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The seismic response of a timber frame with dowel type connections Frenette, Caroline D. 1997

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THE SEISMIC RESPONSE OF A TIMBER FRAME WITH DOWEL TYPE CONNECTIONS by CAROLINE D.FRENETTE B.Eng., Universite de Sherbrooke, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Civil Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1997 ©Caroline D. Frenette, 1997 In p resent ing this thesis in partial fu l f i lment of the requ i remen ts fo r a n " advanced d e g r e e at t h e Univers i ty o f Brit ish C o l u m b i a , I agree tha t t h e Library shall make it f reely available fo r re ference and s tudy. I fu r ther agree that permiss ion f o r extensive c o p y i n g of this thesis f o r scholar ly pu rposes may be g ran ted by the head o f m y d e p a r t m e n t or by his o r her representat ives. It is u n d e r s t o o d that c o p y i n g or pub l i ca t i on o f this thesis f o r f inancial ga in shall n o t be a l l o w e d w i t h o u t m y w r i t t e n permiss ion . D e p a r t m e n t o f C W i u • > ! € € . f t l The Universi ty of Brit ish C o l u m b i a Vancouver , Canada Date flpRtt / > q q 7 DE-6 (2/88) A B S T R A C T The purpose of this study was to analyse the dynamic behaviour of timber structures with moment resisting connections. The response of ductile dowel connectors and their effect on the seismic behaviour of semi-rigid timber frames were exarnined. Experimental results from component and frame tests were compared with analytical predictions from two different analytical models. The modelling of dowel type connections based on fundamental material properties was achieved and its incorporation in a dynamic frame analysis program has been investigated. To observe the overall non-linear dynamic behaviour of semi-rigid timber frames with dowel type connections, a two-storey test specimen was subjected to simulated ground motion produced by a shake table. The test specimen consisted of two planar moment resisting timber frames braced together in the out-of-plane direction. The beams and columns, composed of parallel strand lumber (Parallam®), were connected using steel plates and tightly fitting steel dowels. The behaviour of one typical connection was monitored in detail in addition to data pertaining to the overall response of the system. Prior to this shake table test, materials and connections properties were studied through monotonic and cyclic tests. To enable the modelling of timber frames with dowel type connections, a finite element model that simulates the non-linear behaviour of the connector caused by yielding of the dowel and crushing of the wood was used. This connector model was included in a two-dimensional static frame analysis program to predict the non-linear response of semi-rigid timber frames. Specific attention was paid to the multi-cycle behaviour of a multiple connector joint. This model was developed and calibrated using test results of dowel connections subjected to cyclic loads. The ultimate goal of modelling the dynamic behaviour of any ii timber structure joined with dowel type connectors and steel or wood plates, using basic material properties, was investigated. A comparison of the experimental data with the simulated analytical response for the timber frame enabled the verification of various programs, such as SAP90\and DRAIN-2DX, and aided in the calibration of the connection model under dynamic conditions. iii T A B L E O F C O N T E N T S page ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES viii LIST OF FIGURES ix ACKNOWLEDGEMENT X 1. INTRODUCTION 1 1.1 PROBLEM OVERVIEW 1 1.2 AIMS AND OBJECTIVES 2 1.3 SCOPE 4 1.4 OUTLINE OF THE STUDY 4 2. BACKGROUND 5 2.1 TIMBER: A STRUCTURAL MATERIAL 5 2.2 ENGINEERED WOOD PRODUCTS 8 2.3 SEISMIC PERFORMANCE OF TIMBER STRUCTURES 10 2.3.1 E X P E R I E N C E F R O M PREVIOUS E A R T H Q U A K E S 10 2.3.2 D E S I G N C O D E S 11 2.4 ANALYSIS OF CONNECTIONS 13 2.4.1 S T A T I C P E R F O R M A N C E O F C O N N E C T I O N 14 2.4.1.1 Experimental testing 16 2.4.1.2 Analytical prediction and design 18 2.4.2 D Y N A M I C P E R F O R M A N C E O F C O N N E C T I O N S 19 2.4.2.1 Experimental testing 21 2.4.2.2 Analytical simulation 24 2.5 ANALYSIS OF FRAMES UNDER DYNAMIC LOADING 26 2.5.1 E X P E R I M E N T A L T E S T I N G 26 2.5.2 A N A L Y T I C A L PREDICTION 28 2.6 CONCLUDING REMARKS 29 iv 3. EXPERIMENTAL TESTING • ; 32 3.1 TESTS ON CONNECTIONS 32 3.1.1 BASIC MATERIAL PROPERTIES 32 3.1.2 SINGLE PIN CONNECTION 33 3.1.3 TWO AND FOUR PIN CONNECTIONS 33 3.1.4 EIGHT PIN DOUBLE CONNECTIONS 34 3.2 SHAKE TABLE TEST-EXPERIMENTAL PROCEDURE 34 3.2.1 DESCRIPTION OF THE TEST-FRAME 34 3.2.1.1 Materials 34 3.2.1.2 Design of the Frame 35 3.2.1.3 Construction of the Frame 37 3.2.2 INSTRUMENTATION 37 3.2.2.1 Data Acquisition System 38 3.2.2.2 Monitoring of the Structure 38 3.2.2.3 Monitoring of the Connections 39 3.2.2.4 Procedure to Install the Instruments 40 3.2.3 LOADING PROCEDURE 41 3.2.3.1 Choice of Excitation Record 41 3.2.3.2 Proposed Testing Sequence 42 3.2.3.3 Actual Testing Sequence 43 3.2.4 VIBRATION TESTING AND MODAL ANALYSIS 44 3.2.4.1 Ambient Vibration Testing 44 3.2.4.2 Hammer Blow Testing 45 3.2.4.3 Shake Table Tests 45 3.3 SHAKE TABLE TEST - EXPERIMENTAL RESULTS 46 3.3.1 RESPONSE OF THE FRAME 46 3.3.1.1 Out-of-Plane and Vertical Response 46 3.3.1.2 Integration of Acceleration Records 47 3.3.1.3 Longitudinal Response 48 3.3.1.4 Serviceability 49 3.3.2 CONNECTIONS 50 3.3.2.1 Calculations of Rotations and Relative Displacements 50 3.3.2.2 Moment-Rotation Response 52 3.3.2.3 Observed Damage 54 3.3.3 NATURAL FREQUENCIES OF THE STRUCTURE 54 3.3.4 EVALUATION OF THE MEASURING SYSTEM 55 3.3.4.1 Measuring System for Connections 55 3.3.4.2 Moment in Connections 57 3.3.4.3 Integration of Acceleration Time Histories 57 3.3.4.4 Camcorder 58 3.3.4.5 Damage in the Connections 58 3.4 CONCLUDING REMARKS 58 4. DEVELOPMENT OF AN ANALYTICAL MODEL ' 77 4.1 INTRODUCTION 77 4.2 ANALYSIS OF A CONNECTOR 78 v 4.2.1 T H E FINITE ELEMENT MODEL 79 4.2.1.1 Mcxlelling of the steel 81 4.2.1.2 Modelling of the Wood Foundation 82 4.2.1.3 Virtual Work Equilibrium 84 4.2.2 POSSIBLE IMPROVEMENT 91 4.3 ANALYSIS OF A CONNECTION 92 4.3.1 DESCRIPTION OF THE MODEL 92 4.3.2 COMPARISON WITH TEST RESPONSE 95 4.3.3 POSSIBLE IMPROVEMENT 95 4.4 FRAME ANALYSIS 95 4.4.1 STATIC ANALYSIS - P S A - W O O D F R A M E 96 4.4.1.1 Problems in developing PSA-WoodFrame 97 4.4.2 HYSTERESIS LOOP OF A DOUBLE CONNECTION: M O D E L PREDICTION VS. EXPERIMENTS 99 4.4.3 DYNAMIC ANALYSIS - D P S A - W O O D F R A M E 101 4.5 CONCLUDING REMARKS 102 5. ANALYTICAL PREDICTION 104 5.1 MODELLING OF THE TEST-FRAME 105 5.1.1 W O O D MEMBERS 105 5.1.2 BASE SUPPORTS AND LATERAL BRACING 106 5.1.3 MASSES AND STATIC LOADS 106 5.1.4 CONNECTIONS 107 5.2 EARTHQUAKE EXCITATION 109 5.2.1 INPUT ACCELERATION RECORD 109 5.2.2 CONSECUTIVE EXCITATIONS 110 5.3 DAMPING 110 5.4 ANALYTICAL RESULTS 111 5.4.1 LINEAR ANALYSIS (SAP 90) 111 5.4.1.1 Determination of results 112 5.4.1.2 Comparison of results 112 5.4.2 NONLINEAR ANALYSIS WITH DRAIN-2DX 112 5.4.2.1 Comparison of moment-rotation curves 113 5.4.2.2 Comparison of displacements and rotations 114 5.4.2.3 Comparison of top acceleration 115 5.4.2.4 Comparison of forces 116 5.4.2.5 Possibilities of improvement 116 5.5 CONCLUDING REMARKS 118 6. CONCLUSIONS AND FURTHER STUDIES 133 6.1 CONCLUSIONS 133 6.1.1 EXPERIMENTAL TESTING 133 6.1.2 DEVELOPMENT OF A N ANALYTICAL M O D E L 134 6.1.3 ANALYTICAL PREDICTION 134 6.2 FURTHER STUDIES 135 vi REFERENCES 137 APPENDIX A ; 144 A. 1 Influence of the Beam Curvature on the Frame Response 145 A. 2 Calculation of Moment in Connection 148 APPENDIX B ••• 1 5 0 B. 1 Monitoring of the Connections - Calculations of Relative Displacements 151 B.2 Precision of Measurement for connection response 159 B.3 Calculation of Cahbration Factors 165 B. 4 Calculation of Base Shear for Shake Table Tests 168 APPENDIX C ; 1 6 9 C. 1 Shape Functions for beam elements 170 C. 2 Computer programrning 172 APPENDIX D ' 121 D. 1 Manufacturer Information on Parallam® 180 D.2 SAP90 Input File 181 D.3 DRAIN-2DX Input File 185 vii L I S T O F T A B L E S CHAPTER 3 Table 3.1- Summary of Shake Table Tests Table 3.2 - Shake Table Test Results CHAPTER 5 Table 5.1- Comparison of Maximum Values of the Frame Response viii L I S T O F F I G U R E S page CHAPTER 2 Figure 2.1- Experimental load-slip curves for joints in tension parallel to grain IS Figure 2.2 - Failure modes assumed in European Yield Theory for three-member joints 18 CHAPTER 3 Figure 3.1- Typical stress-strain curve for steel dowel material 62 Figure 3.2 - Single dowel connection cyclic testing 62 Figure 3.3- Cyclic tests on two- and four-dowel connections 63 Figure 3.4 - Eight-dowel connection 63 Figure 3.5- Shake table test frame 64 Figure 3.6 - Hysteretic behaviour of a ductile dowel connection 64 Figure 3.7- Schematic drawing of the test frame 65 Figure 3.8 - Steel bracket for concrete masses 65 Figure 3.9 - Monitoring of shake table test frame 66 Figure 3.10- LVDT measuring system for connection displacement 67 Figure 3.11- Photo of the connection displacement measuring system 67 Figure 3.12- Earthquake acceleration records investigated 68 Figure 3.13- Moment-rotation response for the connection as simulated by DRAIN-2DX 69 Figure 3.14- Accelerometers on test frame for Ambient Vibration Testing 70 Figure 3.15- Displacement of the top floor, Integrated values vs. Measured values 71 Figure 3.16- Top floor relative displacement (0.15g simulations) 72 Figure 3.17- Top floor relative displacement (0.5g simulations) 73 Figure 3.18- Moment-rotation hysteresis loops of the monitored connection 74 Figure 3.19- Connections after shake table tests 75 Figure 3.20 - Observable damage on cut-open connections 76 CHAPTER 4 Figure 4.1 - A finite element and its degrees of freedom (DOFs) 79 Figure 4.2 - Coordinate transformation for each beam element 80 Figure 4.3 - Degrees of freedom of a two element model 81 Figure 4.4 - Stress-strain relationship of steel dowel 81 Figure 4.5 - Parameters of the load-deformation law of the wood foundation 83 Figure 4.6 - Load displacement relationship for wood foundation 83 Figure 4.7 - DOFs of the steel plate and wood member 92 Figure 4.8 - PSA-WoodFrame results compared to cyclic test results for a 8 dowel connection 100 CHAPTER 5 Figure 5.1- Slope parameters for DRAIN-2DX connection models 120 Figure 5.2 - Connection response sensitivity regarding the Modulus of Elasticity of the members, through a frame analysis with DRATN-2DX 121 Figure 5.3- Material properties needed in DPS A input file 121 Figure 5.4 - Moment-rotation curve from DRAIN-2DX compared to cyclic test results 122 Figure 5.5 - Moment-rotation curve from DRAJN-2DX compared to shake table test results 123 Figure 5.6 -Relative displacement of the top floor 124 Figure 5.7- Relative acceleration of the top floor 127 Figure5.8 -Rotation within the mid connection 130 i x A C K N O W L E D G E M E N T S I would like to acknowledge the contribution of those who helped me complete this research project. First of all, I am sincerely grateful to my thesis supervisor, Dr. Helmut G.L. Prion, for his thoughtful guidance and constant encouragement. I also acknowledge my co-supervisor, Dr. Ricardo O. Foschi, for providing excellent technical support. To both of them, my thanks for their unfailing presence during these past few years, as I deepened my understanding of engineering and life. During my studies at UBC, I have appreciated the excellent teaching by many faculty members of the structural group. I particularly thank Dr. C.E. Ventura for sharing his passion for Dynamics principles in such an inspiring way. I gratefully acknowledge the financial and in-kind support by the Natural Sciences and Engineering Research Council, Industry Canada, Forest Renewal BC, Macrnillan-Bloedel Research and the Department of Civil Engineering of the University of British Columbia. I greatly appreciated the help offered by Dr Ario Ceccotti; he provided me with a better understanding of timber structures and analytical predictions. Special thanks to Vincent Latendresse and Howard Nichol for their contribution to the experimental part of this project and helpful technical support. My colleagues, David, Manzar, Jennifer, Ye and Hong were always present and provided me with a creative and supportive work environment. And finally, I would not have accomplished this without the constant encouragement of my special friends, Isabelle, Didier, Elizabeth and Susan, and dear family, Chantal, Jean-Louis and Veronique. Introduction Chapter 1 INTRODUCTION 1.1 Problem overview Timber structures have in the past performed relatively well during major earthquakes. The reason for this is mostly due to the high strength to weight ratio of the material and the relatively simple and proven construction methods used. This belief was, however, severely shaken during recent earthquakes in Northridge (1994) and Kobe (1995), where the collapse of residential timber structures caused a substantial loss of life. It was observed that a large part of the damage was due to the lack of or inconsistency of the lateral load resisting system. The North-American platform construction method using shear wall systems performed well, except when large openings reduced the lateral capacity, often creating a soft storey effect. Recent developments to increase the size of timber buildings and the demand for larger openings in the shear walls requires a careful assessment of the different lateral load supporting systems available in timber structures and their performance under seismic loading. In many cases, moment resisting frames may provide an acceptable solution. Very little information is available about such a system and the development of efficient moment resisting frames is desperately required to fill this gap. By providing lateral resistance, moment resisting frames can be used to prevent large differences in stiffness between storeys well supported by shear walls and other storeys with large openings. In present construction practice, moment resisting frames are avoided when possible because of the difficulty to design moment connections and the uncertainty related to their stiffness and ductility. A fully elastic design, which inevitably leads to large connections that will remain elastic under the higher expected loads, is usually 1 Introduction avoided in seismic design because of the brittle failure caused by splitting of the wood near the connector or along the members. The design approach presendy used for seismic performance includes the creation of ductile connections that will sustain service loads in the elastic range, without large deformation, yet can deform and dissipate energy under larger loads such as a major earthquake. This ductile behaviour is preferred since it limits the force level in the members through hysteretic damping in the connections. Dowel type connections can be used to construct a semi-rigid moment resisting frame. The behaviour of dowel connections under static loading is recognised and often used in large European timber construction. Its cyclic performance has been studied, and good ductility in its hysteresis response was shown. Yet, the performance of such a connection under dynamic loading has not been assessed and its influence on the frame dynamic behaviour has hardly been explored. The connection (iarnage caused by dynamic loading on a moment resisting frame, and the response of the frame to consecutive earthquakes also needs to be investigated. The nonlinear behaviour of moment resisting timber frames with ductile connectors, such as dowels, complicates the prediction of its dynamic response. Even when considering the connection alone, the particular hysteresis curves are difficult to simulate since crashing of the wood happens during the first cycles only, while the yield resistance of the steel is present throughout all the cycles. 