452  Error  --  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 of the 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 U, U,  12000 8000 4000  x  0 0  2  4  6  10  8  12  Midspan lateral deflection in x axis (mm) 20000  z 16000 0  .2 U,  12000  U,  8000 4000 0 0  20  40  60  80  Midspan lateral deflection in y axis (mm)  Figure 5-20. Test results and model predictions of Specimen Group 8  -  100  -  ________  20000  Upperbound  16000 Average max load  C  .2 12000  0  8000 4000  —*-—  mean model  0  o  2  4  6  8  10  Midspan lateral deflection in x axis (mm) 20000  U perbound  16000 .2 C  .2 Cd) Co  12000 8000  0 C) -  4000 model  0 20  0  40  60  80  Midspan lateral deflection in y axis (mm)  Figure 5-21. Test results and model predictions of Specimen Group 9  35000 Upperbound  30000 .2  25000  C  20000 Cl)  15000 E  3  10000  •  5000  —UE-—  mean  —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  100  Midspan lateral deflection in y axis (mm)  Figure 5-22. Test results and model predictions of Specimen Group 10  35000 30000 25000 .2  20000  15000  3  10000 5000 0  5  10  15  20  Midspan lateral deflection in x axis (mm)  35000 30000 2 25000 20000 15000  3  10000 5000 0  5  10  15  20  25  Midspan lateral deflection in y axis (mm)  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 -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  ::: 8000  Averagemaxload  —model  /x’Lowerbound  /ff”’  C-)  400:  2%rule __—Wnterarid  0  200  400  600  800  1000  Lateral bracing force of specimen group 8(N) 20000  Upper bound  16000  -  ——  12000 ,/  owerbound  Averagemaxload mean  8000 —  400:  —Winter  —  0  500  1000  1500  2000  Lateral bracing force of specimen group 9(N)  -  103  -  35000  z  30000  V Co 0  25000  0 0 0  20000  a!  15000  E  0 0  10000  Co  5000 0  0  500  1000  1500  2000  Lateral bracing force of specimen group 10(N)  g 35000 30000  0  25000 20000 0.  E  15000  0 0  10000  5000 0  500  0  1000  1500  2500  2000  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 Group  Critical buckling load (kN) Test  Model  Corresponding lateral bracing force (N)  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 -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 of the 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 89mm 2 (nominally 2 by 4 in. ) and 38x 139 mm 2 2 (nominally 2 by 6 in. ) in 2 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 mm 2 (1 in. ) with each tooth 2 9.5 mm (3/8 in.) in length. •  Twenty-six sheets of four-ply plywood sheathing of 1219 x 2438 mm 2 (4 by 8 ft ) 2 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/m 2 (3.0 lbf/&) on the top chords and 2 (74.6 lbf/&) and a dead load of 0.144 kN/m 0.335 kN/m 2 (7.0 lbf/ft ) on the bottom chords, which were in compliance with Part 9 of 2 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  aa  0  aa  Ii en  cd  v,  0  en ‘-I  0) C,  0  0  ‘a  th CL  C  C  0 C.’ I  0,  0, C.’  en  en  CC  Co  en  a CD  Co  eo  en  CD  en  x en en  U, en  F C.’ en  >< ‘a  0 CC I—  0  9 0  F 0 C 0  CC  -1  0  C.’  en  ‘C,  op  z  F C.’  ‘a en  \\  C F  en en  a C I en  0  0  CC.  Co  en  en en  en  a  to  C-)  ‘LI  0  CL F-  en  0  C.’  F  \\ 0 C-I F  0 F—  to  L C  C  CL CL  CL CC F to  I—i  9-9-0  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—  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 mm 2 (nominally 2 by 4 in. ) dimension 2 lumber and one hundred and forty-two pieces of38x 139 mm 2 (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  supports (mm)  (mm/mm)  400  300  7.2  30  700  600  1.8  30  Width (mm)  Length (mm)  Al  50  A2  50  configuration  Replication  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 of two sheets of plywood, one piece of dimension lumber of 38 x 139 mm 2 (nominally 2 by 6 in. ) in cross-sectional size and eight nails. The specimens 2 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  -  Figure 6-13. Test setup for truss assembly testing  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  -  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  Loading clamps  Top chord  Wire rope  Hydraulic Actuator  Wire rope Wire rope fixed to  fixed  the actuator  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. -  126  -  Figure 6-18. Load cell underneath the wire rope of the loading system  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  -  (top) and truss assemblies (bottom).  