FORCE TRANSFER AROUND OPENINGS IN CLT SHEAR WALLS by Sai Ganesh Sarvotham Pai B.Tech., National Institute of Technology, Karnataka, 2011 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December 2014 © Sai Ganesh Sarvotham Pai, 2014 ii Abstract During an earthquake, shear walls can experience damage around corners of doors and windows due to development of stress concentration. Reinforcements provided to minimize this damage are designed for forces that develop at these corners known as transfer forces. In this thesis, the focus is on understanding the forces that develop around opening corners in cross laminated timber (CLT) shear walls and reinforcement requirements for the same. In the literature, four different analytical models are commonly considered to determine the transfer force for design of wood-frame shear walls. These models have been reviewed in this thesis. The Diekmann model is found to be the most suitable analytical model to determine the transfer force around a window-type opening. Numerical models are developed in ANSYS to analyse the forces around opening corners in CLT shear walls. CLT shear walls with cut-out openings are analysed using a three-dimensional brick element model and a frame model. These models highlight the increase in shear and torsion around opening corners due to stress concentration. The coupled-panel construction practice for CLT shear walls with openings is analysed using a continuum model calibrated to experimental data. The analysis shows the increase in strength and stiffness of walls, when tie-rods are used as reinforcement. Analysis results also indicate that the tie-rods should be designed to behave linearly for optimum performance of the wall. Finally, a linear regression model is developed to determine the stiffness of a simply-supported CLT shear wall with a window-type opening. This model provides insight into the effect of various geometrical and material parameters on the stiffness of the wall. The process of model iii development has been explained, which can be improved further to include the behaviour of anchors. iv Preface This dissertation is original, unpublished, independent work by the author, Sai Ganesh Sarvotham Pai. v Table of Contents Abstract .................................................................................................................................... ii Preface ..................................................................................................................................... iv Table of Contents .................................................................................................................... v List of Tables .......................................................................................................................... ix List of Figures .......................................................................................................................... x Acknowledgements .............................................................................................................. xiv Dedication .............................................................................................................................. xv Chapter 1: Introduction ........................................................................................................ 1 1.1 Motivation ................................................................................................................. 1 1.2 Objectives ................................................................................................................. 2 1.3 Background ............................................................................................................... 2 1.4 Overview of the Thesis ............................................................................................. 8 Chapter 2: Analytical Models for Force Transfer around Openings ............................. 11 2.1 Drag Strut Analogy ................................................................................................. 11 2.2 Cantilever Beam Analogy ....................................................................................... 14 2.3 Coupled Beam Analogy .......................................................................................... 18 vi 2.4 Diekmann Method .................................................................................................. 24 2.5 Numerical Example ................................................................................................ 27 2.6 Finite Element Modeling and Comparison of Results ............................................ 33 2.7 Conclusions ............................................................................................................. 40 Chapter 3: CLT Shear Wall with a Cut-out Opening ...................................................... 41 3.1 In-plane Behaviour of CLT ..................................................................................... 41 3.2 CLT Shear Wall with a Cut-out Opening ............................................................... 47 3.3 Frame Model for a CLT Shear Wall with a Cut-out Opening ................................ 52 3.4 Results from the Frame Models (Model C) ............................................................ 56 3.5 Conclusions ............................................................................................................. 60 Chapter 4: Coupled-Panel CLT Shear Walls .................................................................... 62 4.1 Modeling a Coupled-Panel CLT Shear Wall .......................................................... 63 4.2 Modeling the CLT Panels ....................................................................................... 64 4.3 Connector Modeling ............................................................................................... 66 4.4 Contact Modeling.................................................................................................... 70 4.5 Displacement-based Pushover Analysis in ANSYS ............................................... 71 4.6 Modeling a Coupled Panel CLT Shear Wall with an Opening ............................... 74 vii 4.7 Pushover Analysis of a Coupled Panel CLT Shear Wall with an Opening ............ 76 4.8 Effect of Tie-Rods as Reinforcement on Performance of the CLT Shear Wall ..... 79 4.9 Effect of Anchoring and Opening Layout on Wall Behaviour ............................... 80 4.10 Design Transfer Force from the Diekmann Model and Finite Element Models .... 82 4.11 Effect of Tie-Rod Stiffness on Performance of the CLT Shear Wall ..................... 85 4.12 Conclusions ............................................................................................................. 87 Chapter 5: Linear Regression Model for Stiffness of a Wall ........................................... 89 5.1 General Form of a Linear Regression Model ......................................................... 89 5.2 Model Development................................................................................................ 92 5.3 Model Reduction ..................................................................................................... 97 Chapter 6: Conclusion and Future Work ........................................................................ 107 6.1 Conclusion ............................................................................................................ 107 6.2 Future Work .......................................................................................................... 108 References ............................................................................................................................ 109 Appendices ........................................................................................................................... 113 Appendix A Linear Regression Model for Stiffness of a CLT Shear Wall ...................... 113 A.1 Trials ................................................................................................................. 113 viii A.2 Model Development Calculations..................................................................... 118 ix List of Tables Table 2.1 Transfer force from the finite element model ...................................................... 36 Table 3.1 Material property of CLT panel (Yawalata and Lam 2011) ................................ 49 Table 3.2 Comparison of stress in the glued surface ........................................................... 60 Table 4.1 Homogenized material property of a CLT panel ................................................. 65 Table 5.1 Input parameters of the finite element model ...................................................... 94 Table 5.2 Model parameters ................................................................................................ 96 Table 5.3 Model parameters of the regression model ........................................................ 102 Table 5.4 Model parameters of the reduced regression model .......................................... 105 x List of Figures Figure 1.1 Overview of finite element models ...................................................................... 8 Figure 2.1 Drag strut analogy .............................................................................................. 12 Figure 2.2 Cantilevered beam analogy ................................................................................. 15 Figure 2.3 Analysis of the full-width panel on the left side of the opening ........................ 16 Figure 2.4 Coupled beam analogy ....................................................................................... 19 Figure 2.5 Free-body diagram for coupled beam analogy ................................................... 22 Figure 2.6 Diekmann method .............................................................................................. 25 Figure 2.7 Example shear-wall ............................................................................................ 27 Figure 2.8 Comparison of transfer force from analytical models ........................................ 33 Figure 2.9 Finite element model of a shear wall as a continuum (Model A) ...................... 34 Figure 2.10 Deformed shape of the wall under the action of a lateral load .......................... 36 Figure 2.11 Comparison of design transfer force obtained from different models .............. 37 Figure 2.12 Principal stress vector plot of corner 2 .............................................................. 39 Figure 3.1 Structure and discretization of a CLT panel (Bogensperger et al. 2010) ........... 42 Figure 3.2 Nominal shear stress in a RVSE .......................................................................... 44 Figure 3.3 Torsional shear stress in a RVSE ........................................................................ 45 xi Figure 3.4 Real shear stress distribution in a RVSE ............................................................. 46 Figure 3.5 CLT shear wall with a cut-out opening ............................................................... 47 Figure 3.6 3-dimensional finite element model of a CLT shear wall with a cut-out opening (Model B) ................................................................................................................................ 48 Figure 3.7 Axial stress σX in the panels ................................................................................. 50 Figure 3.8 Principal stress plot for the glued surface diagonally adjacent to the top-right corner of the opening ......................................................................................................................... 51 Figure 3.9 CLT shear wall without an opening .................................................................... 52 Figure 3.10 Modeling of glued surface in the frame model ................................................. 53 Figure 3.11 Frame model for a CLT shear wall without an opening ..................................... 54 Figure 3.12 Frame model for a CLT shear wall with a cut-out opening (Model C) .............. 55 Figure 3.13 Torsion (Nm) in each glued surface of a CLT shear wall .................................. 57 Figure 3.14 Torsion (Nm) in each glued surface of a CLT shear wall with an opening ....... 58 Figure 3.15 Difference in torsional moment in the glued surfaces due to the presence of an opening .................................................................................................................................... 59 Figure 4.1 Coupled-panel CLT shear wall (Gavric et al. 2012) ............................................ 63 Figure 4.2 k-factors from composite theory (Gagnon and Popovski 2011) .......................... 65 Figure 4.3 Effective strength and stiffness calculation (Gagnon and Popovski 2011) .......... 66 xii Figure 4.4 Force-deformation response of the hold-down in tension .................................... 67 Figure 4.5 Force-deformation response of the angle bracket in tension ................................ 67 Figure 4.6 Force-deformation response of the hold-down in shear ....................................... 68 Figure 4.7 Force-deformation response of the angle bracket in shear ................................... 68 Figure 4.8 Force-deformation response of the half-lap joint in X-direction.......................... 69 Figure 4.9 Force-deformation response of the half-lap joint in Y-direction.......................... 69 Figure 4.10 Simulating a compression-only spring ............................................................... 70 Figure 4.11 Force-deformation response of the compression-only contact spring ................ 71 Figure 4.12 Finite element model of the test set-up by Gavric et al. (2012) ......................... 72 Figure 4.13 Deformed shape of the coupled-panel shear wall set-up .................................... 73 Figure 4.14 Force-deformation response of the set-up considered for validation ................. 73 Figure 4.15 Coupled-panel CLT shear wall with an opening (Configuration 1) ................... 74 Figure 4.16 Finite element model of a coupled-panel CLT shear wall with an opening (Model D) ............................................................................................................................................ 76 Figure 4.17 Deformed shape of the coupled-panel CLT shear wall with an opening ........... 77 Figure 4.18 Force-deformation response of the CLT shear wall ........................................... 78 Figure 4.19 Effect of tie-rods on shear wall performance ..................................................... 79 Figure 4.20 Configuration 2 ................................................................................................... 81 xiii Figure 4.21 Force-deformation response for the four wall configurations ............................ 81 Figure 4.22 Force in tie-rod and transfer force ...................................................................... 83 Figure 4.23 Transfer force obtained from Diekmann method and finite element model ...... 84 Figure 4.24 Effect of stiffness of tie-rod on wall performance .............................................. 86 Figure 5.1 Geometry of a shear wall with an opening ........................................................... 93 Figure 5.2 Model prediction versus observation.................................................................. 103 xiv Acknowledgements I begin by sincerely thanking my advisor Dr. Terje Haukaas for hours of fruitful discussions, for his limitless supply of ideas, and for being so patient and generous. I have learnt a great deal from our interactions and I am fortunate to have had an opportunity to work with him. I would also like to express my gratitude in working with my co-supervisor, Dr. Frank Lam, for his unlimited bank of know-how, his constructive ideas and advice during the course of this project. It is my pleasure to thank Dr. Thomas Tannert, who always had interesting questions for me during my different research presentations in the department. I am also indebted to the wonderful instructors at the University of British Columbia, who have been instrumental in providing a comprehensive research environment. I would also like to thank my friends in the office and outside, who have made my experience at UBC a pleasurable one and helped me create a home away from home. More so than anyone, I want to thank my parents for their selfless love and for always doing what was best for me. I sincerely hope I have made them proud. xv Dedication To my beloved parents 1 Chapter 1: Introduction 1.1 Motivation The west coast of Canada experiences high seismic activity with three earthquakes of magnitude greater than 6.0 having occurred along the west coast of Canada in 2014 (“Earthquakes Canada” 2014). This necessitates that buildings along the west coast of Canada be designed for the seismic hazard at the site in which shear walls play a crucial role. However, under the action of an earthquake load, corners of doors and windows in a shear wall experience damage due to stress concentration. This is the reason that during an earthquake evacuation drill, it is always advised not to take shelter close to a window or a door. Reinforcement provided to minimize damage at opening corners are designed for forces that develop at these corners called as transfer forces. In the literature, various methods and models exist to account for the effect of an opening on the performance of a shear wall. These methods have been predominantly studied for application to timber frame shear walls. This thesis focuses on understanding the forces that develop around opening corners in a cross-laminated timber (CLT) shear wall. CLT is a relatively new material in the Canadian market and is gaining prominence due to its good structural performance and low carbon footprint. CST Innovations (2011) is one of the first Canadian companies to develop CLT boards from mountain-pine beetle infested timber, which otherwise is not fit for use directly in structural applications. Manufacturing CLT from this timber provides utility to a resource that otherwise would have been wasted. CLT panels can also be manufactured from high-grade timber for better appearance and strength. CLT panels have higher stiffness and strength compared to other timber products, which makes 2 them suitable for use in construction of mid-rise buildings. However, CLT is a new material and there is a need to understand its behaviour. This project is an attempt at understanding the behaviour of CLT shear walls with openings. 1.2 Objectives Force transfer around openings (FTAO) is a design paradigm that explicitly considers the development of transfer force around opening corners and its effect on the performance of a shear wall. This thesis focuses on understanding the forces that develop around the opening corners in a CLT shear wall with the aid of numerical models developed in ANSYS, a commercial finite element analysis software. The analytical models from the literature to determine the transfer force in timber-frame shear walls has been reviewed in this thesis. The complex layered structure of CLT results in a complicated stress distribution unlike in timber frame shear walls. Therefore, an effort is first made to understand the in-plane behaviour of CLT and its effect on stress redistribution around an opening. Different finite element models have to be developed to study the transfer force development in CLT for different construction practices. Finally, the thesis tries to highlight the need for reinforcing the opening corners against the transfer force for better performance of the CLT shear wall. 1.3 Background In this section, the ongoing research in the field of CLT shear walls has been summarised. The vast amount of research on CLT shear walls is driven by its ability to be a good substitute for concrete in future. There is a research gap of few decades in understanding the behaviour of CLT and concrete and significant efforts are underway to minimize this gap. The literature 3 review in this section highlights the objective of this thesis in addressing one such research gap. Timber-frame shear walls have good seismic performance due to their high strength-to-weight ratio and ductile behaviour. Several experiments and numerical analysis have been conducted to identify the parameters to be considered in seismic design of timber structures (Ceccotti and Karacabeyli 2002; Ceccotti et al. 2000; Dolan 1989; Folz and Filiatrault 2004; Noory et al. 2010; Salenikovich and Dolan 2003). The research conducted on timber-frame shear walls have provided motivation for similar studies on CLT shear walls to characterize their seismic performance. Research in the area of lateral resistance of CLT shear walls has been conducted by Ceccotti and Follesa (2006) and Popovski et al. (2010). Their work brings to forefront the design considerations in a CLT shear wall for improving their seismic performance such as step joints in longer walls and nailed hold-down connections. A seminal work in studying the lateral resistance of CLT shear walls was conducted in the SOFIE project by Ceccotti et al. (2006). This study highlighted the importance of wall-to-floor connector behaviour to the performance of a CLT shear wall. Extensive tests conducted during this project has provided a database of connector responses for use in numerical modeling. Gavric et al. (2012) also carried out tests on CLT shear walls with different wall-to-floor connectors. The research focused on applicability of Euro Code 5 to the design of CLT shear walls and their seismic performance. The tests underlined the need to apply a capacity-based design principle in design of the shear wall to avoid brittle failure of wood. The tests revealed the difference in behaviour of single panel and coupled panel CLT shear walls. More details of this project can be obtained in the thesis by Gavric (2012). 