1.2 Aims and Objectives Several years ago a project was initiated at the University of British Columbia (UBC) to study the seismic behaviour of timber structures with dowel type connections and to establish appropriate analytical models. The aims of this global project were: • to increase the body of knowledge on the dynamic performance of moment resisting timber frames, 2 Introduction 4 • to study the dynamic behaviour of dowel type connections and its influence on the response of semi-rigid timber frames subjected to seismic excitation, • to enable the analytical simulation of dowel type connection using basic material properties of each individual dowel, in order to reduce the amount of costly testing on entire connections, • to enable the dynamic analysis of timber frames using dowel type connections: ; . The first part of the project consisted of the investigation into the specific behaviour of a ductile dowel encased in a wood foundation. Fundamental analysis and basic material properties were used to develop a finite element model producing realistic hysteresis models to predict the response of the connectors under cyclic loads. The details of this model are described in the paper by Foschi and Prion (1994). The project was continued in this specific study which investigated the dynamic performance of timber frames using dowel type connections. The objectives of this study as being part of the global research project were: • to understand the dynamic behaviour of a two storey timber frame using dowel type connections, • to provide experimental data regarding the behaviour of a timber frame with dowel type connections subjected to a known earthquake record, • incorporate a finite element model of a ductile dowel in a dynamic analysis program for timber frames with dowel type connections. The specific tasks involved to achieve these objectives were: - Design and install the monitoring system of a two storey timber frame to capture the overall behaviour of the frame during the dynamic test, as well as specific relative displacement of the connections. - Choose among the available earthquake records an excitation that would create a highly nonlinear response of the tested frame. - Conduct the shake table test and monitor the response of the frame to several consecutive excitations. - Analyse the data, discuss the resulting response, and organise the data for comparison with analytical simulations. 3 Introduction - Implement the finite element model of a ductile connector, in a static analysis program for the entire frame, and develop the theoretical background for its implementation in a dynamic analysis program dealing specifically with timber structures using dowel type connections. - Predict the response of the frame using a linear analysis program, SAP90, and a nonlinear program dealing with semi-rigid connections, DRAIN-2DX. - Compare the results from the different analytical programs to the response of the tested frame. 1.3 Scope In the full development of the procedure for analytical modelling of timber frames under dynamic excitation, only one type of connection was investigated. The connection studied in this project consisted of one hidden steel plate connected to the members by eight tightly fitting steel dowels. The analytical model was developed to accommodate various configurations of dowel type connections, yet the experimental verification only considered this specific connection. 1.4 Outline of the study This thesis presents the different steps followed in the study to achieved its objectives. Chapter 2 presents a background of the problem supported by a literature review related to timber connection and dynamic performance of timber frames. Chapter 3 describes different experimental tests performed during this study including cyclic testing on dowel type connections and a shake table test on a two-storey timber frame. Chapter 4 outiines the theoretical background supporting the development of the analytical model, including the finite element model of a connector, the model for each connection, as well as static and dynamic models for the entire frame. In Chapter 5, an overview of various analytical tools that can be used to estimate the response of timber frames with dowel type connections is given and the results from two specific analytical models are compared with experimental data. Finally, Chapter 6 summarises the conclusions obtained from this study and provides recommendations for further research. 4 Background Chapter 2 BACKGROUND 2.1 Timber: a structural material The interest in wood as a structural material has always been present in human history. It has been the most common material for the construction of houses, small buildings, and churches in many parts of the world. Timber was often used in construction due to its availability, workability, lightness, and ease of construction. For a long time, the design of wooden structures was based on the performance of existing buildings. Yet, some examples of very complex wooden structures, such as old churches in Europe, show the designer's ingenuity and knowledge about this structural material. More recently, with the development of other building materials such as steel and reinforced concrete, the recognition of timber as a structural material for large structures decreased significantly. Presently, wood is, by far, the most used building material for residential construction in North America. Timber is also used in the construction of larger buildings throughout the world, yet this application is not as frequent. The timber construction industry is facing several challenges: (a) in the residential area, the principal engineering challenge is in the preparation of rational designs for structural systems and the technical support required to implement them (Galligan, 1988), (b) the commercial and industrial markets present even more challenges, such as design methodology, material research, computer aids and education that supports the design of such systems. Timber, as a structural material, presents several advantages for certain types of construction, while other aspects of this material limit its usability. The challenge of the design industry is to evaluate the specific needs for each project and to determine which structural material presents the most advantageous 5 Background characteristics. Unfortunately, in the design of larger projects, timber is often not considered as a choice due to a misunderstanding of its capacity as well as a lack of design information. Around the world, civil engineers are well versed to design steel and reinforced concrete structures, yet only few are knowledgeable regarding the design of timber structures. The use of timber as a structural material has several advantages, the most important of which are presented in the following paragraphs. The main advantage is its light weight, thus requiring less heavy load machinery during construction. This characteristic, combined with the softness of the material, allows for simple construction techniques and the possibility for construction with fewer workers. The high strength to weight ratio of timber structures, as well as the ductility of conventional connection methods, often lead to favourable seismic behaviour. Experience has shown, for example, that under earthquake excitation, ductile structures with lower natural frequencies can be more efficient than very stiff structures. Contrary to common belief, the fire resistance of many wood structures is often considered an advantage of this material. For large timber sections, the burning mechanism consists of charring around the cross-section, which insulates the member and actually slows down the rate of loss of wood due to further burning. This mechanism helps the member keep its structural capacity, unlike other materials such as steel, which experiences a loss of stiffness at temperatures as low as 300°C. It has also been reported that the ease of exiting the building as fast as possible is of greatest importance, since smoke and heat often produced by non-structural elements are the most damaging to occupants (CWC, 1994). 6 Background Environmental considerations also favour timber over other building materials because it is a renewable material with a full life cycle requiring much less energy compared to steel or concrete (CWC, 1994). " Considering global carbon emissions, wood is much better than other building materials such as steel or concrete, but an increase in wood can only be sustained if there is a corresponding improvement in sustainable forest management " (Buchanan and Bry Levine, 1996). The natural beauty of timber, which provides possibilities of exposing the structure as an architectural characteristic of the building, is also a definite advantage of this material. Other characteristics of wood need to be understood and considered to optimise its utilisation in structural design. In some countries, the availability and the price of wood material can be a concern. Moisture cycles can affect the dimensional stability and the strength of wood lumber. Unlike steel, wood does not rust, but, combined with other conditions, the presence of moisture can cause decay. The contact of wood with specific chemicals can also result in its degradation. On the other hand, wood is often used because of its resistance to the attack of specific chemicals. Due to its composition, the strength and stiffness properties of wood change with the relative orientation of the applied load to the wood fibres. Wood is characterised as an orthotropic material since the material properties can be defined in three perpendicular directions: longitudinal, transverse and radial. The mechanisms of failure in these orthogonal directions are very different, which cause the strength of the material to vary accordingly. Some loading conditions such as tension perpendicular to grain result in brittle failures, while other types of loading, such as compression parallel to grain, lead to a more ductile behaviour. Many studies have been conducted to deterrrtine the strength of the material at an angle to its 7 Background grain with respect to the strength parallel and perpendicular to the grain. The first known investigation on this subject was conducted by Hankinson in 1921. He proposed a formula to predict the off-axis compressive strength of spruce. Even though his formula is purely empirical, it is relatively accurate for various species of wood and widely accepted in the wood industry. Since then, an effort has been made to develop a failure criterion for wood that can consider planar and three-dimensional stress states (Tsai and Wu, 1971; Liu, 1984; Tan and Cheng, 1993). Since wood is a naturally produced material, variability is an intrinsic characteristic Of this material. In addition to knots and other defects present, factors such as the rate of growth, growing conditions, species, and moisture content also affect its structural behaviour: Extensive research has been undertaken to optimise the utilisation of this natural resource by improving the grading techniques, transforming sub-standard wood into various by-products, and developing new design techniques. 