In-plane Stiffness of Cross-laminated Timber Floors by Sepideh Ashtari B.Sc., Sharif University of Technology, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2012 © Sepideh Ashtari, 2012 Abstract This study investigates the in-plane stiffness of CLT floor diaphragms and addresses the lateral load distribution within buildings containing CLT floors. In practice, it is common to assume the floor diaphragm as either flexible or rigid, and distribute the lateral load according to simple hand calculations methods. Here, the applicability of theses assumption to CLT floor diaphragms is investigated. There is limited number of studies on the subject of in-plane behaviour of CLT diaphragms in the literature. Many of these studies involve testing of the panels or the connections utilized in CLT diaphragms. This study employs numerical modeling as a tool to address the in-plane behaviour of CLT diaphragms. The approach taken to develop the numerical models in this thesis has not been applied so far to CLT floor diaphragms. Detailed 2D finite element models of selective CLT floor diaphragm configurations are generated and analysed in ANSYS. The models contain a smeared panel-to-panel connection model, which is calibrated with test data of a special type of CLT connection with self-tapping wood screws. The floor models are then extended to building models by adding shearwalls, and the lateral load distribution is studied for each building model. A design flowchart is also developed to aid engineers in finding the lateral load distribution for any type of building in a systematic approach. By a parametric study, the most influential parameters affecting the in-plane behaviour of CLT floor diaphragm and the lateral load distribution are identified. The main parameters include the response of the CLT panel-topanel connections, the in-plane shear modulus of CLT panels, the stiffness of shearwalls, and the floor diaphragm configuration. It was found that the applicability of flexible or rigid ii diaphragm assumptions is primarily dependent on the relative stiffness of the CLT floor diaphragm and the shearwalls. iii Table of Contents Abstract .................................................................................................................................... ii Table of Contents ................................................................................................................... iv List of Tables ......................................................................................................................... vii List of Figures ......................................................................................................................... ix Acknowledgements .............................................................................................................. xiii Chapter 1: Introduction ........................................................................................................ 1 1.1 Motivation ................................................................................................................. 1 1.2 Objectives ................................................................................................................. 1 1.3 Scope ......................................................................................................................... 3 1.4 Overview of the Thesis ............................................................................................. 4 Chapter 2: Mechanics of CLT Panels .................................................................................. 6 2.1 Structure of CLT Panels............................................................................................ 6 2.2 Typical Mechanical Properties and Dimensions....................................................... 8 2.3 Mechanical Behaviour of CLT Panels under In-plane Loads ................................. 12 Chapter 3: Analysis of Connections ................................................................................... 23 3.1 Connections in CLT ................................................................................................ 23 3.2 CLT Panel-to-Panel Connections Tested at UBC ................................................... 26 3.2.1 Connection Layouts ............................................................................................ 27 3.2.2 Loading Procedure .............................................................................................. 28 3.2.3 Test Results ......................................................................................................... 32 3.3 Calibrated Connection Model in ANSYS ............................................................... 35 iv 3.3.1 Average Response Curve for Layout E .............................................................. 35 3.3.2 Piecewise-Linear Average Curve for Layout E .................................................. 38 3.3.3 Implementing the Connection Model in ANSYS ............................................... 39 Chapter 4: Analysis of Floor Assemblies ........................................................................... 43 4.1 Finite Element Modelling of CLT Floors in ANSYS ............................................. 43 4.1.1 ANSYS Pre-Processing Phase ............................................................................ 43 4.1.1.1 Floor Configurations ................................................................................... 44 4.1.1.2 Modelling of CLT Panels............................................................................ 45 4.1.1.3 Generating Models of the Floor Configurations ......................................... 48 4.1.2 ANSYS Solution Phase....................................................................................... 50 4.1.2.1 Analysis Method ......................................................................................... 50 4.1.2.2 Boundary Conditions .................................................................................. 51 4.1.2.3 Loading ....................................................................................................... 52 4.1.3 ANSYS Post-Processing Phase........................................................................... 53 4.1.4 Analysis Results .................................................................................................. 55 4.2 Connection Model Validation ................................................................................. 65 Chapter 5: Analysis of CLT Buildings ............................................................................... 69 5.1 Lateral Load Distribution to Shearwalls ................................................................. 69 5.2 Hand Calculation Methods for Lateral Load Distribution ...................................... 70 5.3 Lateral Load Distribution Flowchart ...................................................................... 78 5.4 Finite Element Modelling of CLT Buildings in ANSYS........................................ 83 5.4.1 Analysis Results .................................................................................................. 86 5.4.2 Comparison of the ANSYS Outputs with the Force Distribution Methods........ 92 v Chapter 6: Parametric Study.............................................................................................. 93 6.1 Influential Parameters ............................................................................................. 93 6.1.1 Material Properties .............................................................................................. 94 6.1.2 Number of Connected Panels............................................................................ 107 6.1.3 Dimensions of CLT Panels and Diaphragm Configurations ............................ 108 Chapter 7: Conclusion ....................................................................................................... 110 7.1 Future Work .......................................................................................................... 111 References ............................................................................................................................ 113 Appendices ........................................................................................................................... 117 Appendix A : APDL Codes for Generating Diaphragm Models ...................................... 117 vi List of Tables Table 2.1 Engineering constants of the boards in CLT panels – these are the properties of CLT manufactured by CST Innovation (Yawalata and Lam 2011; Bergman et al. 2010) ...... 9 Table 2.2 Engineering constants computed from the values reported by Gsell et al. (2007) . 11 Table 2.3 Engineering constants assumed for the finite element modelling of CLT panels in the later chapters ..................................................................................................................... 11 Table 2.4 Typical dimensions of CLT panels ......................................................................... 12 Table 3.1 Test results for Layouts A, B, and E (Lam 2011) ................................................... 34 Table 3.2 Force values for distinctive displacements from the test results for the connection Layout E .................................................................................................................................. 37 Table 3.3 Average and piecewise linear curves for the connection Layout E with four sets of connection ............................................................................................................................... 40 Table 3.4 Per-unit-length response curve for the connection Layout E and the corresponding curve for 2cm and 4cm tributary length of COMBIN39 springs ............................................ 42 Table 4.1 CLT diaphragm stiffness values for Configuration 1 and 2, and Configuration 3 and 4 ........................................................................................................................................ 61 Table 4.2 Validation model response for the connection Layout E ........................................ 68 Table 5.1 Lateral load distribution with the tributary area method for the demonstration building ................................................................................................................................... 72 Table 5.2 Material properties for the shearwalls of the demonstration building .................... 73 Table 5.3 Geometrical properties, moments of inertia, shear area and reduction factors for the demonstration building ........................................................................................................... 73 vii Table 5.4 Local stiffness matrices for the shearwalls of the demonstration building............. 74 Table 5.5 Transformation matrices from the local coordinate to the global coordinate, for the shearwalls of the demonstration building ............................................................................... 75 Table 5.6 Global Stiffness matrices for the shearwalls in the demonstration building and the final global stiffness matrix of the whole system ................................................................... 76 Table 5.7 Global force and displacement vectors for the demonstration building ................. 76 Table 5.8 Global force vector for each shearwall in the demonstration building ................... 77 Table 5.9 Comparison of the lateral load distribution to the shearwalls by tributary area and stiffness methods ..................................................................................................................... 77 Table 5.10 Total stiffness values for the four sets of shearwalls and the according stiffness values for the COMBIN14 springs ......................................................................................... 85 Table 5.11 Comparison of the ANSYS model results for lateral load distribution with tributary area and stiffness methods ....................................................................................... 92 Table 6.1 In-plane stiffness of the CLT diaphragm for practical range of the elastic moduli ............................................................................................................................................... 101 Table 6.2 Comparison of the ANSYS analysis results for the lateral load distribution with hand calculations methods .................................................................................................... 105 viii List of Figures Figure 2.1 (a) CLT panel without intended spacing (u=0) (b) with intended spacing (u>0) (Moosbrugger et al. 2006)......................................................................................................... 7 Figure 2.2 (a) CLT panel (b) Representative Volume Element and (c) Representative Volume Sub-Element (Moosbrugger et al. 2006) ................................................................................. 14 Figure 2.3 (a) Decomposition of shear loading into (b) pure shear mechanism and (c) torsionlike mechanism (Moosbrugger et al. 2006) ............................................................................ 14 Figure 2.4 Non-dimensional equivalent shear stiffness for the simplified mechanical model and the finite element model, as proposed by Moosbrugger et al. (2006) .............................. 16 Figure 2.5 Test rig for the in-plane shear test (Gubana 2008) ................................................ 18 Figure 2.6 Force-displacement response for CLT panels under monotonic load (Gubana 2008) ....................................................................................................................................... 19 Figure 2.7 Force-displacement response for CLT panels under cyclic load (Gubana 2008) . 19 Figure 2.8 Two types of connections used in (a) Element 5 and (b) Element 6 (Vessby et al. 2009) ....................................................................................................................................... 21 Figure 2.9 Load-displacement curves for the tested elements used for measuring the overall in-plane stiffness of the panels (Vessby et al. 2009) .............................................................. 22 Figure 2.10 Load-displacement curves for the tested elements loaded to the failure point (Vessby et al. 2009) ................................................................................................................ 22 Figure 3.1 Typical CLT panel-to-panel connections (a) single surface spline (b) half-lapped joints (c) internal spline (d) double surface spline (Mohammad and Munoz 2011) ............... 26 ix Figure 3.2 Connection layouts tested at UBC (a) Layout A (b) Layout B (c) Layout E (Lam 2011) ....................................................................................................................................... 29 Figure 3.3 Connection Layout E (a) the in-plane test setup (b) installation of screws (c) 3D view of the out-of-plane test setup (Lam 2011) ...................................................................... 30 Figure 3.4 Step 1 to 5 of the Loading procedure used for the CLT connection test at UBC .. 31 Figure 3.5 Load-displacement curves for Layout E test specimens (Lam 2011) ................... 34 Figure 3.6 Average and modified load-displacement curves for Layout E test specimens .... 37 Figure 3.7 Average and piecewise-linear load-displacement curves for Layout E test specimens ................................................................................................................................ 38 Figure 3.8 ANSYS connection model for Layout E test specimens ....................................... 42 Figure 4.1 Floor Configuration 1 ............................................................................................ 46 Figure 4.2 Floor Configuration 2 ............................................................................................ 46 Figure 4.3 Floor Configuration 3 ............................................................................................ 47 Figure 4.4 Floor Configuration 4 ............................................................................................ 47 Figure 4.5 Boundary conditions of Configuration 3 modelled in ANSYS ............................. 49 Figure 4.6 Boundary conditions of Configuration 4 modelled in ANSYS ............................. 49 Figure 4.7 Pushover curves for Configuration 1 and 2-total load of 1296 kN ....................... 60 Figure 4.8 Pushover curves for Configuration 3- total load of 4536 kN ................................ 60 Figure 4.9 Pushover curves for Configuration 4- total load of 4012 kN ................................ 61 Figure 4.10 Deformation contour plot for Configuration 1 in the direction of loading.......... 62 Figure 4.11 Deformation contour plot for Configuration 2 in the direction of loading.......... 62 Figure 4.12 Deformation contour plot for Configuration 3 in the direction of loading.......... 63 Figure 4.13 Deformation contour plot for Configuration 4 in the direction of loading.......... 63 x Figure 4.14 Stress contour plot for Configuration 4- normal stress componentσx ................ 64 Figure 4.15 Stress contour plot for Configuration 4- normal stress componentσy ................ 64 Figure 4.16 Stress contour plot for Configuration 4- shear stress componentσxy .................. 65 Figure 4.17 Comparison of the validation model response with the average response curve and the piecewise linear approximation for the connection Layout E .................................... 67 Figure 4.18 Deformed shape and undeformed edge plot for the validation model-Total load of 64kN ................................................................................................................................... 68 Figure 5.1 Center of rigidity and center of mass ................................................................... 80 Figure 5.2 Lateral load distribution flowchart ....................................................................... 81 Figure 5.3 Some typical floor configurations as categorized in FEMA 454 (2006) and Commentary to the SEAOC (1980) ........................................................................................ 83 Figure 5.4 Pushover curves for Configuration 3 building with the 1st set of shearwalls ........ 89 Figure 5.5 Lateral load distribution for Configuration 3 building with the four sets of shearwalls ................................................................................................................................ 89 Figure 5.6 Deformation contour plot in the direction of loading, for the Confiuration3 building with the 1st set of shearwalls ..................................................................................... 90 Figure 5.7 Deformation contour plot in the direction of loading, for the Configuration 3 building with the 2nd set of shearwalls .................................................................................... 90 Figure 5.8 Deformation contour plot in the direction of loading, for the Configuration 3 building with the 3rd set of shearwalls .................................................................................... 91 Figure 5.9 Deformation contour plot in the direction of loading, for the Configuration 3 building with the 4th set of shearwalls .................................................................................... 91 xi Figure 6.1 In-plane stiffness of the CLT diaphragm for different initial stiffness of the panelto-panel connection ................................................................................................................. 96 Figure 6.2 Deformation contour plot for Configuration 2 in the direction of loading –initial stiffness of the panel-to-panel connections reduced to 0.01 of the Layout E-total load of 1296kN.................................................................................................................................... 97 Figure 6.3 Deformation contour plot for Configuration 2 in the direction of loading –initial stiffness of the panel-to-panel connections increased to 1000 times of the Layout E-total load of 1296kN ............................................................................................................................... 97 Figure 6.4 Stress contour plot for Configuration 4- normal stress componentσx ................ 100 Figure 6.5 Stress contour plot for Configuration 4- normal stress componentσy ................ 100 Figure 6.6 Stress contour plot for Configuration 4- shear stress componentσxy.................. 101 Figure 6.7 In-plane stiffness of the Configuration 2 CLT diaphragm for different number of connected panels and two different in-plane shear moduli of panels ................................... 102 Figure 6.8 Deformation contour plot in the direction of loading for Configuration 3 building with the 4th set of shearwalls and rigid panel-to-panel connections ..................................... 104 Figure 6.9 Deformation contour plot in the direction of loading for the concrete slab floor with the 4th set of shearwalls ................................................................................................. 104 xii Acknowledgements I would like to express my deep gratitude to my supervisors Dr.Terje Haukaas and Dr. Frank Lam, for their valuable support, enthusiastic encouragement and constructive suggestions through every step of this research project. Their understanding and willingness to dedicate their time so generously has been a constant motivation for me, to tackle all the obstacles and achieve the standards of the academic research. I also wish to gratefully acknowledge the funding for this research project, which is provided by the Strategic Network on Innovative Wood Products and Building Systems (NEWBuildS). NEWBuildS is a Forest Sector Research & Development Initiative funded by the Natural Sciences and Engineering Research Council of Canada (NSERC). It is my pleasure to thank the members of the Timber Engineering and Applied Mechanics group at UBC, members of the Reliability Group at Civil Engineering Department of UBC, and my friends and colleagues, who helped me with their comments and resources. My thanks also go to all the wonderful staff of the Graduate Office of the Civil Engineering Department of UBC, who were always ready to assist me throughout my studies. Last but not least, I would like to thank my parents, who gave me their true love and support every step of the way. It would not have been possible without them. xiii Dedication To my beloved mother and father xiv Chapter 1: Introduction 1.1 Motivation Floor diaphragms are important structural elements in a building system, with the purpose of carrying both vertical and lateral loads. For buildings subjected to lateral loads, diaphragms are designed to distribute the in-plane loads to the shearwalls, which in turn transfer these loads to the foundation. For all but the simplest cases, the in-plane stiffness of a floor diaphragm is a key parameter in determining the lateral load distribution to the shearwalls. In current practice, the floor diaphragms are considered as either flexible or rigid. Making either of these assumptions will allow engineers to use simplified methods for obtaining lateral forces in the shearwalls. Hence, it is of great interest in the context of lateral load design to identify which of the two assumptions are correct, if any. As a relatively new wood product to the North American wood industry, the behaviour of cross-laminated timber (CLT) as floor diaphragms is not yet clear. There is limited literature on the in-plane behaviour of CLT panels and, and even fewer studies are available on the in-plane behaviour of entire CLT floor assemblies. It is common to assume that light wood-frame floor diaphragms are flexible. In contrast, no guidelines have so far provided clear recommendations on whether CLT diaphragms should also be considered flexible. As a result, the motivation behind this study is to address the question of flexibility of CLT floor diaphragms and to provide designers with a general understanding of the inplane behaviour of these elements. 