4 Various efforts have also been undertaken to evaluate the q-factor or R-factors for CLT, which is a measure of the ductility in the system. Schneider et al. (2012) have provided a review of energy-based damage indices for determining the damage in a CLT shear wall under cyclic loading.. Pei et al. (2012) carried out performance-based seismic design of CLT buildings. They recommend Rd and Ro of 2.5 and 1.5 respectively for CLT structures in Canada with a symmetrical floor plan and an R-factor of 4.5 for ASCE 7. These recommendations have been provided in the CLT Handbook (Popovski et al. 2011) by FP Innovations as well. Pei et al. (2012) carried out component level testing on CLT shear walls and suggested an R-factor of 4.3, which is in good agreement with the performance-based design assessment in the previous study. The results from these studies have been used to assess the possibility of constructing medium to high-rise CLT buildings (Kuilen et al. 2011; Pei et al. 2012). However, there has been no significant research to address damage around an opening in a CLT shear wall. Force transfer around openings is one of three design paradigms, which addresses the effect of an opening on the performance of a shear wall. The two other design paradigms that address this problem are the perforated shear wall method (Line and Douglas 1996) and the segmented shear wall method (Breyer and Ank 1980). These two methods are code accepted procedures for design of shear walls with openings. However, they do not explicitly consider the forces that develop at the corners of the opening and do not provide solutions to minimize the damage at these locations. These two methods focus on the reduction in strength and stiffness of a shear wall due to the presence of an opening. The perforated shear wall method has been shown to provide conservative estimates of the shear wall stiffness (Dolan and Heine 1997). FTAO (Breyer et al. 2007) considers the effect of the opening more explicitly and provides a better understanding of the forces that damage the corners of the opening. 5 FTAO is a design paradigm recommended by the International Building Code (IBC 2006). The code states that the design of shear walls based on FTAO can be done using a rational design philosophy. This rational design philosophy is not an established procedure and four models are available in the literature, which are commonly used in practice. These four models are the drag-strut analogy, the cantilevered beam analogy, the coupled beam analogy and the Diekmann method. The analytical models were developed primarily for application to timber frame shear walls. A joint effort by APA, UBC and USDA conducted experimental testing on twelve shear wall set-ups to evaluate the transfer force in timber-frame shear walls (Skaggs et al. 2010). An in-depth study of the analytical methods for FTAO and comparison of the results obtained with experimental results was presented by Yeh et al. (2011). They observed that due to the presence of an opening, the stiffness and strength of the shear wall decreased and the transfer force was observed to increase with the size of the opening. The transfer force was found to be sensitive to the width of the wall segments on either side of the opening. Their research brought to forefront the discrepancy in the results obtained from the different analytical models and emphasized on the need for better models to determine the transfer force. Li et al. (2012) presented a detailed account of numerical modeling of timber frame shear walls. The modeling was done using a program known as Wall2D, which has the capability to model failure in the wood sheathing. The results from the study provide the global behaviour of the wall and can be used in the future to determine the transfer force. Dujic et al. (2008) conducted tests and numerical analyses of CLT shear walls with openings to assess their performance. They conducted a parametric study on 36 different opening configurations to develop empirical factors for reduction in strength and stiffness due to the 6 opening. Their research showed that the presence of an opening does not significantly alter the strength of a CLT shear wall relative to timber-frame shear walls. This is one of the few studies in the literature that exclusively considers the effect of an opening on the performance of a CLT shear wall. However, the study focuses on characterizing the reduction in strength and stiffness rather than determining the forces that develop at the corners of the opening due to stress concentration. To study FTAO in CLT shear walls it is necessary to study the in-plane behaviour of CLT panels. CLT is a complex material with a laminated plate-like structure and orthotropic material properties. Joebstl et al. (2008), Moosbrugger et al. (2006) and Bogensperger et al. (2010) have studied the in-plane behaviour of CLT panels. The efforts in studying the in-plane behaviour of CLT have targeted the determination of strength and stiffness of CLT panels. Gsell et al. (2007) present a procedure to obtain the homogenized orthotropic linear elastic material property of a CLT board. The procedure is based on minimizing the difference between estimated and measured resonance frequencies of rectangular CLT board specimens. The research on in-plane behavior of CLT panels provides an analogy to determine the stress distribution in a CLT panel under the action of in-plane loads. The recent advances in performance-based earthquake engineering (PBEE) and direct displacement-based design (DDBD) provide an incentive to develop models amenable for use in these paradigms. A procedure to carry out displacement-based design for timber structures was presented by Filiatrault and Folz (2002) and Pang and Rosowsky (2007). They proposed a pancake model (Filiatrault et al. 2003), which could simulate the 3-dimensional seismic response of a timber frame building. Using the pancake model, they were able to provide a validated equation for hysteretic damping input into the DDBD framework. The 7 implementation of PBEE framework for timber structures is an on-going research topic. The CURRE-Caltech Woodframe Project and NEESWood Project have helped shape a framework for performance-based design. The framework developed in this project enables the designer to select multiple hazard and performance expectation combinations (Lindt et al. 2013; Rosowsky 2002). Due to the importance of PBEE in seismic design, the objective in this thesis is to develop models amenable for use in this framework. The literature review points to the research gap in analysing the effect of openings on the performance of CLT shear walls. The lack of understanding of this problem cripples the effort to develop a performance-based design framework for CLT shear walls with openings. In this thesis, numerical models have been developed in ANSYS, which are amenable for use in a PBEE framework. The modeling of a CLT shear wall has two important components. One component is the CLT panel, which can be modeled using plane stress elements with homogenized material properties (Ashtari 2012). The next important component of the shear wall is the connectors, which have significant effect on the response of the wall. The modeling of the connectors in ANSYS is carried out using zero-length spring elements (Blasetti et al. 2006, 2008). Each connector is considered to be composed of a pair of springs acting in perpendicular directions. The springs are assigned the force-deformation response of the connectors. This modeling procedure has been explained in Chapter 4. An alternative to the deterministic models that have been discussed so far is regression modeling (Ang and Tang 2006; Box and Tiao 2001). Regression modeling has been previously used to predict structural responses, such as shear capacity of RC columns (Gardoni et al. 2002), building response and damage (Mahsuli and Haukaas 2013b). The development of a regression model can be done using the multi-model reliability analysis software Rt (Mahsuli 8 and Haukaas 2013a). Regression models are amenable for use in the unified reliability analysis framework (Haukaas 2008), which can be used in the future for design of CLT shear walls with openings. 1.4 Overview of the Thesis Finite element modeling is the primary approach adopted in this thesis to study the development of transfer force in CLT shear walls with openings. A library of finite element models has been developed in this thesis to analyse CLT shear walls with openings constructed using different practices, which is shown in Figure 1.1. Figure 1.1 Overview of finite element models 9 Figure 1.1 presents four finite element models that have been developed in this thesis using ANSYS. Model A is a continuum model developed using quadrilateral 4-node elements for studying the analytical models discussed in Chapter 2. Model B is a 3-dimensional model developed using 8-noded brick elements. Model C is a frame model that has been developed using 2-noded beam elements. The development of Model B and Model C is discussed in Chapter 3 for analysing a CLT shear walls with a cut-out opening. Model D is a continuum model calibrated to experimental data, which has been is discussed in Chapter 4 for analysing coupled-panel CLT shear walls. The thesis is structured to adopt a step-by-step approach for understanding the transfer force development in CLT and corresponding reinforcement requirements. The second chapter reviews the various analytical models available in the literature to determine the transfer force. A finite element model is also presented, i.e., Model A, which can provide the transfer force. The finite element model clearly shows the stress concentration at corners of the opening, leading to the development of transfer force. In the third chapter, the in-plane behaviour of CLT has been first explained. CLT shear walls with cut-out openings have been analysed in this chapter using Model B and Model C. The results from the analysis suggest that stress concentration around opening corners can lead to shear failure in the wood panels for large openings. The fourth chapter focuses on development of a finite element model, i.e., Model D, to study the behaviour of coupled panel CLT shear walls with opening. Tie-rods have been used in this case as reinforcement around the opening corners. The analysis results suggest that the use of tie-rods significantly increases the strength and stiffness of the wall. 10 In the fifth chapter, the development of a regression model for in-plane stiffness of a simply-supported CLT shear wall with a window-type opening is explained. The regression analysis indicates the effect of various geometrical and material parameters on the stiffness of the wall. The sixth chapter provides the conclusion of the thesis with emphasis on highlighting the manner in which this thesis addresses the present research gap. The chapter also provides directions of research that can be explored in future for design of CLT shear walls for the transfer force around opening corners. 11 Chapter 2: Analytical Models for Force Transfer around Openings Force transfer around openings (FTAO) is an approach suggested in the International building code for design of shear walls with openings. The objective of this approach is to ensure that the wall deforms as a single unit. In the presence of an opening, stress concentration occurs around the corners of the opening. These stress concentrations lead to development of tensile and compressive forces at the corners of the opening, which leads to deformation of wall at the opening corners. In FTAO design approach, reinforcements are provided at the corners to carry these forces to minimize the deformations around the opening corners. The tensile and compressive forces developing at the corners are called as transfer forces. In the literature, various analytical models are available to compute this transfer force. In this chapter, few of these models have been explained. Furthermore, the application of these models on an example shear wall is demonstrated. Based on this example and from literature, the advantages and disadvantages of using these models and their applicability to CLT shear walls is studied. 2.1 Drag Strut Analogy Drag strut analogy is the simplest method to determine FTAO (Martin 2005). A drag strut is a structural member that distributes the forces within a shear wall. In this analogy, it is considered that the panels on either side of the opening act as drag struts transferring the load from the full-width Panel A to full-width Panel D as shown in Figure 2.1. Panel B and Panel C are grey shaded in Figure 2.1 to highlight their function as drag struts. The analogy considers that the shear flow is constant in the horizontal direction and varies only at the boundary of the full-width panels with the drag struts. The transfer force develops due to the change in shear flow from one panel to another. Hence, the transfer force is computed as the difference in shear 12 flow across the intersection of a full-width panel and a drag strut, times the length of the intersection. The following discussion considers a sample shear wall and outlines the methodology followed in drag strut analogy. Consider the sample shear wall shown in Figure 2.1. The shear wall has an opening of height h and width w. A lateral load V acts at one end of the shear wall. In this method, the shear flow in drag strut and full-width panels, v and vd respectively, is first sought as shown in Figure 2.1. In this methodology, the shear flow is assumed to be constant in the panels. As a result, the shear flow in the drag struts, v, is obtained by dividing the shear force, V, by the total length that it acts over, namely l1+l2, which gives, )( 21 llVv (1) where, l1 and l2 are defined in Figure 2.1. Figure 2.1 Drag strut analogy 13 Similarly, vd is obtained by dividing the shear force, V, by the total length that it acts over i.e., l1+w+l2, which gives )( 21 lwlVvd (2) where, l1, l2 and w are defined in Figure 2.1. Next, consider the intersection between Panel A and Panel B. The length of intersection between the panels is l1. The shear flow in Panel A is v and shear flow in Panel B is vd. The force developed in Panel A at the intersection is the shear flow, v, times the intersection length, l1. Similarly, the force developed in Panel B at the intersection is the shear flow, vd, times the intersection length, l1. This shows that at the intersection, the forces are not balanced. This difference in force is the transfer force. Hence, the transfer force, Fu, developed at the intersection of Panel A and Panel B is given as, )(1 du vvlF (3) The drag strut, Panel B, is under equilibrium. Hence, by balancing the forces on this panel, the transfer force developing at the intersection of Panel B and Panel D can be shown to be equal and opposite to Fu, as shown in Figure 2.1. On the other side of the opening, transfer force, Fl, develops at the intersection between Panel A and Panel C. The panels overlap with an intersection length l2. The shear flow in panel A and Panel C is vd and v, respectively. The force developed in Panel A at the intersection is the shear flow, v, times the intersection length, l2. Similarly, the force developed in Panel C at the intersection is the shear flow, vd, times the intersection length, l2. As done previously, the 14 transfer force Fl is computed as the difference in force across the intersection. The transfer force, Fl, developed at the intersection of Panel A and Panel B is given as, )(2 vvlF dl (4) The drag strut, Panel C, is under equilibrium. Hence, by balancing the forces on this panel, the transfer force developing at the intersection of Panel C and Panel D can be shown to be equal and opposite to Fl, as shown in Figure 2.1. 2.2 Cantilever Beam Analogy Cantilevered beam analogy is another model for determining FTAO (Martin 2005). For the purpose of analysis based on this model, the wall is divided into panels as shown in Figure 2.2. The transfer force is computed in this model as the reaction force exerted by the full-height panels on either side of the opening on the panels above and below the opening. The computation of this reaction force is complex and the model considers few assumptions to simplify the problem to attain an analytical solution. Firstly, the model assumes an inflection point at the mid-height of the opening. A horizontal section X-X through this point divides the full-height piers on both sides of the opening into Panel A, Panel B, Panel C and Panel D as shown in Figure 2.2. Along this section the bending moment is zero as the shear force does not vary along the height. Hence, these panels act as cantilevered piers with Panel E and Panel F as supports. In the presence of the opening, the shear flow is assumed to be distributed equally to panels above and below the opening. The shear flow from the cantilevered piers to the supports leads to the development of reaction forces in supports. These reaction forces act as a moment-couple as shown in Figure 2.2. The resolution of this moment couple helps to 15 determine the reaction force. The following section considers a single panel and analyses it to highlight the steps involved in the methodology. Figure 2.2 Cantilevered beam analogy Firstly, consider the full-height panel to the left of the opening. As shown in Figure 2.3 (a), this panel is divided into two cantilevered piers, i.e., Panel A and Panel B by the horizontal section X-X. A lateral load V is applied on the shear wall, which leads to the development of a shear force V in the shear wall. The corresponding shear flow that develops varies in horizontal and vertical directions in each pier. But, in this model, the shear flow in each pier is assumed to be constant. At section X-X, the net shear force is V, which acts over a length l1+l2. Hence, the shear flow, v, in the cantilevered piers is given as, )( 21 llVv (5) 16 where, l1 and l2 are the width of the full-height panels on either side of the opening as shown in Figure 2.2. Therefore, the shear force, V1, developed in Panel A and Panel B, of width l1, is given as, )( 2111 lllVV (6) Figure 2.3 Analysis of the full-width panel on the left side of the opening The shear force diagram is shown in Figure 2.3 (b). The bending moment diagram for the piers can be drawn from the shear force diagram as shown in Figure 2.3 (c). The bending moment, MA, along the top edge of the support, Panel E, is given as, uA hhVM 21 (7) where, h and hu are defined in Figure 2.3 (a). The reaction force developed in Panel E due to the load acting on Panel A is FA. The reaction force acts as a moment couple acting along the edges of Panel E as shown in Figure 2.3 (d). This internal force is the transfer force arising at 17 that corner of the opening. The free-body diagram in Figure 2.3 (d) shows the equilibrium of Panel A. The reaction force FA acts at a distance hu from the edge of Panel A. The moment due to the reaction force at the edge of the support is FA times the height, hu. The bending moment at this edge from the bending moment diagram is MA. Equating MA to the moment developed by the reaction force, the reaction force, FA, can be determined as, uuA hhhVF 21 (8) Adopting the same methodology, the free-body diagrams of Panel B and Panel F can be established. Using the procedure described above the transfer force FB is given by, llB hhhVF 21 (9) where, hl, h and the transfer force FB are defined in Figure 2.2. Next, the full height pier on the right side of the opening is considered. The first step is to determine the shear force V2 in this pier. The shear flow in the pier is v, which acts over a length l2. Hence, the shear force V2 is given as, )( 2122 lllVV (10) Following the previously established procedure, the shear force and bending moment diagram of Panel C and Panel D can be established. Then, considering the equilibrium of supports, Panel E and Panel F, the reaction forces FC and FD can be computed as given by, 18 uuC hhhVF 22 (11) llD hhhVF 22 (12) where, h, hl, FC and FD are described in Figure 2.2. The reinforcing straps at the corners of the opening is designed for the largest tensile transfer force occurring at the corner of the opening. For the wall shown in Figure 2.2, FA and FD are the tensile forces and the larger of these two forces will be the design force. In the case the load applied V, is in the opposite direction, then, FB and FC are the tensile forces, which have to be considered in design. 