2.2 Engineered wood products The building industry is facing an increasing demand regarding the quantity, the strength and the reliability of wood products. At the same time, the reduction in the size of available trees, as well as environmental concerns regarding the cutting of old growth forests, have led to a limitation in the size of available sawn lumber. To overcome this challenge, the wood industry has taken advantage of the advances in adhesive chemistry and developed composite wood products. The first products available were plywood and oriented strand board (OSB), which are used as panels in structural walls. Lately, more products have been developed to replace large wooden sections used as posts or beams. Glued laminated timber (Glulam) is manufactured by fmger-jointing dimension sawn lumber, then laminating and gluing it together to achieve a larger section. Since knots and defects are not aligned for all the lamina, load distribution is possible within the section, resulting in less strength variability compared to sawn lumber of the same size. 8 Background This product has been included in timber codes for several years and is used in many large structures throughout the world. More recently, other products such as laminated veneer lumber (LVL) and parallel strand lumber (PSL) were developed (Sharp, 1996). Similar to plywood, LVL is composed of several layers of wood veneer, all placed with the same grain orientation and glued together. It is mostly used for beam elements, but not for columns since the thickness of the section is limited by the manufactoring process. PSL is composed of small strands of wood veneer oriented in the longitudinal direction of the element, pressed and glued together. Much larger sections can be created so that this product can be used as beam or post elements. Since LVL and PSL are composed of smaller pieces of wood glued together, all natural defects of the wood are spread out in the section and have little effect on the overall strength of the member. Therefore, these products have three times the design bending strength of a similar size piece of lumber and are about 30% stiffer (CWC, 1994). These manufactured products have a low moisture content to mirumise shrinkage, deep checks, cracks and twist (Trus Joist Macmillan, 1992). The manufacturer of these products in North America is Trus Joist MacMillan and the registered names are Microllam® for LVL and Parallam® for PSL. Until now, they have mostly been used in very simple applications such as header or supporting beams in residential construction. Their higher design strengths and the range of available dimensions make them desirable for a large range of residential and non-residential construction. Some examples where Parallam® has been used in non-residential applications in British Columbia are the South Surrey Ice Arena (1990), the University of Northern British Columbia.(1994), the Seabird Island School (1992), the Feric Building (1991), and the Forintek Western Research Facility (1990). 9 Background Most of the tests on moment resisting connections for heavy timber construction were done using Glulam members since it has been on the market for several years. LVL and PSL are newer products in the wood industry, and not many studies have been done to date using these products in different applications. Since Parallam® is a proprietary product, the results of studies regarding its different properties are not readily available. It is a relatively new product, and much is still to be learned about its interaction with other components, especially with regards to connection behaviour. Parallam® was used in this study to help overcome this lack of information. Embedment properties were tested as part of this research, but the design strength and modulus of elasticity published by the manufacturer were used in the different analyses. 2.3 Seismic performance of timber structures Earthquakes produce strong motions in the horizontal and vertical directions, and thus affect structures in a three dimensional space. The horizontal motions are typically the only ones considered in the design of structures, since gravity loads are often more critical than vertical seismic loads. For timber structures, the most common systems used to resist these lateral forces are shear walls, braced frames, and moment-resisting frames. It is important to note that, even though timber structures have in the past gained a good reputation regarding their earthquake performance, more recent buildings with longer spans, more storeys, and larger openings offer new challenges to design engineers. 2.3.1 Experience from Previous Earthquakes The performance of timber buildings in several recent earthquakes has been assessed. Following the Lafkas experience (1825), the principle of independently deformed but collaborating parts, as well as the advantage of using strong timber joints with adequate ductile behaviour were recognised (Touliatos, 1991). The insufficiency of let-in bracing, the instability caused by large garage openings, the vulnerability of house construction on sloping sites, the lack of tie-downs for shear walls and moment 10 Background resisting systems, as well as the adequacy of houses built following the present American Building Code (UBC) were the main conclusions from San Fernando (1971) and Loma Prieta (1989) earthquakes (Brown, 1991). Assessments after the Northridge earthquake (1994) detennined that the lack of lateral support for porch columns often results in the collapse of part of the roof, and soft storeys weaken a building and can lead to collapse. It was also reported that failure of structural elements such as studs or joists are unlikely to be the primary cause of collapse, and the weak link in wood frame timber structures is the connections between structural elements (Stieda, 1994). Heavy roofs, soft storeys, and lack of lateral support were the main causes of damage recorded from the Hyogo-Ken Nanbu earthquake (1995) (Prion et al., 1995). Other authors have reported general characteristics of timber structures under seismic loading. Low weight is an incontestable advantage of timber construction, while the relatively low stiffness increases the period, and thus the displacement, of the structure, but also results in more damage to non-structural components. A further advantage is the high hysteresis damping due to the deformations in ductile connections between wood members (Walford, 1984; Williams, 1984; Deam and King, 1994). Amongst all these reports, few investigations about failures of larger timber structures using moment-resisting connections are cited, mainly because this construction method has not been used very much in the past. The increasing demand in larger construction with unconstrained floor space and large window openings, necessitates the pursuit of research on frame systems with moment resisting connections. 2.3.2 Design Codes Observations after large earthquakes, as well as extensive laboratory testing, have provided engineers and researchers with vital irrforrnation to be able to assess the performance of timber structures under seismic loads. The codes and regulations, which direct the engineering design of structures, are generally a compilation of this information. Typical seismic requirements for regular buildings are that no damage 11 Background occurs under moderate earthquakes and that no collapse occurs during large earthquakes, with tolerance for some damage (Walford, 1984). Although a structure could theoretically be designed to remain elastic for all earthquakes, this is not a practical solution due to the uncertainty of seismic excitation. A more practical approach is to assure that a structure can absorb the shock with a ductile response which incorporates significant plastic deformation and dissipates the input energy. In the seismic design of structures, different approaches can be followed to achieve the desired ductility level. The more complex approach is a complete dynamic analysis that simulates the response of the structure to several probable earthquakes. For linear structures, a modal analysis can provide the information required for the design. The nonlinearity of important elements within the structure, however, complicates the analysis since the response of all the structural components has to be simulated in a time history analysis. In many cases, a simplified quasi-static approach is sufficient to determine the maximum base shear that the structure should resist. For this approach, a typical acceleration spectrum, an approximation of the natural period of the structure, an acceleration factor for the given area, and the weight of the structure are needed. The maximum elastic base shear is reduced by a force reduction factor based on the understanding that the ductility of a structure will permit larger deformations for lower associated forces (NBCC 1995, 4.1.9.1). Even though the quasi-static analysis is considerably simpler to use than a dynamic analysis, substantial testing is needed to provide the information required to ensure its accuracy for various types of structures. In this process, many assumptions have to be made and such a simplified method invariably would have to be relatively conservative. The determination of the ductility factor for different materials and construction systems is a very controversial topic. Ceccotti and Vignoli (1987) conducted a study to assess the behaviour of timber structures, and to determine realistic ductility factors. They concluded that 12 Background structures with nails or dowel connectors could justify a ductility factor much higher than structures with glued joints. Blass (1991) also suggested an increase of the ductility factor for timber structures with semi-rigid connections. He based his research on analytical simulations of timber walls made of LVL and connected with screws. Buchanan and Dean (1988) stated the importance of the capacity design approach in order to ensure that brittle elements have the capacity to remain intact while deformations occur in selected ductile elements. This design criterion is very important for timber construction, given the brittle characteristic of the material in comparison to the ductile behaviour of the metallic connectors. More research regarding the response of different timber structures to dynamic loading will permit these ductility factors to be assessed with more accuracy and will help structural engineers appreciate the capacity of timber structures. 2.4 Analysis of Connections The main challenge in the design of timber structures is the design of the connections between the different wooden elements. As mentioned earlier, the properties of wood vary with the orientation of loading, and the failure mechanism is therefore more complex to predict than for isotropic materials. Because the strength of the material in some directions is significantly weaker, certain loading conditions should be minimised to achieve optimal connection behaviour. In the history of timber construction, several connection methods have been used. Traditional connections, such as complex tongue and groove assemblies, often using wooden pegs, have slowly been abandoned due to their cost and the extensive carpentry expertise required. Some of these techniques are still being used in Japanese temples where the culture maintains an emphasis on traditional wooden structures (Prion et al, 1995). Nowadays, the most common connectors can be divided into two categories, namely dowel type 13 Background and skin type (Madsen, 1992). The dowel type connectors, such as nails and bolts, transfer load to a substantial part of the wood cross-section by bearing on the wood and bending within the connector, whereas skin type connectors, such as shear plates, split rings and nail plates, concentrate the stresses near the surface of the member. Extensive experimental research has been conducted to determine the performance of different connections. Using test observations, analytical models were developed to enable the deterrnination of general design rules for many different connection configurations without the need for expensive testing. A summary of the different research areas in the static and dynamic performance of connections, and the various analytical models developed will be presented next.. . ' 2.4.1 Static Performance of Connection The initial concerns regarding timber connections were mainly about the ultimate strength of the joints. The proceedings of the first International Conference on Timber Engineering, Southampton 1961, present various analytical and experimental techniques to evaluate the ultimate strength of nailed and bolted joints, studied by European researchers (Brochard, 1961; Schjodt, 1961; Moe, 1961). Creep and excessive deformation of joints were also considered as a limiting aspect in the design of wood connections (Noren, 1961). In further studies, different conditions influencing the performance of connections were also evaluated. Specific research will be presented in the next section on experimental testing. During the last few decades, several types of connections have been explored. Dowel type connectors, such as nails, small bolts and dowels, have a high ductility capacity but are inadequate to carry high loads. Larger bolts and lag screws have a high load capacity but often lead to brittle failures when splitting of the wood occurs before yielding of the connector. Glued joints can develop relatively high strength but have 14 Background almost no ductility. Shear plate, split ring and Bulldog® connectors have a very high strength capacity but also lead to brittle failures (CWC, 1994). The latter is a patented tooth plate connector that has teeth profonding in both directions and is positioned between wood members with a bolt through its center. For larger connectors with high load capacity, frequent occurrence of brittle failure due to splitting of the wood raised concerns about the suitability of such connection methods. Several techniques were explored to reduce this phenomenon and therefore increase the ductility of the connection, resulting in better load distribution and possibly an increase in the ultimate capacity. Figure 2.1 shows typical load deflection curves for a variety of commonly used connectors. -i 1— 1 1—i i 1 1— O 2 4 6 8 1 0 12 u(mm) Figure 2.1 Experimental load-slip curves for joints in tension parallel to grain: (a) glued joints (12,5 103 mm2), (b) split-ring (100 mm), (c) double sided toothed-plate (62 mm) (Hirashima, 1990), (d) dowel (14 mm), (f) punched plate (104 mm2), (g) nail (4,4 mm). (Ceccotti, 1995) Other techniques were developed recently to provide moment resisting connections for post and beam construction. Some use several small connectors, such as glulam rivets, while others rely on the greater strength of large ductile fasteners such as glued-in rods (Madsen, 1992). The mortise and tenon technique, 15 Background which was used in traditional construction, has also been considered in combination with other fasteners for some studies (Sadakata, 1989; Foschi and Prion, 1992). Much success was achieved recently in Europe in the construction of high capacity connections using large numbers of small diameter dowels. (Aasheim, 1994). In Canada, large moment resisting connections using glulam rivets were studied. A large quantity of these specialised nails were used in combination with steel plates to create a strong, yet ductile, connection for large members (Madsen, 1992). Unfortunately, this connection uses steel plates attached externally and is not very aesthetically pleasing. In the current study, it was decided to investigate dowel type connections that can be used with hidden steel plates. Larger connectors, such as 12 mm dowels made of soft steel, were chosen since they are stronger than nails, yet not too large to allow bending under large loads. The yielding capacity of the steel in combination with the crushing of the surrounding wood provides good ductility as well as excellent behaviour under cyclic loads. Dowels are well known connectors that have several advantages; they are easily available, do not require skilled labour and form an easthetically pleasing connection when used with hidden steel plates. 