D F  --Deflection measurement  0  --Force measurement  --Deflection measurement --Force measurement  Figure 6-19. Measurements of the force and deflection responses  -  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 Side 1 —Side 2 Mean  4000 2000 0 400  600  800  1000 1200 1400 1600 Longitudinal coordinate (mm)  1800  2000  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 mm 2 and 38 x 139 mm 2 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 0  1•  —p  3PWeibulldistribution: Location :8833.65 MPa Scale :1665.54 MPa Shape:1.56  0.8 C  2  0.6  • 0.4 0 Testresults —3PWeibull  0.2 E  D  o  0 7000  8500  10000  11500  13000  14500  16000  17500  MOE of 38x89 mm 2 lumber (MPa)  -  133  -  ci 0 0  2D  0.8 0.6 0.4 0.2  .  o  7000  8500  10000  11500  13000  14500  16000  17500  MOE of 38x139 mm 2 umber (MPa)  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.  -  134  -  C 0 C.) C C 0  1 0.8  3P Weibull distribution: Location :241.2 MPa Scale :8553.5 MPa Shape: 9.18  0.6  .0 C,)  •0 ci) > C  2  D  C-)  0.4 o  0.2  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  0  C  4-  C 0  3PWeibull distribution: Location: 799.4 MPa Scale: 851.0 MPa Shape :2.41  0.8 0.6  .0 .  04 o  I  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 0  2 1 3 4 5 Nail displacement of specimen configuration BI (mm)  6  1200 1000 •0 (U  800  2  600  ac.  400 200  Model  0 0  1 2 4 3 6 7 5 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.  -  136  -  ________  Table 6-2. Material property tests results ofdimension lumber, plywood sheathing and nail connections Material  Configuration  Dimension  2 38x89mm  lumber  38x 139 mm 2  Plywood  Al  sheathing  A2  Property  Unit  MOE  MPa  MOE  MPa  0 m  N  1 m  N/mm  k  N/mm  0 m  N  1 m  N/mm  Nail  k  N/mm  connection  0 m  N  1 m  N/mm  k  N/mm  B1  B2  83  B4  0 m  N  1 m  N/mm  k  N/mm  Weibull distribution parameters *  Mean  St.d.  location  scale  shape  8833.65  1665.54  1.56  10327.7  977.3  8811.21  1982.2  1.5  10595.6  1200.2  241.2  8553.5  9.18  8382.8  1165.3  799.4  851.0  2.41  1558.1  324.2  --  --  --  905.2  141.1  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  --  0.0  0.0  2720.3  2128.9  885.2  169.4  0.0  0.0  2510.6  1509.3  1014.3  105.1  0.0  0.0  1645.0  577.5  1010.3  46.6  0.0  0.0  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, m 1 was assumed to be zero for simplicity.  -  139  -  Table 6-3. Calibrated MPC connection properties based on an individual tooth Parameter k(N/mm)  0 (N) m  Direction*  Mean  AA  3514.14  AE  1312.36  EA  3562.45  EE  1477.22  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% of the loading point deflections applied in the tests. Good agreement was achieved.  -  140  -  5000 —  4000  e  — —  3000 2000 .  1000  —0—FEM model  0 0  1  2  3  4  Bottom chord deflection at node P (mm) 5000 ‘  —  4000  0  3000 .  C.)  2000 1000  FEM model  0 0  1  2  3  4  5  6  Bottom chord deflection at node 0 (mm) 5000 ‘  —  4000  0  3000 0  2000 1000  model  0 0  1  2  3  4  5  6  Bottom chord deflection at node M (mm) 5000 4000 3000 C .  C)  2000  ‘1)  °  1000  —€-• FEM model  0 0  1  2  3  4  5  6  Bottom chord deflection at node K (mm)  -  141  -  5000 4000 3000 .  C.)  2000 1000 0 0  1  2  3  4  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.  -  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  Steel plate  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  _o_ø  where M, 0 and  ( 62  are the bending moment, angle of rotation and curvature, respectively.  Considering a web of 38 x 89 mm 2 in cross-section size, a metal truss plate of 0.9 mm in thickness and MOE of2.Ox io N/mm , k can be calculated for a gap width of 2.0 mm as 2 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  -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 .  C.)  8000 4000 0 0  20  40  60  80  100  120  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.  -  145  -  20000  16000 12000 .  ()  8000 4000  0 0  5  10  15  20  25  Bottom chord midspan deflection (mm)  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.  -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 0  a)  20000 15000 10000 5000 0 0  25  50  75  100  125  Midspan lateral deflection of W2 web (mm)  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  z  50000 40000  0  30000 0  a)  20000 10000 0 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-  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  -  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  E E 0  6 5  C.)  4  ci  3  ci)  0  2  C)  E 0  0  0  1  2  3  4  5  Truss position  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 of the roller support reaction forces measured from the other four -  152  -  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  -  Tension failure at the quarter node  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  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 of the 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. 