2.3 Coupled Beam Analogy The coupled beam analogy is a rigorous mechanics-based approach to determine FTAO (Diekmann 1995). In this analogy, the panels above and below the opening are considered to act as coupling beams. Panel A and Panel B are shaded in Figure 2.4 to highlight their function as coupling beams. They connect the full height panels, Panel C and Panel D, transferring the shear from one panel to another. Unlike drag-strut and cantilevered-beam models, this model considers the variation in shear flow along the width of the wall. Considering this variation, the free-body diagram of the wall is established as shown in Figure 2.5. The transfer force is determined by solving this free-body diagram. To solve the free-body diagram, certain assumptions are made. Firstly, an inflection point is assumed at the mid-point of the coupling beams and mid-point of the side-panels. A horizontal section X-X and a vertical section Y-Y can be drawn through this inflection point as shown in Figure 2.4. The shear flow is assumed 19 to be constant along these sections. Below, the analysis procedure is explained to determine the transfer force based on the coupled-beam model. Figure 2.4 Coupled beam analogy Consider the sample shear wall shown in Figure 2.4, which has an opening of height h and width w. The shear wall is subjected to a lateral load V. To analyse the shear wall using coupled beam analogy, the first step is to determine the shear force that develops in the full-height panels and the coupling beams. The analogy considers that the shear flow is constant along the section X-X and section Y-Y. In Panel C and Panel D, the shear force at the horizontal section X-X is V. This shear force acts over an effective length l1+l2. Therefore, the shear flow in Panel C and Panel D at section X-X is given as, 20 )( 21 llVv (13) where, l1 and l2 are defined in Figure 2.4. In Panel C, the shear flow v acts over a length l1 to develop a shear force V1, which is given as, )( 2111 lllVV (14) Similarly, in Panel D, the shear flow v acts over a length l2 to develop a shear force V2, which is given as, )( 2122 lllVV (15) Next, the shear force in the coupling beams, Panel A and Panel B has to be determined. Prior to determining the shear force in the coupling beams, it is useful to determine the reaction force that develops at the base of the shear wall to prevent overturning. The hold-downs for the shear wall are provided at B and C. The reaction forces at these hold-downs are FT and FC, respectively, as shown in Figure 2.4. The reaction force FT can be calculated by considering the equilibrium of the shear wall about C. The lateral load V acts at distance hu+h+hl from point C. The clockwise moment due to the lateral load is V times the distance hu+h+hl. The reaction force in the hold-down, FT, acts at a distance l1+w+l2 from the point C. The counter-clockwise moment produced by this force is calculated as the reaction force FT multiplied by the distance from point C, i.e., l1+w+l2. As the net moment at point C is zero, equating the moments, FT, can be computed as, 21 lwlhhhVF luT (16) 21 where, hu, h, hl, l1, w and l2 are defined in Figure 2.4. The net force in the vertical direction is zero. This implies that the reaction force at the other hold-down, FC, is equal and opposite of FT. After obtaining the reaction forces, the shear forces in the coupling beams can also be calculated. The horizontal section X-X and vertical section Y-Y divide the shear wall into four quadrants. Consider the top left quadrant of Figure 2.4, which consists of a segment of Panel C and Panel A. The quantity of interest here is the shear force, V3, which develops in the coupling beam, Panel A. The shear force, V3, in this beam is constant along its length w. To compute this shear force, the equilibrium of the quadrant is considered. The quadrant is under stable equilibrium. Therefore, the net moment of all forces acting on this quadrant, i.e., V1 and V3, about any point on the quadrant is zero. Consider the net moment about point A due to the forces V1 and V3. V1 is the shear force in Panel C, which acts at a distance l1+w/2 from the point A. V3, acts at a distance hu+h/2 from the point A. Equating the net moment at A due to V1 and V3 to zero , V3 is given as, 22113 wlhhVVu (17) Next, the shear force, V4, in the other coupling beam, Panel B is determined. To compute this shear force, consider the sum of vertical forces at the section Y-Y shown in Figure 2.4. The net force at the section is the reaction force FT. This implies that the sum of V3 and V4 should be equal to FT. Hence, V4 is given as 34 VFV T (18) 22 After computing the shear force in the panels, the next step is to establish the complete free-body diagram of the wall as shown in Figure 2.5. The wall is divided into 12 elements and Figure 2.5 shows the forces that develop in these elements. These internal forces in the elements can be computed using mechanics-based relationships. In the free-body diagram the transfer forces are indicated as V8, V9, V10 and V11. Figure 2.5 Free-body diagram for coupled beam analogy Here underneath, the procedure to compute the transfer force V8 is outlined. The procedure can be extended to compute the transfer forces at the other corners of the opening as well. To 23 compute V8, consider the element II shown in Figure 2.5. This element is under equilibrium. Therefore, the sum of moments created by all internal forces on this element about any point on it is zero. So, consider the moment of all internal forces in the element about a point E, as shown in Figure 2.5. The force V3 acts at a distance w/2 from the point E and creates a counter-clockwise moment. The transfer force V8 acts at a distance hu from the point E, creating a clockwise moment. As the sum of moments about point E is zero, V8 is given as, uhwVV 238 (19) where, w and hu are defined in Figure 2.5. Similarly, the transfer force V9, V10 and V11 can be obtained. To determine the transfer force V9, consider moment about point F for element III. Considering the equilibrium of element III about point F, the transfer force V9 is given as, uhwVV 239 (20) The transfer force V10 can be determined by considering moment about point G for element X. Considering the equilibrium of element X about point G, the transfer force V10 is given as, lhwVV 2410 (21) Similarly, the transfer force V11 is computed by considering the equilibrium of element XI about point H. The transfer force V10 is given as, 24 lhwVV 2411 (22) A limitation of this method is that the minimum panel height above opening has to be 12 inches (Yeh et al. 2011). A lower height results in large resolved shear forces causing overstressing of the panel. 2.4 Diekmann Method The Diekmann method is a variation of the coupled-beam analogy described in section 2.3. The method was suggested by Edward Diekmann in response to the comparison of methods presented by Martin (Martin 2005). The lateral load applied on the wall produces a horizontal shear, which is resisted by the panels on either side of the opening. This is considered for analysis in the drag-strut analogy and cantilevered beam analogy. But, the hold-down forces produces a vertical shear in the panels above and below the opening. This effect is not considered in drag-strut analogy and cantilevered beam analogy. By considering this effect, the free-body diagram for the wall can be established, which is similar to the one shown in Figure 2.5. The difference between coupled beam analogy and the Diekmann method is the approach used to determine the vertical shear developing in the wall. Below, a simplified methodology is explained to determine the transfer force based on the Diekmann model. Consider the shear wall shown in Figure 2.6. The shear wall has an opening of height h and a width w. The wall is analysed for a lateral load V applied along the top edge. The wall is connected to the floor at its two bottom corners by hold-downs. The hold-downs develop resisting force, which can be determined by equating the overturning moment from the lateral load V to the resisting moment developed by the hold-down forces. The tensile hold-down 25 force is FT and is given by Eq. (16). The compressive force developing at the other bottom corner, FC, will be equal and opposite in nature. This couple of hold-down forces produces a vertical shear in the wall resisted by the panels above and below the opening. The shear flow developing in these panels is denoted by vuo and vlo as shown in Figure 2.6. This shear flow develops due to the vertical shear FT, which acts over an effective height hu+hl and is given as, luTlouo hhFvv (23) where, vuo, vlo, FT, hu and hl are defined in Figure 2.6. Figure 2.6 Diekmann method 26 The shear flow in the panels above and below the opening are assumed to create a boundary force at the top and bottom edge of the opening. This boundary force, FB, is the product of shear flow in the panel, vuo or vlo, and the length that it acts over, i.e., w, which is given as, wvF uoB (24) where, w is the width of the opening over which the unit shear, vuo, acts. The boundary force, FB, is distributed to the corners of the opening based on the relative width of the panels on either side of the opening. This distributed force at the corner is the transfer force. Therefore, the transfer force in the left-hand side top corner of the opening, F1, is given as, 2111 lllFF B (25) where, l1 and l2 are defined in Figure 2.6. Similarly, the transfer force at the right hand top corner of the opening, F2, is given as, 2122 lllFF B (26) The model as previously mentioned, works by developing the free-body diagram. Hence, the other forces in the boundary members can be computed. These forces in boundary members, though not the transfer force, are important in design of the structural members composing the wall. The above sections explain the analytical models available to determine the design transfer force. In the next section, an example shear-wall will be analyzed. The results will be compared with those from a finite element model in order to assess the accuracy of the previously presented analytical methods. 27 2.5 Numerical Example In this section an example shear wall will be analyzed. The geometry of the wall is shown in Figure 2.7. A lateral load of 100kN is applied on the shear-wall. The analytical models previously described are used to determine the transfer force developing at the corners of the opening for the load applied. Figure 2.7 Example shear-wall Drag-strut analogy is the first method employed to determine the transfer force in the wall. The shear flow distribution in the wall for this model is shown in Figure 2.1. The shear flow developing in the panels due to the lateral load is first determined. The shear flow, v, in the panels acting as drag struts is given by Eq. (1) as, mkNllVv /33.3312100)( 21 28 Similarly, the shear flow, vd, in the full-width panels above and below the opening is given by Eq. (2) as, mkNlwlVvd /00.25112100)( 21 The difference in shear flow is assumed to produce the transfer force at the corners of the opening. The transfer force, Fu, at the left-hand top corner of the opening is given by Eq. (3) as, kNvvlF du 67.1600.2533.3321 Similarly, the transfer force, Fl, at the right-hand top corner of the opening is given by Eq. (4) as, kNvvlF dl 33.833.3300.2512 The next model used to compute the transfer force is the cantilevered beam analogy. The division of the wall into panels for analysis using this model is shown in Figure 2.2. The first step is to compute the shear flow in each panel due to the lateral load. The shear flow, v, in the cantilever piers on either side of the opening is given by Eq. (5) as, mkNllVv /33.3312100)( 21 29 By knowing the shear flow in the pier, the shear force and bending moment diagram can be established, which is similar to Figure 2.3. The shear force, V1, in the cantilevered pier A is given by Eq. (6) as, kNlllVV 67.6632100)( 2111 The transfer force, FA, which develops at the top right-hand corner of the opening is given by Eq. (8) as, kNhhhVFuuA 67.1665.05.025.167.6621 Adopting the same procedure, the transfer force, FB, at the top left-hand corner of the opening is given by Eq. (9) as, kNhhhVFllB 67.1161125.167.6621 The shear flow in the cantilever pier B is given by Eq. (10) as, kNlllVV 33.3331100)( 2122 The transfer forces, FC and FD, are computed using Eq. (11) and Eq. (12) as, kNhhhVFuuC 32.835.05.025.133.3322 30 kNhhhVFllD 33.581125.133.3322 The third model employed herein is the coupled-beam analogy. The shear flow in the wall assumed in this model is shown in Figure 2.4. In this method the free-body diagram of the wall is established as shown in Figure 2.5 to determine the transfer force. First, the shear flow, v, in the full-height panels is computed using Eq. (13) as, mkNllVv /33.3312100)( 21 Then, the horizontal shear force in these panels, V1 and V2 can be computed using Eq. (14) and Eq. (15) as, kNlllVV 67.6632100)( 2111 kNlllVV 33.3331100)( 2122 Next, the equilibrium of the panel is considered to compute the hold-down forces. The hold-down force FT, is given by Eq. (16) as, kNlwlhhhVF luT 7511215.15.010021 The hold-down forces produce a vertical shear in the wall, which is carried by the coupling beams shown in Figure 2.4. The vertical shear, V3 and V4, in coupling beams, Panel A and Panel B, is computed using Eq, (17) and Eq (18) as, 31 kNwlhhVVu33.3321225.15.067.6622113 kNVFV T 67.4133.337534 After determining the vertical and horizontal shear in the panels, the free-body diagram is set up as shown in Figure 2.5. The transfer forces developing at the corners of the opening, V8, V9, V10 and V11 are shown in the figure. These forces are determined using Eq. (19) to Eq. (22) as, kNhwVVu33.335.02133.33238 kNhwVVu33.335.02133.33239 kNhwVVl84.2012167.412410 kNhwVVl84.2012167.412411 Diekmann method is the last analytical model used to determine the transfer force. For analysis using this model, the wall is divided into panels as shown in Figure 2.6. The wall develops horizontal and vertical shear in the panels under the action of the lateral load. The horizontal shear in the wall is resisted by the panels on either side of the opening. The vertical shear in the wall is produced due to the action of the hold-down forces. This vertical shear is resisted 32 by the panels above and below the opening. The shear flow in these panels is given by Eq. (23) as, mkNhhFvvluTlouo /5015.075 The shear flow computed above acts over the top and bottom edges of the opening over a length w. This produces a force along the horizontal free edges, known as boundary force, FB, which is given by Eq. (24) as, kNwvF uoB 50150 This force is distributed to the corners of the opening as transfer force. The transfer force at the top left-hand opening corner, F1, is given by Eq. (25) as, kNlllFF B 33.331221502111 Similarly, the transfer force at the top right-hand opening corner, F2, is given by Eq. (26) as, kNlllFF B 67.161211502122 The forces at the bottom edge of the opening are same as the top edge but in the opposite direction. The results obtained from the various models is compared in Figure 2.8. 33 Figure 2.8 Comparison of transfer force from analytical models The analytical models provide a means to compute the transfer force in a simple manner. Also, the analytical models provide insight into the development of forces in the shear wall. However, these models make significant assumptions to simplify the analysis. The force distribution in a shear wall is complex. In the next section, a finite element model is developed to study the stress distribution and development of forces within the wall. 2.6 Finite Element Modeling and Comparison of Results In this section, a finite element model is developed in ANSYS, which has been introduced in section 1.4 as Model A. The objective of this simplified model is to study the stress distribution across the plane of the wall. It also provides a simple means to compute the transfer force. The results from the analytical and finite element models will be compared later in this section. The comparison of results is expected to expose the need for better models to study the forces within a shear wall. 020406080100120140160180Corner 1 Corner 2 Corner 3 Corner 4Transfer force (kN)Drag-strut analogy Cantilever-beam analogy Coupled-beam analogy Diekmann method 34 The example wall shown in Figure 2.7 was considered for analysis. The finite element model assumes that the shear wall behaves as a continuum. As the thickness of the shear wall is relatively small compared to the lateral dimensions, plane stress quadrilateral elements were used to model the wall. The finite element model developed is shown in Figure 2.9. The model represents a single panel wall with an opening in the middle. In the model, four different panels, namely panels A, B, C and D, are joined together around the opening using spring elements to act as a single panel. This approach of modeling the wall with four panels connected by springs enables computation of the transfer force in a simple manner. The panels are meshed using 4-node quadrilateral elements called PLANE42 in ANSYS. The mesh is generated using ANSYS parametric design language (APDL). In this method of mesh generation, the nodes are manually assigned to specific coordinates with ordered numbering. Then, the nodes are joined together to form a mesh of PLANE42 elements. In the model shown in Figure 2.9, the quadrilateral elements have a side length of 0.01m. There are 106,254 nodes in the model, which are connected together to form 105,000 quadrilateral elements. Figure 2.9 Finite element model of a shear wall as a continuum (Model A) 35 The four panels in the model are joined together using spring elements. In Figure 2.9, the vertical lines (shown in yellow) emanating from the corners of the opening show the intersection of the panels. The springs connecting the panels are distributed along these lines. At the intersection, every node from either adjoining panel is connected using a pair of springs. One spring acts in the horizontal direction and the other acts in the vertical direction, carrying the forces that develop between the panels. Both the springs are unidirectional zero length springs with high stiffness in the direction of orientation. The quadrilateral elements used to mesh the panels have two degrees of freedom, translation in X and Y direction denoted as UX and UY. This paired set of springs connect the degrees of freedom of the quadrilateral elements across the intersection. The next aspect considered in modeling is the loading and boundary conditions. The shear wall is considered to be simply-supported at the base as shown in Figure 2.9. A lateral load of 100kN is applied at the top left-hand corner of the wall. The objective of using springs along the intersection of panels is to compute the transfer force. The transfer force is the sum of the axial force exerted by one panel over another around the corner of the opening. For example consider the top left corner of the opening. When a lateral load is applied at the top left-hand corner of the wall, this corner of the opening experiences a tensile force. The intersection height between panels A and B at this corner is hu. If the axial stress distributed along this intersection is σx(y), then the transfer force, F1, at this corner can be computed as, dytyF uh x 01 (27) where, t is the thickness of the wall. When, the springs join the panels at the intersection, this axial force, which varies along the height of the intersection is distributed to the springs. 