2.4.1.1 Experimental testing Much testing has been done, either to define more precisely different aspects of dowel connections (Bodig and Farquar, 1988), or to try out variations using dowel connections in a range of applications (Pellicane, 1991; Rode, 1988). For example, Rodd (1988) explored the possibility of increasing friction around the dowel fastener to avoid splitting of the wood along the grain and improve the embedment strength. Recently, a hollow dowel fastener injected with resin was tested and showed sufficient strength and a high level of ductility (Guan and Rodd, 1996). A number of studies concerned with various factors that affect the performance of dowel type connections will be presented. 16 Background Testing has showed that the moisture content of timber has a significant influence on the strength of dowel connections (Welchert and Hinkle, 1966). In the Canadian design code, the capacity of timber connections under wet conditions is reduced to allow for the loss of strength of the wood and also the possibility of splitting caused by shrinkage during moisture content variations. (CWC, 1994) The group effect is also an important factor to be considered in the analysis of timber connections. For a small group of fasteners, the total strength is approximately equal to the product of the strength of a single fastener with the number of fasteners. Yet for a larger group of fasteners, the capacity of the connection is reduced due to a group effect (Lantos, 1969). The relative rigidity of the wood compared to the fasteners is the main factor influencing the group effect. Under linear elastic conditions, each fastener has to be designed to carry a higher load, since the load distribution in a group of fasteners is very uneven. When yielding of the connectors occurs, it allows for re-distribution of the load throughout all the fasteners, resulting in a more equal load sharing and thus a reduced group effect (CWC, 1994). The rate of loading is another factor that can influence the strength of timber connections. Smith et al. (1988) studied the time dependent deformation of fasteners in order to evaluate an accurate load-slip deformation for short term and long term loading considering temperature and humidity factors. Bodig and Farquhar (1988) determined that nails were more affected by the rate of loading than bolts. The influence of this factor affects the comparability of different studies on connections, as well as the design regulations for diverse types of loading. Komatsu et al. (1990) studied the effect of fire on the resistance of moment-resisting connections. Two types of connections were investigated to connect glulam members: side steel plates with nails, and insert steel plate with drift pins. The nailed connections exhibited poor fire-resistance unless the steel side plates 17 Background were covered with a protective layer, such as a wooden board. The dowel connections with insert-type steel gussets performed much better and did not need protection to satisfy the fire regulations. Corrosion of various configurations of connections were studied. Davis (1994) investigated different factors that influence the corrosion of timber-metal connections and concluded that moisture content is the key factor. The chemical properties of the specific species of wood also influenced the corrosion process. 2.4.1.2 Analytical prediction and design Before design rules can be established, the ultimate load of a joint has to defined. In most cases, empirical values are found based on a large number of tests (Call, 1990). In 1949^  Johansen introduced a theory using equations to predict the ultimate strength of a dowel-type joint when failing in a ductile manner. The European Yield Model based on Johansen's theory is currently used in many timber design codes in several countries (Hilson, 1995; Soltis and Wilkinson, 1991; CWC, 1994; Yasumura, 1988). The model considers several possible modes of failure for a given joint configuration, based on the relative strength between the steel dowels and the wood. The maximum capacity of the joint is based on equilibrium equations derived from the free-body diagram of a bolt in a wood member. Figure 2.2 shows the different failure modes that are assumed in the European Yield Model for a three-member joint (Soltis et al, 1986). This model has been used, verified or revised in several studies since then (McLain et al., 1993; Soltis and Wilkinson, 1991; Wilkinson, 1993; Soltis et al, 1986). Figure 2.2 Failure modes assumed in European Yield Theory for three-member joint: (a) Modela; (b) Mode la; (b) Mode lb; (c) Mode 2; and (d) Mode 4 (Soltis et al, 1986) 18 Background More recently, several researchers have developed more complex analytical models of joints with the intent to provide accurate and reliable simulation methods and avoid frequent and expensive destructive tests. While considering temperature and humidity factors, Smith et al. (1988) investigated the load-slip deformation of short term lateral loading, as well as time dependent deformations. Using Lantos' analysis on group effect, Zahn (1991) developed design equations to maximise the arrangement of fasteners in a given connection and predict the number of fasteners required to achieve a given row capacity. Bodig et al. (1991) used the superposition of non-linear springs to simulate the load-slip behaviour of bolted connections. He compared his predictions to test results on one- or two-bolt connections with loading at different angles to the grain (Pellicane et al., 1991). Smith and Hu (1994) used fracture mechanics theory to predict the brittie fracture modes of bolted timber connections. With the most recent code changes toward reliability-based design, researchers have embarked On studies to determine the safety level of different connections. McLain et al. (1993) proposed new load and resistance factor design criteria based on the European yield theory, whereas Zahn (1992) studied the reliability of present practice in comparison with two new codes and concluded that the new codes maintained the same level of reliability. 2.4.2 Dynamic Performance of Connections In recent years, the importance of seismic performance of structures has resulted in new criteria for the evaluation of connections. In the present seismic design philosophy, the ductility of a structure produces larger deformations for a given associated force, and often leads to greater energy absorption, resulting in lower accelerations. Thus, it is permitted to reduce the design load with the understanding that the structure will undergo large inelastic deformations (NBCC 1995, 4.1.9.1; Moss and Carr, 1988; Buchanan and Fairweather, 1994). In the design of steel structures, where the ductility of connections is 19 Background often considered to be questionable, the members are typically dimensioned to yield before the connections. Beam mechanisms are preferred since such a system would provide sufficient ductility to the structure without creating a mechanism of collapse. For timber structures, however, the failure of wood members in bending or tension is undesirable since it entails brittle behaviour without ductility. To achieve ductile behaviour of the structure, it is essential to include ductile connections that are weaker than the members (Buchanan and Dean, 1988). Since earthquakes produce numerous reversed loading cycles on the structure, it is important to emphasise not only the ductility achieved during the first loading cycle but also the behaviour during subsequent cycles which might include nonlinear deformations. The main parameter governing the static performance of a connection is the maximum load carrying capacity; in seismic design, the entire range of nonlinear behaviour of the connection under different levels of displacement is important and needs to be characterised. The load-displacement curve of a ductile connection under cyclic loading is usually characterised by its hysteresis curve, which typically includes a nonlinear envelope followed by loops with reduced stiffness caused by the wood degradation within the connection. The wood degradation reduces the amount of energy dissipated within the connection, as can be observed in the reduction of the area inside the hysteresis loops. The significance of energy dissipation is still a controversial topic. Chui et al. (1995) mentioned the importance of this parameter since it characterises the damping capability of the structure. Buchanan and Dean (1988) have, however, concluded from recent studies in New Zealand that to resist earthquakes, it is more important for a structure to be able to undergo large displacements without significant loss of strength, rather than provide fat hysteresis loops. Dcam and King (1994) support this conclusion after considering the low damage ratio experienced by timber buildings in past earthquakes. 20 Background 2.4.2.1 Experimental testing In addition to the traditional monotonic static loading test, cyclic tests have been performed on different types of connections to investigate their ductility as well as the effect of damage accumulation. To date, no uniform procedure has been established as a protocol for this type of testing. Some disagreement prevails between researchers regarding the relative importance of strength, large ductility, and energy dissipation in the evaluation of the dynamic performance of timber connections. Chui et al. (1995) attempted to develop a test protocol to characterise the dynamic properties of timber connections. Comparing two draft standards from the USA and Europe, they looked at the displacement sequence, the loading frequency, the energy dissipation, the definition of yield point, and the assessment of connection stiffness. Ceccotti (1995) presented the recommended procedure in Europe for cyclic loading tests. A special meeting was held during the International Wood Engineering Conference (IWEC '96) in New Orleans to discuss a uniform cyclic test procedure that will quantify the behaviour of the elements under real dynamic loading and make comparisons possible between studies across the world. No conclusion has yet been reached regarding the adoption of a uniform protocol. The rate of loading and the cycle sequence are the two parameters that'differ the most between cyclic testing and actual earthquake excitation. In their research on the dynamic properties of timber connections, Chui et al. (1995) looked specifically at the frequency of loading and agreed that it affects the strength and stiffness of timber connections. They observed premature wood failure due to inertial forces from the test machine during high frequency loading tests, but also noticed that "the frequencies at which connections in timber frames oscillate under previous seismic records are generally much lower than the dominant frequencies of those seismic records"(Chui et al., 1995). They concluded that a loading frequency of 0.25 to 0.5 Hz is justified for the testing of connections. Nevertheless, the different researchers involved with cyclic testing have not yet agreed on a standard rate of loading. 21 Background The ability to predict the dynamic behaviour of timber connection using cyclic test results is also controversial. Ceccotti (1995) believes that cyclic tests are sufficient to provide the necessary information to predict the behaviour of a structure in a real earthquake. Yet, dynamic tests need to be performed to demonstrate the influence of the connection nonlinear response on the behaviour of a structure. Studies about pseudo-dynamic or dynamic tests on entire structures will be presented later. The effect of fatigue is usually not an issue during cyclic tests. Abendroth and Wipf (1989) studied the fatigue of bolted connection and found that a rninimum of 2,000,000 cycles was required to create fatigue failures in the bolts. They did not specify the load level reached during these tests. Nevertheless, since the length and the rate of occurrence of earthquakes will produce a much lower number of cycles, fatigue does not seem to be of concern here. The fatigue life of a connector is very much dependent on the stress levels and low cycle fatigue failures can occur when the yield level is exceeded for a large number of cycles (He, 1997). Various types of connections have been tested under cyclic loading. In addition to traditional connections, new types of connections that were designed to meet specific needs for dynamic performance have also been tested, often under proprietary protection. Shear plates, split rings and Bulldog® connectors have very high load carrying capacities but are usually not recommended for seismic design since they lead to a brittle failure caused by shearing of the wood (CWC, 1991). Nevertheless, recent cyclic tests conducted in Japan showed that a reasonable ductility of such connections can be achieved before the final brittle failure of the wood (Ohashi and Sakamoto, 1989; Sakamoto et al., 1992; Ohashi et al., 1994). 