40000  z 30000  I, I-  0 0 .  /  20000 First truss assembly  0.  10000  Second truss assem bly —  Model  0 0  20  40  60  80  100  120  140  Midspan lateral deflection of the W2 webs (mm)  Figure 6-41. Applied load and midspan lateral deflection of the W2 webs of the second truss assembly  -  156  -  40000 —  z 30000  - - - -  .2 a.  20000  - - .-‘  ———-  10000 — 0  -200  100  400  Firsttruss assembly Second truss assembly 2% rule predictions Model  I  I  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 0  5  10  15  20  25  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  -  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 of MPC 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.  Figure 6-44. Load distribution via plywood sheathing  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 6-3 4, 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.  -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 of the influence of a 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: x +/3, 1 x 1 Y=1 +/3 i=1  (71)  +6  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 mm . The configuration of a half truss 2 is shown in Figure 7-1. The notations of the dimensions were organized in the same way as in Figure 6-1. 6x10 MT20 E  7x8 MT2O D 4x8MT20 C 5x6 MT2O B  B A  Q 7x6 MT2O  35x6 MT2O  24 MT2O  I  T  4.  N  0  2210 mm  5x12 MTI8HS  M 81 MT2O  I 1943 mm  1943 mm  6096 mm  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  -  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  -  Rdy  :j  /1  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  -  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  S 1 cS -+-  SI  c”  El E S  c’I  00  k3.5 x  610 mm  N.mm/rad  8540 mm  Figure 7-4. Fixity factor model for one CLB bracing member per W2 web  -  168 -  W2 webs of the three-bay truss assembly —1-- c  T Ig icc  L  7  I Ice  F  —‘-———_.f6  j  /  Compression web  /  610 mm  —  —4-  DLI  7 Diago nal bracing k=3.5 x i0 7 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.  -169-  Table 7-1. Simulation results ofthe fixity factors of CLB bracing members Item  Sub-item  MOE of dimension lumber Nail connection stiffness Fixity factors  --  Type 11 Type2i  Unit  Three-parameter Weibull distribution location  scale.  2 shape  Mean  St.d  GPa  8.834  1.666  1.56  10.328  0.977  N/mm  -.  --  --  824.7  122.0  --  --  --  1371.0  297.1  N/mm  One CLB  N/mm  354.8  70.0  3.22  417.6  21.3  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.  The cumulative distributions and the Weibull distribution fitting results of the fixity factors are shown in Figure 7-6 for comparison.  C  0  0.8 C  o 0. 0  0.4 0.2 ( D  350  400  450  500  550  Fixity factor of one CLB per web (N/mm) C  40.8 .  D 0  0.6 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 of the 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  0 +ax +b y=a x? 1 1=1  5 +C 4 X 0 C 6 +C 4 X 1 6 +6 5 X 2  i ii  +  c +c 7 3 x 0 5 +c 4 x 1 6 +c 4 x 2 6 +6 5 x 3 4 +c 1 x 0 c 5 +c 4 x 1 6 +c 4 x 2 8 +x 4 x 3 6 5 x 4  (7-2) 8 +6 6 X 5 +C  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, b 1 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 Variable  Description  Unit  3-parameter Weibull distribution location  scale  shape’  Mean  St.d.  Basic random variables 1 x  MOE  GPa  8.834  1.666  1.56  10.328  0.977  2 x  MOE  GPa  8.834  1.666  1.56  10.328  0.977  3 x  MOE  GPa  8.834  1.666  1.56  10.328  0.977  4 x  Midspan deflection 2  mm  --  --  --  -0.93  2.42  5 x  Midspan deflection  mm  --  --  --  -0.93  2.42  6 x  Midspan deflection  mm  --  --  --  -0.93  2.42  417.6  21.3  Additional random variables for one CLB per W2 web 7 x  Fixity factor  N/mm  354.8  70.0  3.22  Additional random variables for two CLBs per W2 web  Note:  7 x  Fixity factor  N/mm  391.5  90.0  3.81  472.9  23.6  8 x  Fixity factor  N/mm  391.5  90.0  3.81  472.9  23.6  ‘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, x . 1 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.  Th  2 x oh 3  2 x  33 h  p3  p3  1 x  1 x  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 of the random valuables. The procedures of a reliability analysis using the response surface method are explained with the flowchart in Figure 7-8.  -174-  [start  1  Conduct a complete factorial sampling  4, I  Randomly select 100 samples 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/mm . 