1.2 Objectives The main goal of this thesis is to study the in-plane behaviour of connected CLT floor diaphragms, and to investigate if the assumption of flexible or rigid diaphragms is applicable 1 to them. To achieve this goal, several objectives are pursued in this thesis. The first objective is to develop a detailed numerical model of connected CLT floors, which could serve as a platform for further studies on CLT floor assemblies. This model should utilize mechanical and geometrical parameters as input that can be calibrated by test data. With such a model, various aspects of the in-plane behaviour of CLT floors could be explored, without the need to conduct many expensive full-scale tests on CLT floor assemblies. More specifically, inplane stiffness values could be explicitly obtained based on the outputs of the analyses on the CLT floor models. The second objective is to study the lateral load distribution in a building with CLT diaphragms. Accomplishing such an objective leads to the answer to the key question of the flexibility of CLT floor diaphragms. To do so, shearwalls will be added to the previously generated floor models to simulate buildings. The share of the lateral load absorbed by the shearwalls in each model is compared with the results of the hypothetic flexible and rigid diaphragm assumptions. The flexibility of CLT diaphragms is important for seismic design of CLT buildings, which is the main lateral load considered in this study. In fact in seismic design, it is crucial to have a reasonably accurate estimate of the share of the lateral load attracted to each shearwall, to avoid erroneously designed shear walls and connections. The third objective in this thesis is to carry out a parametric study of the most influential parameters, which affect the in-plane behaviour of CLT floor diaphragms. Both the geometrical and material properties are considered. The outcomes of this parametric study reveal which parameters should be given the most attention when modelling or designing CLT floors. It will also provide insight into the sensitivity of the in-plane behaviour of CLT floors to some of the design parameters. Finally, as part of the background 2 needed to achieve the aforementioned objectives, this thesis tries to provide a broad literature review of related studies and technical information on the subject. 1.3 Scope Numerous factors affect the in-plane behaviour of CLT diaphragms, such as mechanical properties of panels, dimensions of panels, type of panel-to-panel connections, diaphragm configuration, orientation of shearwalls, etc. The variations of these parameters, even when assumed only in their practical and realistic range, produce almost indefinite cases of CLT floors with different in-plane behaviour. However, it is attempted in this thesis to provide some general insight into the overall in-plane behaviour of CLT diaphragms, which is useful for design purposes. Accompanying such general insight, a detailed generic finite element model of CLT floors is offered, for future use in cases where a more rigorous analysis of the in-plane behaviour of CLT floors is sought. To this end, this thesis provides detailed 2D finite element models of a few common CLT floors. Although only certain configurations are studied here, the methodology is applicable to a wide range of CLT floor configurations. The non-linear response parameters of panel-to-panel connections are calibrated to test data for a specific type of CLT connection, as described in Chapter 3. However, it is possible to calibrate these parameters to many other types of panel-to-panel connections, which are not studied in this thesis. To study the lateral load distribution in CLT buildings, linear springs are incorporated to simulate shearwalls in the finite element model. It is obvious that the general behaviour of shearwalls under lateral in-plane loads do not normally follow a linear trend. Nevertheless, the scope of this thesis is limited to the study of the lateral load distribution to the shearwalls in the early stages of loading, before they undergo substantial non-linear deformations. Consequently, assigning linear springs 3 calibrated to the initial stiffness of shearwalls is justifiable. In this thesis, quadrilateral elements with linear elastic orthotropic material behaviour are utilized in the finite element model for CLT panels. It is noted that, as discussed throughout this thesis, the structure of CLT panels are rather complex. Therefore, a 3D model of the panels would be more accurate for a thorough study of CLT panels. However, while that concern is more prominent for the study of out-of-plane bending, the objective here is to study the in-plane behaviour of connected CLT panels that act as diaphragms. For that reason, due to the fact that a 2D model is capable of representing the in-plane behaviour that is of interest in this study, 2D finite element models are exploited for both CLT floor and building models. 1.4 Overview of the Thesis The material presented in the remainder of this thesis is organised into five chapters and the developments in each chapter build upon the material provided in the previous chapters. Because there is limited literature available on the subject of the thesis, the background information is provided separately in each chapter. Chapter 2 primarily explores the structure and mechanical behaviour of CLT panels, with a focus on the in-plane behaviour. Chapter 3 provides details on types of CLT connections, with particular emphasis on panel-to-panel connections. A local finite element model of a specific panel-to-panel connection is generated and validated in ANSYS. This model is calibrated by test data and is utilized in the subsequent chapters. In fact, the local connection model is further expanded in Chapter 4, to global models of CLT floor assembles. The details of the finite element modelling, as well as early analyses results are presented in this chapter. Chapter 5 is dedicated to the study of CLT buildings and the lateral load distribution to the shearwalls. Shearwalls are added to the previously generated floor models of Chapter 4 to simulate 4 buildings. The results of the analyses, along with the lateral load distributions, are exhibited and discussed. Chapter 6 is dedicated to the parametric study of the most influential parameters for the in-plane behaviour of CLT diaphragms, and aims to identify and demonstrate their impact. Finally, after the five chapters that represent the main body of this thesis, the conclusions are provided in Chapter 7, along with comments and suggestions for future work. 5 Chapter 2: Mechanics of CLT Panels The basic components of CLT buildings, and in particular CLT diaphragms, are CLT panels. The first step to study the in-plane behaviour of CLT diaphragms is to understand the mechanics of CLT panels. Therefore, this chapter elaborates on the subject of CLT panels and specifically addresses their mechanical behaviour under in-plane loads. To do so, the literature and available technical information provided by the CLT manufacturers on the mechanics of CLT panels, are reviewed. The structure of CLT panels, their typical dimensions, and mechanical properties are explored first. Next, the behaviour of CLT panels under in-plane loads is addressed. Finally, a set of engineering constants is selected for CLT panels, which will be used in later chapters for modelling the CLT diaphragms in ANSYS. 2.1 Structure of CLT Panels To recognize the mechanical behaviour of CLT panels, it is necessary first to identify the structure of panels. CLT panels are massive multi-layer wooden plates, which are comprised of crosswise layers of wood boards. Two types of CLT panel structures are illustrated in Figure 2.1. As shown in the figure, there are normally 3, 5, 7, or more layers in one CLT panel, arranged symmetrically around the mid layer (Chen 2011; FPInnovations 2011). The boards within each layer are placed parallel to each other but are generally orthogonal to the direction of the neighbouring layers. Layers are typically connected to each other by means of full gluing on common surfaces. Sometimes mechanical connectors, such as nails or screws are utilized to connect the layers, but this is rather uncommon in practice and gluing is much preferred. The narrow faces of the boards are typically not glued, except in particular cases. The gaps between the boards are normally so narrow that some friction may form. Nevertheless, due to shrinkage and fabrication procedures, the frictional and 6 contact conditions of the narrow faces are uncertain. Therefore, to be conservative, the contact between the boards is normally neglected for analysis purposes (Moosbrugger et al. 2006; Jobstl et al. 2008). There are also some rare configurations in which intended gaps have been considered in the board arrangement, as shown in Part(b) of Figure 2.1. In this case, the mechanical behaviour of CLT panels will change with the size of the gaps. However, since these configurations are uncommon, their mechanical behaviour is not described further here. One important aspect of the CLT panels is the orientation of the outer layers. The orientation of the outer layers in CLT panels affects their mechanical behaviour, particularly for out-of-plane bending. The outer layers in wall panels are oriented such that the grain direction in these layers is vertical, to maximize resistance against vertical loads (FPInnovations 2011). For floor and roof panels, the outer layer orientation is usually parallel to the span direction, since the panels are stronger in bending under out-of plane loads, in that direction. Figure 2.1 (a) CLT panel without intended spacing (u=0) (b) with intended spacing (u>0) (Moosbrugger et al. 2006) 7 2.2 Typical Mechanical Properties and Dimensions In order to provide realistic models of CLT diaphragms, it is essential to identify the typical values for mechanical properties and dimensions of CLT panels. Both mechanical properties and dimensions of CLT panels may vary for different manufacturer companies. Of great interest are the properties for the Canadian made CLT panels. As described earlier in the previous section, the structure of CLT panels is made up of crosswise layers. The local mechanical properties of theses layers are not the same as the global panels. The behaviour of wood varies for each three mutually perpendicular directions. This type of behaviour is referred to as “orthotropic” material behaviour. The boards in each layer are placed in the same direction. Therefore each layer obtains an orthotropic material behaviour as well. However, the crosswise orientation of the layers, which is symmetric about the mid layer, causes the panels to possess different mechanical properties as a whole, than those of each layer. Another reason for the difference between the mechanical properties of panels and the boards is the contribution of the glue. In this thesis, the main interest is in the in-plane behaviour of CLT panels only. However, a brief review of the mechanical behaviour of the boards is presented in the following. A description of the mechanical properties of the boards in common CLT panels is provided by Gagnon and Popovski (2011). It is explained that the most important European manufacturers use boards that are stress graded C24 according to EN 338 (2010) and EN 1912 (2006). In Canada, this is almost equivalent to machine stress-rated grade 1650Fb-1.5E lumber with a mean modulus of elasticity of about 10,300 MPa according to CSA 086 (2009). Depending on the application of CLT panels, different quality of boards may be utilized for different layers. Normally low grade lumbers are used for inner layers, while 8 higher grade species are incorporated for the outer layers (Chen 2011). The basic mechanical properties of the boards are primarily defined in terms of the material constants for an orthotropic material behaviour. Elastic orthotropic materials have nine independent material constants for each set of three mutually perpendicular basis vectors. These material constants include three moduli of elasticity E, and three shear moduli G, and the corresponding Poisson’s ratios ν, which are referred to as “engineering constants”. The three mutually perpendicular basis vectors for wood are normally selected in the longitudinal, tangential, and radial directions relative to the grain direction. These directions are recognized by the subscripts L, T, and R assigned to the material constants, respectively. For example EL refers to the elasticity modulus in the longitudinal direction relative to the grain direction of the wood. One common type of wood species used for CLT panels, is spruce-pine-fir (SPF). Table 2.1 provides a summary of the orthotropic engineering constants SPF stud grade, No.2 or better. The moduli of elasticity in this table are based on the vibration tests done at the Timber Engineering Applied Mechanics (TEAM) Laboratory at UBC (Yawalata and Lam 2011) and the rest are taken from the Wood Handbook (Bergman et al. 2010). The abovementioned mechanical properties are the properties of the individual boards in CLT panels. However, as mentioned earlier, CLT panels as a whole possess different mechanical properties than those of the individual boards. Table 2.1 Engineering constants of the boards in CLT panels – these are the properties of CLT manufactured by CST Innovation (Yawalata and Lam 2011; Bergman et al. 2010) Species Group S.P.F. Grade Modulus of Elasticity (GPa) EL ET ER Shear Modulus (GPa) GLR GLT GRT Stud 11.43 0.777 1.166 0.560 0.526 0.057 No.2 or better 10.66 0.725 1.087 0.522 0.490 0.053 Poisson Ratios νLR νLT νRT 0.316 0.347 0.469 9 For generating realistic models of CLT panels, there is a need estimate these mechanical properties. Unfortunately, there is limited test data available on the overall mechanical properties of CLT panels. These properties are either calculated from the properties of the individual layers, or determined experimentally by testing specimen cut from CLT panels (Chen 2011). One example of the latter approach can be found in the research carried out by Gsell et al. (2007). In their study, a fully automated procedure to determine the global elastic properties of full-scale CLT panels is developed. One main conclusion of their work is that orthotropic, homogeneous, linear elastic material behaviour can be assumed to model the overall mechanical behaviour of CLT panels with sufficient accuracy. The elastic constants reported by Gsell and the colleagues, are converted to common engineering constants shown in Table 2.2. The values shown in this table are used as a reference for assigning the mechanical properties of CLT panels in finite element models, which are developed in the next chapters. In this table, subscripts X, Y, and Z refer to the three mutually perpendicular directions; and E0 and E90 refer to the elasticity moduli parallel and perpendicular to the grain direction, respectively. Considering the above outcomes, the assumption of orthotropic material behaviour can be applied to CLT panels, and hence in this thesis, for the sake of accuracy, it is the preferred assumption for modelling of CLT panels. However, since normally the properties of the panels in the two in-plane directions are relatively close, in many cases an isotropic behaviour would simulate the CLT performance sufficiently well (Moosbrugger et al. 2006). Table 2.3 reports the engineering constants selected for modelling of CLT panels in the later chapters. These values are primarily assumed based on the engineering constants reported in Table 2.2, with slight changes. 10 Table 2.2 Engineering constants computed from the values reported by Gsell et al. (2007) Ex (E90) Ey (E0) Ez νyx νzx νyz νxy νxz νzy Gyz Gxz Gxy 4.63 GPa 8.21 GPa 0.5 GPa 0.090 0.040 0.364 0.051 0.380 0.022 0.54 GPa 0.0949 GPa 0.747 GPa Table 2.3 Engineering constants assumed for the finite element modelling of CLT panels in the later chapters ANSYS Model Constants 4.0 GPa Ex (E90) 8.0 GPa Ey (E0) 0.5 GPa Ez 0.07 νxy 0.35 νyz 0.35 νxz 0.6 GPa Gxy 0.5 GPa Gyz 0.1 GPa Gxz Another feature of a realistic modelling of CLT panels is to use typical dimensions. Presently, a variety of CLT panel dimensions are conceivable. As with the material properties, different manufacturers produce panels of different dimensions. Nevertheless, it is possible to obtain a sense of typical dimensions by comparing the products of different manufacturers. Table 2.4 provides some of these typical dimensions. Many of the values in this table are reported by Gagnon and Popovski (2011). It is observed that panel widths up to 4m are possible, but typical widths are probably around 1.2 m. While lengths up to 24 m 11 appears to be possible using finger joints in the individual boards, it likely that practical applications will be limited to, say, 10-12 m. In the forthcoming chapters, the dimensions of CLT panels utilized in the CLT diaphragm models are selected in accord with the abovementioned typical dimensions. Table 2.4 Typical dimensions of CLT panels Boards Panels 2.3 width thickness width thickness length ~63 mm to ~250 mm ~10 mm to ~50 mm 0.6, 1.2, 2.95 m - up to 4 m up to 0.5 m up to 24 m Mechanical Behaviour of CLT Panels under In-plane Loads As part of the background for modelling CLT panels in this thesis, the mechanical behaviour of CLT panels under in-plane loads is reviewed. This background helps to identify the perception of the overall in-plane behaviour of CLT panels among other researchers, and forms a basis, to which the outputs of the numerical analyses of CLT diaphragms can be compared. Nevertheless, this comparison is possible only in terms of the quality of the inplane behaviour of CLT panels. Any quantitative comparison is unjustifiable, due to the differences in the material properties of the panels, assumptions involved in modelling and test procedures, and etc. There are relatively few studies conducted so far with the sole purposes of investigating the in-plane behaviour of CLT panels or CLT diaphragms. Most of the studies related to the mechanical behaviour of CLT panels are focused on their out-of-plane behaviour rather than their in-plane behaviour (Gagnon and Popovski 2011; Guggenberger and Moosbrugger 2006; Jobstl et al. 2006; Stürzenbecher et al. 2010a; Stürzenbecher et al. 12 2010b). One major work on the out-of plane performance of CLT utilized in floor application was recently done at the University of British Columbia by Chen (2011). Her work involves comprehensive description of CLT product, manufacturing process and grading, literature review on the topic. In addition, it includes detailed 3D finite element modelling of CLT panels in the finite element program ANSYS. Therefore it is a profound resource for future studies on topics related to CLT. Among the few works on the in-plane behaviour of CLT panels, examples of both numerical and experimental studies can be found. In the following, first the most prominent numerical studies and afterwards experimental studies are reviewed and briefly discussed. One of the most prominent numerical works on the in-plane behaviour of CLT panels is the study done at Graz University of Technology (Moosbrugger et al. 2006; Bogensperger et al. 2010). In that study, detailed numerical models of CLT panels as well as simplified and yet efficient mechanical models were developed. These models were compatible with the experimental results as well. One unique feature of their research is the approach, which is utilized to address the in-plane stiffness of CLT panels. The in-plane stiffness was primarily defined in terms of the in-plane shear modulus of panels. In order to be able to clearly explain their study, the details are provided in the following. Nevertheless, the general approach used to address the in-plane stiffness of CLT panels in this thesis is different. The proposed mechanical model in the Graz university study was developed based on the periodicity of the internal geometric structure of CLT panels, and uniformity of the applied shear loading on the boundaries. The combination of these two features results in periodically repeated stress and deformation patterns. This unique quality, allows studying a sub-section of CLT panels, called representative volume element (RVE), instead of the whole 13 panel. The representative volume element extends over the entire thickness of CLT panels and it is further divided into sub-elements called representative volume sub-elements (RVSE). Each representative volume sub-element is comprised of two orthogonal neighboring boards with half the thickness. The second subdivision is justified by assuming infinite periodicity in the thickness direction and thus neglecting the effect of a stress-free boundary conditions on the results. Figure 2.2 shows the CLT panel along with the RVE and RVSE as defined in the Graz University study. (a) (b) (c) Figure 2.2 (a) CLT panel (b) Representative Volume Element and (c) Representative Volume SubElement (Moosbrugger et al. 2006) Figure 2.3 (a) Decomposition of shear loading into (b) pure shear mechanism and (c) torsion-like mechanism (Moosbrugger et al. 2006) 14 A fundamental assumption in the Graz University study is the decomposition of the complete state of shear loading into two basic mechanisms, namely pure shear mechanism and a torsion-like behaviour, which is illustrated in Figure 2.3 . Pure shear mechanism is the state in which uniform shear is present across the total thickness of the sub-element. The torsion mechanism corresponds to a complicated state of torsion over the thickness of the sub-element. For the abovementioned sub-element, a simplified model as well as a finite element model was introduced. In the simplified approach, basic solid mechanics equations for each loading mechanism were set up. Consequently, a closed form equation was obtained, which relates the total shear strain to the applied loads and mechanical properties assumed for CLT panels. Based on that equation, a non-dimensional shear stiffness was defined as follows: ( ( ) ) (2.1) where, G* is the equivalent shear modulus for the panel, G*1+2 is the shear modulus that combines the effects from both pure shear and torsion mechanisms, and therefore is equal to the equivalent shear modulus; G is the shear modulus parallel to the grain direction; Geff is the effective shear modulus and is equal to the average of the shear modulus parallel and normal to the grain; (ti/a) is the board-thickness to board-width ratio. The above formula clearly suggest that the equivalent shear modulus for the whole CLT panel depends on the shear moduli parallel and normal to the grain direction, as well as the geometric aspect ratio of the thickness to the width of panels. Since the equivalent shear modulus in the Graz university study is an indicator of the in-plane stiffness of CLT panels, the abovementioned parameters can be considered as influential parameters, which affect the in-plane behaviour 15 of CLT panels directly. Figure 2.4 depicts the change in the dimensionless shear stiffness against the thickness to width ratio of panels (Moosbrugger et al. 2006). The curves in this figure correspond to the results of the finite element modelling, as well as the simplified model relation in Equation (2.1) (Moosbrugger et al. 2006). The analyses were also carried out for the assumption of isotropic behaviour of CLT panels, where G = Gparallel = Geff. In this relation, Gparallel is the shear modulus for parallel to the grain direction. Figure 2.4 suggests that in the practical range of the thickness to width ratio of CLT panels, indicated by the shaded area in the figure, the non-dimensional shear stiffness of panels drops by 40% with increasing 10 times the thickness to the width ratio. As part of the study, it was also shown that increasing board spacing and existence of openings, both will result in reduction of equivalent shear modulus, and thus the in-plane stiffness. Figure 2.4 Non-dimensional equivalent shear stiffness for the simplified mechanical model and the finite element model, as proposed by Moosbrugger et al. (2006) 16 As an alternative to the numerical approach, experimental studies address the in-plane behaviour of CLT diaphragms based on monotonic and cyclic test results. The study at the Graz University of Technology also involved testing of CLT panels under in-plane shear loads (Jobstl et al. 2008; Bogensperger et al. 2007). These tests were mainly focused on developing new procedures to accurately measure the shear capacity of CLT panels under inplane loads. The shear stiffness of the test specimens were also obtained, which were utilized further to modify the results of the finite element modelling described above. Another example of the experimental studies is the research carried out by Gubana (2008). Gubana presented the results of a series of tests carried out on CLT panels, in order to determine their in-plane shear behaviour. The primary objective of his work was to investigate the capability of CLT panels for strengthening ancient wood floors in retrofitting interventions. The CLT panels used in the tests had four layers, instead of common odd number layers. The external layers were 40 mm and the two inner ones were 20 mm in thickness; making a total thickness of 120 mm. The width of the boards was 200 mm. The boards of the external layers were in the direction of maximum length and the inner layers in the orthogonal direction. The timber could be classified as GL24C, after EN 1194 (Eurocode 8 1998) with mean shear modulus of 590 MPa. The test rig design, shown in Figure 2.5, allowed the panels to be stressed with constant shear tension on the lateral sections. This would simulate a seismic loading situation, where the floor diaphragm would be under cyclic in-plane shear tension. Ten couples of panels were tested. Seven tests were monotonic, two cyclic and one with reversed cycles of force. The tests were forced-control and in all of them, only one of the two panels reached the collapse state and the other remained in the elastic range. Figure 2.6 and Figure 2.7 show the force-displacement curves for one of the monotonic and cyclic tests. 