22 Background Connections with tension bolts, which are attached to the top and bottom of the beam and go through the column, have been tested under different arrangements in Japan (Sadakata, 1989; Ohashi et al., 1994). In a study by Sadakata (1989), the tension bolt connections exhibited high ductility compared with nailed connections, and prevented the generation of cracks in the column due to the clamping effect. Testing on connections consisting of a mortise and tenon joint held together with tension bolts (Foschi and Prion, 1992) showed good resistance and ductility of the connection for the first cycles. The response to consecutive cycles was unsatisfactory since the slackness caused by the yielding of the bolts resulted in extremely pinched hysteresis loops with almost zero resistance through the central position. These connection systems are not covered in the Canadian timber code (CWC, 1994). Due to the ease of construction, nailed connections are very popular and have been thoroughly studied in the past decades. Moment resisting connections using nails were tested by Komatsu (1989) and Ohashi et al. (1994) in Japan, Ceccotti et al. (1994), in Italy, and Buchanan and Dean (1988) in New Zealand. These researchers arrived at similar conclusions: moment resisting connections with steel plates and nails provide large ductility under cyclic loading. The hysteresis curve to reversed cycles, however, shows a very pinched behaviour caused by the softness of the nails and the gap formed by the crushing of the wood. Due to the large surface area required to accommodate the large number of nails needed, this type of connection is only suitable for deep, slender glulam members. Buchanan and Dean (1988) also recommended a minimum nail length to avoid premature withdrawal. Glulam rivets have also been tested under cyclic testing (Cheng, 1996) and performed similarly, with pinched hysteresis loops leading to a stable and ductile failure. 23 Background Several projects have involved cyclic testing of dowel and bolt connections (Ceccotti and Vignoli, 1988; Prion and Foschi, 1994; Komatsu, 1989). Connections with dowels smaller than 18mm in diameter showed good hysteretic behaviour. As mentioned by Buchanan and Dean (1988), small bolts and dowels provide good ductility that allows sharing of the load between a number of fasteners, whereas the rigidity of large bolts often leads to uneven load distribution and brittle failure of the wood. To facilitate on-site assembly, Komatsu et al. (1991) from Japan proposed a dowel connection with separate steel plates that are bolted together between the beam and the column. A good hysteretic behaviour of the connection was found. Buchanan and Fairweather (1994) presented various types of unique connections using epoxied rods, steel brackets, dowels, nails or a mix of these fasteners. They stressed the importance of including steel components that can provide a sufficient ductility through yielding. Testing of various arrangements of glued-in steel rods was also conducted by Madsen (1992), and showed very high load carrying capacity and sufficient ductility, provided yielding of the steel components was permitted. 2.4.2.2 Analytical simulation Since this study is directed towards dowel type connections, models for other fastener types will not be discussed here. For dowel type connectors with ductile behaviour, the prediction of the monotonic strength can be approximated using the European Yield Model, as described previously. The prediction of the overall hysteretic behaviour under cyclic loading is another challenge, however. A good simulation of the hysteretic behaviour of nonlinear connections is important to enable the prediction of the response of an entire structure under seismic loading. The first attempts were oriented towards models that could be calibrated empirically to test data and reproduce the hyteresis loop of the 24 Background tested connection when incorporated in a structural analysis program. In these models, the behaviour of individual connectors within the connection is usually not considered. Moss and Carr (1988) used a model with hysteresis loops that they had developed in 1981, but no information was provided regarding the parameters used in the simulation. Ceccotti and Vignoli (1989) developed a computer subroutine to reproduce the moment-rotation hysteresis curve of a tested connection with a four-slope skeleton. This model considers the degrading stiffness and possible slippage in the joint. Their subroutine has been incorporated into the nonlinear analysis program DRAIN-2D. Since 1989, the subroutine has been updated to a six-slope skeleton and more options in the model behaviour have been added. This model will be discussed in further detail in Chapter 5. Another computer program was developed by Deam and King (1994); they extended a model proposed by Dean (1994) to simulate the hysteretic behaviour using a rigid bar with a number of bilinear elasto-plastic springs attached to it. The computer program PhylMas was used to generate the parameter values and quickly adjust the model to the test specimen hysteresis loops. The Bouc-Wen-Baber-Noori Model was used by Foliente (1995) to mathematically simulate the desired hysteresis curve. This model, characterised by a single mathematical form, can reproduce the nonlinear, inelastic behaviour, stiffness degradation, strength degradation, and pinching. Komatsu et al. (1988) have used a finite element model of the connector as an elasto-plastic beam on a nonlinear foundation based on equations by Tsujino and Hirai(1983) and a concept by Foschi (1974). Various models using different fitting parameters are available, yet they always need to be fitted to hysteresis loops found through cyclic testing. Another approach might consider the simulation of the different components of a connection in order to provide a simulation for the behaviour of a connection using basic material properties. As opposed to previous models, this simulation does not require 25 Background expensive testing for every new configuration of a connection. Once the basic parameters are calibrated, this model could simulate the behaviour of any configuration of a known connection using defined connectors and wood parameters. Foschi (1974) developed a model which was initially used to predict the behaviour of nails in a wood foundation. It uses the analogy of an elastic-plastic beam supported by a nonlinear foundation to determine the behaviour of a connector. 2.5 Analysis of Frames under Dynamic Loading As discussed earlier, extensive monotonic and cyclic testing was done on several types of connections: Recently, new technologies such as shake tables allowed the simulation of actual earthquake records for the testing of a variety of structures, perrmtting the deterrnination of the real effect of the different connection parameters on the structural behaviour. 2.5.1 Experimental Testing Testing has been done to determine the seismic behaviour of timber structures, using cyclic tests, pseudo-dynamic tests, and shake table tests. Due to the different types of structures, as well as variability in the testing techniques and problems regarding an uniform testing procedure, comparison between different studies is difficult. In this review, only testing on moment resisting frames will be studied. Even if some of these structures are smaller than reality, the tests are considered equivalent to full size since scaling of the wood fibres is not possible. Ceccotti et al. (1994) tested full-scale timber frames with semi-rigid connections using a pseudo-dynamic test method which combines numerical and experimental analysis. This testing technique can be used to simulate the seismic behaviour of a structure similar to a shaking table test, with the only exception being the time scale. Ceccotti and Vignoli (1988) also performed free vibration dynamic tests on a structure composed of 16 glue-laminated three-hinged arches, with a span of 40 m and spaced 5 m apart. From this 26 Background study, they concluded that free vibration testing provides useful information, while limiting the testing cost to a reasonable level. They recommend, however, that further research is needed using this testing method for timber structures in order to determine the method's accuracy. Touliatis (1991) mentioned a shake table test on a full scale two-storey timber frame but did not specify any results. In Yugoslavia, Gavrilovic and Gramaticov (1990) performed shake table testing on a full scale one-storey wooden truss frame structure which demonstrated the good dynamic behaviour of such a system. Hirashima (1988) studied the structural behaviour of an in-situ two-storey post and beam wood construction, representing a traditional house in Japan. The response of the building was monitored for a period of 3 months during which time 21 actual earthquakes occured. Following this, static testing as well as forced vibration tests with a vibration generator were used to assess the behaviour of the structure using common laboratory techniques. The first natural frequency of the structure was assessed using the data from the actual earthquakes and compared to the response from forced vibration tests; similar results where found. Not many shake table tests have been done on timber frames. This relatively new testing method allows the simulation of recorded or artificial earthquakes, and it provides information on the actual dynamic behaviour of a structural model. Some aspects of this testing technique need careful monitoring, however. Problems are often encountered associated with scaling of the structure, the choice of earthquake record that will represent the most realistic design earthquake in a given area, the scaling of this earthquake record, and the monitoring system used to determine the response of the frame. It is important to recognise that the response of the frame is not just a function of the amplitude of the input force, but also of the main frequencies of the excitation in relation to the natural frequencies of the structure. Other details related to the input of the earthquake record to the shake table actual movement also need to be considered (Blondet etal., 1988). 27 Background 2.5.2 Analytical Prediction Due to the nonlinearity resulting from the ductile connections, the accuracy of the dynamic analysis of a timber structure depends largely on the capacity to simulate the connection behaviour and its incorporation into the structural analysis program. As discussed previously, several models have been used to simulate the particular hysteretic behaviour of various types of connections. Komatsu et al. (1988) proposed a program that simulates the nonlinear behaviour of glulam frames by updating the stiffness matrix of the structure at each iteration. To determine the reaction in the connection, the program uses a finite element model of the connector, simulating an elasto-plastic beam on a nonlinear foundation, which uses the equations developed by Tsujino and Hirai (1983) and a concept by Foschi (1974). He concluded that this behaviour can be predicted using the MoEs of glulam and the arrangement of the connections. Kikuchi (1991) proposed an elastic model and an elasto-plastic model to simulate the behaviour of semi-rigid joints in the analysis of a timber frame. He used the analytical response of a typical frame to review code regulations. Peng and I wan (1992) proposed an analytical method to simulate a specific class of structures, but their hysteretic model did not consider the degradation observed in timber connections. Kataoka (1991) analysed ancient Japanese monuments using a numerical simulation, which replaced the joints with nonlinear springs. As mentioned earlier, Ceccotti and Vignoli (1989) developed a subroutine to simulate the moment-rotation hysteresis curve of a tested connection to be used with the nonlinear analysis program DRALN-2DX. Several simulations have shown the capability of this package to simulate the seismic behaviour of structures (Ceccotti and Vignoli, 1989; Ceccotti et al., 1994; Ceccotti and Vignoli, 1991; Ceccotti and Vignoli, 1988; Blass, 1991). Sakamoto et al. (1992) proposed a seismic response analysis using rotational springs to simulate a structure with shear plate connections. The hyteresis curve of the connection, defined through testing, was reproduced considering nonlinearity and degradation. Deam and King (1994) further developed a model proposed by Dean (1994) 28 Background to simulate the hysteretic behaviour using a rigid bar with a number of bi-linear elasto-plastic springs attached to it. The structural analysis program performed a time-history analysis and generated displacement response spectra. A good review of hysteresis modelling was done by Foliente (1995). In his own analysis, he used the Bouc-Wen-Baber-Noori Model to mathematically simulate the desired hysteretic behaviour. The hysteresis curves had to be calibrated to curves from tests on connections and were not derived from basic material properties. In summary, several models have been developed to reproduce the hysteretic behaviour of tested nonlinear wood connections. Yet, their degree of sophistication and level of integration into user-friendly structural analysis programs vary. None of the models, however, simulates the connection behaviour using basic material properties. In this thesis, the DRAIN-2DX program using the Florence subroutine is used for its ease of utilisation and its capability to reproduce a six-slope hyteresis curve, simulating the degradation of the connection (Ceccotti and Vignoli, 1991). Moreover, the framework for the development of a dynamic analysis program was established, including the model by Foschi (1995), which predicted the ductile mechanism of each fastener within each connection. 