36 Therefore, the transfer force can be computed by summing the force in the springs at the corners of the opening. For instance, the transfer force at the top left-hand corner of the opening is computed by summing the tensile forces in the springs at the intersection between panels A and B. Similarly, using the model, the transfer force at the other corners of the opening can be computed. The transfer force computed at the corners of the opening is summarized in Table 2.1. The notation of corner numbers is shown in Figure 2.9. Corner 1 Corner 2 Corner 3 Corner 4 -18.46 kN 30.32 kN -27.39 kN 12.10 kN Table 2.1 Transfer force from the finite element model Figure 2.10 Deformed shape of the wall under the action of a lateral load 37 The design transfer force is the maximum tensile transfer force that can develop at the corners of the opening. The design transfer force is used to design reinforcing straps at the corners of the opening. These reinforcing straps carry the tensile force preventing damage at the corners of the opening. Figure 2.11 compares the transfer force obtained from the different analytical models and the finite element model. Figure 2.11 Comparison of design transfer force obtained from different models The comparison in Figure 2.11 clearly shows the variation in results from different models. The transfer force obtained from the finite element model can be considered as the most reliable due to the better approximation of stress distribution in this model. Compared to the finite element model, the drag-strut analogy severely under predicts the design transfer force and the cantilever-beam model severely over predicts the design transfer force. This variation in results obtained from these two models is mainly due to the assumptions made in the shear flow distribution. The coupled-beam analogy and the Diekmann model provide reasonably 020406080100120140160180Corner 1 Corner 2 Corner 3 Corner 4Transfer force (kN)Drag-strut analogy Cantilever-beam analogy Coupled-beam analogyDiekmann method FE model 38 good approximation of the transfer force when compared to the finite element model. This improvement in the result is primarily due to more detailed consideration of the shear flow variation at different sections of the wall. The trend observed here is well documented in the literature as well (Li et al. 2012; Robertson 2009). The Diekmann model has been regarded as the most suitable model for determining the transfer force for window-type openings. The coupled-beam analogy and the Diekmann method are based on establishing the free-body diagram. The computation of shear flow in the panels is a pre-requisite for developing the free-body diagram. In the presence of a door-type opening, the coupled-beam analogy is not applicable because in this model, as the height of the panel above or below the opening tends to zero, the transfer force tends towards infinity. The Diekmann model, which is also based on similar stress distribution assumption over predicts the transfer force. Finite element modeling provides an alternative approach to determine the transfer force. With finite element modeling, the stress distribution can be captured in more detail. This aids in predicting the transfer force accurately. Moreover, finite element models can be improved based on availability of data to represent the structure as close to reality as possible. An advantage of finite element modeling is the capability to study the principal stress plot. Figure 2.12 shows the principal stress vector plot of corner 2 of the model. From this plot, the direction of principal stress and the type of stress developing at the corner can be identified. From Figure 2.12, it can be seen that corner 2 is subjected to tensile stress. The length of the principal stress vector is proportional to its magnitude. In the figure, closer to the corner, the length of vectors increases. This indicates that the magnitude of stress is much larger closer to the corner due to stress concentration. The cumulative effect of this stress concentration is the transfer force. The stress developing at the corner is aligned in an inclined direction. This 39 indicates that the reinforcement at the corner designed to carry this stress should ideally be aligned in this direction. But, in a timber shear wall, the diagonal force is decomposed into two different forces along vertical and horizontal directions. The vertical force is generally carried by studs in the wall. The horizontal force is the transfer force, which is carried by the reinforcing straps. In a reinforced concrete shear wall, in the absence of studs, the reinforcement at the corners are generally provided by either using diagonal straps or L-shaped angle straps. This is indicative of the usefulness of finite element modeling in understanding the development of internal forces in the shear wall. Figure 2.12 Principal stress vector plot of corner 2 1 DEC 3 201316:31:43VECTORSTEP=1SUB =1TIME=1SPRIN1PRIN2PRIN3 40 2.7 Conclusions In this chapter, the application of four different analytical models to determine the transfer force around opening corners was explained. Also, a finite element model was developed that could provide the transfer force at the corners of the opening. The finite element model can provide results such as the principal stress vectors, which are useful in determining the type of stress for which reinforcements need to be designed. The Diekmann model was observed to be most suitable analytical model for determining the transfer force around window-type openings. The analytical models fail in their application to walls with multiple openings and door-type openings, which can be analysed using the finite element model. The coupled-beam analogy is not applicable when there is panel missing above or below the opening and the Diekmann model is found to over predict the transfer force (Yeh et al. 2011). 41 Chapter 3: CLT Shear Wall with a Cut-out Opening Construction of CLT shear walls with openings generally follows two different practices and each practice has its own design considerations. This chapter focuses on understanding the reinforcement requirements for the first construction practice in which an opening is cut-out in the wall. The development of two finite element models used to analyse this construction type is discussed in this chapter along with the results obtained from finite element analysis. The first model is a three-dimensional finite element model and the second model is a frame model, both of which help in analysing the stress distribution around an opening. The second construction practice of coupled-panel CLT shear walls with openings is discussed in the next chapter. 3.1 In-plane Behaviour of CLT It is necessary to study the in-plane behaviour of CLT to understand the type of stress and force for which reinforcements need to be designed for at the corners of a cut-out opening. The complex structure of CLT is shown in Figure 3.1. The figure shows a CLT block consisting of five laminates with adjacent laminates aligned orthogonally. This layered arrangement coupled with the orthotropic nature of wood results in complex behaviour of CLT under the action of in-plane loads. An analytical approach to explain the stress distribution in a CLT panel was presented by Bogensperger et al., (2010). 42 Figure 3.1 Structure and discretization of a CLT panel (Bogensperger et al. 2010) In Figure 3.1, the CLT block is composed of five laminates, where each laminate is composed of wood panels aligned in the same direction. There is miniscule spacing between the panels in a laminate, which imparts discrete behaviour to the CLT block. In certain cases, the narrow faces of the panels are glued together to ensure continuum like behavior. However, for the study here, the panels are considered to be glued together only at the interface of the laminates. The analytical approach to determine the stress distribution in CLT employs a tool called as representative volume sub-element (RVSE). The development of an RVSE can be understood by studying the symmetric structure of CLT. In a CLT block, the laminates are glued to one another and a glued surface exists at the overlap of any two wood panels from the orthogonally aligned laminates. Over the face of the wall many such glued surfaces are present, which are 43 responsible for transfer of force from one laminate to another. The CLT block can be discretized into smaller elements called as the representative volume elements (RVE) based on symmetry of the system as shown in Figure 3.1. Each RVE, has planes of symmetry in the thickness direction as CLT consists of numerous layers. In the case considered here, as there are five laminates, there are three planes of a symmetry in each RVE. Therefore, the RVE can be discretized into three smaller repetitive units along these planes of symmetry called as representative volume sub-elements (RVSE). Each RVSE consists of a single glued surface and panels associated with that glued surface. The thickness of each RVSE depends on the laminates it is associated to, the guidelines for which have been outlined by Bogensperger et al., (2010). The RVSE is a useful tool in understanding the stress distribution in a CLT block under the action of an in-plane load. Bogensperger et al., (2010) state that the stress distribution in an RVSE can be assumed to be constituted of two different mechanisms, a shear mechanism and a torsion mechanism. These two mechanisms do not exist independently, but their combined effect helps to analyse the stress distribution in CLT. The shear mechanism is represented in Figure 3.2. The figure shows the glued surface and the panels associated with it in the RVSE, i.e., Panel A and Panel B. Panel A runs vertically and Panel B runs horizontally. Under the action of an in-plane load, the RVSE is subjected to a shear force Vxy,RVSE, which is proportional to the load on the CLT block. This shear force produces a shear stress τo, in the panels, which is given as taV RVSExyo , (28) 44 where, a and t are the dimensions of the RVSE as shown in Figure 3.2. The orientation of the shear stress is shown in the figure. Figure 3.2 Nominal shear stress in a RVSE The other mechanism is the torsion mechanism, which is shown in Figure 3.3. Under the action of an in-plane load, there exists a torsional moment MT, between the panels, Panel A and Panel B. This torsion leads to development of shear stress, τt, in the glued surface. This shear stress varies across the glued surface to a maximum value of τm at the edges of the RVSE. The orientation of the shear stress can be seen in Figure 3.3. The torsional moment MT, in the RVSE is given as, 2atM mT (29) where, t and a are the dimensions of the RVSE as defined in Figure 3.3. τm is the shear stress at the edge of the RVSE. 45 Figure 3.3 Torsional shear stress in a RVSE The combined effect of the two mechanisms described above provides an analytical approximation of the stress distribution in the RVSE, which is shown in Figure 3.4. Panel A runs vertically and has vertical free edges along which the shear stress is zero. Similarly, the shear stress along the horizontal free edges of Panel B is zero. The shear stress along all the edges by the shear mechanism is τo. The shear stress along all the edges is τm from the torsion mechanism. The sum of these two mechanisms provides the net shear stress. As the shear stress at the free edge of the RVSE is zero, the sum of τo and τm is zero. Therefore, the relationship between the two mechanisms is, vo (30) Along the non-free edges of the panels, the shear stress is τv. This shear stress is the sum of the shear stress from the two mechanisms, i.e., τo and τm. Using Eq. (30), the shear stress at the non-free edges is computed to be, 46 mvτ 2 (31) where, τm is the shear stress at the edge of the RVSE from the torsion mechanism. Figure 3.4 Real shear stress distribution in a RVSE The procedure described above brings to forefront the mechanisms underplay in deformation of a CLT panel. It can be seen that by determining the torsional moment in the glued surface, the stress distribution in that glued surface and the associated panels can be established analytically. If the torsional moment at a glued surface is MT, then the average torsional stress, τm, which develops in the glued surface is given as, 2atMτ Tm (32) where, t is the thickness of the RVSE that the glued surface is a part of and a is the width of the glued surface. 47 The maximum torsional stress, τt can be computed from the average torsional stress as, atmt 3 (33) The calculation of torsional moment and stress along with design checks is explained with an example by Bogensperger et al. (2010). The next few sections of this chapter will explain the development of two different finite element models and the results obtained from them. 3.2 CLT Shear Wall with a Cut-out Opening In this section, the development of a 3-dimensional solid model of a CLT shear wall with a cut-out opening will be explained, which is introduced in section 1.4 as Model B. The geometry of the wall considered for modeling is shown in Figure 3.5. Figure 3.5 CLT shear wall with a cut-out opening 48 Figure 3.5 shows a square wall of side 2.84m, composed of three laminates with a total thickness of 0.12m. At the center of the wall, an opening is cut-out of width 1.12m and height 1.12m. Each laminate in the wall is composed of 13 wood panels of width 0.2m, thickness 0.04m and height 2.84m. The spacing between panels in a laminate is assumed to be 0.02m, which is usually due to engineering limitations or provided intentionally. The spacing between the panels is limited by functional and aesthetic requirements. In the wall considered here, the narrow faces of the panels are not considered to be glued together. The three dimensional finite element model developed in ANSYS for the wall is shown in Figure 3.6. Figure 3.6 3-dimensional finite element model of a CLT shear wall with a cut-out opening (Model B) 49 In this model, each panel is exclusively modeled as shown in Figure 3.6. The panels are meshed using 8-node brick elements called as SOLID 82 in ANSYS. The material properties of the constituting wood, which is defined to the SOLID 82 elements is given in Table 3.1. The size of each element is defined to be 0.02m, which implies that each panel has 10 elements across its width and 2 elements along its thickness. The next component to model is the glued contact between the laminates, which is assumed to be rigid in the model. This rigid glue contact is modeled by merging the nodes shared by the panels in contact. The wall is assumed to be fixed at the base, with a lateral load distributed along the top edge of the wall. The results from Model B will help in determining the stress distribution in the glued surface, as well as the redistribution of stress around the corners of the opening. Species group Modulus of Elasticity (GPa) Shear modulus (GPa) Poisson ratios EL ET ER GLR GLT GRT νLR νLT νRT S-P-F 11.43 0.777 1.166 0.56 0.526 0.057 0.316 0.347 0.469 Table 3.1 Material property of CLT panel (Yawalata and Lam 2011) The wall shown in Figure 3.5 was analysed for a load of 50kN applied along the top edge with the base assumed to be fixed to the floor. Figure 3.7 shows the variation of the axial stress in X-direction across the face of the CLT shear wall. The figure shows that there is no definite axial stress concentration at the corners of the opening as observed in the continuum case studied in Section 2.6. This is due to the discontinuous nature of the CLT laminates. The maximum shear stress in the wood panel around the opening corners is found to be 1.94N/mm2. 50 Figure 3.7 Axial stress σX in the panels The stress redistribution in the presence of an opening is reflected in the torsion in the glued surface. Figure 3.8 shows the principal stress vector plot of the glued surface diagonally adjacent to the top-right corner of the opening. The alignment of the principal stress vectors indicates that there is interaction between the shear and torsion mechanism in the glued surface. This observation is coherent with the theory put forward by Bogensperger et al. (2010). 51 Figure 3.8 Principal stress plot for the glued surface diagonally adjacent to the top-right corner of the opening The three-dimensional model helps in determining the stress distribution around the corners of the opening. However, the model is computationally intensive and improving the model to include effect of connectors and other non-linear elements will significantly increase computational cost. A simpler approach is to obtain the torsion in the glued surface and employ the analytical approach suggested by Bogensperger et al. (2010) to determine the stress distribution. To address this concern, a frame model was developed that could quantify the torsion in glued surfaces. The development of the frame model and subsequent analysis is presented in the next section. 52 3.3 Frame Model for a CLT Shear Wall with a Cut-out Opening In this section, the development of a frame model is discussed, which has been introduced in Section 1.4 as Model C. The frame model provides a tool to compute the torsional moment in the glued surface and to subsequently employ the analytical method to determine the stress distribution in the panels. Two different frame models were developed; one for a wall without an opening and another for a wall with an opening. The two frame models help quantify the variation in torsion in the glued surface due to the presence of an opening. First, consider the development of a frame model for a wall without an opening. The wall considered for modeling is shown in Figure 3.9. Figure 3.9 CLT shear wall without an opening 53 The wall shown in Figure 3.9 consists mainly of two components that need to be considered in developing the frame model. The first component are the wood panels in the laminates, which have been modeled using 2-noded beam elements called as BEAM 44 in ANSYS. The length of each element is the center-to-center distance between adjacent glued surfaces, i.e., 0.22m. The material property attributed to the beam elements is the material property of wood given in Table 3.1. The other component in the model is the glued surface, which represents the connection between the laminates. At each glued surface, there are two wood panels intersecting perpendicularly. These intersecting panels are glued together, which is assumed to be a rigid connection. The glued surface is modeled here using zero-length 3-dimensional spring elements called as COMBIN 39, which have high stiffness against translation in X and Y-directions and rotational stiffness. A schematic representation of the connection is shown in Figure 3.10. The output from the spring will provide the forces in X and Y-directions as well as torsion in the glued surface. Figure 3.10 Modeling of glued surface in the frame model The CLT shear wall considered here has three laminates with two glued surfaces on either side of a plane of symmetry in the thickness direction. The torsional moment in the glued surfaces on either sides of the plane of symmetry is the same. Hence, to simplify the model, only one pair of orthogonal laminates is modeled. The finite element model developed is shown in 54 Figure 3.11. The wall is subjected to a lateral load, which is applied on the nodes along the top edge of the wall. The base of the wall is assumed to be fixed to the floor and all the degrees of freedom of the nodes at the base are restrained Figure 3.11 Frame model for a CLT shear wall without an opening The finite element model for the wall without an opening, shown in Figure 3.11, will help in determining the torsional moment in each glued surface in the wall. The next model developed is for a wall with an opening. This model will provide insight into the effect of an opening on the torsional moment in the glued surface after stress redistribution. The geometry of the wall considered is shown in Figure 3.5. The methodology adopted for developing the model is same as explained for the previous frame model. The finite element model developed is shown in 55 Figure 3.12. The loading and boundary conditions are same as applied on the model shown in Figure 3.11. The torsional moment at each glued surface in the wall can be obtained from the model using the COMBIN39 spring elements. A comparison of the torsional moment at each glued surface between the two frame models will provide an insight into the effect of the opening. The stress distribution in the glued surface and across the face of the wall can now be determined using Eq. (31), Eq. (32) and Eq. (33). Failure in the glued surface and wood panels is brittle in nature and damage at these locations need to be avoided. Figure 3.12 Frame model for a CLT shear wall with a cut-out opening (Model C) 56 3.4 Results from the Frame Models (Model C) The frame models discussed in Section 3.3 provide a tool to determine the torsion in the glued surfaces. In this section, the results from the analysis of the two frame models developed is presented and the shear stress obtained is compared to the value obtained using the three-dimensional finite element model. The three dimensional finite element model was subjected to a lateral load of 50kN. As the frame model considers only half the thickness of the wall in analysis, the load applied also needs to be halved. Therefore, a lateral load of 25kN is applied along the top edge of both the frame models and the torsional moment in each glued surface is obtained. First, the results from analysis of the frame model for a wall without an opening is presented. Figure 3.13 shows the torsion in each glued surface across the face of the wall between two CLT laminates for a lateral load of 25kN. The glued surfaces at the base of the wall have no torsion due to the fixed boundary condition assigned to the model. The torsional moment is evenly distributed in the glued surfaces in the middle due to even stress distribution. The torsion in glued surfaces towards the periphery of the wall decreases due to boundary effects. 