17 Figure 2.5 Test rig for the in-plane shear test (Gubana 2008) These figures are taken from the original text, and thus lack in quality. The displacement in these figures is the diagonal displacement of a square inside the panel, indicated by the measurement devices. With some data elaboration, the shear stress vs. shear strain curves were also obtained for each case. Consequently, the in-plane stiffness was defined in terms of the shear modulus G, taken as the slope of the shear stress-strain curve in the elastic range. The shear moduli were calculated based on the data related to 10% and 40% of the shear failure strength as specified in most of the codes such as EN 789 (2005). The main observation from Gubana`s work is the almost linear behaviour of CLT panels under in-plane loads, which continues up to the failure point as suggested by Figure 2.6 and Figure 2.7. The panels which did not collapse, the left panel in Figure 2.6 and Figure 2.7, showed no ductility in their force-displacement response. However, the collapsed panels, the right panels in the figures, showed unexpected ductility. Gubana explains that this ductility is shown on the 18 force-displacement curves, because the rupture in the panels does not occur simultaneously in all the layers. Therefore, CLT panels in general behave almost linearly up to the failure point, showing moderate to low levels of ductility. It is also concluded in his research that the failure strength is not affected by the previous cycles of load. This is also mainly related to the linear elastic behaviour of panels up to the failure point. Figure 2.6 Force-displacement response for CLT panels under monotonic load (Gubana 2008) Figure 2.7 Force-displacement response for CLT panels under cyclic load (Gubana 2008) 19 In-plane stiffness of CLT panels is also investigated experimentally by Vessby et al. (2009). Experimental tests are performed to study the structural performance of CLT panels as wall members under in-plane loading. What is rather unique in their study is the comparison of the stiffness and strength of two different types of panel-to-panel connections. Force-displacement curves, stiffness values and failure loads were compared for the connected panels and panels of the same size but without connection. Four CLT panels, 4955 mm long, 1250 mm wide, and 96 mm thick, were tested during in-plane bending. Each specimen had 5 layers of 19mm thickness. Layers 1, 3 and 5 were in the length direction and the other two in the orthogonal direction. The width of the boards was 120 mm. The boards were made of Norway spruce and expected to be class C24 or higher, according to EN 14081-1 (2005) strength classes. Elements 1 and 2 were first tested at low levels of loading to obtain stiffness values. They were then sawn into two pieces and reconnected with Methods 1 and 2 shown in Figure 2.8, forming Elements 5 and 6, respectively. The first type of connection has been utilized in earlier practical designs. The slot cut into each of the halves was 60 mm deep and 25 mm wide. The screws were, hexagonal-head wood screws, 96 mm long and 8 mm in diameter. In the second type of connection, both gluing and screwing were utilized. A sheet of fiberboard, 300 mm wide and 8 mm thick was connected to both sides of the specimen. A single-component polyurethane adhesive with approximate 12 h curing time was used. The sheets of fiberboards were also mechanically joined by means of hexagonal-head wood screws 50 mm long and 6 mm in diameter. The test setup was comprised of a simply supported beam, with roller supports at both ends, in which each half of the beam would simulate cantilever wall boundary conditions. 20 Figure 2.8 Two types of connections used in (a) Element 5 and (b) Element 6 (Vessby et al. 2009) The tests were under displacement control condition with 2 mm/min rate. The deformations were measured by means of 7 gauges located at the ends and the middle section of the beam. Since the tested specimens could be categorized as deep beams, the deformations observed during the tests were in fact combined deformations of three major contributions, namely bending, shear and local material compression at the loading and supporting areas. In the Vessby and the colleague`s study, the in-plane stiffness was addressed in terms of a combined shear and bending stiffness. This stiffness was defined based on the loaddisplacement curves obtained for the elements tested up to a certain level of load, as shown in Figure 2.9, which was considered as non-destructive to the elements. The displacements in this figure were calculated by subtracting the average deflection at the ends from the average mid-span deflection. The elements were then loaded up to the failure point. Figure 2.10 shows the load-displacement curves for elements tested to failure. These curves again suggest that overall in-plane behaviour of CLT panels is almost linearly up to the 21 failure point. Furthermore, all the failures occurred suddenly, showing no ductile behaviour. In these particular series of tests, two main types of failure were bending failure and local failure at the supports and the loading point. It was also concluded that the first type of panel to panel connection had much lower in-plane stiffness and strength compared to the other type and panels without connections. However the response of the second type of connection was close to that of the panels without the connections, and therefore was superior to the first type in terms of the in-plane performance. Figure 2.9 Load-displacement curves for the tested elements used for measuring the overall in-plane stiffness of the panels (Vessby et al. 2009) Figure 2.10 Load-displacement curves for the tested elements loaded to the failure point (Vessby et al. 2009) 22 Chapter 3: Analysis of Connections The two major components of a CLT floor are CLT panels and CLT connections. Chapter 2 was dedicated to the subject of CLT panels and their mechanical behaviour under in-plane loads. This chapter provides details on types of CLT connections and in particular, panel-to-panel connections and their performance in CLT floor diaphragms. A special type of CLT panel-to-panel connections recently tested at UBC is addressed in Section 3.2, and the test results are further discussed. Section 3.3 expands on modelling the tested connection in the finite element commercial software ANSYS. The connection model generated here is a local model, which is calibrated by the test data. This model forms the basis for the global models developed in the later chapters and therefore has a great significance in this thesis. 3.1 Connections in CLT It is known that the behaviour of connections plays a key role in determining the overall performance of a structural system. Connections maintain the structural integrity of a building and are crucial to strength and stability of the system. In comparison to other types of wood buildings, connections serve an even more important purpose in CLT assemblies. In these buildings, connections are believed to be the main source of ductility and energy dissipation, if not the only (Dujic and Zarnic 2006; Ceccotti et al. 2006; Follesa et al. 2010). Therefore, providing a suitable set of connections is indispensable to the design of CLT structures, especially under lateral loads, where affording proper ductility matters the most. In order to study the behaviour of connections in CLT assemblies, first different types of connections should be identified and categorized. CLT connections vary in composition, type of fasteners, as well as the application of the connection in the structural system. Various types of traditional or innovative fasteners for CLT are utilized in current practice in 23 Europe (Munoz et al. 2010; Mohammad and Munoz 2011). Traditional fasteners include dowel type fasteners such as nails, wood screws, lag screws and bolts, which can be combined with metal plates, brackets and holdowns. Innovative or proprietary fasteners include self-tapping screws and dowels, metal hooks, and glued in rods bearing-type systems. The proprietary products are those products that are sold by a particular company and are protected by a registered trademark. Among the aforementioned fasteners, proprietary long self-tapping screws have gained the most attention, especially for multi storey construction purposes. A considerable number of studies have been carried out so far in Europe, to investigate the behaviour of each type of fasteners and to quantify their fastening capacity in CLT. Nevertheless, characterizing fastening capacity in CLT is more complicated than other wood products. This is mainly due to complex internal structure of CLT panels, crosswise buildup of the layers, and discontinuities due to small gaps between the boards of each layer. In addition, different fasteners and connection details are recommended by different CLT manufacturers. In fact, no clear guidelines on the structural performance of such connection in CLT have been provided yet (Munoz et al. 2010). With regard to the growing interest in establishing CLT in the North American wood industry, and for the aforementioned reasons, there is a need for delivering detailed guidelines on the performance of CLT connections. In general, connections in a CLT building can be categorized into five application groups: wall-to-wall, wall-to-floor, wall-to-foundation, wall-to-roof, and floor-to-floor. In the context of studying the in-plane behaviour of CLT diaphragms, floor-to-floor or more generally panel-to-panel connections (including wall-to-wall connections) are the main interest. This type of connections is generally established in-situ due to production and transportation limitations. Figure 3.1 depicts four common types of panel-to-panel 24 connections, namely single surface spline shown in Part(a), and half-lapped joint shown in Part(b), internal spline shown in Part(c), double surface spline shown in Part(d) (Mohammad and Munoz 2011). Another type of panel-to-panel connection was also tested at UBC, which will be introduced shortly in Section 3.2. This type of connection is the only panel-to-panel connection, which is specifically utilized in the CLT floor models in this thesis. However, to provide a broader view of the choices available for CLT panel-to-panel connections, other common types are demonstrated here as well. Each of these connections differs from each other in joint strength, joint stiffness, ease and speed of execution, and total cost. In a study conducted by Follesa et al. (2010), some typical CLT panel to-panel connections were compared in terms of the above criteria. The comparison was made based on a series of tests conducted with varying geometry, type of panels, and type of fasteners. The three selected types of connection were half-lapped joint, internal spline, and single surface as already introduced in Figure 3.1. For the fasteners, 6 and 8 mm self-tapping screws with or without washers, as well as 3.1 mm smooth nails, and 3.1/3.4 threaded shank nails were utilized. The authors argued that the half-lapped joint has the advantage of being the quickest joint, since it requires only one layer of connections and no additional timber panel is needed. The internal spline though, takes advantage of having two shear planes for each connector as opposed to the single surface spline, in which each connector has only one shear plane. Nevertheless, the internal spline turned out to be least efficient among the three types of connection and it required more accuracy in fabrication, and on site execution. 25 (a) (c) (b) (d) Figure 3.1 Typical CLT panel-to-panel connections (a) single surface spline (b) half-lapped joints (c) internal spline (d) double surface spline (Mohammad and Munoz 2011) 3.2 CLT Panel-to-Panel Connections Tested at UBC In a recent study carried out at the Department of Wood Science at UBC (Lam 2011), a special type of panel-to-panel connections with self-tapping wood screws was tested. This type of connection is used for the local connection model, which is subsequently utilized in the CLT floor models as described in the forthcoming chapters. Furthermore, the connection test results presented in this chapter are the only experimental data used for the calibration of the connection model. Since this type of connection is employed later in this study, a detailed description of the connection layout, test procedure, and the test results are presented here. 26 The tests were carried out by the Timber Engineering Applied Mechanics (TEAM) Laboratory. The objective of the study was to experimentally evaluate the performance of self-tapping wood screws for CLT panel-to-panel connections. Shear transfer capabilities of this type of screws was investigated for both in-plane and out-of-plane loading, in the tested connection configurations. The tested CLT connections were intended to be utilized in the newly constructed Bioenergy Research and Demonstration Project Nexterra Building on the UBC campus. The tested CLT panels were produced by CST Innovations Ltd. The panels were 3-ply, with 35 mm thickness of each ply (Lam 2011). Consecutive layers were placed orthogonal in orientation to the neighboring layers. Originally the panels were 1.4 m in width and 4.2 m in length, with total thickness of approximately 105 mm. The wood was SprucePine-Fir harvested from Mountain Pine Beetle infected forest, and the grade was No. 2 or better according to CSA 086 (2009). The lumber laminas were square-edged, without finger joints, and polyurethane adhesive was used in the manufacturing process. The self-tapping wood screws were 8 mm ASSY VG plus made by SWG Schraubenwerk Gaisbach GmbH. Screw lengths ranging from 180 to 250 mm were available. In cases when the screw lengths were too long and the screws protruded from the surface of the specimens, the protruded portions were cut off. 3.2.1 Connection Layouts Three main connection layouts were tested under in-plane loading, namely Layouts A, B, and E (Layouts C and D were tested only under out-of-plane loading). Figure 3.2 shows these three connection layouts. Each connection test specimen included 3 pieces of CLT cut-offs, which were connected by four sets of screws. In the connection Layout A, the screws were installed at a 45 degree angle and mainly loaded perpendicular to their axis, as 27 shown in Part(a) of Figure 3.2. In connection Layout B, the screws were arranged in 3D with double 45 degree angle incline, resulting in a 45 degree screw-in-angle perpendicular to the top of and bottom layer of the CLT panel, and a 35 degree screw-in-angle parallel to the grain, as illustrated in Part(b) of Figure 3.2. In this layout the screws are mainly loaded in either tension or compression, depending on the direction of loading. Since the lateral load acts in both directions, in the connection Layout E, the number of screws was doubled compared to Layout B, by putting in one more set of screws, which would be in compression while the other set is in tension, as shown in Part(c) of Figure 3.2. For connection Layouts A and B, 10 replicates and for connection layout E, 6 replicates were tested, for which the test results are presented in Section 3.2.3. The test setup for both in-plane and out-of-plane loading of connection Layout E is illustrated in Figure 3.3, Parts (a) and (c) respectively. The Part(c) of the figure is for demonstration purposes only, and is meant to show the arrangement of screws in a 3D format in Layout E. The installation of self-tapping wood screws in the test specimens is depicted in Part(b). 3.2.2 Loading Procedure The loading procedure as described in the TEAM report (Lam 2011) was based on the European standard DIN EN 26891 (1991). The procedure included 6 steps. The first five steps were force-controlled. The last step however, was basically displacement-controlled, which allowed obtaining the failure load and the load-displacement curve for the post-peak response. The steps of the procedure are described below and the first 5 step are schematically shown in Figure 3.4. 28 (b) (a) (c) Figure 3.2 Connection layouts tested at UBC (a) Layout A (b) Layout B (c) Layout E (Lam 2011) 29 (a) (b) (c) Figure 3.3 Connection Layout E (a) the in-plane test setup (b) installation of screws (c) 3D view of the out-of-plane test setup (Lam 2011) 30 Each step is indicated by its number in the figure. Here, Fest refers to the estimated load for the failure load: 1. Constant loading until 0.4×Fest with rate of 0.2×Fest per minute ± 25% 2. Keep load at 0.4×Fest for 30 sec 3. Reduce load until 0.1×Fest 4. Keep load at 0.1×Fest for 30 sec 5. Constant loading until 0.7×Fest with rate of 0.2×Fest per minute ± 25% The last step is not shown in Figure 3.4, since it was displacement-controlled. Beyond 0.7 × Fest a constant displacement speed is required until the failure load or a maximum displacement of 15mm is reached. It is required to reach the failure load or the maximum displacement within 3-5 minutes. The whole testing procedure should be 10 to 15 minutes. Load 0.7 Fest 2 0.4 Fest 5 3 1 4 0.1Fest Time (min) 2.0 2.5 3.0 6.0 Figure 3.4 Step 1 to 5 of the Loading procedure used for the CLT connection test at UBC 31 3.2.3 Test Results The applied loads and the relative movement between adjacent CLT panels were continuously measured during the tests (Lam 2011). For each test, the failure load Fmax, the displacement at the failure load DFmaxx, and the load at 40% and 70% of the failure load, and their associated displacements were specified. The failure load here is the same as the capacity of the connection and it refers to the maximum applied load, at which failure in the connection initiates. The 40% and 70% of the failure load values were used for defining the stiffness of the connection layouts according to DIN EN 26891 (1991): (3.1) where kS is the connection stiffness; D0.7Fmax and D0.4Fmax refer to the displacements corresponding to the load at 40% and 70% of the failure load, respectively. The above equation provides and experimental value for evaluating the connection stiffness. Table 3.1 summarizes the test results for Layouts A, B, and E. The average, standard deviation and coefficient of variation are calculated for kS, Fmax, 0.4 Fmax, and 0.7Fmax and the corresponding displacement values. The results strongly suggest the superior performance of layout E in comparison with both Layouts A and B. In fact, there is an increase of 83% and 35% in the stiffness relative to Layouts A and B, respectively. An increase of 2700% and 433% in the capacity of Layout E relative to Layouts A and B respectively was also observed. According to Table 3.1, the coefficient of variation for Fmax, 0.4 Fmax, and 0.7Fmax is higher for Layout E than the other two layouts. However, for the experimental stiffness kS, this value is lower for Layout E. This fact suggests that, adding the second set of screws 32 causes more variability in the non-linear response, but it reduces the variability in the initial stiffness. One reason for he superioir performance of Layout E is the double angle orientation of the screws, which results in creation of forces along the axis of screws. These types of screws perform considerably better when loaded along their axis, comparing to the case when they are loaded in bearing and bending mode, as in Layout A. Another reason is installing the second set of double inclined screws, which enables the connection to perform better under forces acting in both directions. In this case, for both directions, one set of the screws works in tension while the other works in compression. This type of behaviour will result in a significantly better performance than Layout B, where there is only one set of screws acting in tension or compression. According to the TEAM report, the performance of the connection Layout E is satisfactory under in-plane loads (Lam 2011). Therefore, the behaviour of this type of connection when utilized in CLT floors and its effect on the in-plane behaviour of the floor is investigated throughout the next chapters. The test results in terms of the load-displacement curves for the six tested specimens of Layout E are shown in Figure 3.5. As suggested by the figure, all specimens responded almost linearly in the initial stages of loading. Moreover, the response of all of the specimens is close in the initial part of the load-displacement curves, showing less variation in the results. Specifically the initial slopes of all of the curves are almost the same, suggesting consistent initial stiffness for all the specimens. However, as the nonlinear behaviour triggers the response of different specimens tends to vary more in terms of stiffness, capacity, and the maximum displacement before unloading. 33 Table 3.1 Test results for Layouts A, B, and E (Lam 2011) Layout A Layout B Layout E Average Stdev COV (%) Average Stdev COV (%) Average Stdev COV (%) Fmax (kN) 35.60 4.85 13.64 48.14 4.86 10.09 65.21 10.32 15.83 DFmax (mm) 52.29 6.93 13.26 18.29 2.60 14.20 5.29 0.41 7.75 0.4 Fmax (kN) 14.24 1.94 13.64 19.26 1.94 10.09 26.09 4.13 15.83 D0.4 Fmax (mm) 13.27 4.03 30.35 3.82 1.28 33.56 1.52 0.48 31.58 0.7 Fmax (kN) 24.92 3.40 13.64 33.70 3.40 10.09 45.65 7.22 15.82 D0.7 Fmax (mm) 28.93 3.29 11.39 8.84 2.03 22.97 2.56 0.61 23.83 ks (kN/mm) 0.69 0.14 20.47 2.99 0.70 23.42 18.74 2.17 11.58 90 80 70 E1 Force (KN) 60 E2 50 E3 40 E4 30 E5 20 E6 10 0 0 2 4 6 8 10 12 14 16 Displacement (mm) Figure 3.5 Load-displacement curves for Layout E test specimens (Lam 2011) 34 3.3 Calibrated Connection Model in ANSYS This section describes the modelling of the connection Layout E in the commercial finite element software ANSYS. As mentioned earlier, this type of connection is selected to be implemented in the CLT floors models throughout the next chapters. This model is calibrated with the test data from the connection tests. Therefore, all the global CLT floor models generated in the forthcoming chapters are calibrated with real test data at the local level of connection model. 