2.6 Concluding Remarks Timber is a structural material that possess several advantages, such as its high strength to weight ratio, simple methods of construction, heat resistance, environmental benefits, and its architectural beauty. Yet wood is a limited resource with orthotropic properties and a high variability due to natural defects. The development of engineered wood products, such as parallel strand lumber, permit the maximisation of the available material and a reduction of the variability in the final product. Only limited information from experimental studies on connections using these new composite wood products was available to determine the influence of their higher performance. 29 Background In seismic zones, the capability of structures to resist lateral dynamic loading is very important. Experience from previous earthquakes has demonstrated the need for reliable lateral load resisting systems, as well as the importance of good connections, to obtain good performance in timber structures. The current design codes recommend a very simple quasi-static method to determine the maximum base shear that the structure has to resist. The determination of the various parameters used in this method however are based on an extensive background of experimental studies and analytical simulations for various types of structures. The ductility capacity of the lateral resisting system is one of the main topics of investigation since it can substantially reduce the forces in the structure. Due to the brittle nature of timber failures, the ductile capacity of a wood structure is developed through its connections. Due to their importance in timber design, many studies have been done on connections. At first, different fasteners were tested to determine their load carrying capacity when used with various wood species. Different factors affecting the strength as well as the behaviour of the entire connection were also assessed. Research showed the effect of moisture content on the strength of connections. Other tests established that dowel type connections with hidden steel plates provide good fire resistance. Studies regarding group effect showed that the strength reduction is inversely proportional to the degree of ductility of the connectors within a connection, moreover, a good ductility in combination with required end distance reduces the risk of splitting at the end of connection. Numerical models have been developed to evaluate the strength of fasteners as well as the monotonic performance of entire connections. The European Yield Model is often used to describe the ductile behaviour of dowel type connections. Ductility and good response to several reversed cycles are more important than ultimate strength to evaluate the seismic performance of connections. Various configurations of connections which use 30 Background traditional fasteners, as well as new types of connections such as glued-in rods and epoxied bolts, were developed to respond to these new criteria. There are, however, topics regarding the evaluation of their performance that are still controversial. No international standards have been established to determine the dynamic parameters. Also, the requirements for rounded hysteresis loops and optimum energy dissipation are not consistent. Even if the test procedures are not uniform, different cyclic tests were performed on dowel type connections using small and medium size fasteners thus demonstrating their ductility as well as their capacity to undertake several cycles without failure. In dowel connections, the specific interaction between the yielding of the connector and the crushing of the surrounding wood creates pinched hyteresis loops in the force-displacement curve. The analytical simulation of the dynamic response of frames using these connections are made more complex by their highly nonlinear behaviour. Different models have been formulated to reproduce analytically the moment-rotation curve of nonlinear connections, and enable a time history analysis of the frames. These models include fitting parameters that can be adjusted to hysteresis curves found through cyclic testing of a specific connection. Few methods have been tried which use basic parameters to allow the simulation of these curves for different configurations of a connection without needing to perform individual tests. One objective of this study is to enable such a simulation using the analytical model developed by Foschi (1995) analysing the behaviour of individual dowels within the connection. New testing methods, such as dynamic tests on shake tables, have recently been developed to enable a better understanding of the dynamic response of different structures. Few tests, however, have been done on timber moment resisting frame. This study was undertaken to assess the real influence of the ductile behaviour of connections on the response of a two-storey moment resisting frame using shake table tests. The results from these tests also serve to verify an analytical model for timber structures which simulates the moment resisting connections' behaviour from basic material properties. 31 Experimental Study Chapter 3 EXPERIMENTAL TESTING The experimental part of this study included monotonic tests on the steel dowels and on the parallel strand veneer lumber, cyclic testing on different connections and shake table simulations of a two-storey frame built with the dowel connections. These tests served in the calibration and the verification of different analytical models as described in chapters 4 and 5. The shake table test was the main component of this research to investigate the effect of the nonlinear behaviour of the connections on the frame dynamic response. The responses of the connections to cyclic and dynamic testing were also compared. 3.1 Tests on Connections A test program was undertaken to determine the connection characteristics for the calibration of analytical models. Firstly, tests were conducted to obtain basic material properties needed for the calibration of the finite element model. Secondly, entire connections with one, two, four and eight dowels were tested under cyclic loading, providing data to verify the model. 3.1.1 Basic Material Properties Experimental testing was performed to deterrnine the stress-strain curve of the steel dowel material, as well as the embedment characteristics of Parallam® for different orientations relative to the wood grain. These tests were performed by engineering students enrolled in the course CIVL 321 - "Laboratory Project in Engineering Materials" as part of a term paper (78). Testing was done using Parallam® members and 12.7 mm diameter mild steel dowels. 32 Experimental Study Tensile tests were conducted on 12.7 mm diameter steel dowels to determine the average yield strength and the ultimate strength. In addition, three point bending tests were also performed. A typical stress-strain curve is shown in Figure 3.1. To determine the crushing strength of Parallam® in a connection, embedment tests were performed parallel and perpendicular to grain, as well as parallel and perpendicular to the wood strands. These tests were done by pushing a round bar into a half-round groove in the wood. The groove was made by drilling a hole in a block of wood and then cutting it in half. The test results were not very satisfactory, however, since the action of the wood surrounding the dowel was not considered. The tests on single pin connections as described below provided more realistic results. 3.1.2 Single Pin Connection To obtain realistic embedment characteristics of a dowel connection, single pin connection tests were performed under cyclic loading, applied parallel to the wood grain (Figure 3.2a,b). The pin was oriented either parallel or perpendicular to the strands. These tests provided more satisfactory information on the embedment characteristics and these results were used to determine the wood parameters in the calibration of the analytical model through a best fit approach. The load-displacement curve of the connection exhibited the typical pinched hysteresis behaviour that is associated with wood connections (Figure 3.2c). This is due to the yielding of the steel combined with unrecoverable crushing of the surrounding wood. The final failure occurred when the steel pin sheared off after extensive yielding. No wood splitting was observed. 3.1.3 Two and Four Pin Connections To determine the group effect of connectors in a moment connection, two and four pin connections between a wood member and a steel plate were tested (Figure 3.3). Similar to the single pin connection, 33 Experimental Study the moment-rotation curves showed pinched hysteretic responses with a ductile failure mode when some of the dowels sheared off. A secondary goal of these tests was to amfirm whether the end and edge distances were sufficient to assure ductile behaviour. Results from these tests were also used to verify the analytical model of the connection. 3.1.4 Eight Pin Double Connections As a verification of the connection for the frame to be tested on the shake table, and to determine the amount of load sharing, a double connection was tested (Figure 3.4). The moment-rotation curve of this eight pin connection was found through cyclic testing conducted by Latendresse (1994) (Figure 5.1). The hysteretic characteristics of the connection were also used in the analytical simulation of the response of the frame, as discussed in section 5.4. 3.2 Shake Table Test • Experimental Procedure The main experimental part of this research was a shake table test on a two-storey moment resisting timber frame structure (Figure 3.5). The major objective of this test was to observe the response of the frame to earthquake excitation and to verify analytical modelling procedures. 3.2.1 Description of the Test-Frame 3.2.1.1 Materials The test frame was made of 177 x 177 mm parallel strand lumber (Parallam ®) column members and 300 x 177 mm Parallam® beam members, connected with 12 mm steel plates and 12.7 mm mild steel dowels. More information regarding Parallam® is available in section 2.2. One reason for using Parallam® is that this relatively new product has rarely been used in experimental studies before. More research considering this product in different types of applications would help to define its behaviour with more accuracy. 34 Experimental Study Most importantly, however, the consistency of the material was an overriding consideration for its use, especially since only one frame could be tested on the shake table. Hot rolled mild steel rods with well defined elastic and plastic load deformation behaviour were used to fabricate the dowels for the connections. No special requirements were considered for the connector plates, and grade 300W steel was used with properties similar to the dowel material. 3.2.1.2 Design of the Frame The frame was designed to serve the objective of two different projects: this study related to the behaviour of the timber frame and a project to study the influence of specimen nonlinearity on the performance of the shake table. The specimen used for the shake table test was a two-storey structure consisting of two planar moment-resisting frames braced together in the out-of-plane direction with steel angles. The frame had a height of 2.7 m (8'-10") and a width of 2.6 m (8'-6"). The connections between beams and columns were fabricated from steel plates, hidden inside the members, and connected to the wood by eight tightly fitting 12.7 mm diameter mild steel dowels (Figure 3.4). These connections were similar to the ones tested previously (section 3.1.4). The size of the dowels had to be selected to maximise the load carrying capacity while ensuring ductile behaviour created by yielding of the steel as well as crushing of the wood (Figure 3.6). A non-ductile dowel that would remain elastic while crushing the wood could lead to a brittle failure. Even if no splitting of the wood occurred, crushing of the wood without yielding of the steel would create a severely pinched hysteresis behaviour of the connection with unacceptable slackness after several cycles. The thickness of the steel plate used within the connection was chosen to provide adequate stiffness and minimal 35 Experimental Study deformation around the dowel holes: The Parallam® members were overdesigned to ensure sufficient stiffness and strength. Each storey of the structure supported a concrete mass of 1.7 ton (3400 lb.) to simulate the mass of the structure. The size and the weight of the frame had to conform to the limitations of the shaking table, which are a maximum footprint of 3 m x 3 m, a maximum height clearance of 6.S m and a maximum payload capacity of 14 000 kg. Cross braces made of 6 mm steel rod were used during the tuning of the shake table, which was done at low amplitude level and served to adjust the input motions for the table (Figure 3.7). A lateral bracing system fabricated with 100 mm x 100 mm steel angles was used to connect the two wooden frames in the out-of-plane direction. The results from a linear dynamic analysis were used to design the size of the connections and the members, as well as the applied masses for the shake table test specimen. SAP90 analyses with and without the temporary cross braces were done to estimate the initial natural frequency of the frame and to obtain an approximate maximum response for various earthquake records. The connections were simulated with flexible elements with a constant stiffness equal to the initial slope from the response of the eight pin double connection. For the unbraced frame, the response was not expected to be accurate since the nonlinearity of the system was not considered. Nevertheless, this information was very useful to estimate the behaviour of the frame and to ensure that the level of excitation provided by the shake table would be sufficient to achieve the desired ductile response. Latendresse presented the results of this linear analysis in an internal report (Latendresse, 1994). A comparison of the applied mass to realistic building design is made in section 3.3.1.4. 