57 Figure 3.13 Torsion (Nm) in each glued surface of a CLT shear wall The frame model with an opening provides the torsional moment in each glued surface, which is shown in Figure 3.14. The load and boundary conditions on the wall are same as in the previous case. The torsional moment is not equally distributed over the face of the wall due to stress redistribution around the opening. The glued surfaces along the free edges of the opening experience significantly lower torsional moment due to the free boundary condition. The lower torsion in these glued surfaces results in increased torsion in the glued surfaces adjacent to them. This effect is most profound in the glued surfaces located at the corners of the opening, which have been highlighted in red in Figure 3.14. 58 Figure 3.14 Torsion (Nm) in each glued surface of a CLT shear wall with an opening The redistribution in stress around the opening is reflected in the variation in torsion in each glued surface. Figure 3.15 shows the difference in torsional moment experienced by the glued surfaces in the bottom left quarter of the wall due to the presence of the opening. The difference in torsional moment in each glued surface for the two cases studied is indicated by the grey and white bars. The grey bars in the figure indicate an increase in torsional moment in the presence of an opening, while a white bar represents decrease in torsional moment. The red bar represents the glued surface adjacent to the corner of the opening, which experiences maximum increase in torsion. This significant increase in torsion at opening corners brings to 59 forefront the possibility of damage at these locations. Larger the opening, greater will be the increase in the torsion, which may cause problems in walls with large garage-type openings. Figure 3.15 Difference in torsional moment in the glued surfaces due to the presence of an opening The stress in the glued surface at the corners of the opening is calculated using Eq. (31), Eq. (32) and Eq. (33) and presented in Table 3.2. The RVSE comprising this glued surface has a side length, a, of 0.2m and thickness, t, equal to 0.02m. In Table 3.2, the maximum shear stress observed in the case with and without opening is compared. The maximum shear stress in the wood panels increased by 96% for the opening size considered here. The strength of the glued connection in torsion is 1.8N/mm2 and the strength of wood in shear is 3.6N/mm2. The stress levels for the studied case are within limits, but for a larger opening, such as in a garage there 60 is a possibility of damage. Therefore, the opening corners are a potential location for damage that have to be reinforced for shear and torsion when necessary. Without opening With opening Load, P (kN) 50 50 MT (kNm) 0.48 0.94 τm (N/mm2) 0.6 1.18 τt (N/mm2) 0.18 0.35 τv (N/mm2) 1.2 2.35 Table 3.2 Comparison of stress in the glued surface The frame model predicts that the maximum shear stress in wood is 2.35N/mm2 whereas the three-dimensional model from Section 3.2 predicts that the maximum shear stress in wood is 1.95N/mm2. The frame model over-predicts the shear stress within acceptable limits while significantly saving computation time compared to the three dimensional model. 3.5 Conclusions This chapter presented two different types of finite element models to simulate the behaviour of a CLT shear wall with a cut-out opening. The three dimensional finite element model (Model B) developed showed that due to discrete nature of CLT, axial stress concentration does not occur around the corners of the opening. The principal stress vectors obtained from Model B also showed that load is transferred between laminates through torsion in the glued surface. The second type of finite element model developed is the frame model, which has been introduced as Model C. This model provides the torsion in a glued surface, which can then be used an input into the analytical model presented by Bogensperger et al. (2010) to determine the stress distribution in a glued surface. The results from the frame model developed in this 61 chapter showed that there is stress redistribution as indicated by increase in torsion in the glued surface adjacent to opening corners. The subsequent calculation of stress also showed that there is a possibility of shear failure in wood around the opening corners. The results though are not conclusive as the stress values have been obtained using an analytical model. Experimental testing of such wall set-ups need to be conducted in future to understand the reinforcement requirements in more detail. Nonetheless, the results from the finite element models indicate the type of stresses that develop at the corners of the opening for which reinforcements have to be designed. 62 Chapter 4: Coupled-Panel CLT Shear Walls In this chapter, the focus is on CLT shear walls with coupled-panels around openings, which is a commonly observed construction practice in North America. The analysis of this construction practice is carried out using a finite element model calibrated to experimental data, which has been introduced in section 1.4 as Model D. This chapter explains the development of the finite element model and the results obtained from it. This construction approach is popular because of three reasons. First, the use of smaller panels to form a wall with an opening reduces the wastage of material, which amounts to substantial savings in large-scale projects. Secondly, the process of cutting an opening in a CLT shear wall, which was described in the previous chapter, is difficult and requires skilled labour and expensive tools for cutting. Thirdly, by coupling the panels together using metal connectors, the wall has an additional source of ductility. Ductility in timber walls has been observed to significantly increase the reliability and robustness of the walls (Kirkegaard et al. 2011) making them more viable for use in high seismic zones. In the coupled-wall construction practice, under the action of a lateral load, the deformation in the wall is mainly concentrated to the connectors. Due to this characteristic, the torsional mechanism and related failure in CLT discussed in the previous chapter will not be of significant concern. In the next section, the development of an experimentally calibrated finite element model will be discussed. The model development procedure will be first validated by simulating an experiment from literature. 63 4.1 Modeling a Coupled-Panel CLT Shear Wall In this section, the development of a finite element model for a coupled-panel CLT shear wall will be discussed. This model focuses on simulating the connection between the panels and establishing a modeling procedure. The wall considered for modeling is shown in Figure 4.1, which has a dimension of 2.95m x 2.95m and is composed of a 5-layered CLT panel with a total thickness of 85mm. The wall consists of two panels connected together using a half lap joint. The wall panel is connected to the floor of the set-up by two hold-downs at the edges of the wall and four angle brackets as shown in the figure. This test set up was analysed by Gavric et al. (2012) to study the seismic performance of the CLT shear walls. However, the results from the test have been used here to validate the modeling approach by conducting a push-over analysis. Figure 4.1 Coupled-panel CLT shear wall (Gavric et al. 2012) 64 The CLT shear wall shown in Figure 4.1 has different components, such as the CLT wall panel, wall-to-floor connection, panel-to-panel connection, wall-to-floor contact and panel-to-panel contact. These components have been modeled explicitly and calibrated to experimentally determined behaviour. In the next few sections of this chapter, the modeling of each of these components will be explained. 4.2 Modeling the CLT Panels Under the action of a lateral load, the CLT panels in the wall undergo only in-plane deformation. Hence, plane stress quadrilateral elements called as PLANE42 were used to model the CLT panels. The CLT panel is composed of laminates, which exhibit orthotropic material behaviour that has been presented previously in Table 3.1. This material behaviour is incorporated into the model by homogenizing the material property over the numerous layers. The CLT Handbook (Gagnon and Popovski 2011) describes a procedure for homogenization of the material property using the concept of k-factors from composite theory. Figure 4.2 and Figure 4.3 present snippets from the CLT handbook, which explain the calculation of the k-factors and the effective homogenized properties of the CLT panel. k3 and k4 are the k-factors that have to be computed in this case for determining the effective panel properties under the action of an in-plane load. In the equations for the k-factors, Eo and E90 are the moduli of elasticity in the lateral and longitudinal directions. ai is the thickness of the ith laminate in the CLT board. The homogenized material property for the CLT laminate manufactured by Structurlam (Structurlam n.d.) is presented in Table 4.1, which is the material model input for the plane stress elements. 65 Species group Modulus of Elasticity (N/m2) Shear modulus (N/m2) Poisson ratios EX EY ER GXR GXY GRY νXR νXY νRY S-P-F 9500x106 9500x106 500x106 950x106 950x106 50x106 0.03 0.03 0.2 Table 4.1 Homogenized material property of a CLT panel Figure 4.2 k-factors from composite theory (Gagnon and Popovski 2011) 66 Figure 4.3 Effective strength and stiffness calculation (Gagnon and Popovski 2011) 4.3 Connector Modeling The next component is the wall-to-floor connection and the panel-to-panel connection. The wall-to-floor connectors are the hold-downs and angle brackets. The hold-downs used in the test set-up are WHT540 and the angle brackets used are BMF 90x116x48x3 mm. Under the action of a lateral load, the connectors exhibit two different mechanisms of deformation. In the Y-direction, the connectors are subjected to tension, while in the X-direction they experience shear deformation. These two deformation mechanisms are incorporated into the model by using individual springs for each mechanism, which act in unison. The springs used are unidirectional zero-length elements with non-linear capability called as COMBIN39 in ANSYS. The tension behavior is modeled using a spring with only tension capability. The force-deformation response of the tension-only hold-down and angle bracket springs is shown in Figure 4.4 and Figure 4.5. Similarly, the shear behaviour is modeled with a unidirectional spring but the spring is capable of both positive as well as negative displacement. The force- 67 deformation response of the shear spring for the hold-down and angle bracket is shown in Figure 4.6 and Figure 4.7. Figure 4.4 Force-deformation response of the hold-down in tension Figure 4.5 Force-deformation response of the angle bracket in tension 68 Figure 4.6 Force-deformation response of the hold-down in shear Figure 4.7 Force-deformation response of the angle bracket in shear The panels are joined together with a half-lap joint with an overlap length of 50mm using self-tapping screws type HBSΦ8x80mm at 150mm c/c spacing. The screws in this joint were modeled using COMBIN39 springs with unidirectional behaviour. Each screw was modeled using a pair of zero-length springs, one for deformation in X-direction and the other for 69 deformation in Y-direction. The experimentally determined force-deformation response of the screws in either direction is shown in Figure 4.8 and Figure 4.9. Figure 4.8 Force-deformation response of the half-lap joint in X-direction Figure 4.9 Force-deformation response of the half-lap joint in Y-direction 70 4.4 Contact Modeling In this section, the modeling approach used to simulate the contact between the wall and the floor as well as the contact between the panels in the wall is explained. Contact modeling is one of the most important aspects of this model and imperative for convergence of the push-over analysis. Traditionally, the contact is modeled by defining a high-stiffness compressive curve to all connectors that deform in tension. For example, the tension-only springs used to model the hold-downs would have a high-stiffness compression component in the force-deformation input for the element. However, in the model developed here, the contact has been modeled explicitly using a set of compression-only springs distributed along the boundary of contact. The compression-only springs used here are unidirectional zero-length COMBIN39 springs. ANSYS does not have an option for defining the COMBIN39 springs as compression only springs, but they can be defined as tension-only springs. In order to simulate compression-only behaviour, the COMBIN39 springs are defined as tension-only springs and the node order is reversed as shown in Figure 4.10. In Case 1, the relative deformation of the spring is positive and hence the spring is assumed to behave in tension. In Case 2, the relative deformation of the spring is negative implying the spring is in compression. However, ANSYS reads the relative deformation between the nodes as positive due to the order of node numbering. Figure 4.10 Simulating a compression-only spring 71 These compression-only springs are provided with high-stiffness to simulate the contact, which in reality has infinite stiffness. The force-deformation response input for these springs is shown in Figure 4.11. These springs are distributed along the contact between the wall and the floor as well as the contact between the coupled CLT panels. In the wall-to-floor contact, zero-length compression-only springs connect the nodes of the wall panel to the floor nodes. The floor nodes are generated at the base of the wall, but all the degrees of freedom of these nodes are constrained. At the contact between the coupled panels, the nodes at the intersecting boundary of the two panels are connected using the compression-only springs. Figure 4.11 Force-deformation response of the compression-only contact spring 4.5 Displacement-based Pushover Analysis in ANSYS In order to carry out a displacement-based non-linear pushover analysis in ANSYS, the user must define the number of load steps, increment in displacement to be applied in each step and maximum displacement to be applied. A displacement-based pushover analysis in ANSYS does not stop at the point where the first connector fails, but runs till the point when a connector 72 experiences negative force for a positive displacement. Therefore, the user must always check back the solution obtained to determine the exact load step at which the wall failed. For the test set-up considered here, a maximum displacement of 35mm is applied. Figure 4.12 shows the finite element model that has been developed to simulate the test set-up shown in Figure 4.1 under the action of a lateral load. Figure 4.12 Finite element model of the test set-up by Gavric et al. (2012) The deformation of the wall is shown in Figure 4.13. The shear deformation between the coupled-panels is clearly visible in the figure. The pushover curve obtained through the simulation is shown in Figure 4.14. The force-deformation response obtained matches to the experimental response observed by Gavric et al. (2012). This indicates that the modeling procedure adopted here appropriate and can be extended to simulate the behaviour of a coupled-panel CLT shear wall with an opening. 73 Figure 4.13 Deformed shape of the coupled-panel shear wall set-up Figure 4.14 Force-deformation response of the set-up considered for validation 74 4.6 Modeling a Coupled Panel CLT Shear Wall with an Opening In this section, the modeling of a coupled-panel CLT shear wall with an opening will be explained. The purpose of developing this model is to simulate the behaviour of a coupled-panel CLT shear wall with an opening under the action of a lateral load. This model will also provide insight into the forces that develop at the corners of the opening. Results from this model can be used to assess the necessity of reinforcements at the corners of the opening. The set-up considered for modeling is shown in Figure 4.15. Figure 4.15 Coupled-panel CLT shear wall with an opening (Configuration 1) Figure 4.15 shows a wall of height 3m and width 2.4m, with an opening of width 1.2m and height 1.5m. The wall is composed of three-layered CLT panels with a total thickness of 99mm. The wall is connected to the floor using four hold-downs and four angle brackets. The hold-downs have been represented in the figure using vertical downward arrows highlighting their primary function as anchors resisting the tensile forces. The angle brackets have been depicted 75 in the figure as double-sided horizontal arrows indicating their primary function to resist the shear between the floor and the wall. The panels are connected to one another using a half-lap joint and self-tapping screws at 150mm c/c. One important additional component in the wall is the steel tie-rods. These tie-rods serve as reinforcement to resist the tensile transfer force that develops at the corners of the opening. The tie-rods here are considered to stretch half-way into panels on either side of the corners of the opening. The tie-rods extends half-way into the full-height panels on either side of the opening, i.e., 0.3m. The tie-rods extend for 0.50m into the panels above and below the opening. The tie-rods on either side of the opening have a gap of 200mm between them. The total length of each tie-rod is 0.80m and the cross-section of the tie-rods is 0.05mx0.02m. The tie-rods are modeled using beam elements called as BEAM188 in ANSYS. The length of every beam element is defined as 0.05m, therefore each tie-rod has 16 beam elements. The tie-rod is screwed to the CLT panel externally with center-to-center distance between the connections being 100mm. So, after every 100mm in the model, the node from the tie-rod is merged with the corresponding node of the CLT panel. This also provides an estimate for the minimum size of the mesh to be adopted for the CLT panels. However, in the model here, the side-length of a quadrilateral element used to model the CLT panel is defined to be 0.05m. This size can be further reduced, but it was found that a further decrease in the mesh size significantly increases the computation time without any improvement in the solution obtained. The finite element model developed is shown in Figure 4.16. The finite element model developed is capable of simulating the behavior of the wall under the action of a lateral load. The model will serve as a useful tool in designing test set-ups for experimental work and in understanding the function of tie-rods as reinforcement. The next 76 sections in this chapter will explain the results that have been obtained using this finite element model. Figure 4.16 Finite element model of a coupled-panel CLT shear wall with an opening (Model D) 4.7 Pushover Analysis of a Coupled Panel CLT Shear Wall with an Opening This section presents the results from pushover analysis of the model developed in Section 4.6. The finite element model shown in Figure 4.16 is capable of simulating the response of a CLT shear wall under the action of a lateral load. The objective of studying the response of this model is to understand the effect of the tie-rods as reinforcement around the corners of the opening. The deformed shape of the wall subjected to a lateral displacement is shown in Figure 4.17. 77 Figure 4.17 Deformed shape of the coupled-panel CLT shear wall with an opening Figure 4.17 shows the deformation of the wall under the action of a lateral load applied along the top edge of the wall. The hold-downs and angle brackets resist uplift of the wall and sliding between the wall and the floor. The tie-rods, as can be seen in the figure, hold the panels together by resisting the tensile forces developing at the corners of the opening. The tie-rods are also subjected to shear that develops between the coupled panels. The force-deformation response of the wall as obtained from the analysis is shown in Figure 4.18. 78 Figure 4.18 Force-deformation response of the CLT shear wall The wall begins to yield at 68.3 kN due to yielding in tension of the hold-down at the left-side corner of the wall. As the load increases, other connectors in the wall begin to yield and the wall resists a maximum load of 100 kN. The deformation in the CLT panels is observed to be negligible and the majority of the deformation occurs at the wall-to-floor connectors, i.e., the hold-downs and the angle brackets. The strength and stiffness of the wall shown in Figure 4.15 is greater than the wall of similar size shown in Figure 4.12, which does not have an opening. This difference in performance of the walls is due to the greater fixity provided at the base of the wall in Figure 4.15 from two extra hold-downs. The wall behaviour is observed to be significantly affected by the wall-to-floor connectors, which has been observed previously as well by Ceccotti et al. (2006) and Popovski et al. (2010). The next few sections in this chapter will address the need for tie-rods as reinforcement and the parameters that affect the tie-rod performance. 