2 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 of the 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  = 0.77 2 R  110.5  7  1:.:  Ezzz  2 qrcf=-0.001 NImm 9.0 9.0  9.5  10.0  11.0  10.5  Sampled critical buckling load y  Figure 7-9. Sampled and predicted critical buckling loads of the truss assembly (zero CLB)  18 = 0.84,, 2 R  B i:  16.5  --‘  a)  0  0 2 qref=-O.OO1 N/mm  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  29  C 0 D .0  28  CU 0 0 V a) C) V ci)  27  0  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 Index  Zero CLB  a  One CLBs  c  1 a  b  Two CLBs  c  1 a  0  8.143  -0.006  11.471  0.000  34.290  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  0.000  --  3  -0.241  0.015  4  -0.004  -0.002  5  0.035  0.007  6  0.019  0.003  --  --  --  --  --  --  7 8  --  --  --  --  --  0.621  -0.029  -0.004  0.003  -0.023  0.003  -0.020  -0.007  0.00 1  0.000  --  --  --  --  --  --  --  --  0.004  1.418  -0.064  0.000  0.091  -0.002  -0.007  0.128  0.002  -0.001  0.261  0.002  -0.009  0.000  0.013  0.000  --  --  --  The coefficients in the table are the results of minimizing the fitting error of the -  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,, andy 1 are the predicted and sampled critical buckling load coefficients, respectively, , 3 corresponding to the  ith  , of the input random variables, i = 1 to n, and n is the 1 sample values, x  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  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 14 2% rule  I U-  1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0  6 4 2 0  —  o  0.2  0.4  0.6  0.8  1  1.2  Frequency Cumulative % 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  1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0  2% rule  C.) C  10 U-  6 4 2 0  Frequency Cumulative %  I  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 bracing members  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-  fixity factors of the 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)  where  qref is  —  (74)  Jext  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, q , were uniformly 1  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/mm , which is the design value of the live load of the 2 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, 1 u, 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, Level  Mean (10 N/mm ) 2  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  qext  With the explicitly expressed performance function, the probability of buckling 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 and probabilities 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  a2.  x 0  .9  10 9 8 7 6 5 4 3 2  cov=1.o COVO.5  —e-- COV=O.2  0 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 of the 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. . 6 55 2 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  th, 5  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 of wood 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 of plane 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 of Agricultural 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 of the 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: 1 =--(1-),N N 2 =--(1+)  (1)  1 1 21 3 — 2 M = l[-—(+1)——(+l) L +—(+1)] 3 M  =  3 L  =  O.75(+1)2 —O.25(+1)  4 =L M 4  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’ +p 858’)dV—JGp8’p58’dV 2  Ignore the terms of higher order and note that fzwdV  1Ese1’&t’  =  fyv”dV  =  0  w”” +y 2 +z v”” +u’(w’’ +v’’ +p 2 8’SO’) 2 (4)  + I[(w’)2 + 2 (v’) + (p8) 2  5u’}dV  —  8’58’dV 2 JGp  Replacing the element displacement,  u, v, w  and 8 in Equation (4) by the kinematic  matrix, B, and nodal displacement vector, a, yields: =  + z2BTB + y2B0TB)dv  _aT  TB + 3 (B TB + p2BTB)aB + BaaT(B B 3 + p2B T aT 2 B + BTB T  (5)  )]dVöa  —  —  TBldVoa 4 aT fGp2B  =  Therefore, the internal force vector, P, can be calculated by: *  =  a T K + Pt  (6)  where the element stiffness matrix, K, and nonlinear internal force vector, P, can be calculated by: KSEsec(BBfz2B2TBy2TB+BiTB4dI7  E3e01  Pt  =  v  +—h 1 , 2 B c B ÷—b 2  T3 2 aT TB +B  3 B B )+Gp B 3 4 B ]Det(J)dd77dK 4  TB 4 3 +B T +p2B ) ]Det(J)dd1ldJc B  -  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  +p 0 2 ’09)dV _jGp2eoOdV  =  2 +aTBTB —yB —zB 3 +p2aTBTB)dVOa_jGp2aTTdVOa  Consequently, the internal element force P 1 can be expressed as: +aTBTB —zB TB 3 2 ÷aTB  irnt  3 +p2aTBTB)dV+fGp2aTBTBdV —yB  =  =  2TT 2 2 where c=Bi+aTB —zB B + a B 3 + —yB 3 B p a B 4 B . 