3.3.1 Average Response Curve for Layout E To model the connection Layout E in ANSYS, a response curve is needed. Response curve in this context refers to the load- displacement curves obtained during the connection tests as described in the previous section. Due to the variation of the load-displacement curves for the six test specimens an average curve, which is representative of the overall behaviour of the connection could be utilized here. To obtain the average response curve a simple method is used. The average response curve is obtained by averaging the force values of the six response curves over a set of displacement values. For instance, as illustrated in Figure 3.6, to obtain the average force value for the displacement of 5mm, the force values of the six curves at 5mm displacement are first indicated. This figure will be discussed more, later in this section. Then the average of the six force values is plotted at 5mm displacement for the average response curve. By choosing a proper number of displacement values, at which the averaging is done, an average curve is attained. Before the process of averaging, the six response curves from the connection tests should be modified to remove the initial slip at the beginning of each test. The initial part of each curve contains a section with much lower slope. This part indicates that displacement 35 had occurred in the test specimens without a noticeable change in the force value. In the test results shown in Figure 3.5, the slip part is especially significant for the E6 curve. To eliminate the slip from the test results, a simple procedure is utilized here. The idea is to shift each curve to the left so that the slip is eliminated and the curves pass through the origin with a slope equal to the slope of the curve at the initial stages of loading. As mentioned earlier, the load-displacement curves of the test specimens are almost linear at the initial stages of loading. The stiffness of the load-displacement curve at this linear part can be evaluated with Equation (3.1). To estimate the slip in each curve a line is drawn, which would pass through the points on the curve corresponding to 0.4Fmax and 0.7Fmax introduced Section 3.2.3. These two points were used to define the experimental stiffness value defined in Equation (3.1), and therefore the line passing through these points idealizes the linear part of the curve. Next the slip can be estimated by obtaining the displacement intercept of the drawn line. Reducing the estimated amount of slip from all the displacement values on the load-displacement curve, shifts the curve to the left, and thus the desired modified curve is attained. Figure 3.6 shows the average response curve for the connection Layout E, along with the experimental curves. It is noticeable that the curves are close to the average curve in the initial linear part of the curve. However, the deviations from the mean response increase as the non-linear behaviour appears, and are significant in the peak and post-peak regions. It is of great importance to realize that few measurements are recorded for the post-peak behaviour. For that reason, the average curve is limited to 9mm displacement. Table 3.2 shows the selected displacements and their corresponding force values along with the average force values for the connection Layout E. 36 90 80 70 E1 Force (KN) 60 E2 50 E3 40 E4 30 E5 20 E6 Average 10 0 0 2 4 6 8 10 12 14 Displacement (mm) Figure 3.6 Average and modified load-displacement curves for Layout E test specimens Table 3.2 Force values for distinctive displacements from the test results for the connection Layout E Displacement 1mm 2mm 3mm 4mm 5mm 6mm 7mm 8mm 9mm E1 21.96 38.79 46.65 50.84 51.89 50.91 47.35 41.36 35.32 E2 16.33 35.36 51.30 61.41 64.21 63.44 62.46 57.89 53.85 E3 20.15 37.17 50.18 55.38 56.43 55.90 52.13 47.74 44.00 E4 15.27 35.51 52.11 63.86 71.31 72.32 64.16 55.73 49.59 E5 22.09 43.96 57.89 64.45 65.20 63.44 59.90 53.25 46.56 E6 24.16 48.15 67.27 78.26 79.60 78.45 76.73 72.72 68.38 Average 19.99 39.82 54.23 62.37 64.77 64.08 60.46 54.78 49.62 37 3.3.2 Piecewise-Linear Average Curve for Layout E The average response curve for the connection Layout E was obtained in the previous section. To model the connection in the finite element software ANSYS, the average response curve should be defined for the program. A simple and yet convenient way to do so is to utilize a piecewise-linear approximation of the average curve. The piecewise linear curve utilized here is tri-linear up to the failure point and has two segments in the post-peak region. This curve is fitted to the average response curve by means of trial and error. A more systematic approach would be to use least square regression instead of trial and error to fit the piecewise linear curve. Figure 3.7 shows the piecewise-linear average curve for the connection Layout E. 70 60 50 Force (KN) 40 30 20 10 0 0 2 4 6 8 10 Displacement (mm) Average Curve Piecewise-Linear Approx Figure 3.7 Average and piecewise-linear load-displacement curves for Layout E test specimens 38 As evident in the figure, the piecewise linear curve obtained by trial and error approximates the average response curve closely. The difference between the two curves is small compared to the large variability in the test results. Consequently, the use of the least square regression method would yield no particular advantage over the trial and error method, in terms of improving the accuracy of the approximation. The values of the average response curve and the piecewise-linear curve for the connection Layout E are listed in Table 3.3. The ten points on the average response curve is reduced to 6 points on the piecewise linear curve. The two curves coincide at the origin, 2 mm, peak of the curve at 5 mm, and the maximum displacement at 9mm. The force values for 3.6 mm and 6.6 mm are obtained by linear interpolation of the force values of the average response curve at 3 mm and 4 mm, and 5 mm and 6 mm, respectively. It should be noted that the values in this table are based on the test results for four sets of screws as described in Section 3.2.1 and depicted in Part(c) of Figure 3.2. The distance between the installed screws was 900 mm. To obtain the loaddisplacement curves for each set of screws the values in Table 3.3 should be divided by four. 3.3.3 Implementing the Connection Model in ANSYS In this section, modelling of the connection Layout E in ANSYS is described. The connection model is a local model, since it will be utilized in the next chapters, as part of the global CLT diaphragm models. Therefore, prior to generating the diaphragm models, the connection model should be developed in ANSYS. To model the connection Layout E in ANSYS, each connection is replaced with two sets of springs, one acting in the direction of loading, and the other acting in the perpendicular direction. The number of the springs and their properties should be defined in a way that in total they yield the same in-plane behaviour as a single connection. 39 Table 3.3 Average and piecewise linear curves for the connection Layout E with four sets of connection Average Curve Piecewise Linear Approximation Δ (mm) F (kN) 0 0 2 39.82 3.6 60.18 Δ (mm) 0 1 2 F (kN) 0 19.99 39.82 3 54.23 5 64.77 4 62.37 6.6 62.63 5 6 7 8 9 64.77 64.08 60.46 54.78 49.62 9 49.62 The approach of replacing connections or fasteners with two unidirectional longitudinal springs has been used by other researchers as well and examples can be found in the literature. One example is the study done by C. Hite and Shenton (2002) on modelling the non-linear behaviour of wood frame shearwalls. One way of defining the properties of the springs for the connection Layout E, is to calibrate the springs in the direction of loading with the piecewise linear curve. In the perpendicular direction, rigid links are needed, which ensures that the connection does not open up under in-plane loading. In fact in practice, the CLT panels of floor diaphragm will not open up under in-plane loads, due to the restraining effect of the chord members, beams, and shearwalls acting in the perpendicular direction. Due to the non-linearity in the piecewise linear curve, non-linear springs are utilized in the direction of loading. For modelling the rigid links linear springs with a high stiffness value are employed. One option for the nonlinear springs in ANSYS is COMBIN39. This type of spring is a unidirectional element with nonlinear force-displacement capability that can be used in any type of analysis. The element has longitudinal or torsional capability in 1-D, 2-D, or 3-D 40 applications. The longitudinal option, which is used here, is a uniaxial tension-compression element (ANSYS 14.0 Help 2011b). In the perpendicular direction, COMBIN14 could be utilized. COMBIN14 shares many features with COMBIN39 except that it is linear. Figure 3.8 is a snapshot of the finite element model of the test specimen for the connection Layout E generated in ANSYS. The test specimen was already shown in Part(c) of Figure 3.2, and Part(a) of Figure 3.3. The finite element model includes three CLT panel cut-offs, which are connected by a series of lines representing the connections springs. Each of these short lines includes one COMBIN39 and one COMBIN14 spring. Each node on one side of the panels is linked to the corresponding node on the side of the neighbouring panel with a COMBIN39 and a COMBIN14 spring. In this manner, the COMBIN39 springs form a parallel system of springs in the direction of loading. In other words, the stiffness of these springs adds up to each other in the direction of loading. It should be noted that the wood between consecutive springs will also deform under the imposed in-plane load. However, the deformation in the wood is relatively small compared to COMBIN39 deformations in the direction of loading. This is due to the much higher stiffness of the wood relative to the stiffness of COMBIN39 springs. For that reason, the assumption of parallel springs is proper. Consequently, the response curve for each of the COMBIN39 springs could be obtained by dividing the force values of the piecewise-linear curve by the number of COMBIN39 springs. A response curve per unit length could be also obtained for the COMBIN39 springs, by dividing the ordinates of the piecewise linear curve by the total length of the connections. The length of the connection is the length at which the two CLT panels are connected. In the connection test each two panels are connected over a length of 300 mm, resulting in a total connection length of 600 mm. Then the response curve for each of the COMBIN39 curves 41 could be obtained by multiplying the ordinates of the per-unit-length-response curve by the “tributary length” of each spring. The tributary length is the length along the edge of panels, on which each spring acts. Here the tributary length is equal to the size of quadrilateral elements, which are used to mesh the CLT panels. Table 3.4 shows the response curve values for the per-unit-length and for the tributary length of 20 mm and 40 mm. The tributary length of 20 mm and 40 mm are the two sizes of quadrilateral elements, which are used to mesh the CLT panels in the next chapters, and otherwise has no particular significance. Table 3.4 Per-unit-length response curve for the connection Layout E and the corresponding curve for 2cm and 4cm tributary length of COMBIN39 springs Δ (mm) 0 2 3.6 5 6.6 9 Per-unit-length F (kN/mm) 0.000 0.066 0.100 0.108 0.104 0.083 20 mm Tributary Length F (kN) 0.000 1.327 2.006 2.159 2.088 1.654 40 mm Tributary Length F (kN) 0.000 2.655 4.012 4.318 4.175 3.308 Figure 3.8 ANSYS connection model for Layout E test specimens 42 Chapter 4: Analysis of Floor Assemblies This chapter is dedicated to the numerical modelling of CLT floor diaphragms and expands on the details of the finite element models generated in ANSYS. Selected floor diaphragm configurations are modelled in ANSYS for investigations on the in-plane behaviour of CLT floor diaphragms. For convenience, the naming of the subsequent sections follows the sequence of modelling in the ANSYS computer program. The results of the analyses are presented and discussed. In the last section, the connection model described in Section 3.3 is validated by checking the results of the finite element model of the connection test specimen against the test results. The last section is presented in this chapter instead of Chapter 3, since the necessary steps to create a finite element model in ANSYS is first described here. 4.1 Finite Element Modelling of CLT Floors in ANSYS The ANSYS finite element toolbox is one the most powerful commercial computer programs available for analyzing various boundary value problems. The various features of ANSYS for structural mechanics enable engineers to develop advanced and detailed models of structural and mechanical problems. This section provides step by step procedure of generating CLT floor models in ANSYS. The three major steps in building a model in ANSYS are pre-processing, solution, and post-processing, which are described shortly. The descriptions are provided such that they allow the reader to replicate the models for future studies. 4.1.1 ANSYS Pre-Processing Phase At the preprocessor phase in ANSYS, all the geometric and material properties are defined and the finite element model is generated (Madenci and Guven 2005). In general, 43 two approaches are possible for creating a finite element model. The first option is solid modelling, and the second is direct generation. In solid modelling, geometric shapes are predefined, and nodes and elements are automatically generated based on user specifications. In contrast, in the direct generation approach, the user defines the coordinates of every node and the connectivity of every element. As a result, in the direct generation approach, the user is in charge of controlling node and element numbering, while these numbers are assigned automatically when meshing objects in the solid modelling approach. The two approaches may be utilized in combination, offering advantages of the both methods. In this study the finite element models are created using the direct generation approach. ANSYS Parametric Design Language (APDL) is utilized to facilitate the generation of various CLT diaphragm models with varying input parameters. APDL is a scripting language, which enables the automation of common tasks and the creation of models with user-defined parameters. All ANSYS commands can be used as part of this scripting language, and the APDL commands encompass a wide range of features, such as do-loops, if-else branching, and macros. 4.1.1.1 Floor Configurations Four floor configurations are studied in this chapter. The phrase floor configuration means the shape, size, and proportions of the diaphragm. These four configurations, referred to as Configuration 1, Configuration 2, Configuration 3, and Configuration 4, are depicted in Figure 4.1 to Figure 4.4. For comparison purposes, all models have similar outer dimensions. The thickness of the CLT floor is 0.15 m in all cases. This thickness is almost equal to the thickness of 5-ply CLT panels produced by the Canadian manufacturer CST Innovations. It 44 should be noted that generally the thickness of floor is selected based on the out-of-plane design requirements and could be higher than the value assigned here. From floor Configuration 1 to floor Configuration 4, each configuration is evolved from the previous one with a number of extra features. In this manner, it is possible to compare the configurations to study the particular effect of those added features. Configuration 1 depicted in Figure 4.1, is a basic model, which includes a large CLT panel attached to two Shearwalls A and D at the ends. It should be recognized that the dimensions assigned to the single CLT panel in Configuration 1 are not common in practice. Configuration 2 shown in Figure 4.2, is essentially like Configuration1, which is instead covered by nine smaller CLT panels. The CLT panels in Configuration 2 are 1.2 m wide and 5 m long, which are common dimensions in practice. The panels are connected to each other by the panel-to-panel connection Layout E introduced earlier in Section 3.2 and 3.3. In Configuration 3, as illustrated in Figure 4.3, two more shearwalls are added to the floor Configuration 2. The four shearwalls in this model divide the floor into three bays, and are oriented in the direction of loading. In Configuration 4, depicted in Figure 4.4, an opening is introduced in the middle bay of Configuration 3, and otherwise is the same as Configuration 3. 4.1.1.2 Modelling of CLT Panels Due to complexity of the internal structure of CLT panels, a detailed 3D finite element model, such as the one developed at Graz University (Guggenberger and Moosbrugger 2006; Bogensperger et al. 2010), appears to be the best solution. Nevertheless, since the panels behave almost linearly under in-plane loads up to the failure point, a simpler 2D model would be a good approximation to capture the in-plane behaviour of CLT panels. 45 5.00m 10.80m A D Figure 4.1 Floor Configuration 1 10.80m 5.00m 1.20m 1 2 3 A 4 5 6 7 8 9 D Figure 4.2 Floor Configuration 2 46 10.80m 5.00m 1.20m 1 2 3 4 3.60m 5 6 7 2.40m A 8 9 4.80m B C D Figure 4.3 Floor Configuration 3 2.40m 10.80m 1.20m 2 3 4 5 6 7 8 9 5.00m 1 3.60m A 2.40m B 4.80m C D Figure 4.4 Floor Configuration 4 47 In ANSYS there are several elements available for 2D modelling of plate type members. To model the CLT panels, the simplest 2D element, namely the PLANE42 is selected herein. This element is defined by four nodes, each having two translational degrees of freedom (ANSYS 14.0 Help 2011b). PLANE42 can be utilized either as element plane stress or plane strain element. If the member to be modelled is a flat thin 2D member, with the thickness being much smaller than the other two dimension, and if the elements are under in-plane loads only, then the plane stress assumption is applicable (Madenci and Guven 2005). The above conditions apply to the study of the CLT panels under in-plane loads. Therefore, the element behaviouris chosenfrom theelement optionstabas“planestress with thickness ” The material assigned to these elements should simulate the overall behaviour of CLT panels. As already discussed in Section 2.2 and 2.3, the overall behaviour of CLT panels under inplane loads can be considered as linear elastic orthotropic. For the mechanical properties of this type of material behaviour, the values provided for the engineering constants of CLT panels in Table 2.3 are used. 4.1.1.3 Generating Models of the Floor Configurations The above four floor configurations are modelled in ANSYS. These models are utilized again in the next chapters for generating the CLT building models. To build the CLT panels, a mesh as depicted in Figure 4.5 and Figure 4.6, is generated through APDL commands. Main advantage of using this type of mesh is its simplicity and easy data access. As mentioned earlier, the mesh generation here is controlled by the user and is not automatic. All the panels are meshed with 40×40 mm PLANE42 elements. The panels are connected to each other by the connection model described in Section 3.3.3. This model was calibrated with the test data of connection Layout E introduced in Section 3.2. 48 Figure 4.5 Boundary conditions of Configuration 3 modelled in ANSYS Figure 4.6 Boundary conditions of Configuration 4 modelled in ANSYS 49 The COMBIN39 and COMBIN14 springs are also added to the models through APDL commands. At this stage, the floor models are generated, and the boundary conditions will be introduced to each model in the solution phase. 4.1.2 ANSYS Solution Phase After generating the floor models in the preprocessing phase, the type of analysis, solution controls, and boundary conditions are defined in the solution phase in ANSYS. The boundary conditions include displacement boundary conditions, and nodal, surface, and body loads. 4.1.2.1 Analysis Method The analysis method herein is non-linear static monotonic analysis. The non-linear analysis option should be selected because of the non-linear COMBIN39 springs, which are utilized in the connection model. As already shown and described in Section 3.2 and Section 3.3, the test data used for calibration of these non-linear springs are from monotonic tests. Therefore, the analysis is also selected to be static and monotonic. Since no geometric non-linearity is involved, the small displacement static option should be selected from the Solution Controls window. There are three levels in a nonlinear solution in ANSYS, namely load steps, substeps, and equilibrium iterations. Within each load step, the solution is obtained by applying the load incrementally in substeps. In each substep, convergence is accomplished by performing several equilibrium iterations. Although increasing number of substeps ensures higher accuracy in solution, it also means higher computational effort and thus time of analysis. To achieve an accurate solution within a reasonable amount of time, the Automatic Time 50 Stepping option should be turned on in the Solution Controls window (Madenci and Guven 2005). Here the number of substeps and its minimum and maximum are entered as 30, 10, and 3000 respectively. All solution outputs are selected to be written to the results file at every substep. In this manner, sufficient solution points are obtained for generating forcedisplacement response curve of the model. In the Nonlinear tab, the line search should also be also turned on. This feature enhances the convergence of Newton-Raphson solver (Lilley et al. 2001). In the same tab, the maximum number of equilibrium iterations is set to 1000 or higher. 4.1.2.2 Boundary Conditions The boundary conditions should be applied to the floor models, following the main objective in this chapter, which is to study the pure in-plane behaviour of CLT floor diaphragms. In other words, only the in-plane behaviour of connected CLT panels in the floor diaphragms is studied in this chapter. Therefore, the effect of the attached shearwalls on the response of the floor models should be avoided. One way to achieve this objective is to apply displacement constraints to the nodes at the location of shearwalls. These constraint are applied to all degrees of freedom in those nodes, which include translations in X and Y directions (in-plane directions). It should be recognized that imposing such boundary conditions to the floor Configuration 3 and Configuration 4, already described in Section 4.1.1.1, causes the floor to be separated into three independent bays. More clearly, the deformation of the floor in each bay is not affected by the other bays, and each bay deform like a deep beam clamped at the two ends. In fact each bay could be modelled separately in ANSYS, which in that case would be similar to Configuration 2 with less 51 number of panels. However, these floor models are to be utilized again in the building models, and therefore, they are generated and used in this chapter as well. In ANSYS, both types of boundary conditions, namely displacement constrains and imposed forces, can be defined from Solution > Define Loads > Apply > Structural, in form of displacements or forces. The location at which these boundary conditions are imposed should be clarified. There are four shearwalls in the floor configurations described Section 4.1.1.1. The two outer shearwalls exist in all of the four configurations, while the two middle ones exist only in Configuration 3 and Configuration 4. For the outer shearwalls (Shearwalls A and D) the nodes on the outermost sides of the end panels are constrained. At the location of the middle shearwalls (Shearwalls B and C), there are two CLT panels connected to each other. The displacement constraints are applied to only one side of one of these two panels. For both Configuration 3 and Configuration 4, depicted in Figure 4.3 and Figure 4.4, these are applied to the right edge of the third panel and to the left edge of the sixth panel for Shearwall B and C respectively. Figure 4.5 and Figure 4.6 show all the applied displacement and force boundary conditions for the two diaphragm models. 4.1.2.3 Loading In this study the force-displacement response of the floor diaphragms models under in-plane loads is of are great interest. The force-displacement curves reflect the in-plane behaviour of the modelled CLT floor diaphragms and provide a measure for their in-plane stiffness. In order to obtain a force-displacement curve in ANSYS, the applied static load should be increased incrementally until the load is fully applied. As mentioned earlier in Section 1.2, the main lateral load considered in this thesis is seismic load. To simulate the seismic load, a uniform pressure load is applied along the lower edge of the floor models. 52 Figure 4.5 and Figure 4.6 show the pressure load applied to the Configuration 3 and Configuration 4. For Configuration 4 with an opening in the middle bay, the static pressure load is scaled down by the ratio of the omitted CLT area to the whole CLT area between Shearwall A and B. It should be noted that, the seismic load is in fact an inertia force, which acts on the whole mass of the structure. Therefore, a better representation of the seismic force is a uniformly distributed body load, which acts on the whole floor diaphragm. However, to be able to extend the results of this study to other lateral loading situations, the uniform pressure load is preferred over the body load here. The pressure value is increased until one of the non-linear springs acting in the direction of loading reaches its capacity, i.e. the peak of the piecewise-linear average curve. In this thesis, this is assumed to be the failure point for the floor diaphragm model. The failure of is detected when the analysis fails to converge. It should be considered that in a real situation, the diaphragm failure does not necessarily occur at this point, and the connections undergo no-linear deformation after the peak point. Indeed, a number of connections should fail before the failure of the whole diaphragm occurs. This fact is observed in the tests carried out on full-scale CLT structures (Ceccotti et al. 2006; Ceccotti 2008; Ceccotti et al. 2010). Nevertheless, assuming the failure point of the floor diaphragm being the same as the capacity point of one connection is on the safe side, and therefore is reasonable. 