79 4.8 Effect of Tie-Rods as Reinforcement on Performance of the CLT Shear Wall Under the action of a lateral load, tensile forces are generated at the corners of the opening, which render the panels to separate at the corners of the opening. Tie-rods resist these forces to maintain the structural integrity of the system and ensure the wall behaves as a single unit. In the absence of tie-rods, these tensile forces are resisted by the screws in the joint between the panels. The screws closest to the corners of the opening will experience high tensile forces and yield at relatively low lateral loads. This is shown by the comparison presented in Figure 4.19. Figure 4.19 Effect of tie-rods on shear wall performance Figure 4.19 presents the results from analysis of two cases, one in which only screws are used to join the panels and in the other tie-rods and screws are used. In the case when tie-rods were used as reinforcement around the corners of the opening, the wall begins to yield at a load of 80 69 kN and observes a displacement of 0.05m for a load of 100 kN. The wall fails at a load of 94.5 kN for a lateral displacement of 0.06m. In the case when only screws were considered to join the panels together around the opening, the wall begins to yield at a relatively low load of 24.4 kN. The wall only carries a load of 55.6 kN at a lateral displacement of 0.06m, which was the maximum deformation in the previous case. The decrease in strength and stiffness of the wall observed clearly shows the importance of reinforcing the corners of the opening so that the wall deforms as a single unit. 4.9 Effect of Anchoring and Opening Layout on Wall Behaviour Anchoring of the wall to the floor has significant effect on the behaviour of the wall (Ceccotti et al. 2006). The anchoring layout affects the rotation and deformation of the coupling panels in the CLT shear wall. This in turn affects the stiffness and strength of the wall and the transfer force that develops at the corners of the opening. In this section, two wall configurations have been compared to observe the effect of anchoring layout on the force-deformation response of the wall. The first configuration considered, i.e., Configuration 1, is the wall set-up from Section 4.6, which is shown in Figure 4.15. In this configuration, the CLT shear wall is connected to the floor using four hold-downs and four angle brackets. This anchoring layout limits the rotation of the wall panels, which will be reflected in the force-deformation response observed for the wall. Configuration 2 is shown in Figure 4.20, which has a wall set-up similar to Configuration 1 but with a different anchoring layout. In Configuration 2, the shear wall is connected to the floor using two hold-downs and four angle brackets. The decrease in number of hold-downs will allow for greater rotation of the wall panels, resulting in reduced strength and stiffness of 81 the wall. Pushover analysis for the two configurations of the wall set-up was carried out and the force-deformation response obtained from the analysis is shown in Figure 4.21. Figure 4.20 Configuration 2 Figure 4.21 Force-deformation response for the four wall configurations 82 Figure 4.21 shows that the anchoring layout significantly affects the strength and stiffness of the CLT shear wall. Configuration 1 has a yield strength of 68kN and maximum strength of 100 kN, whereas Configuration 2 has a yield strength of 52 kN and a maximum strength of 72 kN. The increase in strength and stiffness of Configuration 1 compared to Configuration 2 is due to the reduced rotation of the CLT wall panels. The two additional hold-downs in Configuration 1 can resist greater uplift and shear forces and reduce the deformation in the anchors, which in turn leads to reduced rotation and displacement of the wall panels. The study in this section shows that the anchoring layout plays a significant role in determining the maximum base shear the wall will be subjected to. As the transfer force is a function of this base shear, the design transfer force will depend on the anchoring layout. Moreover, the anchoring layout will affect the shear force that develops between the panels, which will be resisted by the tie-rods and panel-to-panel connectors. 4.10 Design Transfer Force from the Diekmann Model and Finite Element Models In this section, the effect of anchoring layout on the design transfer force is studied. Also, the Diekmann model has been used to determine the design transfer force in the tie-rods to assess its applicability for the two wall configurations introduced in the previous section. The Diekmann method provides the transfer force, FT, which develops at the edge of the opening. The tie-rod however, is located at an edge-distance, e, from the free-edge of the opening as shown in Figure 4.22. Therefore, the transfer force obtained using the Diekmann model has to be corrected for this edge-distance to determine the force in the tie-rods, FTR. 83 Figure 4.22 Force in tie-rod and transfer force Assume that the height of the panel above or below the opening is hp as shown in Figure 4.22. The transfer force that develops at the corner of the opening forms a couple with a net moment FT multiplied by hp. The force in the tie-rods, FTR, also forms a couple with net moment FTR multiplied by the lever arm hp-e. Equating these two moments, we obtain the force in the tie-rod as a function of the transfer force, FT, which is given as, ehhFFppTTR (34) Eq. (34) is a correction that has to be made to the transfer force obtained from the Diekmann method for determining the force in the tie-rods. The comparison of design force in the tie-rods obtained using the Diekmann model and the finite element models is shown in Figure 4.23. The wall set-up in Configuration 1 has a maximum strength of 100 kN, for which the design transfer force from the finite element model is computed to be 63 kN. The wall set-up in Configuration 2 has a maximum strength of 72 kN, for which the design transfer force from the finite element model is computed to be 46 kN. 84 Figure 4.23 Transfer force obtained from Diekmann method and finite element model In Figure 4.23, the transfer force variation with respect to the base shear is same for Configuration 1 and Configuration 2. This is because the transfer force is a function of the base shear and geometry of the wall. As Configuration 1 and Configuration 2 have the same wall geometry, the relationship between the transfer force and base shear is the same. However, the design transfer force, which is the maximum transfer force that the tie-rod will be subjected to is different for both the wall configurations. The finite element model results show that as the wall begins to yield beyond its maximum load and the base shear begins to decrease, the transfer force also decreases. However, as the base shear decreases, the transfer force does not trace back along the loading path. This is due to the yielding of the connectors at the base of the wall and change in redistribution of forces over the face of the wall. 85 Figure 4.23 also shows a comparison between the transfer forces obtained using the finite element models and the Diekmann model. For Configuration 1, the Diekmann model predicts a design transfer force of 75 kN, whereas the finite element model predicts a design transfer force of 63 kN. Similarly, for Configuration 2, the Diekmann model predicts a design transfer force of 54 kN, whereas the finite element model predicts a design transfer force of 46 kN. This indicates that the Diekmann model over predicts the design transfer force with an error of 19% and 17% for Configuration 1 and Configuration 2, respectively. As the transfer force from the Diekmann model is over-predicted within reasonable limits, it can be used for design of tie-rods for Configuration 1 and Configuration 2 and other such configurations. However, for other opening layouts such as door-type openings and garage-type openings, there is a need for further investigation. 4.11 Effect of Tie-Rod Stiffness on Performance of the CLT Shear Wall In this section, the effect of tie-rod stiffness on the performance of the shear wall is studied. The objective is to identify the relationship between the stiffness of the tie-rod and performance of the wall. The previous finite element model presented has explicit modeling of the wall-to-floor connectors, which significantly affects the performance of the shear wall. In order to isolate the effect of tie-rod stiffness on the shear wall performance, the wall was considered to be simply supported at the base in the analysis here. Five different trials were conducted, wherein the thickness of the tie-rods was varied from 1cm to 5cm, to assess the effect of tie-rod stiffness on the shear wall behaviour. The force-deformation response for the five trials with varying tie-rod thickness is presented in Figure 4.24. 86 Figure 4.24 Effect of stiffness of tie-rod on wall performance Figure 4.24 shows the response of the shear wall for five different thickness of the tie-rod. When the tie-rod has a thickness of 0.05m, the tie-rod does not yield and the wall behaves linearly. Next, the tie-rod thickness is reduced to 0.04m and then to 0.03m. The wall still exhibits minimal change in stiffness and strength though the tie-rod stiffness has been reduced. However, even with the lower thickness and stiffness, the tie-rod hasn’t yielded yet. When the tie-rod thickness is 0.02m, the tie-rod begins to yield at a load of 150kN, which corresponds to yielding of the wall. However, the effect of yielding of the tie-rod is not substantial and the performance of the wall is within reasonable limits of the previous cases with thicker tie-rods at the maximum applied displacement of 50mm. In the final case, the tie-rod thickness was reduced to 0.01m. In this case, the tie-rod begins to yield at 100kN and the wall stiffness decreases significantly as the tie-rod yields for the maximum applied displacement of 50mm. This change in wall stiffness is due to two factors. Firstly, the reduced stiffness of the thin tie- 87 rods contributes directly to lower the stiffness of the wall. Secondly, as the tie-rod begins to yield, its stiffness decreases significantly, which results in large deformations at the corners of the opening. This in turn prevents the wall to deform as a single unit, further decreasing the stiffness of the wall. Therefore, it is preferred to design the tie-rods to behave linearly at loads lower than the desired strength of the wall. This will ensure that even at the maximum load of the wall, the system will deform as a single unit. Consider the wall presented in Section 4.6, which had a maximum load of 100kN. For this wall, a tie-rod of thickness 0.01m would be sufficient as the tie-rod begins to yield at this load. The study in this section shows that the tie-rods should preferably be designed to behave linearly for the design load on the wall. 4.12 Conclusions In this chapter, the use of tie-rods as reinforcement around opening corners in a coupled panel CLT shear wall has been discussed. A finite element model is developed that can simulate the behaviour of a coupled panel CLT shear wall with an opening. The finite element model was then used to highlight the increase in strength and stiffness of the wall when tie-rods are used as reinforcements around the opening corners. The modeling approach developed was then used to simulate the behaviour of two different wall configurations, which highlighted the effect of anchoring on the performance of the shear wall. The design transfer force for the two wall configurations was also observed to be significantly affected by the anchoring layout. The Diekmann method was used to determine the design transfer force and found to provide reasonable estimates for Configuration 1 and Configuration 2. Therefore the Diekmann model can be used to determine the design transfer force for reinforcing corners of a simple window-type opening in a CLT shear wall. 88 The effect of tie-rod thickness on the wall behaviour was also studied and it was found that over designing the tie-rods has minimal impact on the wall performance. Also, yielding of the tie-rods significantly reduced the wall stiffness and strength. The results from the finite element analysis indicate that a capacity-based design of the tie-rods to ensure their linear behaviour is suitable for optimum performance of the wall. 89 Chapter 5: Linear Regression Model for Stiffness of a Wall In this chapter, the development of a linear regression model to predict the stiffness of a CLT shear wall with a window-type opening is explained. The previous chapters in this thesis focused on the forces around corners of an opening and reinforcement requirements. The objective in this chapter is to develop a regression model that provides insight into the effect of material property and geometry of the wall on its stiffness. As this probabilistic model provides a physical quantity as output, it can also be incorporated into the unified reliability analysis (URA) (Haukaas 2008) framework for performance-based earthquake engineering (PBEE). An important assumption in this model is that the wall is simply supported at the base. As the wall-to-floor connection contributes significantly to the force-deformation response of a wall, the regression model developed here cannot be used to predict the wall behaviour. 5.1 General Form of a Linear Regression Model A linear regression model is a predictive model, which predicts a response y based on certain physically measurable variables. The general form a linear regression model is, kk xxxxy ...332211 (35) where, y is the response that the regression model predicts, sometimes called as response or output. xi are the physically measurable variables called as regressors or explanatory variables. θi are the model parameters, which are also called as regression coefficient. ε represents the model error, i.e., the difference model prediction and observed response. The model is linear with respect to the model parameters and not the explanatory variables. This implies that in a linear regression model the model parameters have unit power, but the explanatory variables can take the form (x1)0.5 or (x2/x3). 90 Let x denote the k-dimensional vector of explanatory variables. Each trial of the experiment can be represented by a vector x of the explanatory variables and response y. For n such trials, the observations of these trials can be collected in an n-by-k dimensional matrix, X and an n-dimensional vector of y. Similarly, the model parameters can be collected in an n-by-k dimensional matrix θ. The discrepancy between model prediction and observation can be represented by a vector ε. The information collected in n-trials can be contained in the following system of equations, εθXy (36) where, ε represents the discrepancy between the observed response y and the value predicted by the regression model. Eq. (36) forms the basis for determining the characteristics of θ and ε, which is the primary objective in regression analysis. The inference of the model parameters θ and the error ε is first carried out using the method of ordinary least squares to obtain point estimates and then using Bayesian inference. The method of ordinary least squares provides point estimates of model parameters θ and ε (Box and Tiao 2001). This does not yield a probabilistic model but provides useful insight and serves as an input for the Bayesian approach. In this approach, the point estimate of θ is obtained by minimizing the sum of squared errors. The point estimate of model parameters, θ, is denoted by θˆ . The sum of squared errors is given as, 222212 .... n ε (37) By introducing Eq. (36), the point estimate θˆ is given as, 22 minargmin argˆ Xθyεθ (38) 91 The solution can be obtained by equating the derivative of the objective function with respect to θ to zero, which is given as, 0222 XθXyXθXθy TT (39) Solving Eq. (39) the point estimate θˆ is given as, yXXXθ TT 1ˆ (40) The model error, ε, in the method of ordinary least squares is calculated as the difference between each observation, y, and the corresponding prediction, kk xθxθxθ ˆ....ˆˆ 2211 . It is assumed that each error is a random variable with zero mean and standard deviation σ, i.e., ε~(0, σ2In). The observed error, ε, is given as the difference between the observed response y and the predicted response X θˆ . Therefore, based on classical statistics, the estimate of the standard deviation, σ, is given as, θXyθXy ˆˆ12 Tkns (41) where, s is the estimate of σ, often called as the standard error. θXy ˆ is the observed error ε. k is the number of explanatory variables in the model and n is the number of trials of the experiment. After determining the point estimates of the model parameters and the standard errors, a Bayesian approach can be adopted to determine the same in a probabilistic manner. In the Bayesian approach, the model parameters, θ, and the model error, ε, are treated as random variables. In a classical regression model, the model error, ε, is normally distributed with zero mean and standard deviation σ. The objective of Bayesian inference is to determine the probability distribution of θ as well as σ. Assuming non-informative priors for the 92 distribution of θ and σ, given the observations of y, the posterior distribution of θ is the multivariate t-distribution, which is given as, kvTTkkTkvsvvskvf( 2122ˆˆ1212121)θθXXθθXXθ (42) and the posterior distribution of σ is given by the inverse chi-squared distribution as, 22 ) vvsf( (43) where, v is the degrees of freedom given as the difference between n and k. θˆ and s are important quantities in the posterior probability distributions. This implies that error in estimation of θˆand s will affect the Bayesian estimates as well. The explanation in this section provides necessary insight into the procedure adopted to determine the model parameters and model error. 5.2 Model Development The regression model is a predictive model based on statistical analysis of an observed response. In order to carry out the statistical analysis, repeated trials of an experiment need to be observed. In this chapter, the experiment is the lateral loading of a simply-supported CLT shear wall with an opening and the observed response is the stiffness of the wall. The trials of this experiment is carried out using the finite element model, Model A, whose development has been explained in Section 2.6. The regression model takes regressors as input, which in this case are the geometrical and material parameters of the CLT shear wall, which are summarized in Table 5.1. The 93 geometrical parameters in the regression model are defined as shown in Figure 5.1. In Figure 5.1, a single panel wall is assumed to be discretized into four different panels and their geometry is considered to serve as input into the regression model. The material properties are assumed to be normal random variables with mean value as given in Table 4.1 and standard deviation are suggested by the JCSS Handbook. The geometrical parameters are also assumed to be normal random variables and sampling is carried out to determine the input values of all regressors for different trials of the finite element model. The values of different regressors used as input in the trials of the finite element model are summarized in Section A.1. Figure 5.1 Geometry of a shear wall with an opening 94 Ex Modulus of elasticity in X-direction Ey Modulus of elasticity in Y-direction Gxy Shear modulus L Width of the wall H Height of the wall t Thickness of the wall l1 Width of panel on left side of the opening l2 Width of panel on right side of the opening w Width of the opening hu Height of panel on above of the opening h Height of the opening hl Height of panel on below of the opening Table 5.1 Input parameters of the finite element model The process of computing the model parameters based on a set of observations comprises the statistical component of model development. The first aspect to be considered is the minimum number of observations necessary for the development of a regression model. One criterion for minimum number of observations is the number of regressors. An assumption while developing a linear regression model is that number of observations, n, is greater than the number of explanatory variables, k. Hence, the minimum number of trials is the number of explanatory variables. If the relationship between all the individual regressors and the response was linear then for each regressor two trials would suffice. This follows from the concept that to draw a straight line, the minimum requirement is to have knowledge about two points on it. However, in the model, the relationship between the regressor and response as well as the interaction between regressors is not yet identified. In order to account for this, a rule of thumb is to consider ten to twenty trials per regressor (Harell 2001). 