4  Partially differentiating  with respect to the nodal displacement vector, a, yields:  T IGP2aTB d V] B $dv+ 4 8Pint  Ktan  =  V  V  V  TB 4 ÷B T +p2B J dV+ B 4 T fGP2B d B V 2 3 —---Cdv+ fJ[BTB a8aa V  V  Notethat ão/ô6=Etan andã8/8a=C, =  2 + BTB + 4 T p2B J dV B+ 4 TV fGp2B d B JEtCTCdV + JO.(BTB V  V  (7)  TB 4 T Gp2B ] Det(J)dd?ldK B + JBTB 2 +3 TB + p2B B 4 +4  =  V.[  where Etan is the tangent modulus of elasticity 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): 1 (,i) N  (,i) 1 L  =  1(1 + )(1  2 (, ij) N  =  (,i) 3 N  =  4 (, ) N  =  L (, i) =1(1 )(1 + )  1 M  =  0.25 —0.375+ 0.125 —O.375i + 0.125  2 (, i) L  -  =  =  +  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)! (,i7)=(0.125—0.125—0.125 +0.125 —0.1875+0.0625i +0.187517+0.18752,7_0.187537 3 M 2 —  0.0625  —  0.06252173 + 0.06254)  ()=0.0625(1—— 4 M 2  6 (,i) M  —  _3 _23 _32  __  =  0.25+ 0.375—0.125 —0.375i +0.125 —0.5625 + 0.1875 + 0.18757  =  (0.125 + 0.1875  —  0.0625  0.125i _0.125172 +0.125’ —0.1875 _0.1875172 +0.0625  —  +0.187577 +0.06253172 7 (,i) M —  =  (—0.125 —0.125+ 0.1252 + 0.125 +0.1875 —0.062577 + 0.1875 _0.187527 —0.1875 +  0.0625  0.06252773 +  )  _3  ()=0.0625(—1—+ 8 M 2  =  0.25 + 0.375— 0.125  =  —  +  0.375i —0.125  _3_22 _3 23 _32  +  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  i7 _0.06253173)! 3 0.0625 +0.1875 2  -  200  -  =  +  (—0.125— 0.125 + 0.1252 +0.125 —0.1875i + 0.0625 —0.1875 + 0.187527 + 0.1875  0.0625  0.06252773  —  —  (, 1 M 2 )=0.0621+— 77 2  0.062577 ) _  +_  -  _3  _3772772 _3 23 3772 3773)  13 = 0.25— 0.375 +0.125 +0.37577 —0.125 —0.5625ç + 0.18757 + 0.1875 M ) 77 (, 14 M (,77)  —  0.062577  (—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 +0.06254 3 i 2 7  (,77)=0.062—1++ 1 M 2 6  —  _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 =  _f(De)T 5dV  =  at a virtual strain vector &  _1TDdV  Substituting the expressions of the stresses in Equation (3-18) yields c5Wjt  vE yx  E  vE  6L58,dV s68dV— j vxyvyx Cö6dV— v _ 1 v _j vxyvyx 1 _J vxyyyx 1 v  (8)  8yôSydV_fGxyrxyôrxydV  =_1(klsxo6x  where k 1 3 k  =  =  E 1(1  +k 6 88 +Gxyyxyc5yxy)JV cyö6 +k 2 +k 6xô8y 3 2  E 1(1  —  —  2 k  =  1(1  —  =  1(1  —  and  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+ k 1 [(ut Xw )8(w )+ fk2  -  [(  (w  )2  o(u )]dv  [  )+ (  Jk3 [(v), )(u  —  --  )]+ k ( ) )+ ( o( )]dv )+ (u )o(v )÷ (zw )(zw, )+ (zwy, )5(zw )J1v  V -  (9)  fk3 [(vp Xw )o(w )+ (u Xwy )(wy )+  _fG[(u + )o(  (w  )+ ! (wy  )2  Replacing the element displacements, and the kinematic matrix, _aT  aT  —  —  —  B,  =  u, v and w,  by the nodal displacement vector, a,  yields:  2+3 yyTB z2(B x x+3 [B,7Bi + BlTB 3 Jic y ,B y)jiVöa JB  aT fo[BlTBl + 2 2 +B XTBIYIjVöa 2 xTB + BlTB B x  XaaTB 1 Ikl[B X X 3 TB +  x aTB x 3 aBlx]d TB V  —  [ Jk y 2 y 3 y aaTB TB B +2 y -aTB y 3 y ]dVöa TB aB  [ fk y 2 x 3 x aaTB TB B + Biaa y -aTB y aBlx TB + 2 x -aTB x 3 y ]dV3a TB aB 3 +3 B T T B ) + B1 Xa(B aTB Y 3 Y TB + B xaT(B 2 B x y TB + 3 1 +3 fG[(B )JiVea = 2  Therefore, the internal force vector, Pint  +v)Iiv  [B k T B+ jki [BIxTBlx + z2BTBJ+ 2  —  —  V  +v)÷4z2(w)5(w)}jv  _$G[(u + vxX(wx)3(wy)+(wy)5(wx))+(wxXwy)5(uy  =  o(u  aTK  —  can be calculated by: (10)  +  where the stiffness matrix, K, and the nonlinear internal force vector,  can be expressed  as: K  =  [B k T B +3 T z2B j B2 + T Z2B J ,Y1jV B XTB + 3 [B X 11 fk +  3 TB fk  yyTB z2(B x x +3 xy)11V B y xTB xTB + 3 1 +B y 2  YTBIY 2 1 +fG[B 2 +B TB ÷BlTB +B TBl,jIV 2  -  202  -  xaaTB 1 fkl[B X X 3 TB +  =  --  y]dV aTB y 3 y TB aB x aTB x 3 aBlx]dV TB + 3 [ 1k Y 2 Y Y aaTB B TB + 2  [ 1k Y 2 x X aaTB TB B+T Y .aTB Y aBIX TB + 2 x]dV .aTB x 3 y TB aB + 3 B Bixaa B ’y y +3  x)IaT(B 2 B X y TB + 3 T+ B ) B3 x aTB y a(Bly TB + B + fG[(B 1+3 )JiV 2  The tangent stiffness matrix of the plate element was also obtained by differentiating the internal force,  Ktan  with respect to the nodal displacement, a as:  =  ãPjfl  =  xTB BlXTaTB x 13 fk  =  (aK+Pint)K  t 1 a1  (11)  where 1flt 81  +  x +B x aBlx}IV TB B3 T 3 BiaB  yTaTB 2 [B y y TB 3 yTB 2 +B y y aB 2 YIJV +B Y 3 Y TB aB +1k 3 2 xTaTB 1 B y y TB + 3 xIJV 1 B y y TB aB [ +1k Y 2 X X 3 TaTB B TB + 3 aB + 3 2 B T x +B x 3 y ljV+ TB aB jG[(B x)TaT(B 2 +B x y TB 3 T +B ) }IV B + 1 1 3 [TB 1k y y 3 X aB B2  XTB 3 (B Y Y +B ) X Y +(B TB 3 Y(Bl +B X TB +B +B 23 )V 2  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 iso parametric 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  -  ( 0 N ) 7 =(1+)(1+i)/4 ,i N ( 1 ,i)=(1—)(1+i)/4 (,i) = (1—)(1—i)I4 2 N (,i) =(1+)(1—i)/4 3 N where and  (12)  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  38  38  38  38  Width (mm)  38  38  Depth (mm)  89  89  139  139  139  139  Length (mm)  2450  3050  3674  2443  3054  4903  Type E  Type F  Table B-2. Vibration test results (MOE) No.  