4.1.3 ANSYS Post-Processing Phase After the solution phase, the user can review the results in either the General Postprocessor or the Time History Postprocessor. The General Postprocessor provides access to the analysis results over the whole or a portion of the model for a specifically defined load combination, at a single time or frequency. On the other hand, the Time History 53 Postprocessor displays the analysis results at specific locations in the model as a function of time, frequency, or some other parameter related to time (ANSYS 14.0 Help 2011a). To study the in-plane behaviour of CLT diaphragms, the following analysis outputs are of interest: 1. Push-over Curve: or simply force-displacement curves, as already mentioned, are capable of demonstrating the in-plane behaviour of CLT diaphragms and can be exploited to obtain a measure of their in-plane stiffness. They also provide a good source for design purposes. The pushover curves are obtained through performing a static nonlinear analysis, which in this context is also referred to as push-over analysis. In ANSYS there is not an immediate option or a specified procedure to obtain push-over curves compared to some other commercial softwares such as SAP2000. In order to perform a pushover analysis, multiple loading substeps should be defined for a static non-linear analysis and the outputs should be recorded at each substep, which is already done in Section 4.1.2.3. Introducing such substeps in the analysis, will allow the Time-History Postprocessor to display the recorded results of analysis for the parameter of interest versus amount or ratio of the imposed load. If this parameter is the in-plane displacement of the CLT diaphragm, then the desired push-over curve is obtained. The displacement parameter is assumed to be in-plane displacement of the mid-span at the upper edge of each floor bay, in the direction of loading. For Configuration 3 and Configuration 4, as already discussed in Section 4.1.2.2, the in-plane behaviour of each bay is independent of the other bays for the imposed boundary conditions. Therefore, for each bay a separate pushover curve is obtained. 54 2. Reaction forces at the location of shearwalls: In order to study the distribution of the lateral load to the shearwalls, the total reaction forces in the shearwalls, in the direction of loading should be obtained. In this chapter, displacement constraints are applied to the nodes at the location of shearwalls. As a result, the sum of the reaction forces at these nodes yields the total amount of shear forces attracted to each shearwall and hence the force distribution. In ANSYS the reaction forces could be accessed from General Postprocessor > List Results > Reaction Solution. 3. Stress and Deformation Plots: The deformation plot illustrates the overall deformation pattern of the CLT floor diaphragms under the imposed load. The deformation pattern provides information on the location and amount of the maximum in-plane deflection in the diaphragms. To describe the state of stress in an orthotropic material, normally the plots of all six components of stress, namely normal components σx, σy, σz, and shear components σxy, σyz, and σxz are provided. But for the floor models developed here, since only in-plane loads are applied and the panels are modelled with plane stress assumption, only three components are non-zero. These three components are σx, σy, and σxy. The stress plots are less accurate than the displacement plots. However, they provide information on the possible location of stress concentrations and failures in the floor diaphragm models. The stress and deformation plots are available from General Postprocessor > Plot Results > Contour Plots > Nodal Solution, and General Postprocessor > Deformed shape. 4.1.4 Analysis Results The pushover analysis was performed for the four CLT floor Configurations. Pushover curves for the mid-span of each floor bay were obtained for each model, which are 55 shown in Figure 4.7, 4.8 and Figure 4.9. The load axis in these curves is divided by the total applied load to demonstrate the displacement at each load ratio. The total in-plane load that caused failure was 1296 kN for Configuration 2, 4536 kN for Configuration 3, and 4012 kN for Configuration 4. The definition of failure was already presented in Section 4.1.2.3. For Configuration1 a total load of 1296 kN was also applied. Nevertheless, since the whole model is linear elastic in this configuration, no failure is observable in the analysis. Based on the literature review provided in Chapter 2, the overall in-plane behaviour of CLT panels follows a linear trend. To be able to assess the applicability of this assumption for the connected CLT panels, a linear regression is obtained for each pushover curve in Microsoft Excel. In Figure 4.7 to Figure 4.9, the linear trendline is depicted in each graph with a dashed-line along with the equation of the fitted line. The slope of the trendline could be considered as a measure of the in-plane stiffness of the CLT diaphragm in each configuration. To obtain the stiffness value, the slope of the linear trendline should be multiplied by the amount of the total exerted force in that floor bay. The resulting stiffness values for each configuration are summarized in Table 4.1. The shape of the pushover curves for the configurations that contain panel-to-panel connections is almost tri-linear up to the failure point. This is related to the shape of the piecewise-linear average curve utilized to calibrate the connection models, which was trilinear up to the failure point. In fact since the springs in the connection model form an almost parallel system, they experience almost the same deformation at each stage of loading. This means that at each stage of loading, the springs in the connection model are at almost the same point on the piecewise-linear average response curve of the connection. Therefore, for this type of CLT panel-to-panel connection, it is possible that all the connections reach the 56 capacity and then the failure point quite at the same time. If that is the case, then designers should be cautious about the possibility of such type of failure in CLT floor diaphragms with Layout E panel-to-panel connections. Nevertheless, it should be noted that due to variability in the load-displacement response of the connection, the connections are more likely to fail at different times. By comparing the pushover curves and the linear trendline fitted to each curve, it is evident that due to the non-linearity present in the pushover curves, the linear regression does not provide a proper representation of the overall in-plane behaviour of connected CLT panels. However, the slope of the linear regression could be still used to provide some measure of the overall in-plane stiffness of the floor diaphragm. The amount of non-linearity in the pushover curve is case dependent. More clearly, depending on the non-linear behaviour of the panel-to-panel connections, and the location and number of connections, the resulting pushover curve may vary. Figure 4.7 shows the pushover curves for Configuration 1 and Configuration 2, under the same total in-plane load. The difference between these two curves indicates how the inplane behaviour of the CLT diaphragm has changed, when the single CLT panel in Configuration 1 is cut into smaller panels and reconnected by connection Layout E in Configuration 2. According to the values in Table 4.1, the stiffness of Configuration 2 is about 25% less than Configuration 1. Furthermore, Figure 4.7 suggests that adding the panelto-panel connection Layout E has increased the floor in-plane deformation by 40%. This additional deformation is in fact due to the deformations in the connections and is in the form of slip between the adjacent CLT panels. It should be recognized that the change in the inplane behaviour of CLT panels due to addition of panel-to-panel connection depends on the 57 properties of the connection. For instance, if the spacing between the screws was smaller in the connection Layout E, then the connection would become stiffer. Utilizing such stiffer connection, would result in a stiffer CLT floor diaphragm, which undergo smaller in-plane deformations compared to the floor with the connection Layout E. The deformation contour plots for Configuration 1 and Configuration 2 in the direction of loading are shown in Figure 4.10 and Figure 4.11. The maximum deflection in both cases is at the midspan of the floor, which is trivial since the floor in both cases is like deep beam clamped at the two ends. The slip between the panels is maximum for the panels closer to the constrained edges, where the shear forces are higher than the mid-section. Therefore, if the failure of the diaphragm triggers in the panel-to-panel connections, these locations are the most vulnerable within the diaphragm. Figure 4.8 and Figure 4.9 show the pushover curves for the floor Configuration 3 and Configuration 4. As expected, only the pushover curves of the 2nd floor bay in the models are different, and the pushover curves for the other two floor bays are identical. As already discussed in Section 4.1.2.2, the in-plane behaviour of each bay is independent from the rest, if the floor bays are clamped at the two ends. Hence, the 1st and the 3rd bays in the two configurations are identical. The introduction of the opening in the 2nd bay has reduced the in-plane stiffness of that bay by 53%. The reduced area was 52% of the area in that bay. This indicates that the reduction in the in-plane stiffness has been directly proportional to the opening area ratio. Figure 4.12 and Figure 4.13 show the deformation contour plots for Configuration 3 and Configuration 4, respectively. As with the first two configurations, the maximum deflection has occurred at the midspan of the longest bay, which is the 3rd bay in in the two configurations. The 2nd bay in the two models is connected to the constrained nodes 58 with a set of connections at both sides. That is the reason for the observed large slip between the panels and the constrained nodes at both sides of the 2nd bay. By comparing the push over curves for the four configurations, it is observed that by increasing the number of connected panels, the pushover curves tend to soften. One immediate reason for that is the increased floor span of the diaphragm, which deflects more under the same uniform pressure. Another reason is the addition of the panel-to-panel connections, which allow additional deformations due to slip between the panels. Figure 4.14 to Figure 4.15 show the stress components contour plots for Configuration 4. The normal stress components in X and Y directions, are depicted in Figure 4.14 and Figure 4.15, respectively. The σx component is almost uniform across the floor diaphragm with only variations being at the corners of each bay. Furthermore, the value of σx is very low everywhere. The above two observations suggest that the floor diaphragm has not experienced much flexure under in-plane loads, and the share of the flexural deformations in the overall in-plane deformation of the floor diaphragm is small. The σy component plot suggest that the there is some axial deformation present in the Y direction, which is higher at location of the load application (the lower edge of the diaphragm). According to the plot, σy is negative in almost the entire floor diaphragm. Only at the upper edge of the side panels in the 3rd bay some tension stresses have developed. Figure 4.15 shows the in-plane shear component contour plot. The in-plane shear stress σxy is higher close to the clamped ends and is the least at the mid-section of each floor bay. The maximum shear stress is observed at the lower corners of each floor bay. Therefore, it is conceivable that if the failure occurs in the CLT panels, these locations are the most likely locations to develop failures under in-plane shear stresses. 59 1.0 0.9 0.8 y = 0.1003x + 2E-06 0.7 Load Ratio 0.6 0.5 Configuration1 y = 0.0755x + 0.0161 0.4 Configuration2 0.3 0.2 0.1 0.0 0 5 10 15 20 Displacment of the Midspan (mm) Figure 4.7 Pushover curves for Configuration 1 and 2-total load of 1296 kN 1.0 0.9 y = 0.3366x + 3E-06 0.8 y = 0.2291x + 0.0498 Load Ratio 0.7 0.6 1st bay 0.5 2nd bay 0.4 3rd bay y = 0.1488x + 0.0298 0.3 0.2 0.1 0.0 0 2 4 6 8 10 Displacment of the Midspan (mm) Figure 4.8 Pushover curves for Configuration 3- total load of 4536 kN 60 1.0 0.9 y = 0.3366x + 3E-06 0.8 y = 0.2219x + 0.0416 0.7 Load Ratio 0.6 1st bay 0.5 y = 0.1488x + 0.0298 2nd bay 0.4 3rd bay 0.3 0.2 0.1 0.0 0 2 4 6 8 10 Displacment of the Midspan (mm) Figure 4.9 Pushover curves for Configuration 4- total load of 4012 kN Table 4.1 CLT diaphragm stiffness values for Configuration 1 and 2, and Configuration 3 and 4 Configuration 3 Configuration 4 Stiffness (kN/mm) Stiffness (kN/mm) Stiffness (kN/mm) 1st bay 508.8 1st bay 508.0 Configuration 1 130.0 2nd bay 230.9 2nd bay 107.4 Configuration 2 97.8 3rd bay 300.0 3rd bay 300.0 61 Figure 4.10 Deformation contour plot for Configuration 1 in the direction of loading Figure 4.11 Deformation contour plot for Configuration 2 in the direction of loading 62 Figure 4.12 Deformation contour plot for Configuration 3 in the direction of loading Figure 4.13 Deformation contour plot for Configuration 4 in the direction of loading 63 Figure 4.14 Stress contour plot for Configuration 4- normal stress component σx Figure 4.15 Stress contour plot for Configuration 4- normal stress component σy 64 Figure 4.16 Stress contour plot for Configuration 4- shear stress component σxy 4.2 Connection Model Validation The connection model introduced in Section 3.3 should be validated. In other words, the response of the numerical model should be checked against the experimental data. To do so, a 2D finite element model of the test specimens of the connection Layout E with the same boundary conditions is generated in ANSYS. The connection model included a set of nonlinear springs calibrated to the test data, which were acting in the direction of loading; and also a set of almost rigid linear springs acting in the perpendicular direction. This model is incorporated in the finite element model of the connection test specimen to simulate the test conditions. Therefore, the resulting load-displacement response of the finite element model includes the combined effect of the springs and the CLT panels cut-offs. If the load- 65 displacement response of the finite element model corresponds to that of the average response curve of the connection tests, then the model is validated. The model generation follows the same procedure as described in Section 4.1 for the diaphragm models. In fact this is the main reason of including this section in the current chapter, instead of Chapter 3. The implementation of the connection model in ANSYS was already described in Section 3.3.3. For the CLT panels, due to the scale of the test, a finer mesh with 20×20 mm PLANE42 elements are used. This mesh size is half the size of the mesh size used for the diaphragm models. The response curve of the COMBIN39 springs with 20 mm tributary length is shown in Table 3.4. The dimensions of the CLT cut-offs in the test specimen depicted in Figure 3.2, Pat(c) are rounded off in the finite element model to match the 20 mm size of the mesh. A uniform pressure load is applied to the upper edge of the middle panel. The amount of the pressure load is calculated based on a total force of 62 kN and a 200×100 mm cross sectional area. This load is close to the capacity of the connection from the average response curve. The pushover analysis is performed as described in Section 4.1 and the results are summarized in Table 4.2. The response of the validation model is compared with the average response curve and the piecewise linear approximation for the connection Layout E in Figure 4.17. From the results, it is evident that the model response agrees well with the piecewise linear approximation. The initial stiffness of the validation model in general is 5-6% less than the other two curves. This difference is partly due to the choice of mechanical properties, which may not be the same as those of the tested specimens, and partly due to the slight deformation of the wood in the direction of loading. Nevertheless this difference is negligible with regard to the existing uncertainty in the average response curve. Therefore, the performance of the connection model is 66 acceptable and it is validated. Figure 4.18 shows the deformed shape plot of the validation model under the applied load of 62 kN, along with the undeformed edge. The figure suggests that almost the entire deformation in the modelled test specimen has taken place in the connection springs, and it is in the form of slip between the CLT panel cut-offs. This further confirms the agreement of the model outputs and the test results. 70 60 50 Load (KN) 40 30 20 10 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Displacement (mm) Validation Model Response Piecewise Linear Approx Average Curve Figure 4.17 Comparison of the validation model response with the average response curve and the piecewise linear approximation for the connection Layout E 67 Figure 4.18 Deformed shape and undeformed edge plot for the validation model-Total load of 64kN Table 4.2 Validation model response for the connection Layout E Displacement (mm) 0.110 0.221 0.386 0.634 Load (kN) 2.07 4.13 7.23 11.88 0.965 18.08 1.296 24.28 1.627 1.958 2.376 2.882 3.389 3.663 4.350 30.48 36.68 42.88 49.08 55.28 58.64 62.00 68 Chapter 5: Analysis of CLT Buildings The main focus of this chapter is to study the distribution of lateral loads in CLT buildings. The previously generated CLT floor models in Chapter 4 are expanded here to include shearwalls and form CLT building models. In Section 5.2 two common hand calculation methods for determining the lateral load distribution to the shearwalls, namely the tributary area method and the stiffness method, are introduced with an example. A systematic approach for finding the lateral load distribution in any type of building is presented in Section 5.3. The finite element models of the CLT buildings are developed next in Section 5.4, and the results of the analyses are demonstrated and discussed in the rest of the chapter. 5.1 Lateral Load Distribution to Shearwalls One of the challenges involved in designing buildings for the lateral loads, is to find the share of the lateral load attracted to each shearwall. One key factor in determining the distribution of the lateral loads onto the shearwalls is the assumption made for the in-plane behaviour of floor diaphragms. Two common assumptions exist for the in-plane behaviour of a floor diaphragm, in case a full finite element analysis is not available (Prion and Lam 2003); namely a flexible diaphragm assumption and a rigid diaphragm assumption. In case of a flexible diaphragm, the lateral load is distributed to the shearwalls according to the tributary area associated with each shearwall. On the other hand for a rigid diaphragm, the diaphragm is assumed to move as a rigid body and the lateral load is distributed to the shearwalls according to their relative stiffness. Additionally, the torsional effects are taken into account. The above two assumptions enable designers to utilize simple hand-calculation methods to distribute the lateral load to shearwalls. Two of the renowned hand calculation 69 methods are tributary area and stiffness methods. As suggested by the names, the tributary area method is applicable to completely flexible diaphragms, while the stiffness method is suitable for completely rigid diaphragms. It is not known yet whether CLT floor diaphragms should be considered as flexible or rigid for design purposes. Since CLT panels are one type of wood products, it is useful to compare the in-plane behaviour of CLT floor diaphragms with other types of wood floor diaphragms. The in-plane behaviour of wood diaphragms has been a research topic for a considerable number of recent studies (Filiatrault et al. 2002; Bott 2005; Brignola et al. 2008; Pathak 2009; Pang and Rosowsky 2010). In practice, wood diaphragms are generally assumed to be flexible (Canadian Wood Council 2010). But the validity of this assumption is questionable. This due to the fact that in most cases the behaviour of wood diaphragms is rather semi-rigid. Therefore, neither of the above-mentioned two methods provides an accurate estimate of the lateral load distribution (Bott 2005; Pang and Rosowsky 2010). Considering the above issue, the validity of the flexible or rigid diaphragm assumption should also be also investigated for CLT floor diaphragms. This subject is addressed in the following section and the next chapter. 5.2 Hand Calculation Methods for Lateral Load Distribution As mentioned earlier the two hand calculation methods for lateral load distribution to the shearwalls are tributary area and stiffness methods. To demonstrate the methods, a simple CLT building is examined here. To be able to assess the accuracy of the methods, the building is modelled in ANSYS as well, and the results of the hand calculation methods are compared with the finite element model outputs. 70 The one-story building has the same floor plan as Configuration 4, depicted in Figure 4.4. The examined building is referred to as the “demonstration building” here. The story height of the demonstration building is 3 m. The floor diaphragm has a thickness of 150 mm. This thickness is selected because it matches the thickness of the floor configurations of Chapter 4. There are four shearwalls in the assumed direction of loading (Y direction). The outer walls (A and D) and the inner walls (B and C) have thickness of 150 mm and 100 mm, respectively. All shearwalls have rectangular cross-section and are 5 m long. They are assumed to have a 10000 MPa Young’smodulusofelasticityandPoissonratioof 5 A uniform in-plane pressure of 2.8 MPa is imposed to the lower edge of the diaphragm resulting in a total force of 4012 kN. This load is the same as the failure load for the floor Configuration 4 in Section 4.1.4. In the tributary area method, the lateral load attracted to each shearwall is calculated by the following formula: ( ) (5.1) Where Fi is the lateral load in Shearwall i; Ai is the tributary area associated with Shearwall i; A is the total area of the diaphragm; and F is the total lateral load. For the floor Configuration 4, each shearwall gets half of the total force in the corresponding bay. In this way, Shearwall A and D each get half of the total applied forces in the 1st and the 3rd bay, Shearwall B gets shares of the lateral force from the 1st and 2nd bay, and Shearwall C from the 2nd and the 3rd bay. Table 5.1 shows the lateral load in each shearwall for Configuration 4, calculated using Equation (5.1). 71 Table 5.1 Lateral load distribution with the tributary area method for the demonstration building Shearwall A B C D Ai (m2) 9.00 11.88 14.88 12.00 Ai/A 0.19 0.25 0.31 0.25 Fi (kN) 756.0 997.9 1249.9 1008.0 Lateral load share (%) 18.84 24.87 31.16 25.13 In the stiffness method, first the stiffness matrices of the shearwalls are defined for the existing degrees of freedom. Each shearwall has three degrees of freedom, two translations in the X and Y directions, and one rotation around the Z axis. The local stiffness matrix for each of the shearwalls could be obtained as follows: ki (5.2) [ Where ki is the stiffness matrix for Shearwall i; ] i is a reduction factor; Ix and Iy are the second moments of area around the X and Y axis; J is the polar moment of inertia for the shearwall cross section; H is the height of the shearwall; and E and G are moduli of elasticity and shear for the shearwalls, respectively. The first two diagonal entries in the above matrix are in fact the bending stiffness of the shearwall in the X and Y directions, while the third entry is the rotational stiffness of the shearwall for the Z direction. The parameters of the stiffness matrix could be obtained using the following expressions: (5.3) 72 (5.4) (5.5) (5.6) In the above expressions, Avx is the shear area of the shearwall cross section; L, and t are the length, and thickness of the shearwall respectively. It should be recognized that y is zero, since here, shear deformation is not an issue in that direction. Table 5.2 shows the material properties assigned to the shearwalls of the demonstration building. It is assumed that the shearwalls are made out of wood with isotropic material behaviour. Therefore, isotopic material properties commonly assumed for wood are assigned to the shearwalls. Table 5.3 shows the parameters used in the stiffness matrix of Equation (5.2). These values are calculated using the values provided in Table 5.2 and the relations in Equation (5.3) to Equation (5.6). The resulting local stiffness matrices for the four shearwalls of the demonstration building are shown in Table 5.4. Table 5.2 Material properties for the shearwalls of the demonstration building Shearwall properties E (Pa) 1.0E+10 ν 0.35 G (Pa) 3.7E+09 Table 5.3 Geometrical properties, moments of inertia, shear area and reduction factors for the demonstration building Wall T(m) L(m) Ix (m4) Iy (m4) J (m4) Avx (m2) A B C D 0.15 0.1 0.1 0.15 5 5 5 5 1.563E+00 1.042E+00 1.042E+00 1.563E+00 1.406E-03 4.167E-04 4.167E-04 1.406E-03 5.625E-03 1.667E-03 1.667E-03 5.625E-03 7.50E-01 5.00E-01 5.00E-01 7.50E-01 x 7.50E+00 7.50E+00 7.50E+00 7.50E+00 y 0 0 0 0 73 Table 5.4 Local stiffness matrices for the shearwalls of the demonstration building kA= 8.17E+08 0 0 0 6.25E+06 0 0 0 6.94E+06 kB= 5.45E+08 0 0 0 1.85E+06 0 0 0 2.06E+06 kC= 5.45E+08 0 0 0 1.85E+06 0 0 0 2.06E+06 kD= 8.17E+08 0 0 0 6.25E+06 0 0 0 6.94E+06 The local stiffness matrix calculated with Equation (5.1), should be first transformed to the global coordinates. The origin of the local coordinate systems for each shearwalls is located at the center of that shearwall. The bottom-left corner of Configuration 4 is selected here as the origin of the global coordinates. The transformation matrices between the two systems are shown in Table 5.5 for each shearwall. Then the global stiffness matrix is obtained by adding the transformed stiffness matrices of all the shearwalls: ∑ i ki i (5.7) In the above formula, KG is the global stiffness matrix; Ti is the transformation matrix for Shearwalls i; and ki is the local stiffness matrix of the Shearwall i. Table 5.5 contains the global stiffness matrices calculated for each shearwall and the whole system. In the next step, the global force vector should be obtained by calculating the applied force to the building for each degree of freedom, with respect to the global coordinate system. 