36 Experimental Study 3.2.1.3 Construction of the Frame The frame was built by Vincent Latendresse during the summer of 1994. The precision of the connections was the most important part of the construction, and went as follows. Slots were cut out of the end of the wood members with a bandsaw to insert the steel plates. Each frame was pre-assembled horizontally on the floor using small steel brackets. A temporary steel plate template with eight 7/16" holes (slightly smaller than the dowels) was attached to the side of the wood members at the connection locations using wood screws. A magnetic base drill was attached to the template to drill the 7/16" holes in the wood member. The connection steel plates were put in place and the position of the holes were punched individually for each connection. The steel plates were then removed and eight 1/2" holes were drilled in each plate. Finally, the steel plates were inserted into the slots and the 1/2" dowels were hammered through the undersized holes in the wood and through the holes in the steel. The whole operation required very precise measurement to provide a tightly fitting connection. The two Parallam® frames were then erected and braced together with steel angles. The concrete masses, which were available from previous tests on the shake table, were attached to the beams using specially adapted steel brackets. To avoid slippage of the masses, the steel brackets were epoxied into grooves on the top of the beam and 1/2" lag screws were used to restrain the brackets from vertical movement (Figure 3.8). The frame was connected to the table with pinned supports, which were also available from a previous test. The upper half of the hinge was epoxied into the end of the column, and the lower half was bolted to the top of the shake table. 3.2.2 Instrumentation Extensive instrumentation was installed to monitor the dynamic behaviour of the frame, as well as the relative rotation within the connections. Two video cameras were used to capture the response of the 37 Experimental Study frame under earthquake loading. The first camera recorded the motion of a middle connection, while the second camera recorded the response of the entire frame at an angle of about 45°. 3.2.2.1 Data Acquisition System The information monitored during the shake table tests were recorded by two independent data acquisition systems. The first system, Lab view, contained all the recorded information from the accelerometers and the displacement transducers, as well as the base force applied to the shake table. The secondary system, Global lab, formed part of the control system and duplicated some of the information, such as the table acceleration and displacement, and the top floor acceleration. During the monitoring of the shake table tests, the number of filtered channels available to record the data limited the number of recording devices. There were two sets of data collection banks, each of which had 16 filtered channels. A different type of filter was connected to each of these two sets. To reduce differences between signals, one set was used for the connection displacements and the displacement of the frame, and the other for accelerometers. Following each test, all the recorded channel outputs were viewed on a monitor to verify the data. This quick verification was essential to detect recording problems and aided in changing input motions to suit the objectives of the test. 3.2.2.2 Monitoring of the Structure Strong motion accelerometers were placed at each level in the north-south, east-west and vertical directions to characterise the overall response of the frame (Figure 3.9). Although vertical, transverse and torsional motions were not expected, it was necessary to verify these assumptions. The accelerometers used had a range of linearity of ±5g and a precision of 0.1%. Displacement transducers calibrated for a 38 Experimental Study range of ±100 mm with an accuracy of 0.1% were monitoring the longitudinal absolute displacement at each level. -3.2.2.3 Monitoring of the Connections The response of the connections was a very important part of this research. To help in the analytical modelling phase, the relative displacements of the different elements within a connection had to be monitored. Since the, steel plate could be considered to be near-rigid throughout the test, it was possible to derive the position of the dowels from the relative displacements between the steel plate and the two wooden members. The relative rotation between the plate and a member was measured by 2 Linear Variable Differential Transducers (LVDTs) fixed to the member and reading the displacement of the plate at two points (Figure 3.10). Three LVDTs were needed to monitor the relative displacement between the beam and the column at a joint. The measuring system had to be stiff enough to give reliable information regarding the vertical, horizontal and rotational motions without inducing errors due its own vibration. Moreover, the instrumentation had to be fixed to the member outside the connection area to avoid disturbance due to splitting of the wood. The final arrangement consisted of three LVDTs mounted on telescopic aluminium tubes. Each end of the device was attached to a member. This arrangement formed two triangles that could be solved to determine the three relative motions between the beam and column (Figure 3.10). A total of seven LVDTs were needed to monitor the overall behaviour of a given connection. Since two connections were monitored, one at each level, all the available instruments were used. Due to different 39 Experimental Study instrument ranges and sensitivities, some measurements were more precise than others. It was decided to concentrate the better measurement system on the middle connection to be assured of satisfactory results from at least one connection. For the middle connection, the best four 6mm LVDTs were used to measure the beam-plate and plate-column rotations. In the triangular system, the horizontal displacement was measured by a 6 mm LVDT, and the angled measurements were done using 12 mm LVDTs. For the top connection, the remaining four 6 mm LVDTs were used for beam-plate and plate-column displacement measurements, and a 25 mm LVDT and two 100 mm potentiometers were used for the beam-column relative displacements. 3.2.2.4 Procedure to Install the Instruments To ensure precise measurements, metal supports were used to hold the LVDTs in place. For the member-plate measurement, the LVDT holders were attached to a piece of wood bolted between two steel angles attached to the member (Figure 3.11). To nuhimise friction interference, thin pieces of plastic were glued to the connection steel plates at the contact point with the LVDTs. The triangular measuring system, between beam and column, was also attached to the members using steel angles and wood screws (Figure 3.11). The calibration of each LVDT served to define its linear range and calibration factor that was used to transform the measured voltage to millimetres of displacement. Once calibrated, the LVDTs were installed in their supports and adjusted close to their zero position to permit the use of the whole range of linearity in both directions of displacement. A verification of the calibration was done once the LVDTs were connected to the data acquisition system. Accelerometers were also calibrated before installation. The calibration factors and the positive directions were verified by measuring the voltages produced by the gravitational acceleration when the accelerometers were turned vertically. 40 Experimental Study 3.2.3 Loading Procedure For the earthquake simulations on the shake table, displacements were controlled by a real time computer control system to reflect the acceleration input data from prescribed earthquake records. Only motions in the longitudinal direction of the frame were applied to provide excitation to the wooden frame in the principal direction of stiffness of the dowel connections. 3.2.3.1 Choice of Excitation Record The shake table input acceleration record, which was chosen from amongst existing earthquake records, had to provide a sufficient excitation to produce a highly nonlinear response of the dowel connections. To achieve this, it was necessary to select an acceleration record with a main frequency of excitation close to the first longitudinal natural frequency of the frame, as well as maximal length and amplitude of the excitation. An elastic dynamic analysis of the test frame was performed with SAP90 by Latendresse (1994), using the initial stiffness of the tested connection to model the dowel joint. The first natural frequency of the frame was found to be 1.4 Hz. With this in mind, two well known earthquake records were selected for further investigation, namely Imperial Valley at El Centro station and Landers at Joshua Tree station (Figure 3.12). In the linear analysis simulating the elastic response of the frame for these two earthquake records, the moments in the connections reached a maximum of 5.4 kNm for the El Centro record and 11 kNm for the Joshua Tree record. According to cyclic test results, these two levels of excitation would be sufficient to create non linear behaviour in the connections. These moments were the results of a linear response of the frame and would be reduced by the nonlinear behaviour of the frame. Later in the project, a nonlinear time-step analysis using DRALN-2DX, performed by the author, helped to determine with more precision the nonlinear response of the frame to different excitations. As for the linear analysis, the Joshua Tree 41 Experimental Study record produced higher moments and larger displacements in the connections than the El Centro record (Figure 3.13). The DRAIN-2DX analysis also showed several nonlinear cycles for this earthquake due to its unusual length, as well as its appropriate frequency content. The record of the Landers Earthquake from the Joshua Tree station (California) in the East-West direction was chosen for all the test runs. To maximise the response of the frame, the earthquake record was factored to the maximum displacement of 65 mm allowed by the shake table. The acceleration record was multiplied by a factor of 1.5 and filtered to eliminate the very low frequencies. The factored record produced a maximum acceleration of the table of 0.5g. Even though the storey heights of the tested frame were half that of regular buildings, this test was considered to be equivalent to a full scale test since the connections were full size considering the wood fibres, the Parallam8 strands, the dowels and the plate sizes. For this reason, no scaling of the earthquake record was considered. A further increase of the nonlinear response could have been achieved by increasing the concrete mass at each storey, but it was judged unnecessary. 3.2.3,2 Proposed Testing Sequence To allow the comparison between the response of the structure and analytical results, the high amplitude excitation was planned to be performed on an undamaged structure. The tuning of the program controlling the motion of shake table, however, required excitation at low intensity to determine the interaction between the table and the structure. A DRAIN-2DX analysis showed no damage in the connections for a level of excitation corresponding to 30% of the full excitation, and shake table simulations at 5% and 30% of the intensity were planned to precede the 100% excitation. It was understood that a small intensity excitation of the frame would also provide information regarding the behaviour of the structure in its linear 42 Experimental Study range. It was also planned that sine sweep tests would follow the high amplitude destructive test, completing the study on the performance of the shake table. 3.2.3.3 Actual Testing Sequence Due to problems encountered with the data acquisition, the actual tests differed from the proposed sequence. The structure was fully installed on the shake table and all the instrumentation was working on March 18th, 1996. A "Chirp" test that input a very low intensity signal with a broad frequency content was performed as a first step in the tuning of the shake table. On March 19th, the first excitation at 5% amplitude was performed without any conclusive results since the amplitude of excitation was too low. The structure was then subjected to the 30% excitation. Unfortunately, only the second half of the response was recorded due to human error. Comparison of the results from ambient vibration tests done before and after this simulation confirmed that the structure was still reasonably undamaged. On March 20th, the structure was subjected to another 30% excitation to verify that all systems were set correctly. During the following test at full amplitude, a memory overload of the data acquisition computer occurred. The only data acquired during this test was the acceleration of the table and the acceleration of the top of the structure which was recorded by the Global Lab data acquisition system. The chance to monitor the nonlinear dynamic behaviour of the frame with undamaged connections was therefore lost. A second test at full amplitude was then done to collect data regarding the behaviour of the connections even though these had already been damaged. The results from this test were also incomplete: the accelerations at the different levels were available, but no displacements were recorded since the power supplies were turned off. On March 21st, another test at 30% intensity served to determine the reaction of the damaged frame to a low intensity excitation and also to verify that all systems were functioning correctly. A third full intensity excitation was then performed on the structure and all the information was recorded. In the following days, other 100% excitations were monitored. 43 Experimental Study Test Excitation Maximum Acceleration Recorded Data Date Chirp Test no data March 18 0 1st - 5% •02g too low amplitude March 19 1 1st - 30% • 15g all data - second half of the EQ March 19 2 2nd - 30% • 15g all data March 20 3 1st - 100% •5g Long, accelerations of table and top floor March 20 4 2nd - 100% •5g Only accelerations March 20 5 3rd - 30% • 15g All data March 21 6 3rd - 100% •58 All data March 21 7 4th - 100% .58 All data March 22 Table 3.1 - Summary of Shake Table Tests 3.2.4 Vibration Testing and Modal Analysis Vibration tests were performed to detect the natural frequencies of the frame. These frequencies served to verify the analytical model of the undamaged frame. The variation in fundamental natural frequency of the frame also gave some indication of damage suffered by the structure. 