Type A  Type B  Type C  Type D  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 12431  10  11145  10355  10845  10290  11880  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  9450  11325  18  10670  10860  11115  9925  19  9755  9230  10665  (endofTypeD) (endofTypeE)10690  20  9635  10310  11135  11087 10339  21  10570  10230  11645  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  8795  10685  10598  31  9730  32  10345  12530  11730  10688  33  9555  10455  10460  10203  34  9375  11570  10945  8814 9587  35  10085  9760  10485  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.  Type A  Type B  Type C  61  9340  10195  62  9585  11730  63  9035  10915  64  10110  10320  65  11205  11150  66  9665  10180  67  9145  11680  68  10380  10370  69  11380  11875  70  10170  10830 (end of Type C)  71  12850  72  9375  73  9005  74  12660  75  10305  76  10320  77  8795  78  10295  79  9225  80  9670  81  9975  82  10005  83  10385  84  9785  85  11055  86  9400  87  10965  88  10295  TypeD  TypeE  Type F  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.  -  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.  C67  C61  F18  D3  F5  Figure B-i. MOE profile of Truss No. 1  C2  C12  8  Fli  D12  F17  Figure B-2. MOE profile of Truss No. 2  C28  C20  Fl  D4  F20  Figure B-3. MOE profile of Truss No. 3  -208-  C59  C66  F25  D7  F26  Figure B-4. MOE profile of Truss No. 4  C49  C50  F4  D17  F12  Figure B-5. MOE profile of Truss No. 5  C57  C53  F33  D8  F27  Figure B-6. MOE profile of Truss No. 6  -  209  -  C65  D5  F34  F2  Figure B-7. MOE profile of Truss No. 7  C33  C29  D14  F16  F19  Figure B-8. MOE profile of Truss No. 8  C41  C46  F24  D6  F6  Figure B-9. MOE profile of Truss No. 9  -210-  Cl  C4  FlO  D18  Fl  Figure B-b. MOE profile of Truss No. 10  Cs’  C22  F14  D13  F9  Figure B-il. MOE profile of Truss No. ii  C58  C3  F13  D1O  F8  Figure B-12. MOE profile of Truss No. 12  -  211  -  C9  c55  F32  D2  F35  Figure B-13. MOE profile of Truss No. 13  C17  C64  F3  Dl  P36  Figure B-14. MOE profile of Truss No. 14  C25  19  C70  F15  Dli  F28  Figure B-15. MOE profile of Truss No. 15  -  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.  E  D  C B  R A  Q  P  0  N  M  Figure C-i. Metal truss plate placement measurements  -213-  Table C-i Metal truss plate placement measurements Truss No.  Node  Node M  Q  Node B  Node D  x (mm)  x (mm)  y (mm)  x (mm)  y (mm)  x (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  y (mm)  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  1  1.0  1.0  1.0  G5  G6  G7  G8  G9  0.5  1.0  0.5  2  1.0  0.2  1.0  0.5  0.5  0.5  3  1.0  0.5  1.0  1.0  1.0  1.0  4  0.5  0.5  0.5  2.0  0.5  0.5  GlO  Gil  Gi2  G13  1.0  1.0  0.5  1.0  5  1.0  0.2  1.0  1.0  1.0  0.5  2.0  0.5  1.5  6  0.5  1.0  1.5  1.0  1.5  0.2  1.0  0.5  0.5  2.0  2.0  0.5  1.0  7  0.5  1.0  2.0  0.5  1.0  1.0  1.0  1.0  1.5  0.5  0.5  1.5  2.0  8  2.0  0.5  2.0  1.0  1.0  0  2.0  1.0  1.0  3.0  1.0  0.5  0.5  9  1.0  0.5  0.5  2.0  1.0  0.5  1.0  0.5  0.5  3.0  1.0  0.5  0.2  10  3.0  0.2  1.0  0.2  1.0  0.5  1.0  0.5  1.0  1.0  1.0  0.2  1.0  Ii  1.0  0.2  1.0  1.0  0.5  0  12  1.0  0.5  1.0  1.0  1.0  0.5  1.0  0.5  0.2  0.5  1.0  13  1.0  1.5  1.0  3.0  0.5  0.5  1.5  0.5  0.5  5.0  0.2  0.2  0.5  14  1.0  1.0  1.0  2.0  1.5  0.5  3.0  0.5  0.5  1.0  1.0  1.0  0.5  15  2.0  2.0  2.0  2.0  0.5  0.5  0.2  0.5  0.5  1.0  1.0  1.0  1.0  1.0  2.0  3. Initial 1’lidspan Lateral Dellections of W2 Webs  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  11  -1.4  -2.6  1  1.5  -2.5  6  -2  -3  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 <0.0001  x2  2  11.037  5.518  403.42  x3  2  0.696  0.348  25.43  <0.0001  x4  2  3.272  1.636  119.6  <0.0001 <0.0001  x5  2  19.873  9.937  726.41  x6  2  0.727  0.364  26.58  <0.0001  5 4 x  4  0.427  0.107  7.81  <0.0001  6 4 X  4  0.728  0.182  13.3  <0.0001  6 5 X  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  7 3 X  4  0.158  0.040  3.29  0.0158  5 4 X  4  0.480  0.120  9.97  <0.0001  6 4 X  4  0.721  0.180  14.98  <0.0001  6 5 X  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  Model  40  48.39  Error  59  1.03  xl  2  0.081  x2  2  8.029  x3  2  0.548  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  4 1 x  4  0.151  0.038  3.31  0.0163  5 4 x  4  0.391  0.098  8.55  <0.0001  6 4 x  4  0.853  0.213  18.65  <0.0001  8 4 x  4  0.158  0.040  3.46  0.0133  6 5 x  4  0.226  0.056  4.94  0.0017  8 6 X  4  0.153  0.038  3.34  0.0156  .  