74 Table 5.5 Transformation matrices from the local coordinate to the global coordinate, for the shearwalls of the demonstration building TA= 1 0 0 0 1 0 2.5 0 1 TB= 1 0 0 0 1 0 2.5 -3.6 1 TC= 1 0 0 0 1 0 2.5 -6 1 TD= 1 0 0 0 1 0 2.5 -10.8 1 The linear system of equations is solved for the unknown global displacements uG, by inversing the global stiffness matrix and multiplying it by the global force vector FG as follows: u (5.8) Table 5.6 shows the global force and displacement vectors for the demonstration building. The quantity of interest, which is the force in each shearwall, now can be obtained by multiplying the global stiffness matrix of each shearwall FGi, by the global displacement vector: i gi u ( i ki i ) u (5.9) 75 Table 5.6 Global Stiffness matrices for the shearwalls in the demonstration building and the final global stiffness matrix of the whole system KAG= 8.17E+08 0 2.04E+09 0 6.25E+06 0 2.04E+09 0 5.11E+09 KBG= 5.45E+08 0 1.36E+09 0 1.85E+06 -6.67E+06 1.36E+09 -6.67E+06 3.43E+09 KCG= 5.45E+08 0 1.36E+09 0 1.85E+06 -1.1E+07 1.36E+09 -1.11E+07 3.47E+09 KDG= 8.17E+08 0 2.04E+09 0 6.25E+06 -6.8E+07 2.04E+09 -6.75E+07 5.84E+09 KG= 2.72E+09 0.00E+00 6.81E+09 0.00E+00 1.62E+07 -8.53E+07 6.81E+09 -8.53E+07 1.79E+10 Table 5.7 Global force and displacement vectors for the demonstration building FG= 0 4.01E+06 -2.20E+07 uG= 0.006 0.236 -0.002 The lateral load absorbed by each shearwall, is the entry in the force vector, which corresponds to the lateral load direction (in this case Y direction). Table 5.8 contains the global force vectors for each shearwall. The highlighted entry in each vector is the lateral load attracted to that shearwall. 76 Table 5.8 Global force vector for each shearwall in the demonstration building FAG= 0.00E+00 1.47E+06 -1.54E+04 FBG= 0.00E+00 4.52E+05 -1.63E+06 FCG= 0.00E+00 4.62E+05 -2.77E+06 FBG= 0.00E+00 1.62E+06 -1.76E+07 The results of force distribution based on the tributary area and stiffness methods are summarized in Table 5.9. It is an immediate observation that the share of the lateral load received by each shearwall as well as the distribution pattern varies significantly for the two methods. While for the tributary area method the middle shearwalls B and C take the most of the lateral load, it is the opposite for the stiffness method. In fact, since the exterior Shearwalls A and D are stiffer than the middle Shearwalls B and C, they attract more of the lateral loads in case of the rigid diaphragm assumption. Table 5.9 Comparison of the lateral load distribution to the shearwalls by tributary area and stiffness methods Share of the lateral force in each shearwall (% of the total force) Wall A B C D Tributary Area method 18.8 24.9 31.2 25.1 Stiffness method 36.7 11.3 11.5 40.5 77 5.3 Lateral Load Distribution Flowchart Due to the differences in the in-plane behaviour of floor diaphragms and other involving factors, the lateral load distribution within a structure is case dependent. However, it is possible to provide designers with a systematic approach to find the lateral load distribution within any type of building, including CLT buildings. The step by step procedure is based on the idea of categorizing floor diaphragms as flexible, rigid, or semi-rigid diaphragms. The analysis results of this chapter and the next one help to identify which category best describes the in-plane behaviour of CLT floor diaphragms. This procedure is demonstrated by a flowchart in Figure 5.2 at the end of this section, and it includes the following steps: 1. The first step is a preparation step, in which the building configuration should be established. The plan view of the wall system and the location of shearwalls relative to each other and to the floor diaphragm are determined. This is normally done with regards to the limitations imposed by the architectural design of the building and the structural performance considerations. The building configuration is significantly important for the lateral load design of the building. FEMA 454 (2006) states that although the total imposed lateral load is obtained based on code requirements, the distribution of the forces in the structure is determined by the building configuration. 2. In the second step it should be identified whether the structural system of the building is statically determinate or not. If the structural system is statically determinate, then the lateral load distribution is not dependent on the floor diaphragm rigidity. The distribution of the lateral load in this case can be found by the tributary area method. A common case is a one-bay floor diaphragm supported on two identical shearwalls, 78 similar to Configuration 1 and Configuration 2 shown in Figure 4.1 and Figure 4.2. For both configurations the lateral load is distributed in the same way, although the floor systems are significantly different. In fact, in both cases the floor diaphragm acts like a simply supported deep beam and therefore each shearwall receives half of the lateral load. However, it should be recognized that even for the above cases the floor diaphragm rigidity matters, if the in-plane deformation of the floor diaphragm is an issue for the serviceability design of the building. 3. If the system is statically indeterminate, then the floor diaphragm rigidity affects the lateral load distribution to the shearwalls. At this stage, it should be identified whether the floor diaphragm experiences torsion or not. This is a key question, since additional torsional forces are exerted to the shearwalls, which are not distributed in the same way as the lateral forces. To determine if the torsional forces are induced within the structure, the center of rigidity concept is normally used. The center of rigidity of a structural system is a point, at which the application of the lateral load will cause no rotation. This point is determined based on the stiffness of the shearwalls and their locations in the plan view of the floor diaphragm. Figure 5.1 demonstrates the center of rigidity and the center of mass for a simple rectangular floor with three shearwalls. In this figure CR and CM are the center of rigidity and the center of mass, respectively; e is the eccentricity between the two centers; and F is the lateral load. The lateral load acts through the center of mass and it generates a torsional force equal to F×e. The location of the center of rigidity can be found using the following simplified expressions: 79 ∑ ∑ ∑ ∑ (5.10) Where xCR and yCR are the coordinates of the center of rigidity with respect to an arbitrary origin in the plane of the floor diaphragm; xi and yi are the coordinates of the center of Shearwall i with respect to the same origin; kxi and kyi are the in-plane stiffness of the Shearwall i in the X and Y directions. It should be noted that the accuracy of the location of center of rigidity is dependent on the accuracy of the stiffness of shearwalls. In case such information is not available, engineering judgment may be used to assign estimate stiffness values to the shearwalls. F CR CM e Figure 5.1 Center of rigidity and center of mass 4. If the center of mass coincides with the center of rigidity, no additional torsional forces are induced in the shearwalls. To distribute the lateral load, it should be determined, if the floor diaphragm is flexible, or rigid, or semi-rigid. As mentioned before, if the diaphragm is flexible then the lateral load is distributed according to the tributary area method. If on the other hand the floor diaphragm is rigid, the stiffness method is used. 80 Establish the plan view of the wall system Is the system statically determinate? Diaphragm rigidity is immaterial. Distribute forcesby“tributaryarea” Yes No Does the center of mass of the diaphragm coincide with its center of rigidity? Yes Is diaphragm flexible? Yes Distribute forces by“tributaryarea” Yes Distribute forces by “stiffness method” No No Is diaphragm rigid? No The distribution of forces is found by “structural analysis” taking into account accurate rigidity of the diaphragm, and torsional forces Diaphragm rigidity is important. The distribution of forces are found by “structuralanalysis”takingintoaccount accurate rigidity of the diaphragm Figure 5.2 Lateral load distribution flowchart 81 In case the floor diaphragm does not fit into any of the above categories, then it is semi-rigid. For a semi-rigid floor diaphragm, having an accurate estimate of the diaphragm rigidity is important for identifying the lateral load distribution. In this case, a structural analysis is needed, for which the floor diaphragm in-plane behaviour is modelled accurately. 5. If the center of mass does not coincide with the center of rigidity, then the torsional forces should be accounted for as well. In this case, a complete structural analysis is necessary to find the distribution of the lateral forces. The floor diaphragms should be modelled with accurate in-plane behaviour. However, if it is known in advance that the floor diaphragm is completely flexible or rigid, then simplifications may be applicable to the analysis. As mentioned earlier, the floor diaphragm undergoes torsion if there is an eccentricity between the center of mass and the center of rigidity. Floor configuration irregularities and the asymmetric arrangement of shearwalls are the main cause of such eccentricities, and thus invoke torsion. Some floor configurations with plan irregularities, which are more likely to produce such effect, are depicted in Figure 5.3. In the Seismic Design Handbook (Naeim 2001), it is argued that the irregularities depicted in Figure 5.3, are part of the types of “irregular structures or framing systems” introduced in the Commentary to the SEAOC (1980). Design of such structures needs extra analysis and dynamic considerations compared to the code requirements. Therefore, for these types of floor diaphragms, the accurate modelling of the in-plane behaviour of the floor diaphragm is even more critical for the lateral load design of the building. 82 (1) (4) (2) (5) (3) (6) Figure 5.3 Some typical floor configurations as categorized in FEMA 454 (2006) and Commentary to the SEAOC (1980) 5.4 Finite Element Modelling of CLT Buildings in ANSYS The finite element models of the CLT floors were developed in Section 4.1. In that section, a procedure to generate 2D models of the floor diaphragms in ANSYS was introduced and non-linear static analyses were done to obtain the desired push-over curves. In this section, the CLT floor models are further extended to CLT building models. In other words, it is desired to build 2D finite element models of CLT buildings based on the previously developed floor models. The objective is to study the in-plane behaviour of CLT floors when they are attached to the shearwalls and to investigate the lateral load distribution to the shearwalls. To accomplish such an objective, the shearwall is assumed to act as a system of parallel springs with zero length. These springs are distributed along the entire length of the shearwall. They are attached at one end to the nodes of the floor model at the corresponding shearwall location, and at the other end are constrained from movement in both horizontal X and Y directions. Linear springs are used for this purpose, since the main 83 interest is to find the distribution of the lateral loads to the shearwalls, prior to initiation of non-linear deformations in the shearwalls. Nevertheless, non-linear springs calibrated to the response of real CLT shearwalls could also be utilized here. The stiffness value assigned to the linear springs of the shearwalls should in general include the effects of the diaphragmshearwall connections, the shearwall itself, and the shearwall-base connections. Two separate sets of linear COMBIN14 springs are utilized in place of each shearwall, one acting in the X direction and one in the Y direction.COMBIN14 springs were already introduced in Section 3.3.3. Since the linear springs form a parallel system, the stiffness of each spring is obtained by dividing the total stiffness of the shearwall by the number of springs for both X and Y directions, as follows: , (5.11) where kx and kx are the stiffness of the COMBIN14 springs in the X and Y directions respectively; Kx and Kx are the total stiffness of the shearwall in the X and Y directions respectively; and ns is the number of springs in each direction. The number of springs for both of the directions is equal, and is the same as the number of nodes at the location of each shearwall. The total stiffness of the shearwall can be determined based on the available test results, if specific type of shearwall is to be used in the model. Herein however, four sets of stiffness values are assigned to the shearwalls, which cover a wide range of stiffness values reasonable for different shearwalls. The four sets represent the cases of significantly stiff, moderately stiff, normal, and flexible shearwalls respectively. It should be noted that, the above terms are selected to show how the stiffness values of each set is compared to the other, and otherwise they possess no absolute meaning. The last two sets of stiffness values 84 are quite close to the values reported by Popovski et al. (2010; 2011) for the in-plane stiffness of CLT shearwalls. The total stiffness of each shearwall for each set, and the assigned values of stiffness to the linear COMBIN14 springs, are summarized in Table 5.10. The stiffness values of the COMBIN14 springs are calculated from the total stiffness values of the shearwalls using Equation (5.11). Table 5.10 Total stiffness values for the four sets of shearwalls and the according stiffness values for the COMBIN14 springs Set 1 2 3 4 Shearwall A and D B and C A and D B and C A and D B and C A and D B and C Total stiffness of shearwalls (kN/mm) KX KY 1.5 1700 0.5 1200 0.2 450 0.02 250 0.01 150 0.001 100 0.003 50 0.0007 30 Stiffness of COMBIN14 springs (kN/mm) kX kY 0.012 13.492 0.004 9.524 0.002 3.571 0.0002 1.984 0.0001 1.190 0.00001 0.794 0.00002 0.367 0.00001 0.245 The floor model of Configuration 3 is selected here to be further expanded to include the shearwalls shown in Figure 4.3. The resulting building model is referred to here as “Configuration building ”Each of the four sets of shearwalls is added to Configuration 3 in a separate analysis. A uniform pressure load of 2.8 MPa is imposed to the lower edge of the floor diaphragm, resulting in a total force of 4536 kN. This force is in fact the same as the failure load for the floor Configuration 3 in Section 4.1.4. 85 5.4.1 Analysis Results Non-linear static analysis is performed for Configuration 3 building with four different sets of shearwalls. The pushover curves for the building with the 1st set of shearwalls are shown in Figure 5.4. The dashed lines in this figure are the pushover curves for the floor Configuration 3, already presented in Chapter 4. Adding the shearwalls evidently results in increased total in-plane displacements of the diaphragms, which is indicated in the figure by the horizontal stretch of the dashed pushover curves to the solid ones. Figure 5.4 suggests that the response of shearwalls affect the in-plane response of floor diaphragms. For the particular case presented here, this effect is a simple stretch of the pushover curve with no change in the shape of the curve. This is due to the assumed linear behaviour of the shearwalls. However, in reality the behaviour of shearwalls is normally nonlinear. Consequently, the shape of the pushover curve for the in-plane behaviour of the floor diaphragms may change based on the non-linear response of the shearwalls. In other words, the in-plane response of floor diaphragms is not independent of the response of shearwalls. It should be reminded that here, shearwall response includes the response of the floor-toshearwalls and shearwall-to-base connections as well. The main quantity of interest here, which is the lateral load in each shearwall, is obtained by adding the reaction forces in the direction of loading at the location of that shearwall. The lateral load is also distributed for Configuration 3 building, using the tributary area and stiffness methods. Comparing the results of the finite element model and the two hand calculations methods for the lateral load distribution, shows whether the floor diaphragm should be considered as flexible or rigid. If the distribution of the lateral load is close to the results of the tributary area method, then the floor diaphragm is flexible; and if 86 on the other hand the lateral load matches the results of the stiffness method, the floor diaphragm is rigid. In case the distribution of the lateral load falls between the results of the two methods, then the floor diaphragm can be considered as semi-rigid. However, as already discussed in Section 5.3, for a statically determinate structural system the lateral load distribution is not dependent on the in-plane behaviour of the floor diaphragm. Therefore, the comparison of the lateral load distributions for different approaches is only reasonable if the system is statically indeterminate; which is the case for the Configuration 3 building. Figure 5.5 compares the lateral load distribution for the finite element model and the two hand calculations methods. The share of the lateral load in percentage of the total lateral load is shown for each shearwall. By investigating the figure, it seems that for the utilized type of connections and CLT panels, the CLT diaphragm distributes the load almost according to the tributary area method. For the four sets of the shearwalls utilized in the building model, as the shearwalls becomes more flexible from the 1st set to the 4th set, the lateral load distribution tends to differ more from the tributary area results. For all the four sets of the shearwalls the floor diaphragm model was identical. Therefore, this observation suggests that, whether the floor diaphragm is flexible or rigid depends on the relative stiffness of the floor diaphragm and the shearwalls. With the stiffer shearwalls for the Configuration 3 building, the floor diaphragm is more flexible relative to the shearwalls, than the case where more flexible shearwalls are utilized. Nevertheless, the four sets of shearwalls used for the building model, encompass a wide range of practical stiffness values for shearwalls. Consequently, it is justifiable to state that the CLT diaphragm model at hand is flexible to semi-rigid. 87 Based on the above observations, it is possible to find the lateral load distribution for CLT floor diaphragms based on the tributary area method, and then apply a safety factor to the load distribution results to account for the semi-rigid diaphragm behaviour. For the Configuration 3 building, changing the stiffness values of shearwalls from the very stiff case to the very flexible ones has changed the lateral load in each shearwalls by maximum of only 4%. However, it should be recognized that if the floor diaphragm is expected to have a much higher stiffness than the shearwalls, then performing structural analysis is necessary for finding the accurate lateral load distribution. Figure 5.6 to Figure 5.9 show the deformation contour plot in the direction of loading for Configuration 3 building. Each figure corresponds to the building with a different set of shearwalls, starting with the 1st set in Figure 5.6 to the 4th set in Figure 5.9. By comparing the figures, it is evident that for stiffer shearwalls, shown in Figure 5.6 and Figure 5.7, the inplane deformation of the CLT floor diaphragm in each bay is quite independent from the rest. As already discussed in the previous chapter, this type of deformation pattern indicates that the floor diaphragm is acting as flexible diaphragm. On the other hand, for more flexible shearwalls, shown in Figure 5.8 and Figure 5.9, the CLT floor diaphragm tends to deform more or less as a single unit. The defamations of the floor bays are related to each other and follow the overall deformed shape of the floor diaphragm. This type of deformation pattern implies that the floor diaphragm is acting as a rigid diaphragm. Moreover, the amount of the slip between the CLT panels has changed from Figure 5.6 to Figure 5.9. In Figure 5.9, the slip deformation between the CLT panels closer to the middle of the floor diaphragm has decreased, and the panels has deformed fairly as an integrated body. 88 1.0 0.9 1st bay with shearwalls 2nd bay with shearwalls 3rd bay with shearwalls 1st bay w/o shearwalls 2nd bay w/o shearwalls 3rd bay w/o shearwalls 0.8 0.7 Load Ratio 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 2 4 6 8 10 Displacment of the Midspan (mm) Figure 5.4 Pushover curves for Configuration 3 building with the 1st set of shearwalls 40.50 22.62 22.22 22.07 24.43 27.05 32.89 tributary area method 1st set 2nd set 3rd set 11.50 20 11.30 25 27.34 30 25.41 23.30 27.78 27.98 35 30.78 27.34 33.33 33.48 36.70 40 16.67 16.47 17.16 19.38 22.32 Shear Force (% of the total lateral force) 45 15 4th set stiffness method 10 5 0 a b c d Shearwall Figure 5.5 Lateral load distribution for Configuration 3 building with the four sets of shearwalls 89 Figure 5.6 Deformation contour plot in the direction of loading, for the Confiuration3 building with the 1st set of shearwalls Figure 5.7 Deformation contour plot in the direction of loading, for the Configuration 3 building with the 2nd set of shearwalls 90 Figure 5.8 Deformation contour plot in the direction of loading, for the Configuration 3 building with the 3rd set of shearwalls Figure 5.9 Deformation contour plot in the direction of loading, for the Configuration 3 building with the 4th set of shearwalls 91 5.4.2 Comparison of the ANSYS Outputs with the Force Distribution Methods The demonstration building examined in Section 5.2 is modelled in ANSYS. The stiffness of the linear springs utilized in place of the shearwalls is calculated based on the properties of the shearwalls described in Section 5.2. The obtained stiffness values in this case are similar to the 1st set of shearwalls. Therefore, the 1st set of shearwalls is incorporated in the building model. The results of the lateral load distribution extracted from the ANSYS outputs along with results of the tributary area and stiffness methods are summarized in Table 5.11. As suggested by the table, the share of the lateral load for each shearwall is quite similar for the ANSYS outputs and the tributary area method. Therefore, as with the Configuration 3 building, the CLT floor in the demonstration building distributes the load almost according to the tributary area method. Table 5.11 Comparison of the ANSYS model results for lateral load distribution with tributary area and stiffness methods Wall Tributary Area method Stiffness method ANSYS output A 18.8 36.7 18.5 % of total force in each shear wall B C 24.9 31.2 11.3 11.5 25.2 31.5 D 25.1 40.5 24.8 92 Chapter 6: Parametric Study Several parameters affect the in-plane behaviour of CLT diaphragms. The main focus of this chapter is to identify the most influential parameters and to specify their impact on the in-plane behaviour of CLT diaphragms and the lateral load distribution in CLT buildings. 