95 The wall as shown in Figure 5.1 is discretized into four panels on sides of the opening. Length of the wall, L and height of the wall, H, are not independent explanatory variables as they are related to the length and height of each individual panel. Therefore, there are a total of ten independent explanatory variables, which implies around 200 trials are necessary for developing the regression model. An initial regression analysis can be performed with these parameters as regressors. The response of the model is stiffness of the wall, Kw, which is given as the lateral load necessary for unit lateral deformation of the wall, i.e., P/∆. The regression analysis is carried out in Rt, an in-house computer program developed for multi-model probabilistic analysis. The observations are input into the program to obtain the model parameters and the standard error, which can be studied to provide insight into effect of various regressors on the stiffness of the wall. The algorithm employed to determine the model parameters is explained in Section 5.1. The regression model for stiffness of a CLT shear wall is given as, thhhwllGEEKluxyyxw1098762514321 (44) where, Kw is the stiffness of the CLT shear wall. The regressors in the model are explained in Table 5.1. The distribution of the model parameters obtained from the regression analysis is shown in Table 5.2. The standard deviation of the model error is normally distributed with mean 1.25x1006 and standard deviation 6.45x1004. 96 Mean c.o.v (%) θ1 159.78 114.42 θ2 -0.61 31479.9 θ3 9127.18 21.28 θ4 7.75e+06 5.25 θ5 7.00e+06 5.92 θ6 2.06e+06 14.54 θ7 -5.76e+06 6.83 θ8 -7.42e+06 5.55 θ9 -9.09e+06 3.12 θ10 1.59e+08 2.29 Table 5.2 Model parameters The correlation between the model parameters is given by the correlation matrix, which is computed to be, 115.011.003.007.005.010.005.007.017.015.0116.012.009.001.000.007.025.004.011.016.0146.0025.01.009.010.025.004.003.012.046.0101.009.021.010.03.010.007.009.003.001.0135.015.006.016.012.005.001.010.009.035.0144.004.019.012.010.000.009.021.016.045.0104.023.010.005.007.010.010.006.004.004.0117.042.007.025.025.029.016.019.023.017.0162.017.004.004.010.012.012.010.042.062.01jiȡ (45) where, jiȡis the correlation between model parameter θi and θj. The correlation matrix provides insight into relationship between the information contained in two explanatory variables corresponding to the model parameters. The importance of each explanatory variable in the model can be analysed based on the corresponding model 97 parameters. The coefficient of variation of a model parameters is an indicator of the information contained in the corresponding explanatory variable (Gardoni et al. 2002). A large coefficient of variation implies that the information contained in the corresponding explanatory variable is not significant and can therefore be excluded from the model. The coefficient of variation of each model parameter is shown in Table 5.2. The model parameter θ2 has the highest coefficient of variation, whose corresponding explanatory variable is the modulus of elasticity in Y-direction, Ey. This implies that the explanatory variable Ey does not significantly affect the in-plane stiffness of the wall and can be excluded from the model. The relatively low coefficient of variation of the model parameters corresponding to Ex and Gxy indicate that these parameters significantly affect the in-plane stiffness of the wall. The explanatory variables whose model parameters show a high degree of correlation can be merged into a single regressor as explained by Gardoni (2002). In this section, the important explanatory variables have been identified. In the next section, the stiffness of individual panels around the opening will used as regressor to develop a regression model with reduced number of regressors. 5.3 Model Reduction Gardoni (2002) states that model development is an art and the artistic aspect of model development is in understanding the physics of the problem and incorporating it into the model. The shear wall considered in this chapter is assumed to be composed of four panels around a window type opening as shown in Figure 5.1. Under the action of a lateral load, each panel deforms in flexure and shear. The net stiffness of each panel will be a function of its flexural and shear stiffness, but the contribution of each mechanism to the net deformation is not 98 known. In this section, the stiffness of each panel for both mechanisms will be considered as a regressor. As, there are four panels and each panel is associated with two different mechanisms, there will be a total of eight regressors in the model. First, consider Panel A on the left side of the opening. This panel has a height hu+h+hl and width l1 as shown in Figure 5.1. The thickness of the panel as shown is t. The shear stiffness of the panel, Kva, is a function of the shear modulus of CLT and the dimensions of the panel and is given as, luaxyva hhhtlGK 1 (46) where, Gxy is the shear modulus of the CLT panel. l1, t and hu+h+hl are the width, thickness and height of the panel, respectively, as explained in Figure 5.1. The constant, αa, represents the boundary condition of the panel and in this case has a value of 1. This is because the nature of the boundary condition and the subsequent value of αa is incorporated in the regression model through the corresponding model parameter. The panel also deforms due to flexure and the flexural stiffness of panel A, Kfa, is given as, 3luaaxfa hhhIEK (47) where, Ex is the modulus of elasticity of the CLT panel. hu+h+hl is the height of the panel defined in Figure 5.1. The constant, βa, represents the boundary condition of the panel. Herein, the constant has a value 1, because the nature of the boundary condition is again incorporated in the statistically determined model parameter for this regressor. Ia is the moment of inertia of panel A, which is given as, 99 1231ltI a (48) where, l1 and t are the width and thickness of the panel respectively, as shown in Figure 5.1. Similarly, the shear and flexural stiffness of panel B, C and D can be established. Panel B has a shear stiffness, Kvb, which is given as, ubxyvb htwGK (49) where, Gxy is the shear modulus of the CLT panel. w, t and hu are the width, thickness and height of panel B respectively as defined in Figure 5.1. The constant, αb, represents the boundary condition of the panel and in this case has a value of 1. The flexural stiffness of panel B, Kfb, is given as, 3ubbxfb hIEK (50) where, Ex is the modulus of elasticity of the CLT panel and hu is the height of the panel as defined in Figure 5.1. The constant, βb, representing the boundary condition of the panel, has a value of 1. Ib is the moment of inertia of panel A, which is given as, 123wtI b (51) where, w and t are the width and thickness of the panel respectively, as shown in Figure 5.1. Next, consider the deformation of panel C. The panel has a shear stiffness, Kvc, which is given as, 100 lcxyvc htwGK (52) where, Gxy is the shear modulus of the CLT panel. w, t and hl are the dimensions of panel B defined in Figure 5.1. The constant, αc, representing the boundary condition of the panel has a value of 1. The flexural stiffness of panel B, Kfc, is given as, 3lccxfc hIEK (53) where, Ex is the modulus of elasticity of the CLT panel and hl is the height of the panel as defined in Figure 5.1. The constant, βc, representing the boundary condition of the panel, has a value of 1. Ic is the moment of inertia of panel A, which is given as, 123wtI b (54) where, w and t are the width and thickness of the panel respectively, as shown in Figure 5.1. Next, consider the deformation of the full-height panel to the right-side of the opening, which is denoted as panel D in Figure 5.1. The panel has a shear stiffness, Kvd, which is given as, ludxyvd hhhtlGK 2 (55) where, Gxy is the shear modulus of the CLT panel. l2, t and hl+h+hl are the dimensions of panel B as defined in Figure 5.1. The constant, αd, representing the boundary condition of the panel has a value of 1. The flexural stiffness of panel D, Kfd, is given as, 3luddxfd hhhIEK (56) 101 where, Ex is the modulus of elasticity of the CLT panel and hl+h+hl is the height of the panel as defined in Figure 5.1. The constant, βd, representing the boundary condition of the panel, has a value of 1. Id is the moment of inertia of panel A, which is given as, 1232ltI d (57) where, l2 and t are the width and thickness of the panel respectively, as shown in Figure 5.1. This completes the definition of the regressors to be included in the regression model for stiffness of the wall. Eq. (46) to Eq. (57) can be used to determine the regressors in the model for input into Rt to carry out a model inference analysis. The regression model for stiffness of a CLT shear wall with a window type opening is given as, vdfdvcfcvbfbvafaw KKKKKKKKK 87654321 (58) where, Kw is the stiffness of the CLT shear wall with a window-type opening. The regressors in the model are the flexural and shear stiffness of each panel shown in Figure 5.1. The model parameters obtained from the model inference analysis in Rt is shown in Table 5.3. The model error has zero mean and a standard deviation σ, which is a random variable. The standard deviation of the model error is normally distributed with mean 1.52x1006 and standard deviation 7.86x1004. The model parameter of a regressor in the model includes the factor for contribution of the mechanism to the net deformation of the wall as well as the factor for the boundary condition of each panel. A look at the coefficient of variation of the model parameters in Table 5.3 shows that each model parameter is important to the model and has significant information. 102 Mean c.o.v (%) θ1 0.16 46.06 θ2 0.41 10.14 θ3 0.00091 94.75 θ4 -0.01 96.38 θ5 -0.00064 97.34 θ6 0.021 40.76 θ7 0.07 71.91 θ8 0.37 9.00032 Table 5.3 Model parameters of the regression model The correlation between the model parameters can also be obtained from the regression analysis in the form of a correlation matrix, which is given as, 18502603402302504003408501210290180220300290260210188037024023012034029088012301501600902301803702301910420370250220240150910139033040030023016042039018803402901200903703308801........................................................ȡji (59) where, jiȡis the correlation between model parameter θi and θj. The regression model shown in Eq. (58) has an R-factor of 0.97, which indicates that there is good fit between predicted and observed data. This is also shown by the model prediction versus observation plot shown in Figure 5.2. 103 Figure 5.2 Model prediction versus observation Figure 5.2 shows that there is good correlation between the model prediction and observation. Therefore, the model presented by Eq. (58) can be used to predict the in-plane stiffness of a simply-supported CLT shear wall with a window-type opening. In the future, this model can be extended to incorporate uncertainty and non-linearity in the behaviour of the connectors at the base of the wall. However, this will increase the number of regressors in the model and more data points will be needed to arrive at a satisfactory regression model. The model presented in Eq. (58) can be reduced further by utilizing the correlation between the model parameters. The correlation between two model parameters is an indicator of the information shared between the two model parameters. The correlation between the model parameters is given by the correlation matrix, which is given in Eq. (59). Gardoni (2002) states that when two model parameters have a correlation greater than 0.7, the model parameters can be merged into a single term. This merger of model parameters helps to 104 decrease the number of terms in the model. Next, the procedure to decrease the number of model parameters based on the correlation matrix is explained. Consider two model parameters θi and θj have a correlation coefficient greater than 0.7. Then θi can be replaced by, jjijii ji ȝθ ˆ (60) where, iȝandi are the posterior mean and standard deviation of model parameter θi, respectively. jandj are the posterior mean and standard deviation of model parameter θj, respectively. jiis the correlation between θi and θj. Eq. (60) provides the best linear predictor of θi in terms of θj (Stone 1996). From Eq. (59), it can be seen that the pair of model parameters θ1 and θ2, θ3 and θ4, θ5 and θ6, θ7 and θ8 have correlation greater than 0.7. Hence, these pairs of model parameters can be merged into a single term using Eq. (60). Model parameter θ2 can be replaced in the regression model by the linear predictor 𝜃2, which is given as, 16.049.041.016.016.046.041.01.0)88.0(41.0ˆ112θ (61) Similarly, the model parameter θ4 can be replaced by the predictor 4θˆ , which is given as, 000902100100009000090950010960910010ˆ334.θ...θ....).(.θ (62) 105 The model parameter θ6 can be replaced by the predictor 6θˆ , which is given as, 0006.04.1202.00006.00006.097.002.041.0)88.0(02.0ˆ556θ (63) The model parameter θ8 can be replaced by the predictor 8ˆθ , which is given as, 07.056.037.007.007.072.037.009.0)85.0(37.0ˆ778θ (64) Using Eq. (61) to Eq. (64) the regression model in Eq. (58) can be re-written in the reduced form as, vdfdvcfcvbfbvafawKKKKK.θ..KKKK07.056.037.00006.04.1202.00009021001016.049.041.077553311 (65) The reduced regression model shown in Eq. (65) has only four model parameters. The mean and standard deviation of σ is 1.62147x1006 and 8.2745x1004. The increase in the value of σ for the reduced model is not substantial and hence the reduced model can be accepted. The distribution of the model parameters is summarized in Table 5.4. Mean c.o.v (%) θ1 0.13 50.51 θ2 0.0009 21.41 θ3 -0.00076 20.16 θ4 0.10 45.02 Table 5.4 Model parameters of the reduced regression model 106 The correlation between the model parameters is given as, 1006.007.076.0006.0132.033.007.032.0127.076.033.027.01jiȡ (66) Therefore, Eq. (58) and Eq. (65) can be used to determine the in-plane stiffness of a simply-supported CLT shear wall with a window-type opening. In the future, Eq. (65) can be improved by obtaining more data, preferably experimental data and incorporating the behaviour of connectors as regressors in the model. This will significantly increase the confidence in the predictions from the regression model. 107 Chapter 6: Conclusion and Future Work This chapter presents a broader perspective of the results from this thesis. The objective of this chapter is to demonstrate the manner in which this thesis addresses present research gap and provides a foundation for future work in this field. 6.1 Conclusion The introduction chapter highlighted the lack of research carried out in applying the FTAO design paradigm to CLT shear walls. The literature review also highlighted the need for better models to determine the transfer force at the corners of an opening. The analytical models in literature for FTAO were developed for designing reinforcements around the corners of openings in timber frame shear walls. This thesis addresses the development of transfer force around opening corners in CLT shear walls. The focus of this thesis is to study the development of transfer force in CLT as opposed to the more widely researched field of timber-frame shear walls. The objective is to understand the type of transfer force that develops in CLT for different construction practices. The methodology adopted to analyse CLT shear walls is development of finite element models in ANSYS. Numerical models developed address the difference in stress redistribution observed for different construction practices. The library of numerical models presented help in simulating the behaviour of CLT shear walls with different types of openings. The models are also amenable for use in different seismic design frameworks such as PBEE, DDBD and force-based design. The models can be used for capacity-based design of shear walls and to design appropriate connection layouts. 108 6.2 Future Work The scope for the future in this project lies in two directions. Firstly, there has been no experimental work conducted in this project. It will be an asset to conduct an experimental investigation to corroborate the predictions from the numerical models. Also, analysis of the cut-out opening type construction practice indicates that shear in wood panels around opening corners could be a potential source of failure, which needs to be experimentally verified. Tie-rods have been recommended for reinforcing the corners of an opening in a coupled-panel CLT wall. The Diekmann method can be used to design the tie-rods for a wall set-up to experimentally verify the importance of reinforcing opening corners. The second aspect to be considered in the future would be improvement of the models developed in this thesis. In the future, the models can be improved to capture visual damage in connectors, CLT panels and the glued surface. Connection models can be improved to incorporate cyclic behaviour for use in dynamic analysis. The finite element models can be used in a unified reliability analysis framework to optimize the cross-section, length and location of the tie-rods as well as design of the wall anchors. The linear regression model presented in this thesis can be refined in future to incorporate the non-linearity and uncertainty in behaviour of anchors and other wall connectors George E. P. Box in his book Empirical Model-Building and Response Surfaces wrote, “Essentially, all models are wrong, but some are useful.” The scope for the future is to make use of the models in this thesis, to develop a PBEE framework for design of CLT shear walls based on force transfer around openings. 109 References ANSYS. (2012). ANSYS Mechanical APDL Command Reference. ANSYS. (2012). 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Evaluation of Force Transfer Around Openings – Experimental and Analytical Studies. 113 Appendices Appendix A Linear Regression Model for Stiffness of a CLT Shear Wall A.1 Trials Note: All values are in SI Units K Ex Gxy H L Xla Xld Wo Ylb Ylc Ho t 17102300 9.50E+09 5.95E+08 3 3 1 1 1 1 1 1 0.099 17148800 9.50E+09 5.95E+08 3 3 1.1 0.9 1 1 1 1 0.099 17183800 9.50E+09 5.95E+08 3 3 1.2 0.8 1 1 1 1 0.099 17207200 9.50E+09 5.95E+08 3 3 1.3 0.7 1 1 1 1 0.099 17218400 9.50E+09 5.95E+08 3 3 1.4 0.6 1 1 1 1 0.099 17216100 9.50E+09 5.95E+08 3 3 1.5 0.5 1 1 1 1 0.099 16712500 9.50E+09 5.95E+08 3 3 1 0.9 1.1 1 1 1 0.099 16236100 9.50E+09 5.95E+08 3 3 0.9 0.9 1.2 1 1 1 0.099 15612100 9.50E+09 5.95E+08 3 3 0.7 1 1.3 1 1 1 0.099 14486000 9.50E+09 5.95E+08 3 3 0.6 0.9 1.5 1 1 1 0.099 17525000 9.50E+09 5.95E+08 3 3.1 1 1 1.1 1 1 1 0.099 17973000 9.50E+09 5.95E+08 3 3.2 1.1 0.9 1.2 1 1 1 0.099 18383300 9.50E+09 5.95E+08 3 3.3 1.2 0.8 1.3 1 1 1 0.099 18750200 9.50E+09 5.95E+08 3 3.4 1.3 0.7 1.4 1 1 1 0.099 19066200 9.50E+09 5.95E+08 3 3.5 1.4 0.6 1.5 1 1 1 0.099 19321100 9.50E+09 5.95E+08 3 3.6 1.5 0.5 1.6 1 1 1 0.099 18682100 9.50E+09 5.95E+08 3 3.6 1 0.9 1.7 1 1 1 0.099 18467100 9.50E+09 5.95E+08 3 3.8 0.9 0.9 2 1 1 1 0.099 17716800 9.50E+09 5.95E+08 3 3.9 0.7 1 2.2 1 1 1 0.099 16320600 9.50E+09 5.95E+08 3 4 0.6 0.9 2.5 1 1 1 0.099 17043400 9.50E+09 5.95E+08 3 3 0.9 1.1 1 1 1 1 0.099 16970900 9.50E+09 5.95E+08 3 3 0.8 1.2 1 1 1 1 0.099 16882400 9.50E+09 5.95E+08 3 3 0.7 1.3 1 1 1 1 0.099 16775200 9.50E+09 5.95E+08 3 3 0.6 1.4 1 1 1 1 0.099 16645900 9.50E+09 5.95E+08 3 3 0.5 1.5 1 1 1 1 0.099 16653500 9.50E+09 5.95E+08 3 3 0.9 1 1.1 1 1 1 0.099 15295300 9.50E+09 5.95E+08 3 3 0.9 0.7 1.4 1 1 1 0.099 15833100 9.50E+09 5.95E+08 3 3 1 0.7 1.3 1 1 1 0.099 14753800 9.50E+09 5.95E+08 3 3 0.9 0.6 1.5 1 1 1 0.099 15785900 9.50E+09 5.95E+08 3 3 0.9 0.8 1.3 1 1 1 0.099 20984800 9.50E+09 5.95E+08 3 3.5 1 1.5 1 1 1 1 0.099 114 K Ex Gxy H L Xla Xld Wo Ylb Ylc Ho t 17191000 9.50E+09 5.95E+08 3 3 1 1 1 1.1 0.9 1 0.099 17280000 9.50E+09 5.95E+08 3 3 1 1 1 1.2 0.8 1 0.099 17371500 9.50E+09 5.95E+08 3 3 1 1 1 1.3 0.7 1 0.099 17467400 9.50E+09 5.95E+08 3 3 1 1 1 1.4 0.6 1 0.099 17569800 9.50E+09 5.95E+08 3 3 1 1 1 1.5 0.5 1 0.099 16752700 9.50E+09 5.95E+08 3 3 1 1 1 1 0.9 1.1 0.099 3783510 9.50E+09 5.95E+08 3 3 1 1 1 0.9 0.9 1.2 0.023 14214700 9.50E+09 5.95E+08 3 3 1 1 1 0.7 1 1.3 0.09 13862800 9.50E+09 5.95E+08 3 3 1 1 1 0.6 0.9 1.5 0.094 10245600 9.50E+09 5.95E+08 3.1 3 1 1 1 1 1 1.1 0.063 9246000 9.50E+09 5.95E+08 3.2 3 1 1 1 1.1 0.9 1.2 0.06 10088200 9.50E+09 5.95E+08 3.3 3 1 1 1 1.2 0.8 1.3 0.069 13193400 9.50E+09 5.95E+08 3.4 3 1 1 1 1.3 0.7 1.4 0.095 2642500 9.50E+09 5.95E+08 3.5 3 1 1 1 1.4 0.6 1.5 0.02 4654100 9.50E+09 5.95E+08 3.6 3 1 1 1 1.5 0.5 1.6 0.037 9870010 9.50E+09 5.95E+08 3.6 3 1 1 1 1 0.9 1.7 0.083 8655770 9.50E+09 5.95E+08 3.8 3 1 1 1 0.9 0.9 2 0.084 1859190 9.50E+09 5.95E+08 3.9 3 1 1 1 0.7 1 2.2 0.02 7040960 9.50E+09 5.95E+08 4 3 1 1 1 0.6 0.9 2.5 0.086 15809600 9.50E+09 5.95E+08 3 3 1 1 1 0.9 1.1 1 0.092 4273570 9.50E+09 5.95E+08 3 3 1 1 1 0.8 1.2 1 0.025 9684460 9.50E+09 5.95E+08 3 3 1 1 1 0.7 1.3 1 0.057 14513500 9.50E+09 5.95E+08 3 3 1 1 1 0.6 1.4 1 0.086 13901300 9.50E+09 5.95E+08 3 3 1 1 1 0.5 1.5 1 0.083 2524190 9.50E+09 5.95E+08 3 3 1 1 1 0.9 1 1.1 0.015 10631000 9.50E+09 5.95E+08 3 3 1 1 1 0.9 0.7 1.4 0.068 14385100 