Mean Square  F value  Pr>F  2.02  147.38  <.0001  --  --  0.01 0.040  3.54  0.0352  4.015  351.22  <0.0001  0.274  23.96  <0.0001  *  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 . The results for the truss assembly with zero, one and two CLB members per 2 of 0.001 N/mm 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  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  2.951  -0.930  9.742  No.  Coeff. --  9  10.572  12.613  8.439  2.891  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  5.240  9.840  16  8.190  12.613  8.439  -4.751  2.951  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 10.051  23  12.953  8.530  8.439  2.891  2.951  -7.100  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  -0.930  -7.100  9.584  5.240  9.657  27  10.572  8.530  12.704  -4.751  28  8.190  8.530  8.439  -0.930  -4.811  29  8.190  8.530  12.704  -0.930  -4.811  5.240  9.922 10.002 10.171  30  12.953  8.530  12.704  2.891  2.951  -7.100  31  8.190  12.613  12.704  -4.751  2.951  -0.930  -  219  -  Coeff.  RVI  RV2  RV3  RV4  RV5  RV6  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  -7.100  10.036  No.  --  37  8.190  12.613  12.704  2.891  -0.930  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  -7.100  9.625  40  12.953  12.613  8.439  -0.930  -4.811  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  2.951  -7.100  10.051  47  12.953  8.530  8.439  2.891  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  -0.930  5.240  9.888  51  12.953  8.530  8.439  -0.930  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  -4.811  -7.100  9.637  61  12.953  10.572  10.572  -4.751  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  2.951  -7.100  9.934  65  8.190  8.530  12.704  -0.930  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  GPa  GPa  GPa  mm  mm  mm  74  12.953  12.613  8.439  -0.930  -4.811  -0.930  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 10.067  No.  Coeff. --  9.594  90  12.953  12.613  8.439  -0.930  -0.930  5.240  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  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  No.  Coeff. --  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  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 16.187  No.  Coeff. --  30  8.852  8.986  12.138  -0.930  4.054  -0.930  463.801  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  417.600  17.783  34  8.852  12.157  9.005  2.218  -5.914  2.571  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 17.642  37  10.572  12.157  9.005  -4.078  -0.930  -0.930  371.399  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 17.129  48  12.291  10.572  12.138  2.218  -5.914  -0.930  463.801  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  -  No.  RV1  RV2  RV3  RV4  RV5  RV6  RV7  Coeff.  GPa  GPa  GPa  mm  mm  mm  N/mm  53  12.291  12.157  9.005  2.218  4.054  -4.431  463.801  17.484  54  10.572  12.157  9.005  -4.078  4.054  2.571  463.801  17.484  ——  55  8.852  8.986  9.005  -4.078  -0.930  -4.431  371.399  16.190  56  10.572  10.572  10.572  -0.930  -5.914  -0.930  37 1.399  17.086  57  10.572  10.572  10.572  -0.930  4.054  2.571  463.801  16.824  58  8.852  8.986  9.005  2.218  -5.914  2.571  371.399  16.279  59  12.291  10.572  10.572  -4.078  -0.930  -4.431  417.600  16.958  60  10.572  12.157  9.005  2.218  -0.930  -4.43 1  371.399  17.633  61  10.572  12.157  9.005  -4.078  4.054  2.571  463.801  17.484  62  12.291  8.986  9.005  -0.930  -5.914  2.571  371.399  16.324  63  10.572  10.572  10.572  -0.930  -5.914  2.571  417.600  17.075  64  12.291  10.572  12.138  -4.078  4.054  -0.930  371.399  16.883  65  10.572  10.572  10.572  -4.078  -5.914  -0.930  371.399  17.096  66  8.852  8.986  9.005  2.218  4.054  -0.930  417.600  16.085e  67  10.572  8.986  12.138  2.218  4.054  -0.930  463.801  16.190  68  8.852  8.986  12.138  -0.930  -0.930  -4.43 1  417.600  16.277  69  10.572  8.986  9.005  -4.078  4.054  -0.930  371.399  16.127  70  10.572  12.157  9.005  -4.078  -5.9 14  2.571  463.801  17.783  71  8.852  12.157  12.138  -4.078  4.054  2.571  371.399  17.552  72  12.291  12.157  9.005  -4.078  4.054  -0.930  371.399  17.494  73  8.852  12.157  9.005  -0.930  4.054  -0.930  371.399  17.471  74  8.852  12.157  12.138  -4.078  -5.9 14  2.571  417.600  17.886  75  10.572  10.572  10.572  -0.930  4.054  -4.431  463.801  16.824  76  12.291  10.572  10.572  2.218  -0.930  -0.930  371.399  16.956  77  12.291  8.986  12.138  2.218  4.054  2.571  371.399  16.193  78  12.291  8.986  9.005  -4.078  -0.930  2.571  417.600  16.247  79  8.852  10.572  