6.1 Influential Parameters In order to do a parametric study, first the most likely contributing parameters are identified and nominated for the evaluation. In the next step, a reasonable range for each of the nominated parameters is considered for the analysis. In case of a numerical model of a structural system, the most influential parameters are nominated for the parametric study, based on the target structural behaviour that the model aims to capture. For the previously generated ANSYS models, the target behaviour is the in-plane behaviour of the CLT floor diaphragms and the lateral load distribution to the shearwalls. In this context, the contributing parameters can be classified into the following two categories: 1. Material Properties: These parameters include: mechanical properties of the CLT panels, the parameters defining the non-linear response of the CLT panel-to-panel connections, and stiffness of the shearwalls. 2. Geometrical Parameters: These parameters include: floor diaphragm configuration, number of connected CLT panels, dimensions of the panels. 93 In the subsequent sections, each of the above parameters is addressed separately. The models analysed in the previous chapters, are selected here as the base cases for the parametric study. The contributing parameters of the base cases are varied in a reasonable range and the results of the new ANSYS analyses are compared to the base cases. In this way it is possible to identify the impact of varying each parameter on the structural behaviour of interest. 6.1.1 Material Properties Assumptions made for the material properties of the elements incorporated in the ANSYS models, have a great impact on the outcome of the analyses. One major set of the contributing material properties are the parameters, which define the non-linear response of the panel-to-panel connections. For the connection layout E described in Section 3.2, which was utilized in the ANSYS CLT floor models, the average non-linear response of the connection is described by a piecewise linear curve. The piecewise linear curve has three segments up to the failure point, which can be described by four points and has three independent stiffness values for each segment (Figure 3.7). Since the main interest here is to capture the initial stiffness of the CLT diaphragms, only the stiffness of the first segment of the piecewise linear curve will be considered for the parametric study. To recognize the effect of the initial stiffness of the panel-to-panel connections on the in-plane stiffness of CLT diaphragms, the floor Configuration 2 shown in Figure 4.2, is exploited again. Five cases are compared, for which the floor model and the boundary conditions are the same as the Configuration 2 floor model, but the initial stiffness of the panel-to-panel connection is changed. The base case is the Configuration 2 floor model with the connection Layout E. The other four cases include two models with an initial stiffness of the connections reduced to 94 0.1, and 0.01 of the Layout E stiffness; and two more models with the initial stiffness increased to 10, and 1000 times of the Layout E stiffness. The initial stiffness of the connection Layout E varies due the change in the type, size, and density of the connectors, i.e. the distance and number of the self-tapping wood screws utilized in the connection. Therefore, the two extreme cases, where the initial stiffness of the connection has changed to 0.01 and 1000 of the Layout E stiffness, are not practical. However, studying these extreme cases allows finding the trend of the change in the in-plane stiffness of the CLT floor diaphragm with the change in the initial stiffness of the panel-to-panel connection; or in other words the change in the connector density, type, and size for Layout E. The in-plane stiffness of the diaphragms is obtained for each case with the same procedure as for the previous analyses. The results are plotted in the logarithmic scale in Figure 6.1, where K1 is the initial stiffness of the connection Layout E. The in-plane stiffness values are shown in the data labels as well. The figure shows how the in-plane stiffness of the floor Configuration 2 has changed against the initial stiffness of the panel-to-panel connections in terms of K1. It is evident from the figure that the initial stiffness of the panelto-panel connection affects the in-plane stiffness of the CLT diaphragm significantly. It is observed that by increasing the initial stiffness of the panel-to-panel connection, the in-plane stiffness of the floor diaphragm also increases. However, the relation between the two stiffness values is not linear. In fact the curve in Figure 6.1 is non-linear in the logarithmic scale, and it demonstrates faster changes for lower values of the initial stiffness. For instance, by 10 and 1000 times increasing the initial stiffness of the panel-to-panel connection, the inplane stiffness of the floor diaphragm has increased by factor of 2.2 and 2.6 respectively; 95 In-plane Stiffness of the CLT Diaphrgam (KN/m) 1000 247.7 218.8 251.2 97.8 100 17 10 1.3 1 0.01 0.1 1 10 100 1000 Intitial Stiffness of the Panel-to-Panel Connection (× K1) Figure 6.1 In-plane stiffness of the CLT diaphragm for different initial stiffness of the panel-to-panel connection And by 10 and 100 times reducing the initial stiffness of the panel-to-panel connection, the in-plane stiffness has dropped by factor of 5.8 and 75 respectively. A practical example for the above observation is that by changing the spacing between the screws in the connection Layout E, the in-plane stiffness of the CLT floor diaphragm would not be change by the same factor. Figure 6.2 and Figure 6.3 depict the deformation contour plots for the two extreme initial stiffness of the panel-to-panel connection. Figure 6.2 shows that for lower initial stiffness of the panel-to-panel connection, the integrity of the diaphragm is diminished and the CLT panels are allowed to move freely relative to each other. In this case, a significant portion of the in-plane deformation of the floor diaphragm is in the form of slip between the panels and takes place in the connections. 96 Figure 6.2 Deformation contour plot for Configuration 2 in the direction of loading –initial stiffness of the panel-to-panel connections reduced to 0.01 of the Layout E-total load of 1296kN Figure 6.3 Deformation contour plot for Configuration 2 in the direction of loading –initial stiffness of the panel-to-panel connections increased to 1000 times of the Layout E-total load of 1296kN 97 This condition results in excessively low values of in-plane stiffness and large in-plane deformations for CLT diaphragms, which would entail serviceability design issues. On the other hand, increasing the initial stiffness of the panel-to-panel connection decreases the slip between the panels and will add to the in-plane stiffness. For the case shown in Figure 6.3, it is observed that there is no slip between the panels and all the deformations takes place in the CLT panels. By comparing the two figures and considering the above observations, it can be concluded that for softer panel-to-panel connections, most of the deformation takes place in the connections. Therefore the in-plane behaviour of the CLT floor diaphragm is controlled by the panel-to-panel connection behaviour. However, for stiffer panel-to-panel connections, most of the in-plane deformations occur in the CLT panels and therefor the in-plane behaviour of the CLT floor diaphragm is controlled by the CLT panels. This statement would also explain the trend observed previously in Figure 6.1. In this figure, the in-plane stiffness of the floor diaphragm changes much faster for lower values of the initial stiffness of the panel-to-panel connection, because the in-plane behaviour of the diaphragm is controlled by the connections. On the other hand, the change in the in-plane stiffness of the floor diaphragm for stiffer connections is smaller. This is due to the fact that in these cases the inplane behaviour of the floor diaphragm is controlled by the CLT-panels, which are similar for both cases. Therefore, increasing the initial stiffness of the connection has insignificant effect on the in-plane stiffness of the floor diaphragm. Another set of the contributing material properties are the mechanical properties of the CLT panels. As discussed earlier in Sections 2.2, the overall behaviour of CLT panels can be considered as linear elastic orthotropic with nine independent engineering constants. Here, 98 the in-plane shear modulus Gxy and the modulus of elasticity in the strong direction of panels or Ey, and in the weak direction Ex are considered for the parametric study. To study the effect of these parameters on the in-plane stiffness of CLT diaphragms, the floor Configuration 2 is used for the analysis. In one attempt, the values of Ey and Ex of the CLT panels are varied for Configuration 2. Each modulus is varied in a practical range reasonable for CLT panels. The in-plane stiffness values are extracted from the ANSYS analyses outputs and are summarized in Table 6.1. The results suggest that in a practical range of elastic moduli for CLT panels, the in-plane stiffness of the modelled CLT diaphragm is in-sensitive to change in Ey and Ex. This observation is rather trivial, since as discussed in Section 4.1.4, the share of the flexural deformations in the in-plane deformation of the modelled floor diaphragms is small. To substantiate the above statement, stress contour plots of Configuration 2 are shown in Figure 6.4 to Figure 6.6. In this case the Ey and Ex were 12000 and 4000 MPa respectively helowvaluesofσx and its uniformity shown in Figure 6.4, confirms again that the share of the in-plane flexural deformations is small. Figure 6.5 shows the distribution of σy. The figure suggests that as the diaphragm deforms under the in-plane load, the panels may press slightly against the neighbouring panels or slightly open up. This is more observable for the panels closer to the clamped ends. Nevertheless, the effect is quite slight and therefore is negligible. The shear component of the stress is shown in Figure 6.6. As expected, the figure points out that the panels experience high shear stresses and undergo significant in-plane shear deformations. 99 Figure 6.4 Stress contour plot for Configuration 4- normal stress component σx Figure 6.5 Stress contour plot for Configuration 4- normal stress component σy 100 Figure 6.6 Stress contour plot for Configuration 4- shear stress component σxy Table 6.1 In-plane stiffness of the CLT diaphragm for practical range of the elastic moduli Ex (MPa) Stiffness (kN/mm) Ey (MPa) 8000 12000 104.36 104.37 4000 4000 104.3 8000 - 108.51 - 12000 - 110.28 - In another attempt, Gxy is decreased to an extreme value of 100 MPa. The pushover analysis is done for Configuration 2 with the new value of Gxy, and the results are compared in Figure 6.7 with the base case, where Gxy is 600 MPa. The figure demonstrates the change in the in-plane stiffness of the CLT diaphragm against the number of the connected CLT panels, for each of the two values. The number of connected panels is a geometrical parameter, which is discussed in the next section. 101 In-plane Stiffness (KN/mm) 600 500 400 300 200 100 0 2 4 6 8 10 Number of Connected CLT Panels Gxy=100MPa Gxy=600MPa Figure 6.7 In-plane stiffness of the Configuration 2 CLT diaphragm for different number of connected panels and two different in-plane shear moduli of panels The number of CLT panels in Configuration 2 is changed, and for each case a separate model is generated and analysed in ANSYS. It is evident that the change in the shear modulus of the CLT panels affects the in-plane stiffness of the CLT floor diaphragm significantly. Figure 6.7 is referred to again and explained more thoroughly, once the number of the connected panels is addressed in the next section. The effect of panel-to-panel connections properties andpanels’mechanicalproperties on the in-plane deformation of CLT diaphragms is also demonstrated here by comparing three different cases. The first case, shown in Figure 5.9, is the Configuration 3 building with the 4th set of shearwalls, as described in Section 5.4. The second case, shown in Figure 6.8, shows the same model but with almost rigid panel-to-panel connections in the direction of loading. In this case a high value is assigned to the initial stiffness of the panel-to-panel connection model, which simulates the effect of rigid connections. The third case, shown in 102 Figure 6.9, is identical to the previous two models in the floor configuration and type of the shearwalls. However, the floor diaphragm is made of concrete with isotropic material properties. The assigned material properties include modulus of elasticity, E, of 30000 MPa and Poisson’s’ ratio, ν, of 0.25. By comparing the three figures, it is evident that from the first case to the third one, the in-plane deformation of the floor diaphragm has decreased. As mentioned before in this section, by making the panel-to-panel connections rigid the slip between the panels, and thus a major part of the in-plane deformation of the floor diaphragm is eliminated. The deformation pattern the CLT floor diaphragm with rigid connections in Figure 6.8, is close to the deformation pattern of the concrete floor in Figure 6.9. In both cases the in-plane deformation of the diaphragm is insignificant compared to the translational displacement of the whole diaphragm as a rigid body. Therefore, by eliminating the slip between the CLT panels, it seems that the floor diaphragm in-plane behaviour tends to become close to a rigid diaphragm. However, to substantiate the above statement, it is necessary to investigate the lateral load distribution for the three cases as well. As already discussed in Chapter 5, if the lateral load distribution conforms to the tributary are method results, then the diaphragm can be considered as flexible. On the other hand, if the distribution is close to the stiffness method results, then the floor diaphragm is rigid. The effect of shearwalls stiffness on the lateral load distribution in a CLT building system was investigated earlier in Section 5.4.1. Table 6.2 shows some of the previous results from Chapter 5, along with a number of additional cases, which allow a more comprehensive understanding of the in-plane behaviour of CLT floor diaphragms. 103 Figure 6.8 Deformation contour plot in the direction of loading for Configuration 3 building with the 4th set of shearwalls and rigid panel-to-panel connections Figure 6.9 Deformation contour plot in the direction of loading for the concrete slab floor with the 4th set of shearwalls 104 Table 6.2 Comparison of the ANSYS analysis results for the lateral load distribution with hand calculations methods Tributary Area Method Stiffness Method Concrete Floor (1st set of shearwalls) Concrete Floor (4th set of shearwalls) CLT Floor (1st set of shearwalls) CLT Floor (4th set of shearwalls) CLT Floor (4th set + rigid panel-to-panel connections) Share of the Force in each Shearwall (% of the Total Force) a b c d 16.7 27.8 33.3 22.2 36.7 11.3 11.5 40.5 21.2 24.8 27.6 26.4 27.6 19.9 20.6 32.0 16.5 28.0 33.5 22.1 22.3 23.3 27.3 27.0 24.7 22.1 23.7 29.4 The additional cases are all variations of the Configuration 3 building model. Two of the cases include the Configuration 3 building with concrete floor utilizing the 1st and the 4th set of the shearwalls. Two other cases are the Configuration 3 building model with the 1st and the 4th set of the shearwalls. And the last case is the Configuration 3 building model, with rigid pane-to-panel connections. Three of these cases are the same cases, which were just addressed in this section for studying the effect of the material properties on the deformation pattern of CLT floor diaphragms. Comparing the values in Table 6.2, the following could be stated: 1. The lateral load distribution for the concrete floor with both the 1st and the 4th set of shearwalls is close to the rigid floor assumption. However, the results match better in case of utilizing the 4th set of shearwalls, which is more flexible than the 1st set of shearwalls. This suggests that the concrete floor acts as a rigid diaphragm in both cases. Nevertheless, this type of in-plane behaviour is accentuated when more flexible shearwalls are utilized in the building model. 105 2. The distribution of the lateral load has noticeably changed for the Configuration 3 CLT floor model with changing the stiffness of the shearwalls and rigid panel-to-panel connections. The CLT floor tends to act as a flexible diaphragm when stiff shearwalls are incorporated in the building model. In fact as pointed pout in Section 5.4.1, the lateral load distribution of the CLT floor with 1st set of shearwalls matched the flexible floor assumption. However, with more flexible shearwalls and rigid panel-to-panel connections, the same CLT floor tends to distribute the load more or less according to the stiffness method. This observation suggests that whether the CLT floor diaphragm is flexible or rigid depends entirely on the relative stiffness of the floor diaphragm and the attached shearwalls. The stiffness values of the 4th set of shearwalls, which represents the case of much flexible shearwalls, are close to the stiffness of CLT shearwalls, as mentioned in Chapter 5. Normally for the lightwood frame shearwalls, the stiffness values are an order of magnitude less than CLT shearwalls. Therefore, it is expected that CLT floors would generally behave as rigid floor diaphragms in light wood frame buildings. However, if the same CLT floor is used with concrete shearwalls, then it tends to act as a flexible diaphragm. 3. The lateral load distribution of the CLT floor with the 4th set of shearwalls and rigid panel-to-panel connection is close to the concrete floor with the 4th set of shearwalls and the stiffness method results. Therefore, the previous statement about the rigidity of the floor diaphragm, which was concluded from the deformation patterns of the two cases, is valid. In other words, in both cases the floor diaphragm can be considered as a rigid diaphragm. 106 6.1.2 Number of Connected Panels This parameter refers to the number of connected CLT panels, which forms a single bay CLT diaphragm. The objective here is to study the change in the in-plane stiffness of a CLT diaphragm with increasing the number of connected panels and thus increasing length of the diaphragm. As already shown in Figure 6.7, the number of CLT panels is varied from 3 to 9 in Configuration 2 floor model, and the in-plane stiffness of the CLT floor diaphragm is obtained for each number of panels. The analyses are done for two different values of inplane shear modulus of CLT panels, representing significantly flexible panels and regular panels. From Figure 6.7, it is evident that in-plane stiffness decreases with an increase in the number of panels, in a nonlinear trend. The change is more significant for smaller number of panels. Moreover, the in-plane stiffness of the CLT diaphragm with stiffer CLT panels, changes more rapidly with the number of connected panels compared to the more flexible ones. This behaviour can be explained with regards to the type of deformations that occur within the floor diaphragm. For smaller number of panels and thus shorter length of the diaphragm, the diaphragm acts as a deep short beam deforming almost entirely in shear. In this case, changing the in-plane shear modulus of CLT panels affects the in-plane deformations quite significantly. With increasing the length of the diaphragm and thus adding the number of panel-to-panel connections, the share of the deformations of connections in the total in-plane deformation of the CLT diaphragm, becomes more significant. In other words most of the deformation takes place in the connections. In this case, changing again the in-plane shear modulus of CLT panels increases the in-plane deformation of the panels. However, this effect is insignificant compared to the added in107 plane deformations in the panel-to-panel connections. Therefore, it is observed that the change in the shear modulus of the CLT panels has less effect on the in-plane stiffness of the CLT diaphragms with higher number of connected CLT panels. It should be noted that if the panel-to-panel connections are rigid or much stiffer than the CLT panels, then the deformations in the connections becomes quite smaller. Therefore, the change in the stiffness of the CLT diaphragm with the added number of panels becomes less significant. This explains the smaller slope of the curve for shear modulus of 100 MPa than the curve for 600 MPa. In fact with softer panels, the same panel-to-panel connections become stiffer relative to the CLT panels. Therefore, the deformations in the connections form smaller portion of the total in-plane deformation of the floor diaphragm. 6.1.3 Dimensions of CLT Panels and Diaphragm Configurations The dimensions of CLT panel affect the in-plane stiffness of the panels, and therefore the in-plane behaviour of the CLT diaphragm. The significance of this parameter is highlighted when the share of the shear and flexural deformations are compared to each other. For the previously studied models in this thesis, the CLT diaphragm formed a deep beam, for which the shear deformations in the CLT panels were drastically higher than the flexural deformations. However, for narrow CLT panels, where the length is much larger than the width of the panel, the flexural deformations may form a greater portion of the inplane deformations. Moreover, utilizing larger CLT panels in a floor diaphragm indicates that there are fewer connections within the unit length of the diaphragm compared to the floor with smaller CLT panels. This in turn will reduce the non-linear behaviour of the CLT diaphragm and affects the in-plane behaviour of the CLT diaphragm as already discussed in the previous section. 108 The importance of the diaphragm configuration on the lateral load distribution within a building was addressed in Chapter 5. As already mentioned in Section 5.3, irregularities of the floor diaphragm configuration in the cases where the building is statically indeterminate and undergoes torsional deformations, affects the lateral load distribution to the shearwalls. However, to address this parameter thoroughly, a broad range of floor diaphragm configurations with varying dimensions and aspect ratios should be studied, which is outside the scope of this thesis. 109 Chapter 7: Conclusion The in-plane behaviour of CLT floor diaphragms has been studied in this thesis. The approach taken to investigate this subject is a universal approach, which is applicable to the study of other types of floor diaphragms as well. In particular, detailed 2D numerical models of selected CLT floor diaphragms were developed in the finite element software ANSYS. Although the number of the floor diaphragm configurations is limited, they are capable of representing several important aspects of the in-plane behaviour of the CLT floors. One major advantage of the developed numerical models is their simplicity, without compromising accuracy. A simple smeared CLT panel-to-panel connection model was utilized in the ANSYS floor models. The connection model proved to efficiently demonstrate the response of the modelled panel-to-panel connection. Although this model is calibrated to a special type of connection with self-tapping wood screws, it can be calibrated to other types of panel-to-panel connections as well. One specific feature of this study is the emphasis on addressing the in-plane behaviour of floor diaphragm in the context of the lateral load distribution to the shearwalls. The results of the lateral load distribution were compared to the results of the two common hand calculation methods for distributing the lateral load to the shearwalls, the so-called tributary area and stiffness methods. In this manner, the accuracy of the flexible or rigid diaphragm assumption was assessed for the studied CLT floor diaphragm models. As part of this effort, a design flowchart for the lateral load distribution was developed. This flowchart aids engineers to find the lateral load distribution systematically within any type of building, including CLT buildings. 