9.50E+09 5.95E+08 3 3 1 1 1 1 0.7 1.3 0.089 4709090 9.50E+09 5.95E+08 3 3 1 1 1 0.9 0.6 1.5 0.031 16850800 9.50E+09 5.95E+08 3 3 1 1 1 0.9 0.8 1.3 0.105 19242500 9.50E+09 5.95E+08 3 3 1.1 0.9 1 1.1 0.9 1 0.11 13316000 9.50E+09 5.95E+08 3 3 1.2 0.8 1 1.2 0.8 1 0.076 9681330 9.50E+09 5.95E+08 3 3 1.3 0.7 1 1.3 0.7 1 0.055 20274100 9.50E+09 5.95E+08 3 3 1.4 0.6 1 1.4 0.6 1 0.115 7036280 9.50E+09 5.95E+08 3 3 1.5 0.5 1 1.5 0.5 1 0.04 16327100 9.50E+09 5.95E+08 3 3 1 0.9 1.1 1 0.9 1.1 0.099 15248300 9.50E+09 5.95E+08 3 3 0.9 0.9 1.2 0.9 0.9 1.2 0.099 13713400 9.50E+09 5.95E+08 3 3 0.7 1 1.3 0.7 1 1.3 0.099 10994400 9.50E+09 5.95E+08 3 3 0.6 0.9 1.5 0.6 0.9 1.5 0.099 16494900 9.50E+09 5.95E+08 3.1 3.1 1 1 1.1 1 1 1.1 0.099 16031600 9.50E+09 5.95E+08 3.2 3.2 1.1 0.9 1.2 1.1 0.9 1.2 0.099 115 K Ex Gxy H L Xla Xld Wo Ylb Ylc Ho t 15542300 9.50E+09 5.95E+08 3.3 3.3 1.2 0.8 1.3 1.2 0.8 1.3 0.099 15001100 9.50E+09 5.95E+08 3.4 3.4 1.3 0.7 1.4 1.3 0.7 1.4 0.099 14378300 9.50E+09 5.95E+08 3.5 3.5 1.4 0.6 1.5 1.4 0.6 1.5 0.099 13643600 9.50E+09 5.95E+08 3.6 3.6 1.5 0.5 1.6 1.5 0.5 1.6 0.099 11361900 9.50E+09 5.95E+08 3.6 3 1 0.9 1.1 1 0.9 1.7 0.099 9301220 9.50E+09 5.95E+08 3.8 3 0.9 0.9 1.2 0.9 0.9 2 0.099 7693690 9.50E+09 5.95E+08 3.9 3 0.7 1 1.3 0.7 1 2.2 0.099 5604160 9.50E+09 5.95E+08 4 3 0.6 0.9 1.5 0.6 0.9 2.5 0.099 17428600 9.50E+09 5.95E+08 3 3.1 1 1 1.1 0.9 1.1 1 0.099 17769100 9.50E+09 5.95E+08 3 3.2 1.1 0.9 1.2 0.8 1.2 1 0.099 14779300 9.50E+09 5.95E+08 3 3.3 1.2 0.8 1.3 0.7 1.3 1 0.082 4069180 9.50E+09 5.95E+08 3 3.4 1.3 0.7 1.4 0.6 1.4 1 0.021 16438800 9.50E+09 5.95E+08 3 3.5 1.4 0.6 1.5 0.5 1.5 1 0.088 9971390 9.50E+09 5.95E+08 3 3.6 1.5 0.5 1.6 0.9 1 1.1 0.053 2101000 9.50E+09 5.95E+08 3 3.2 1.1 0.9 1.2 0.9 0.7 1.4 0.013 5116800 9.50E+09 5.95E+08 3 3.3 1.2 0.8 1.3 1 0.7 1.3 0.029 14912900 9.50E+09 5.95E+08 3 3.4 1.3 0.7 1.4 0.9 0.6 1.5 0.094 3798470 9.50E+09 5.95E+08 3 3.5 1.4 0.6 1.5 0.9 0.8 1.3 0.021 14479300 1.00E+10 5.95E+08 3 3.6 1.5 0.5 1.6 1 0.9 1.1 0.076 12291500 9.90E+09 5.95E+08 3 3.6 1 0.9 1.7 0.9 0.9 1.2 0.070 7020270 9.80E+09 5.95E+08 3 3.8 0.9 0.9 2 0.7 1 1.3 0.043 2211700 9.70E+09 5.95E+08 3 3.9 0.7 1 2.2 0.6 0.9 1.5 0.017 11192200 9.60E+09 5.95E+08 3.1 4 0.6 0.9 2.5 1 1 1.1 0.073 7053800 9.40E+09 5.95E+08 3.2 3 0.9 1.1 1 1.1 0.9 1.2 0.047 11131100 9.30E+09 5.95E+08 3.3 3 0.8 1.2 1 1.2 0.8 1.3 0.077 12256700 9.20E+09 5.95E+08 3.4 3 0.7 1.3 1 1.3 0.7 1.4 0.089 12679500 9.10E+09 5.95E+08 3.5 3 0.6 1.4 1 1.4 0.6 1.5 0.099 3843350 9.00E+09 5.95E+08 3.6 3 0.5 1.5 1 1.5 0.5 1.6 0.032 10886900 8.90E+09 5.95E+08 3.6 3.6 1 0.9 1.7 1 0.9 1.7 0.087 16159200 8.00E+09 5.95E+08 3 3 1 0.9 1.1 0.9 0.9 1.2 0.099 14377500 1.01E+10 5.95E+08 3 3 0.9 0.9 1.2 0.7 1 1.3 0.099 12720400 1.02E+10 5.95E+08 3 3 0.7 1 1.3 0.6 0.9 1.5 0.099 13370700 7.90E+09 5.95E+08 3.1 3 0.6 0.9 1.5 1 1 1.1 0.099 14880900 7.80E+09 5.95E+08 3.2 3.1 1 1 1.1 1.1 0.9 1.2 0.099 15442500 9.50E+09 5.95E+08 3.3 3.2 1.1 0.9 1.2 1.2 0.8 1.3 0.099 15013300 9.50E+09 5.95E+08 3.4 3.3 1.2 0.8 1.3 1.3 0.7 1.4 0.099 18648500 9.50E+09 5.95E+08 3 3.4 1.3 0.7 1.4 0.6 1.4 1 0.099 18878300 9.50E+09 5.95E+08 3 3.5 1.4 0.6 1.5 0.5 1.5 1 0.099 12989100 9.50E+09 5.95E+08 3.6 3.6 1.5 0.5 1.6 1 0.9 1.7 0.099 15711300 9.50E+09 5.95E+08 3 3 0.9 0.9 1.2 1 0.9 1.1 0.099 116 K Ex Gxy H L Xla Xld Wo Ylb Ylc Ho t 14380500 9.50E+09 5.95E+08 3 3 0.7 1 1.3 0.9 0.9 1.2 0.099 12202300 9.50E+09 5.95E+08 3 3 0.6 0.9 1.5 0.7 1 1.3 0.099 12492400 9.50E+09 5.95E+08 3 3.1 1 1 1.1 0.6 0.9 1.5 0.085 11738000 9.50E+09 5.95E+08 3.1 3.2 1.1 0.9 1.2 1 1 1.1 0.07 7289530 9.50E+09 5.95E+08 3.2 3.3 1.2 0.8 1.3 1.1 0.9 1.2 0.045 11692600 9.50E+09 5.95E+08 3.3 3.4 1.3 0.7 1.4 1.2 0.8 1.3 0.075 4023260 9.50E+09 5.95E+08 3.5 3.5 1.4 0.6 1.5 1.4 0.6 1.5 0.027 2103700 9.50E+09 5.95E+08 3.6 3.6 1.5 0.5 1.6 1.5 0.5 1.6 0.014 11524900 9.50E+09 5.95E+08 3.6 3.6 1 0.9 1.7 1 0.9 1.7 0.094 12066100 9.50E+09 6.00E+08 3 3 0.9 0.7 1.4 0.9 0.7 1.4 0.091 7117820 9.50E+09 6.10E+08 3 3 1 0.7 1.3 0.9 0.9 1.2 0.047 10621700 9.50E+09 6.20E+08 3 3 0.9 0.6 1.5 0.7 1 1.3 0.081 7152040 9.50E+09 6.30E+08 3 3 0.9 0.8 1.3 0.6 0.9 1.5 0.054 4658650 1.00E+10 6.40E+08 3.1 3 1 0.9 1.1 1 1 1.1 0.028 12434700 9.90E+09 6.50E+08 3.2 3 0.9 0.9 1.2 1.1 0.9 1.2 0.081 3292410 9.80E+09 6.90E+08 3.3 3 0.7 1 1.3 1.2 0.8 1.3 0.023 4680990 9.10E+09 7.00E+08 3.4 3.5 1.4 0.6 1.5 1.3 0.7 1.4 0.028 13717700 9.00E+09 5.50E+08 3.5 3.6 1.5 0.5 1.6 1.4 0.6 1.5 0.099 12504000 8.90E+09 5.40E+08 3.6 3.6 1 0.9 1.7 1.5 0.5 1.6 0.099 10402000 8.00E+09 5.70E+08 3.6 3 0.9 0.9 1.2 1 0.9 1.7 0.099 13409900 1.01E+10 5.80E+08 3 3 0.7 1 1.3 0.7 1 1.3 0.099 10183800 1.02E+10 5.00E+08 3 3 0.6 0.9 1.5 0.6 0.9 1.5 0.099 12344700 7.90E+09 4.50E+08 3.1 3.1 1 1 1.1 1 1 1.1 0.088 2168170 7.80E+09 8.00E+08 3.2 3.2 1.1 0.9 1.2 1.1 0.9 1.2 0.012 7146740 9.50E+09 3.00E+08 3.3 3.3 1.2 0.8 1.3 1.2 0.8 1.3 0.069 11176400 9.50E+09 4.40E+08 3.4 3.4 1.3 0.7 1.4 1.3 0.7 1.4 0.087 2364930 9.50E+09 4.10E+08 3.5 3.5 1.4 0.6 1.5 1.4 0.6 1.5 0.02 9245670 9.50E+09 7.20E+08 3.6 3.6 1.5 0.5 1.6 1.5 0.5 1.6 0.059 13976800 9.50E+09 7.30E+08 3.6 3.6 1 0.9 1.7 1 0.9 1.7 0.099 26032700 9.88E+09 6.06E+08 2.8 3.2 0.8 1.4 1 1.5 0.5 0.8 0.119 9212300 9.27E+09 5.77E+08 3.8 2.7 1 1.1 0.6 1.5 0.9 1.4 0.084 10295800 9.82E+09 5.54E+08 3.4 2.8 1 0.5 1.3 1.1 1.3 1 0.088 14671100 9.28E+09 5.94E+08 3.1 3.2 1.4 0.5 1.3 1 0.8 1.3 0.096 20178000 9.77E+09 5.68E+08 3.2 3.5 1.2 1.3 1 0.8 1.4 1 0.107 25214400 9.74E+09 5.36E+08 2.7 3.7 1.3 1.5 0.9 0.7 0.7 1.3 0.113 19776100 1.02E+10 6.40E+08 3 3.2 1 1.5 0.7 0.5 1.2 1.3 0.101 20698400 8.88E+09 5.33E+08 3.2 3.6 1.4 1.5 0.7 0.9 1.3 1 0.102 10959100 8.71E+09 6.13E+08 3.5 2.8 1.1 1 0.7 1 1.4 1.1 0.082 17776600 1.02E+10 6.14E+08 2.3 2.1 0.5 0.9 0.7 1.1 0.5 0.7 0.106 19850500 9.37E+09 5.88E+08 2.8 3.3 0.9 1.2 1.2 0.9 0.7 1.2 0.104 117 K Ex Gxy H L Xla Xld Wo Ylb Ylc Ho t 19063700 1.00E+10 6.34E+08 3.9 3.6 1.5 1 1.1 1.5 1.1 1.3 0.119 16480200 9.77E+09 5.33E+08 2.7 2.7 0.5 1.3 0.9 1 0.8 0.9 0.1 23800100 1.01E+10 6.40E+08 2.2 2.8 0.9 1.3 0.6 0.5 0.9 0.8 0.091 13984100 8.75E+09 5.38E+08 3.8 3.4 1.2 1.4 0.8 1.2 1.1 1.5 0.102 28911800 1.01E+10 5.86E+08 2.1 3.3 1 1.1 1.2 0.6 1 0.5 0.098 8105220 9.33E+09 5.71E+08 3.2 2.2 1.1 0.5 0.6 0.6 1.2 1.4 0.084 13877500 9.94E+09 6.09E+08 3.2 2.6 0.8 0.9 0.9 1.1 0.8 1.3 0.112 12745500 9.54E+09 5.67E+08 4 3.4 1.1 1.2 1.1 1.4 1.3 1.3 0.097 19417800 1.01E+10 6.07E+08 2.4 2.9 0.8 0.6 1.5 1.3 0.5 0.6 0.092 17009500 9.04E+09 6.04E+08 3.1 3.8 1.5 0.9 1.4 1 0.6 1.5 0.09 28779100 9.76E+09 6.48E+08 1.9 2.6 0.9 0.5 1.2 0.8 0.6 0.5 0.111 21199200 1.02E+10 5.78E+08 3.2 3.8 1.1 1.4 1.3 0.8 1.5 0.9 0.103 12217300 8.73E+09 5.78E+08 3.1 3.1 0.7 1 1.4 0.9 1.2 1 0.082 18978500 9.32E+09 5.49E+08 3.1 3 1.1 1.1 0.8 1.3 1.2 0.6 0.111 9677690 9.78E+09 5.78E+08 3.3 2.3 1 0.5 0.8 1 1.3 1 0.092 27517700 9.79E+09 6.25E+08 2.2 3.1 1.5 1 0.6 1 0.5 0.7 0.094 19785800 9.45E+09 6.09E+08 3 3.5 1.2 1.1 1.2 0.8 1.1 1.1 0.097 17990400 8.94E+09 5.32E+08 2.6 3.1 1.2 1.4 0.5 1.2 0.5 0.9 0.082 20011000 9.14E+09 5.76E+08 2.8 2.8 1.2 1 0.6 0.9 0.5 1.4 0.115 21989200 9.47E+09 6.36E+08 2.6 3 0.9 1 1.1 1.1 0.7 0.8 0.1 16771100 9.91E+09 6.02E+08 3.3 3.4 1.2 1.2 1 1.5 1.3 0.5 0.084 24127200 8.83E+09 5.76E+08 1.9 3.7 0.7 1.5 1.5 0.7 0.6 0.6 0.08 12288800 9.91E+09 6.07E+08 2.7 2.9 0.6 1 1.3 0.7 0.5 1.5 0.098 16514100 8.84E+09 5.64E+08 2.9 3 1.2 0.9 0.9 1.2 0.8 0.9 0.093 21513800 8.77E+09 5.75E+08 2.6 2.9 0.9 0.6 1.4 1.2 0.9 0.5 0.108 8571390 9.10E+09 5.57E+08 3.4 2.2 0.7 0.6 0.9 1 1 1.4 0.114 24933100 9.98E+09 5.70E+08 1.7 2.8 1 0.8 1 0.5 0.5 0.7 0.091 36197100 1.03E+10 5.91E+08 1.8 2.8 1.1 0.7 1 0.8 0.5 0.5 0.12 20386700 8.96E+09 6.50E+08 2.6 2.8 0.7 1.4 0.7 1.4 0.6 0.6 0.092 8067240 8.76E+09 5.57E+08 3.2 2 0.6 0.7 0.7 0.8 1.5 0.9 0.097 15254900 1.03E+10 5.97E+08 3.7 3.4 1.2 1.3 0.9 1 1.5 1.2 0.094 18632100 9.11E+09 5.55E+08 2.9 3.5 1 1.1 1.4 1.2 0.7 1 0.097 8070180 1.02E+10 6.42E+08 3.4 2.1 0.5 1.1 0.5 0.7 1.5 1.2 0.09 16016100 9.73E+09 6.46E+08 3.4 3.6 0.7 1.4 1.5 1.4 0.9 1.1 0.091 19234000 9.87E+09 6.50E+08 3.5 3.6 1.3 1.2 1.1 0.6 1.4 1.5 0.11 26013700 9.21E+09 5.73E+08 1.8 3.6 0.6 1.5 1.5 0.6 0.6 0.6 0.088 19547800 9.37E+09 6.20E+08 3.2 4.1 1.1 1.5 1.5 1.2 1.2 0.8 0.081 13459200 8.92E+09 6.15E+08 3 2.4 0.9 0.9 0.6 0.5 1.1 1.4 0.113 15681200 9.92E+09 6.13E+08 3 2.4 0.5 1.1 0.8 1.3 1.1 0.6 0.114 16140300 8.76E+09 6.37E+08 3.4 3.2 0.9 1.2 1.1 1 1.4 1 0.1 118 K Ex Gxy H L Xla Xld Wo Ylb Ylc Ho t 18190000 9.49E+09 6.17E+08 3.3 3.1 1 0.8 1.3 1.4 0.8 1.1 0.118 15335100 9.14E+09 5.34E+08 2.6 3.1 1 0.8 1.3 0.9 0.9 0.8 0.08 14336600 9.70E+09 6.28E+08 2.9 3.2 0.8 0.9 1.5 0.6 1.2 1.1 0.088 8245050 9.00E+09 5.95E+08 3.7 2.7 0.8 0.6 1.3 1.3 1 1.4 0.089 13646000 9.11E+09 6.16E+08 2.7 2.9 0.8 0.9 1.2 0.5 0.9 1.3 0.093 22539100 8.84E+09 5.39E+08 2.5 3.2 1 1.3 0.9 0.7 1 0.8 0.1 26146000 1.02E+10 5.61E+08 2.7 4.2 1.4 1.4 1.4 1 0.7 1 0.099 10799800 8.89E+09 5.43E+08 4.1 3 0.8 1.5 0.7 1.2 1.5 1.4 0.1 14977200 1.00E+10 5.92E+08 3 3 0.6 1.2 1.2 0.9 0.8 1.3 0.104 A.2 Model Development Calculations Note: All values are in SI Units Kw Kfa Kva Kfb Kvb Kfc Kvc Kfd Kvd 17102300 2902778 19635000 78375000 58905000 78375000 58905000 2902778 19635000 17148800 3863597 21598500 78375000 58905000 78375000 58905000 2116125 17671500 17183800 5016000 23562000 78375000 58905000 78375000 58905000 1486222 15708000 17207200 6377403 25525500 78375000 58905000 78375000 58905000 995652.8 13744500 17218400 7965222 27489000 78375000 58905000 78375000 58905000 627000 11781000 17216100 9796875 29452500 78375000 58905000 78375000 58905000 362847.2 9817500 16712500 2902778 19635000 1.04E+08 64795500 1.04E+08 64795500 2116125 17671500 16236100 2116125 17671500 1.35E+08 70686000 1.35E+08 70686000 2116125 17671500 15612100 995652.8 13744500 1.72E+08 76576500 1.72E+08 76576500 2902778 19635000 14486000 627000 11781000 2.65E+08 88357500 2.65E+08 88357500 2116125 17671500 17525000 2902778 19635000 1.04E+08 64795500 1.04E+08 64795500 2902778 19635000 17973000 3863597 21598500 1.35E+08 70686000 1.35E+08 70686000 2116125 17671500 18383300 5016000 23562000 1.72E+08 76576500 1.72E+08 76576500 1486222 15708000 18750200 6377403 25525500 2.15E+08 82467000 2.15E+08 82467000 995652.8 13744500 19066200 7965222 27489000 2.65E+08 88357500 2.65E+08 88357500 627000 11781000 19321100 9796875 29452500 3.21E+08 94248000 3.21E+08 94248000 362847.2 9817500 18682100 2902778 19635000 3.85E+08 1E+08 3.85E+08 1E+08 2116125 17671500 18467100 2116125 17671500 6.27E+08 1.18E+08 6.27E+08 1.18E+08 2116125 17671500 17716800 995652.8 13744500 8.35E+08 1.3E+08 8.35E+08 1.3E+08 2902778 19635000 16320600 627000 11781000 1.22E+09 1.47E+08 1.22E+09 1.47E+08 2116125 17671500 17043400 2116125 17671500 78375000 58905000 78375000 58905000 3863597 21598500 16970900 1486222 15708000 78375000 58905000 78375000 58905000 5016000 23562000 119 Kw Kfa Kva Kfb Kvb Kfc Kvc Kfd Kvd 16882400 995652.8 13744500 78375000 58905000 78375000 58905000 6377403 25525500 16775200 627000 11781000 78375000 58905000 78375000 58905000 7965222 27489000 16645900 362847.2 9817500 78375000 58905000 78375000 58905000 9796875 29452500 16653500 2116125 17671500 1.04E+08 64795500 1.04E+08 64795500 2902778 19635000 15295300 2116125 17671500 2.15E+08 82467000 2.15E+08 82467000 995652.8 13744500 15833100 2902778 19635000 1.72E+08 76576500 1.72E+08 76576500 995652.8 13744500 14753800 2116125 17671500 2.65E+08 88357500 2.65E+08 88357500 627000 11781000 15785900 2116125 17671500 1.72E+08 76576500 1.72E+08 76576500 1486222 15708000 20984800 2902778 19635000 78375000 58905000 78375000 58905000 9796875 29452500 17191000 2902778 19635000 58884298 53550000 1.08E+08 65450000 2902778 19635000 17280000 2902778 19635000 45355903 49087500 1.53E+08 73631250 2902778 19635000 17371500 2902778 19635000 35673646 45311538 2.28E+08 84150000 2902778 19635000 17467400 2902778 19635000 28562318 42075000 3.63E+08 98175000 2902778 19635000 17569800 2902778 19635000 23222222 39270000 6.27E+08 1.18E+08 2902778 19635000 16752700 2902778 19635000 78375000 58905000 1.08E+08 65450000 2902778 19635000 3783510 674382.7 4561667 24977138 15205556 24977138 15205556 674382.7 4561667 14214700 2638889 17850000 2.08E+08 76500000 71250000 53550000 2638889 17850000 13862800 2756173 18643333 3.45E+08 93216667 1.02E+08 62144444 2756173 18643333 10245600 1674163 12091935 49875000 37485000 49875000 37485000 1674163 12091935 9246000 1449585 11156250 35687453 32454545 65157750 39666667 1449585 11156250 10088200 1520021 12440909 31611690 34212500 1.07E+08 51318750 1520021 12440909 13193400 1913503 16625000 34232286 43480769 2.19E+08 80750000 1913503 16625000 2642500 369290.6 3400000 5770165 8500000 73302469 19833333 369290.6 3400000 4654100 627822.1 6115278 8679012 14676667 2.34E+08 44030000 627822.1 6115278 9870010 1408358 13718056 65708333 49385000 90134888 54872222 1408358 13718056 8655770 1211911 13152632 91220850 55533333 91220850 55533333 1211911 13152632 1859190 266918.4 3051282 46161322 17000000 15833333 11900000 266918.4 3051282 7040960 1063802 12792500 3.15E+08 85283333 93392775 56855556 1063802 12792500 15809600 2697531 18246667 99908551 60822222 54720761 49763636 2697531 18246667 4273570 733024.7 4958333 38655599 18593750 11453511 12395833 733024.7 4958333 9684460 1671296 11305000 1.32E+08 48450000 20539372 26088462 1671296 11305000 14513500 2521605 17056667 3.15E+08 85283333 24811710 36550000 2521605 17056667 13901300 2433642 16461667 5.26E+08 98770000 19469136 32923333 2433642 16461667 2524190 439814.8 2975000 16289438 9916667 11875000 8925000 439814.8 2975000 10631000 1993827 13486667 73845450 44955556 1.57E+08 57800000 1993827 13486667 14385100 2609568 17651667 70458333 52955000 2.05E+08 75650000 2609568 17651667 4709090 908950.6 6148333 33664838 20494444 1.14E+08 30741667 908950.6 6148333 16850800 3078704 20825000 1.14E+08 69416667 1.62E+08 78093750 3078704 20825000 19242500 4292886 23998333 65426997 59500000 1.19E+08 72722222 2351250 19635000 13316000 3850667 18088000 34818673 37683333 1.18E+08 56525000 1140938 12058667 120 Kw Kfa Kva Kfb Kvb Kfc Kvc Kfd Kvd 9681330 3543002 14180833 19818692 25173077 1.27E+08 46750000 553140.4 7635833 20274100 9252531 31931667 33178450 48875000 4.21E+08 1.14E+08 728333.3 13685000 7036280 3958333 11900000 9382716 15866667 2.53E+08 47600000 146604.9 3966667 16327100 2902778 19635000 1.04E+08 64795500 1.43E+08 71995000 2116125 17671500 15248300 2116125 17671500 1.86E+08 78540000 1.86E+08 78540000 2116125 17671500 13713400 995652.8 13744500 5.02E+08 1.09E+08 1.72E+08 76576500 2902778 19635000 10994400 627000 11781000 1.22E+09 1.47E+08 3.63E+08 98175000 2116125 17671500 16494900 2630828 19001613 1.04E+08 64795500 1.04E+08 64795500 2630828 19001613 16031600 3183506 20248594 1.02E+08 64260000 1.86E+08 78540000 1743633 16567031 15542300 3768595 21420000 99646918 63813750 3.36E+08 95720625 1116621 14280000 15001100 4380976 22522500 97888484 63436154 6.27E+08 1.18E+08 683966.6 12127500 14378300 5016000 23562000 96397823 63112500 1.22E+09 1.47E+08 394845.5 10098000 13643600 5669488 24543750 95118222 62832000 2.57E+09 1.88E+08 209981 8181250 11361900 1679848 16362500 1.04E+08 64795500 1.43E+08 71995000 1224609 14726250 9301220 1041248 13951184 1.86E+08 78540000 1.86E+08 78540000 1041248 13951184 7693690 453187.4 10572692 5.02E+08 1.09E+08 1.72E+08 76576500 1321246 15103846 5604160 264515.6 8835750 1.22E+09 1.47E+08 3.63E+08 98175000 892740.2 13253625 17428600 2902778 19635000 1.43E+08 71995000 78375000 58905000 2902778 19635000 17769100 3863597 21598500 2.65E+08 88357500 78375000 58905000 2116125 17671500 14779300 4128945 19395178 4.13E+08 90049041 64514773 48487945 1223391 12930119 4069180 1392965 5575330 2.17E+08 30021007 17118822 12866146 217472.4 3002101 16438800 7085055 24451430 1.88E+09 1.57E+08 69714460 52395920 557715.7 10479184 9971390 5284819 15887836 2.38E+08 56490084 1.73E+08 50841076 195734.1 5295945 2101000 509831.6 2850090 24514820 10363963 52102926 13325095 279239.1 2331892 5116800 1515225 7117570 52014836 23132104 1.52E+08 33045863 448955.6 4745047 14912900 6070503 24297136 2.81E+08 87220488 9.48E+08 1.31E+08 947739 13083073 3798470 1727882 5963141 78711815 21296931 1.12E+08 23959048 136014 2555632 14479300 7965474 22749393 2.61E+08 72798058 3.58E+08 80886732 295017.5 7583131 12291500 2150299 13957394 3.91E+08 79091899 3.91E+08 79091899 1567568 12561654 7020270 963353.8 7798578 8.32E+08 74272171 2.85E+08 51990520 963353.8 7798578 2211700 173185.9 2341453 6.72E+08 36794268 1.99E+08 24529512 504915.3 3344933 11192200 422390 8386128 9.1E+08 1.08E+08 9.1E+08 1.08E+08 1425566 12579192 7053800 808380.1 7762487 27299831 25090868 49843724 30666616 1475931 9487484 11131100 852966 11142815 34646539 38303426 1.17E+08 57455139 2878760 16714222 12256700 600161.2 10988541 31302605 41056088 2.01E+08 76247021 3844181 20407291 12679500 378301.3 10100159 27365542 42083994 3.48E+08 98195986 4805827 23567037 3843350 64300.41 2644444 7111111 12693333 1.92E+08 38080000 1736111 7933333 10886900 1382995 14379167 3.17E+08 88000500 4.35E+08 97778333 1008203 12941250 16159200 2444444 19635000 1.21E+08 71995000 1.21E+08 71995000 1782000 17671500 14377500 2249775 17671500 4.2E+08 1.01E+08 1.44E+08 70686000 2249775 17671500 121 Kw Kfa Kva Kfb Kvb Kfc Kvc Kfd Kvd 12720400 1069017 13744500 8.56E+08 1.28E+08 2.54E+08 85085000 3116667 19635000 13370700 472552.1 11400968 2.2E+08 88357500 2.2E+08 88357500 1594863 17101452 14880900 1963806 18407813 64350000 58905000 1.17E+08 71995000 1963806 18407813 15442500 2902778 19635000 78375000 58905000 2.65E+08 88357500 1589876 16065000 15013300 3445756 20790000 78375000 58905000 5.02E+08 1.09E+08 1020965 13860000 18648500 6377403 25525500 9.96E+08 1.37E+08 78375000 58905000 995652.8 13744500 18878300 7965222 27489000 2.12E+09 1.77E+08 78375000 58905000 627000 11781000 12989100 5669488 24543750 3.21E+08 94248000 4.4E+08 1.05E+08 209981 8181250 15711300 2116125 17671500 1.35E+08 70686000 1.86E+08 78540000 2116125 17671500 14380500 995652.8 13744500 2.36E+08 85085000 2.36E+08 85085000 2902778 19635000 12202300 627000 11781000 7.71E+08 1.26E+08 2.65E+08 88357500 2116125 17671500 12492400 2492284 16858333 4.15E+08 92720833 1.23E+08 61813889 2492284 16858333 11738000 2475902 14779032 95760000 49980000 95760000 49980000 1356072 12091935 7289530 1878662 10040625 58804001 31643182 1.07E+08 38675000 556640.6 6693750 11692600 3629877 17579545 94285301 52062500 3.18E+08 78093750 566703.5 9465909 4023260 1368000 6426000 26290315 17212500 3.34E+08 40162500 107685.1 2754000 2103700 801745.8 3470833 13451062 8885333 3.63E+08 26656000 29694.29 1156944 11524900 1595007 15536111 3.66E+08 95081000 5.02E+08 1.06E+08 1162760 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Force transfer around openings in CLT shear walls Pai, Sai Ganesh Sarvotham 2014
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Title | Force transfer around openings in CLT shear walls |
Creator |
Pai, Sai Ganesh Sarvotham |
Publisher | University of British Columbia |
Date Issued | 2014 |
Description | During an earthquake, shear walls can experience damage around corners of doors and windows due to development of stress concentration. Reinforcements provided to minimize this damage are designed for forces that develop at these corners known as transfer forces. In this thesis, the focus is on understanding the forces that develop around opening corners in cross laminated timber (CLT) shear walls and reinforcement requirements for the same. In the literature, four different analytical models are commonly considered to determine the transfer force for design of wood-frame shear walls. These models have been reviewed in this thesis. The Diekmann model is found to be the most suitable analytical model to determine the transfer force around a window-type opening. Numerical models are developed in ANSYS to analyse the forces around opening corners in CLT shear walls. CLT shear walls with cut-out openings are analysed using a three-dimensional brick element model and a frame model. These models highlight the increase in shear and torsion around opening corners due to stress concentration. The coupled-panel construction practice for CLT shear walls with openings is analysed using a continuum model calibrated to experimental data. The analysis shows the increase in strength and stiffness of walls, when tie-rods are used as reinforcement. Analysis results also indicate that the tie-rods should be designed to behave linearly for optimum performance of the wall. Finally, a linear regression model is developed to determine the stiffness of a simply-supported CLT shear wall with a window-type opening. This model provides insight into the effect of various geometrical and material parameters on the stiffness of the wall. The process of model development has been explained, which can be improved further to include the behaviour of anchors. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2014-12-23 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 2.5 Canada |
DOI | 10.14288/1.0165581 |
URI | http://hdl.handle.net/2429/51632 |
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 | 2015-02 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/2.5/ca/ |
AggregatedSourceRepository | DSpace |
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