12.138  2.218  -0.930  -0.930  417.600  16.962  80  12.291  12.157  9.005  -0.930  4.054  2.571  463.801  17.484  81  10.572  10.572  10.572  -4.078  -5.9 14  -4.431  463.801  17.096  82  8.852  12.157  9.005  -0.930  4.054  -4.43 1  417.600  17.484  -0.930  -5.9 14  -0.930  417.600  17.123  83  8.852  10.572  12.138  84  12.291  8.986  9.005  -4.078  4.054  -4.431  371.399  16.136  85  8.852  8.986  9.005  2.2 18  -5.9 14  2.571  371.399  16.279  86  8.852  10.572  12.138  -4.078  4.054  2.571  371.399  16.837  87  8.852  12.157  12.138  2.218  -0.930  -4.431  417.600  17.717  88  8.852  8.986  12.138  -0.930  4.054  -4.431  417.600  16.188  89  8.852  10.572  10.572  -0.930  -5.914  -4.431  463.801  17.364  90  10.572  12.157  9.005  2.218  -5.9 14  2.571  417.600  17.774  91  12.291  10.572  12.138  2.218  -5.914  -4.431  417.600  17.129  92  8.852  8.986  9.005  -4.078  4.054  -4.431  463.801  16.127 16.968 17.717  93  8.852  10.572  12.138  -0.930  -0.930  -0.930  371.399  94  8.852  12.157  12.138  -4.078  -0.930  2.571  463.801  -  223  -  RV1  RV2  RV3  RV4  RV5  RV6  RV7  GPa  GPa  GPa  mm  mm  mm  N/mm  95  10.572  12.157  9.005  -4.078  -0.930  -0.930  417.600  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  No.  Coeff. --  17.633  Table D-6. Sampling results ofthe truss assembly with two CLB bracing members per W2 web RV1  RV2  RV3  RV4  RV5  RV6  RV7  RV8  GPa  GPa  OPa  mm  mm  mm  N/mm  N/mm  11.769  9.038  8.314  -7.460  3.900  -0.930  406.980  5 14.097  27.674  2  9.374  12.105  8.314  -7.460  -0.930  2.399  406.980  472.900  27.887  3  11.769  10.572  10.572  -7.460  -0.930  -4.259  406.980  472.900  27.311  4  10.572  10.572  12.830  5.600  3.900  -0.930  406.980  5 14.097  28.238  406.980  431.703  28.419  No. 1  Coeff. —-  5  11.769  9.038  12.830  -0.930  3.900  2.399  6  9.374  10.572  10.572  -7.460  3.900  -0.930  538.820  431.703  28.091  7  10.572  9.038  8.314  -0.930  -0.930  2.399  406.980  431.703  27.3 15  8  10.572  12.105  8.314  5.600  3.900  -4.259  406.980  472.900  28.964  9  10.572  12.105  8.314  -0.930  -0.930  -0.930  406.980  472.900  28.049  10  9.374  10.572  10.572  5.600  -5.760  2.399  538.820  43 1.703  27.405  11  10.572  9.038  12.830  -7.460  -0.930  -0.930  538.820  514.097  27.098  12  9.374  9.038  8.314  5.600  -0.930  2.399  406.980  43 1.703  27.680  13  10.572  10.572  10.572  -7.460  -5.760  -0.930  406.980  43 1.703  26.837  14  11.769  9.038  8.314  -7.460  -5.760  2.399  406.980  514.097  26.568  15  9.374  10.572  10.572  -0.930  3.900  -4.259  406.980  431.703  28.301  16  11.769  9.038  12.830  -7.460  -5.760  -4.259  406.980  5 14.097  26.507  17  11.769  9.038  8.314  5.600  3.900  2.399  538.820  5 14.097  27.854  18  11.769  10.572  10.572  -7.460  -0.930  -0.930  538.820  472.900  27.473  2.399  406.980  472.900  29.235 28.964  19  10.572  12.105  12.830  -0.930  3.900  20  11.769  12.105  12.830  5.600  3.900  -0.930  406.980  5 14.097  21  9.374  12.105  8.314  5.600  3.900  -0.930  538.820  5 14.097  28.791  22  9.374  10.572  10.572  -7.460  3.900  2.399  406.980  431.703  28.328  23  11.769  9.038  8.314  -0.930  -0.930  -0.930  406.980  431.703  27.185  24  9.374  12.105  12.830  -7.460  -5.760  -4.259  472.900  514.097  27.311  25  10.572  12.105  8.314  -7.460  -5.760  -0.930  538.820  514.097  27.317  26  -7.460  -5.760  2.399  472.900  472.900  27.037 28.385  9.374  10.572  12.830  27  9.374  12.105  8.314  5.600  3.900  2.399  472.900  431.703  28  10.572  12.105  8.314  -7.460  -0.930  2.399  406.980  431.703  27.886  29  11.769  12.105  8.314  5.600  3.900  -4.259  406.980  5 14.097  29.006  30  10.572  12.105  8.314  5.600  3.900  -4.259  538.820  43 1.703  29.030  31  10.572  12.105  12.830  5.600  -0.930  -0.930  538.820  43 1.703  28.492 -  224  -  RV1  RV2  RV3  RV4  RV5  RV6  RV7  RV8  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  No.  Coeff. --  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  3.900  2.399  406.980  431.703  28.328  39  9.374  10.572  10.572  -7.460  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  3.900  -0.930  472.900  472.900  28.529  53  10.572  12.105  8.314  -7.460  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 28.297  68  9.374  12.105  8.314  -7.460  3.900  -4.259  406.980  472.900  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  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 27.382  No.  Coeff. --  81  9.374  9.038  8.314  -7.460  3.900  -4.259  472.900  431.703  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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