110 7.1 Future Work To be able to extend the results of this study to broader range of CLT floor diaphragms, parametric studies have been done, and the most influential parameters affecting the in-plane behaviour of CLT diaphragms were identified. The parametric study presented in this thesis provides a deep insight into the effect and importance of the panel-to-panel connection response on the overall in-plane behaviour of CLT diaphragms. Moreover, it highlights the significance of shear and flexural in-plane deformations in the total in-plane deformation of CLT diaphragms. This study explicitly indicated that the in-plane behaviour of a floor diaphragm must be studied in the context of a building system. In other words, whether the floor diaphragm is flexible or rigid depends on the building system and the relative stiffness of the floor diaphragm and the shearwalls. Although this thesis addresses the in-plane behaviour of CLT floor diaphragms, there are a number of limitations involved. One major limitation in this study is the lack of real test data on the overall in-plane behaviour of CLT diaphragms. Other limitations are related to the modelling: 1) the number of CLT floor diaphragm configurations studied here is limited; 2) the effect of diaphragm configuration irregularities on the lateral load distribution is not addressed in this thesis; 3) the lateral load applied to the models is static in all cases; 4) the shearwalls are assumed to behave linearly, and are not capable of demonstrating the rocking or uplifting of the shearwalls; 5) the shearwall connections to the floor diaphragm and to the foundation are merged into the diaphragm model; and 6) the CLT panels are modelled as a linear material, eliminating the cases where the failure of the diaphragm occurs in form of rupture in the panels. 111 Considering the above limitations, it is recommended for future studies on the related topics to extend the floor models developed here to include more realistic representation of shearwalls and the shearwall connections. 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"Study of Lateral Resistance of Massive X-Lam Wooden Wall System Subjected to Horizontal Loads." International Workshop on "Earthquake Engineering on Timber Structures". EN 14081-1. (2005). "Timber Structures. Strength graded structural timber with rectangular cross section. General requirements." Rep. No. BS EN 14081-1:2005, BSI. EN 1912. (2006). "Structural Timber. Strength classes. Assignment of visual grades and species." Rep. No. BS EN 1912:2004, BSI. EN 26891. (1991). "Timber Structures ; Joints made with mechanical fasteners; general principles for the determination of strength and deformation characteristics (ISO 6891:1983)." Rep. No. DIN EN 26891:1991, DIN. EN 338. (2010). "Structural Timber - Strength Classes." Rep. No. BS EN 338:2009, BSI, . EN 789. (2005). "Timber Structures - Test Methods - Determination of mechanical properties of wood panels." Rep. No. DIN EN 789:2005, DIN. Eurocode 8. (1998). "Design of structures for earthquake resistance." Rep. No. BS EN 1998, BSI. FEMA 454. (2006). "Risk Management Series, Designing for Earthquakes : A Manual for Architects." Rep. No. 454, Federal Emergency Management Agency;, Earthquake Engineering Research Institute (EERI) of Oakland, California. Filiatrault, A., Fischer, D., Folz, B., and Uang, C. (2002). "Experimental parametric study on the in-plane stiffness of wood diaphragms." Canadian Journal of Civil Engineering, 29(4), 554-566. Follesa, M., Brunetti, M., Cornacchini, R., and Grasso, S. (2010). "Mechanical In-plane Joints Between Cross Laminated Timber Panels." 11th World Conference on Timber Engineering. FPInnovations. (2011). "Chapter 1: CLT - Introduction to cross-laminated timber." CLT Handbook : cross-laminated timber, S. Gagnon, and C. Pirvu, eds., FPInnovation. Gagnon, S., and Popovski, M. (2011). "Chapter 3: Structural - Structural design of crosslaminated timber elements." CLT Handbook : cross-laminated timber, S. Gagnon, and C. Pirvu, eds., FPInnovation. 114 Gsell, D., Feltrin, G., Schubert, S., Steiger, R., and Motavalli, M. (2007). "Cross-Laminated Timber Plates: Evaluation and Verification of Homogenized Elastic Properties." Journal of Structural Engineering, 133(1), 132-138. Gubana, A. (2008). "Chapter 108. Cross laminated timber panels to strengthen wood floors (Proceedings of the VI International Conference on Structural Analysis of Historic Construction, 2008, Bath, United Kingdom)." Structural Analysis of Historic Construction, E. Fodde, ed., CRC Press;, 949-955. Guggenberger, W., and Moosbrugger, T. (2006). "Mechanics of Cross-Laminated Timber Plates under Uniaxial Bending." 9th World Conference on Timber Engineering. Hite, M. C., and Shenton, H. W. (2002). "Modeling the Non-linear Behavior of Wood Frame Shear Walls." 15th ASCE Engineering Mechanics Conference. Jobstl, R. A., BOGENSPERGER, T., and Schickhofer, G. (2008). "In-plane Shear Strength of Cross Laminated Timber." CIB W-18. Jobstl, R. A., Moosbrugger, T., BOGENSPERGER, T., and Schickhofer, G. (2006). "A Contribution to the Design and System Effect of Cross Laminated Timber." CIB W-18. Lam, F. (2011). "Cross Laminated Timber (CLT) Connection Tests for UBC Bioenergy Research & Demonstration Project Nexterra Building (Report prepared for CST Innovation Ltd.)." Timber Engineering and Applied Mechanics (TEAM) Laboratory. J. Lilley, Fyfe, K., Jones, S., Mckay, B., Muradali, A., Skoczylas, P., Toogood, R. and Johnson, K. (2001). "University of Alberta - ANSYS Tutorials." (http://www.mece.ualberta.ca/tutorials/ansys/index.html 2012). Madenci, E., and Guven, I. (2005). The Finite Element Method and Applications in Engineering Using ANSYS. Springer. Mohammad, M., and Munoz, W. (2011). "Chapter 5: Connections - Connections in crosslaminated timber buildings." CLT Handbook: cross-laminated timber, S. Gagnon, and C. Pirvu, eds., FPInnovations. Moosbrugger, T., Guggenberger, W., and Bogensperger, T. (2006). "Cross-Laminated Timber Wall Segments under Homogenous Shear - with and without Openings." 9th World Conference on Timber Engineering. Munoz, W., Mohammad, M., and Gagnon, S. (2010). "Lateral and Withdrawal resistance of typical CLT Connections." 11th World Conference on Timber Engineering. Naeim, F. (2001). The Seismic Design Handbook. Springer. 115 Pang, W. C., and Rosowsky, D. V. (2010). "Beam–spring model for timber diaphragm and shear walls." Structures and Buildings, 163(SB4), 227-244. Pathak, R. (2009). "The Effects of Diaphragm Flexibility on the Seismic Performance of Light Frame Wood Structures". Doctor of Philosophy. Virginia Polytechnic Institute and State University. Popovski, M., Karacabeyli, E., and Ceccotti, A. (2011). "Chapter4: Seismic - Seismic performance of cross-laminated timber buildings." CLT Handbook: cross-laminated timber, S. Gagnon, and C. Pirvu, eds., FPInnovations. Popovski, M., Schneider, J., and Schweinsteiger, M. (2010). "Lateral Load Resistance of Cross-Laminated Wood Panels." 11th World Conference on Timber Engineering. Prion, H. G. L., and Lam, F. (2003). "Shear Walls and Diaphragms." Timber Engineering, S. Thelandersson, and H. J. Larsen, eds., J. Wiley, 2003, 383-408. SEAOC. (1980). "Recommended Lateral Force Requirements and Commentary." Seismology Committee;Structural Engineers Association of California. Stürzenbecher, R., Hofstetter, K., and Eberhardsteiner, J. (2010a). "Cross Laminated Timber: A Multi-layer Shear Compliant Plate and Its Mechanical Behavior." 11th World Conference on Timber Engineering. Stürzenbecher, R., Hofstetter, K., and Eberhardsteiner, J. (2010b). "Structural design of Cross Laminated Timber (CLT) by advanced plate theories." Composites Sci.Technol., 70(9), 1368-1379. Vessby, J., Enquist, B., Petersson, H., and Alsmarker, T. (2009). "Experimental study of cross-laminated timber wall panels." Eur. J. Wood Prod., 67 211-218. Yawalata, D., and Lam, F. (2011). "Development of Technology for Cross Laminated Timber Building Systems." Research report submitted to Forestry Innovation Investment Ltd., BC, University of British Columbia. Vancouver, BC. 116 Appendices Appendix A : APDL Codes for Generating Diaphragm Models In this appendix simple APDL codes used for generating the CLT floor models of the thesis are presented and explained with an example. As mentioned in Section 4.1.1, the models are created using direct generation approach. Therefore, the user should define and control the numbering scheme for the nodes and elements of the models. Here, the numbering is done based on the dimensions of the CLT panels and the selected mesh size for PLANE42 elements. As an example, the Configuration 3 floor model shown in Figure 4.3 is selected here to be generated through APDL commands. First the CLT panels and the panel-to-panel connections are created. To do so, the following APDL code in entered in the command line: /PREP7 ANTYPE,STATIC PSTRES,ON X_Length = Y_Length = X_division Y_division 1.2 5 = 30 = 125 ! All in meter Th = 0.15 ! Thickness Dth = Y_Length/Y_division !************************************************************************* ! Element Types and Material Properties ET,1,PLANE42 KEYOPT,1,3,3 R,1,Th ! Plane Stress with Thickness ! THICKNESS MP,EX,1,8e9 MP,EY,1,9e9 MP,EZ,1,1e9 MP,PRXY,1,0.35 117 MP,PRYZ,1,0.3 MP,PRXZ,1,0.47 MP,GXY,1,0.45e9 MP,GYZ,1,0.5e9 MP,GXZ,1,0.45e9 !MP,DENS,1,7800 ET,2,COMBIN14 KEYOPT,2,2,1 !KEYOPT,2,3,2 R,2,1e10 ! KEYOPT(2)=1 (UX only) ! KEYOPT(3)=2 (2D DOFs) ! Spring Constant ET,3,COMBIN39 KEYOPT,3,3,2 ! KEYOPT(3)=2 (UY only) KEYOPT,3,6,1 ! KEYOPT(6)=1 (prints force-deflection table) R,3,0,0,0.002,2654,0.0036,4012,0.005,4318 !************************************************************************* ! First Panel W = 0 L = 0 ! Starting point (X) ! Starting point (y) *DIM, S, Array, X_division+1, 1 S(1) = W *Do, i, 2, X_division+1, 1 S(i) = S(i-1)+X_Length/X_division *enddo *Do, i, 1, X_division+1, 1 N,i,S(i), L K,i,S(i), L *enddo *Do, i, 1, Y_division, 1 *Do, j,1,X_division+1,1 x = S(j) y = L+Dth*i N,,x,y K,,x,y *if,j,GT,1,Then E,(i-1)*(X_division+1)+(j-1), (i-1)*(X_division+1)+j, i*(X_division+1)+j, i*(X_division+1)+(j-1) *Endif *enddo *enddo 118 !------------------------------------------------------------------------! Second Panel W = 1.205 L = 0 ! Starting point (X) ! Starting point (y) *DIM, S, Array, X_division+1, 1 S(1) = W *Do, i, 2, X_division+1, 1 S(i) = S(i-1)+X_Length/X_division *enddo *Do, i, 3907, 3906+X_division+1, 1 N,i,S(i-3906), L K,i,S(i-3906), L *enddo *Do, i, 1, Y_division, 1 *Do, j,1,X_division+1,1 x = S(j) y = L+Dth*i N,,x,y K,,x,y *if,j,GT,1,Then E,3906+(i-1)*(X_division+1)+(j-1), 3906+(i-1)*(X_division+1)+j, 3906+i*(X_division+1)+j, 3906+i*(X_division+1)+(j-1) *Endif *enddo *enddo !------------------------------------------------------------------------! Third Panel W = 2.41 L = 0 ! Starting point (X) ! Starting point (y) *DIM, S, Array, X_division+1, 1 S(1) = W *Do, i, 2, X_division+1, 1 S(i) = S(i-1)+X_Length/X_division *enddo 119 *Do, i, 7813, 7812+X_division+1, 1 N,i,S(i-7812), L K,i,S(i-7812), L *enddo *Do, i, 1, Y_division, 1 *Do, j,1,X_division+1,1 x = S(j) y = L+Dth*i N,,x,y K,,x,y *if,j,GT,1,Then E,7812+(i-1)*(X_division+1)+(j-1), 7812+(i-1)*(X_division+1)+j, 7812+i*(X_division+1)+j, 7812+i*(X_division+1)+(j-1) *Endif *enddo *enddo !------------------------------------------------------------------------! Fourth Panel W = 3.615 L = 0 ! Starting point (X) ! Starting point (y) *DIM, S, Array, X_division+1, 1 S(1) = W *Do, i, 2, X_division+1, 1 S(i) = S(i-1)+X_Length/X_division *enddo *Do, i, 11719, 11718+X_division+1, 1 N,i,S(i-11718), L K,i,S(i-11718), L *enddo *Do, i, 1, Y_division, 1 *Do, j,1,X_division+1,1 x = S(j) y = L+Dth*i N,,x,y K,,x,y *if,j,GT,1,Then 120 E,11718+(i-1)*(X_division+1)+(j-1), 11718+(i1)*(X_division+1)+j, 11718+i*(X_division+1)+j, 11718+i*(X_division+1)+(j1) *Endif *enddo *enddo !------------------------------------------------------------------------! Fifth Panel W = 4.82 L = 0 ! Starting point (X) ! Starting point (y) *DIM, S, Array, X_division+1, 1 S(1) = W *Do, i, 2, X_division+1, 1 S(i) = S(i-1)+X_Length/X_division *enddo *Do, i, 15625, 15624+X_division+1, 1 N,i,S(i-15624), L K,i,S(i-15624), L *enddo *Do, i, 1, Y_division, 1 *Do, j,1,X_division+1,1 x = S(j) y = L+Dth*i N,,x,y K,,x,y *if,j,GT,1,Then E,15624+(i-1)*(X_division+1)+(j-1), 15624+(i1)*(X_division+1)+j, 15624+i*(X_division+1)+j, 15624+i*(X_division+1)+(j1) *Endif *enddo *enddo !------------------------------------------------------------------------! Sixth Panel W = 6.025 L = 0 ! Starting point (X) ! Starting point (y) 121 *DIM, S, Array, X_division+1, 1 S(1) = W *Do, i, 2, X_division+1, 1 S(i) = S(i-1)+X_Length/X_division *enddo *Do, i, 19531, 19530+X_division+1, 1 N,i,S(i-19530), L K,i,S(i-19530), L *enddo *Do, i, 1, Y_division, 1 *Do, j,1,X_division+1,1 x = S(j) y = L+Dth*i N,,x,y K,,x,y *if,j,GT,1,Then E,19530+(i-1)*(X_division+1)+(j-1), 19530+(i1)*(X_division+1)+j, 19530+i*(X_division+1)+j, 19530+i*(X_division+1)+(j1) *Endif *enddo *enddo !------------------------------------------------------------------------! Seventh Panel W = 7.23 L = 0 ! Starting point (X) ! Starting point (y) *DIM, S, Array, X_division+1, 1 S(1) = W *Do, i, 2, X_division+1, 1 S(i) = S(i-1)+X_Length/X_division *enddo *Do, i, 23437, 23436+X_division+1, 1 N,i,S(i-23436), L K,i,S(i-23436), L *enddo *Do, i, 1, Y_division, 1 122 *Do, j,1,X_division+1,1 x = S(j) y = L+Dth*i N,,x,y K,,x,y *if,j,GT,1,Then E,23436+(i-1)*(X_division+1)+(j-1), 23436+(i1)*(X_division+1)+j, 23436+i*(X_division+1)+j, 23436+i*(X_division+1)+(j1) *Endif *enddo *enddo !------------------------------------------------------------------------! Eighth Panel W = 8.435 L = 0 ! Starting point (X) ! Starting point (y) *DIM, S, Array, X_division+1, 1 S(1) = W *Do, i, 2, X_division+1, 1 S(i) = S(i-1)+X_Length/X_division *enddo *Do, i, 27343, 27342+X_division+1, 1 N,i,S(i-27342), L K,i,S(i-27342), L *enddo *Do, i, 1, Y_division, 1 *Do, j,1,X_division+1,1 x = S(j) y = L+Dth*i N,,x,y K,,x,y *if,j,GT,1,Then E,27342+(i-1)*(X_division+1)+(j-1), 27342+(i1)*(X_division+1)+j, 27342+i*(X_division+1)+j, 27342+i*(X_division+1)+(j1) *Endif *enddo 123 *enddo !------------------------------------------------------------------------! Ninth Panel W = 9.64 L = 0 ! Starting point (X) ! Starting point (y) *DIM, S, Array, X_division+1, 1 S(1) = W *Do, i, 2, X_division+1, 1 S(i) = S(i-1)+X_Length/X_division *enddo *Do, i, 31249, 31248+X_division+1, 1 N,i,S(i-31248), L K,i,S(i-31248), L *enddo *Do, i, 1, Y_division, 1 *Do, j,1,X_division+1,1 x = S(j) y = L+Dth*i N,,x,y K,,x,y *if,j,GT,1,Then E,31248+(i-1)*(X_division+1)+(j-1), 31248+(i1)*(X_division+1)+j, 31248+i*(X_division+1)+j, 31248+i*(X_division+1)+(j1) *Endif *enddo *enddo !************************************************************************* ! Rigid Links COMBIN14 TYPE,2 REAL,2 MAT,1 ! Turn on Element 2 ! Turn on Real constants 2 ! Turn on Material 1 *Do, i, 1, Y_division+1, 1 124 E,i*(X_division+1),3907+(i-1)*(X_division+1) *enddo !----------------------------------------------------------*Do, i, 1, Y_division+1, 1 E,3906+i*(X_division+1),7813+(i-1)*(X_division+1) *enddo !----------------------------------------------------------*Do, i, 1, Y_division+1, 1 E,7812+i*(X_division+1),11719+(i-1)*(X_division+1) *enddo !----------------------------------------------------------*Do, i, 1, Y_division+1, 1 E,11718+i*(X_division+1),15625+(i-1)*(X_division+1) *enddo !----------------------------------------------------------*Do, i, 1, Y_division+1, 1 E,15624+i*(X_division+1),19531+(i-1)*(X_division+1) *enddo !----------------------------------------------------------*Do, i, 1, Y_division+1, 1 E,19530+i*(X_division+1),23437+(i-1)*(X_division+1) *enddo !----------------------------------------------------------*Do, i, 1, Y_division+1, 1 E,23436+i*(X_division+1),27343+(i-1)*(X_division+1) *enddo !----------------------------------------------------------*Do, i, 1, Y_division+1, 1 E,27342+i*(X_division+1),31249+(i-1)*(X_division+1) 125 *enddo !************************************************************************* ! Nonlinear Springs COMBIN39 TYPE,3 REAL,3 MAT,1 ! Turn on Element 3 ! Turn on Real constants 3 ! Turn on Material 1 *Do, i, 1, Y_division+1, 1 E,i*(X_division+1),3907+(i-1)*(X_division+1) *enddo !----------------------------------------------------------*Do, i, 1, Y_division+1, 1 E,3906+i*(X_division+1),7813+(i-1)*(X_division+1) *enddo !----------------------------------------------------------*Do, i, 1, Y_division+1, 1 E,7812+i*(X_division+1),11719+(i-1)*(X_division+1) *enddo !----------------------------------------------------------*Do, i, 1, Y_division+1, 1 E,11718+i*(X_division+1),15625+(i-1)*(X_division+1) *enddo !----------------------------------------------------------*Do, i, 1, Y_division+1, 1 E,15624+i*(X_division+1),19531+(i-1)*(X_division+1) *enddo !----------------------------------------------------------*Do, i, 1, Y_division+1, 1 E,19530+i*(X_division+1),23437+(i-1)*(X_division+1) *enddo 126 !----------------------------------------------------------*Do, i, 1, Y_division+1, 1 E,23436+i*(X_division+1),27343+(i-1)*(X_division+1) *enddo !----------------------------------------------------------*Do, i, 1, Y_division+1, 1 E,27342+i*(X_division+1),31249+(i-1)*(X_division+1) *enddo In the above code, first the nine CLT panels are created and next the COMBIN14 and COMBIN39 springs of the connection model are added to the panels. Here, each of the nine panels and the connections between the panels are generated separately. Since the panels are similar, it is possible to utilize do-loops in order to avoid repetition. However, using the above format reduces the chance of error in numbering the nodes and elements. Furthermore, it allows direct access to the properties and dimensions of each panel, which is useful in case the panels are not similar. After generating the panels and the panel-to-panel connections, displacement boundary conditions and loading should be applied to the model at the solution phase. The displacement boundary conditions are defined using the following code: /SOLU X_Length = Y_Length = X_division Y_division 1.2 5 = 30 = 125 ! All in meter ! Define Displacement Constraints on Nodes (D command) 127 *Do, i, 1, Y_division+1, 1 D, D, D, D, 1+(i-1)*(X_division+1),ALL,0 7812+i*(X_division+1),ALL,0 19531+(i-1)*(X_division+1),ALL,0 31248+i*(X_division+1),ALL,0 *enddo And the loading is applied using the simple code shown below: /SOLU X_Length = 1.2 ! All in meter Y_Length = 5 Y_Length2 = 2.4 X_division = 30 Y_division = 125 Y_division2 = 60 Th = 0.15 ! Thickness Elesize = 0.04 ! Element Size Pres = 1 ! Pressure in Mpa ForceN = Pres*1000000*Th*Elesize/2 ! Define Forces on Nodes (F command : F, NODE, Lab, VALUE, VALUE2, NEND, NINC) *Do, i, 1, X_division+1, 1 F,i,FY,ForceN F,3906+i,FY,ForceN F,7812+i,FY,ForceN F,11718+i,FY,ForceN F,15624+i,FY,ForceN F,19530+i,FY,ForceN F,23436+i,FY,ForceN F,27342+i,FY,ForceN F,31248+i,FY,ForceN *enddo As with the first piece of code, the displacement boundary conditions and the loading are also defined for each panel separately. At this point, the generation of the floor model is complete. Other analysis options are set using the ANSYS interface as described in Chapter 4. 128 In order to generate the Configuration 3 building model, the CLT panels and the panel-to-panel connections are first created using the same code as presented before. However, extra nodes and springs are added to the model to include shearwalls. The extra nodes are created in the plane of the floor diaphragm using the following piece of code: /PREP7 ANTYPE,STATIC PSTRES,ON X_Length = Y_Length = X_division Y_division 1.2 5 = 30 = 125 ! All in meter !************************************************************************* ! Shearwall a *Do, i, 1, 126, 1 N,i+35154,0, 0+(i-1)*0.04 K,i+35154,0, 0+(i-1)*0.04 *enddo !************************************************************************* ! Shearwall b *Do, i, 1, 126, 1 N,i+35280,3.61, 0+(i-1)*0.04 K,i+35280,3.61, 0+(i-1)*0.04 *enddo !************************************************************************* ! Shearwall c *Do, i, 1, 126, 1 N,i+35406,6.025, 0+(i-1)*0.04 K,i+35406,6.025, 0+(i-1)*0.04 129 *enddo !************************************************************************* ! Shearwall d *Do, i, 1, 126, 1 N,i+35532,10.84, 0+(i-1)*0.04 K,i+35532,10.84, 0+(i-1)*0.04 *enddo Then the shearwall springs are added to the model. The shearwall springs in the Y direction can be defined using the following code: /PREP7 ANTYPE,STATIC PSTRES,ON X_Length = 1.2 Y_Length = 5 X_division = 30 Y_division = 125 Height1 = 3 ! All in meter th1 = 0.15 ! Thickness of the exterior shearwalls th2 = 0.1 ! Thickness of the interior shearwalls MoE1 = 1e10 IoX1 = 1/12*th1*(Y_Length*Y_Length*Y_Length) IoX2 = 1/12*th2*(Y_Length*Y_Length*Y_Length) K1 = (3*MoE1*IoX1/(Height1*Height1*Height1))/ (Y_division+1) K2 = (3*MoE1*IoX2/(Height1*Height1*Height1))/ (Y_division+1) !************************************************************************* ! Element Types and Material Properties ET,4,COMBIN14 KEYOPT,4,2,2 R,4,K1 R,5,K2 ! KEYOPT(2)=2 (UY only) ! Spring Constant (exterior shearwalls) ! Spring Constant (interior shearwalls) !************************************************************************* ! Shearwall a 130 TYPE,4 REAL,4 MAT,1 ! Turn on Element 4 ! Turn on Real constants 4 ! Turn on Material 1 *Do, i, 1, Y_division+1, 1 E,1+(i-1)*(X_division+1),35155+(i-1) *enddo !*************************************************** ! Shearwall b !TYPE,4 REAL,5 MAT,1 Turn on Element 4 ! Turn on Real constants 5 ! Turn on Material 1 *Do, i, 1, Y_division+1, 1 E,7843+(i-1)*(X_division+1),35281+(i-1) *enddo !*************************************************** ! Shearwall c !TYPE,4 REAL,5 MAT,1 Turn on Element 4 ! Turn on Real constants 5 ! Turn on Material 1 *Do, i, 1, Y_division+1, 1 E,19531+(i-1)*(X_division+1),35407+(i-1) *enddo !*************************************************** ! Shearwall d !TYPE,4 REAL,4 MAT,1 Turn on Element 4 ! Turn on Real constants 4 ! Turn on Material 1 *Do, i, 1, Y_division+1, 1 E,31279+(i-1)*(X_division+1),35533+(i-1) *enddo 131 The shearwall springs in the X direction are introduced to the model using the same code as above, but with different real constants and key options defining the direction of the springs. The loading for the building model is similar to the floor model. However, the displacement constraints for the building model are applied to the shearwalls end nodes, which were just added to the floor model: /SOLU ! Define Displacement Constraints on Nodes (D command) *Do, i, 1, 504, 1 D, 35154+i,ALL,0 *enddo ! 504 = 4 * (Y_division+1), total number of added nodes 132
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In-plane stiffness of cross-laminated timber floors Ashtari, Sepideh 2012
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Title | In-plane stiffness of cross-laminated timber floors |
Creator |
Ashtari, Sepideh |
Publisher | University of British Columbia |
Date Issued | 2012 |
Description | This study investigates the in-plane stiffness of CLT floor diaphragms and addresses the lateral load distribution within buildings containing CLT floors. In practice, it is common to assume the floor diaphragm as either flexible or rigid, and distribute the lateral load according to simple hand calculations methods. Here, the applicability of theses assumption to CLT floor diaphragms is investigated. There is limited number of studies on the subject of in-plane behaviour of CLT diaphragms in the literature. Many of these studies involve testing of the panels or the connections utilized in CLT diaphragms. This study employs numerical modeling as a tool to address the in-plane behaviour of CLT diaphragms. The approach taken to develop the numerical models in this thesis has not been applied so far to CLT floor diaphragms. Detailed 2D finite element models of selective CLT floor diaphragm configurations are generated and analysed in ANSYS. The models contain a smeared panel-to-panel connection model, which is calibrated with test data of a special type of CLT connection with self-tapping wood screws. The floor models are then extended to building models by adding shearwalls, and the lateral load distribution is studied for each building model. A design flowchart is also developed to aid engineers in finding the lateral load distribution for any type of building in a systematic approach. By a parametric study, the most influential parameters affecting the in-plane behaviour of CLT floor diaphragm and the lateral load distribution are identified. The main parameters include the response of the CLT panel-to-panel connections, the in-plane shear modulus of CLT panels, the stiffness of shearwalls, and the floor diaphragm configuration. It was found that the applicability of flexible or rigid diaphragm assumptions is primarily dependent on the relative stiffness of the CLT floor diaphragm and the shearwalls. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-10-19 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0073342 |
URI | http://hdl.handle.net/2429/43504 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2012-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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