3.2.4.1 Ambient Vibration Testing Ambient vibration testing provided useful information regarding the natural frequency of the frame. This method of testing utilises accelerometers to monitor the response of the structure to the very low intensity signals caused by the vibrations from its environment. The frequency response function, which considers the Discrete Fourier Transform of the input and output signals for a given structure, is a function relating the given input to the given response in the frequency domain. This function is a characteristic of the structure and its analysis using an input signal with a full frequency bandwidth, such as ambient vibrations, is used to deterrriine the natural frequencies of the given structure (Felber, 1993). The first natural frequency was considered the most important for the frame studied since the flexibility of the connections and relatively stiff members create a response of the frame following mostly its first mode of vibration. 44 Experimental Study The setting of the accelerometers for the ambient vibration test is shown in Figure 3.14. They were placed to capture the motion at each storey in the three principal orthogonal directions: longitudinal, transverse and torsional. The longimdinal motion at the shake table level was used as input signal, while the motions of the frame at each level were consecutively used as output signal. Ambient vibration tests were performed on the structure before the first simulation, and between simulations to allow the assessment of the damage in the connection which would be reflected by a change of the fundamental natural frequency. 3.2.4.2 Hammer Blow Testing Hammer blow testing was performed after the first ambient vibration test to verify its results. For this test, a hammer blow was applied to the structure in the longimdinal direction at mid4evel. The force induced to the structure was much larger than the ambient vibrations input and it could be measured directly using an instrumented hammer. The response of the structure was also analysed in the frequency domain to determine its natural frequencies (Villemure, 1995). The results from this particular test were not as clear as for the ambient vibration test, and it was decided that only ambient vibration would be used for subsequent tests. 3.2.4.3 Shake Table Tests A complete modal analysis of the shake table test data was not useful since the nonlinear behaviour of the frame changed the stiffness and natural frequencies of the structure during the tests. It is, however, possible to use a window technique to find the natural frequency of the structure for a small lapse of time during the excitation. 45 Experimental Study 3.3 Shake Table Test - Experimental Results During the data analysis from the shake table test, emphasis was put on the connection behaviour and its influence on the in-plane longitudinal response of the frame. The recorded data was analysed with a Grapher worksheet since the data files were too large for regular spreadsheets. All the data for a given simulation were included in a single worksheet of 30 columns and 17 000 rows. Different operations were needed to transform the recorded voltages to displacement or acceleration measurements. Firstly, the first hundred data points of each Column were averaged to deterrnine the initial voltage associated with zero displacement of a given instrument. This quantity was then subtracted from the recorded voltages for the entire column. The level of noise for each measuring device was detenriined through the extreme values of these first data points. Secondly, each column was multiplied by its corresponding calibration factor determined during the calibration of the mstrument. The initial averaged values were recalculated for each test assurning that no important residual displacement needed to be considered. This assumption was reasonable for the response of the entire frame. The connection measurements showed a larger difference between the initial voltages of different tests, yet it was impossible to differentiate changes due to instrumentation errors versus real residual displacements in the connections. 3.3.1 Response of the Frame 3.3.1.1 Out-of-Plane andVerticalResponse Out-of-plane and vertical motions were not expected during the shake table tests, yet accelerometers were used in these directions to verify this assumption. The out-of-plane accelerations of the frame could not be differentiated from the noise recorded by the instruments during all the 0.15g simulations, and were very 46 Experimental Study small compared to the longimdinal acceleration during the 0.5g simulations (in the order of 10%). For all the tests, no vertical acceleration was significant enough to be recorded. 3.3.1.2 Integration of Acceleration Records As mentioned before, due to unforeseen circumstances, the relative and absolute displacements of the structure were not recorded during the first two 0.5g simulations. To enable the comparison between the response of the tested frame and the analytical results, the relative longitudinal displacement at each level of the structure therefore had to be calculated. Since the only data available were the absolute acceleration at these levels, a Mathcad worksheet was used to approximate the required displacements by integrating the acceleration records. This calculation included (i) appending zero value data to the beginning and end of the acceleration record, (ii) using a high pass filter to eliminate large displacements that could be created by low frequencies, and (iii) integrating the record twice to produce an absolute displacement record. This integrated displacement was then compared with the displacements recorded during the 3rd 0.5g simulation to verify its coherence. To calculate the relative displacement at each floor, the table displacement had to be subtracted from the absolute displacement at each level. The addition of zeros to the beginning of the acceleration record, previous to the integration, changed the alignment between the two records. To enable the subtraction, the two records had to be superposed and visually aligned. The approximation during this visual superposition caused additional errors on the relative displacement records of the floors. Figure 3.15 shows a comparison between integrated and measured results for the time histories of absolute and relative displacements during the third test at 0.15g. 47 Experimental Study 3.3.1.3 Longitudinal Response Accelerations and absolute displacements were monitored extensively in the longitudinal direction since it was the direction of main interest. The dowel connections and the lateral dynamic loading produced by the shake table were oriented in this direction. During the design of the test, the assumption was made that the longitudinal response of the structure would be caused mostly by the deformation within the connections due to the large stiffness differences between the connections and members. The data analysis proved that the influence of the member curvature on the displacement at the top of the frame was approximately 10% and depended on the level of damage within the connections. This conclusion was reached by comparing the curvature required in the connection to achieve the relative displacement at each level with the actual rotation measured in the connection. The storey drift to height ratio was also compared between the two storeys. The contribution from the members was higher than 15% for the first two 0.15g simulations and decreased to about 8% after the first 0.5g simulation. This reduction can be attributed to a change in the stiffness ratio between the members and the connection due to damage within the connections (Appendix A). No significant damage was caused to the connections during the first two 0.15g simulations, and the time histories of the displacement at the top floor were almost identical (Figure 3.16). This level of excitation generated a maximum acceleration at the top floor of .28g and a total relative displacement of 20 mm. The first 0.5g simulation resulted in a highly nonlinear response with the maximum acceleration at the top floor of 0.75g and a total relative displacement of 192 mm. This last displacement is an approximation since, as explained above, it was integrated from the acceleration record. The resistance of the damaged structure to subsequent earthquakes was observed during the two additional tests at 0.5g. The maximum 48 Experimental Study response of the top floor increased by 20% for the acceleration and 12% for the relative displacement at the second 0.5g simulation. No noticeable differences were observed between the responses of the last two O.Sg excitations (Figure 3.17). It was concluded that an equilibrium was reached between the energy input and the energy dissipation. This is discussed in more details in section 3.3.2.2. The fifth test had a maximum base acceleration of 0.15g and occurred after two 0.5g simulations. Compared with the first simulation, the maximum acceleration of the top floor increased only by 20%, but the relative horizontal displacement increased by 400%, due to the reduced stiffness of the damaged connections (Figure 3.16). 3.3.1.4 Serviceability The frame tests were done mainly to verify analytical models and were not specifically aimed at simulating an actual size frame supporting realistic masses and subjected to expected earthquakes in a given area. Nevertheless, the comparison between a realistic building design and the experimental set-up could help to highlight some aspects of serviceability. Firstly, the wooden members of the tested frame were highly overdesigned compared to the connections. For example, the moment capacity of the members was 105 kNm for the beams and 39 kNm for the columns compared to a moment capacity of the connections of 8 kNm for a rotation of 0.1 rad, or 6.5 kNm for a more realistic rotation of 0.05rad. The latter were measured during cyclic tests. While the capacity design philosophy recommends ductile connections which are weaker than the members, such a large difference would not be required. In actual design, these members would probably be linked by larger connections to increase the resistance of the structure although the ductility of the section might be compromised somewhat. Secondly, each level of the frame was half the height of a regular building storey and supported an additional mass of 13.3 kN per frame, which would represent a equivalent tributary width of 1.7 m considering a live load of 2.4 kPa and a dead 49 Experimental Study load of 0.5 kPa. This would be a smaller tributary area than what is normally supported by frames in most buildings. Lastly, the peak horizontal ground acceleration for 10 percent probability of exceedance in 50 years suggested in design is 0.21g for most of the Vancouver area. The 0.15g simulation produced a maximum relative displacement of the structure of 10 mm, which represent a drift to height ratio of 1/269, while the first 0.5g simulation generated a maximum relative displacement of the structure of 98 mm, or 1/27. Due to these large differences between the test frame and actual buildings, the only important conclusion relates to the capacity of the frame under large earthquakes. An important issue is also the nonlinear behaviour of the connections, which results in large displacements, and which will need to be considered when other lateral resisting systems are used in the same structure. These large displacement are also important in the design of nonstructural elements. Several tests at large accelerations showed that an equilibrium state is reached where the response of the frame does not deteriorate for subsequent excitations. No damage or failure was visually observed on the frame following these tests. The only indication of deterioration was a reduction of the stiffness of the connections. 3.3.2 Connections As discussed earlier, the response of the connections during the simulations was the most important factor influencing the behaviour of the frame 3.3.2.1 Calculations of Rotations and Relative Displacements It was possible to determine the relative displacements within a given connection using the response recorded simultaneously by the set of seven LVDTs installed. A FORTRAN program was written to 50 Experimental Study calculate the relative displacements between beam and column, and the relative rotations between the steel plate and each of the wood members. Due to the trigonometric transformations required, some calculated displacements were more sensitive to the precision of the measurement than others. The calculation regarding the plate-member relative rotation were very simple and implied only two LVDTs and one measured distance. Given the assumption that the steel plate was very stiff and had rninimal deformation, the precision of these small rotations was only related to the response of the LVDTs. The system measuring the beam-column displacements, however, was much more complex to solve and more parameters had to be measured which also resulted in a higher probability of imprecision (see section 3.3.4.1). The relative rotations between the beams and columns were the most important information regarding the connections, which was obtained from the three measured relative rotations: beam-column, column-plate and plate-beam. The horizontal and vertical displacements of the beam relative to the column were also calculated using trigonometric transformations. These displacements were very small in proportion to the rotation, however, and the lack of precision of these results considerably limited their usefulness. For the first two simulations at 0.5 g, the problems encountered with the data acquisition system resulted in a loss of all results related to the connections. An approximation of the rotations was calculated using the relative storey displacements of the structure. A very poor approximation was possible for the first 0.5g simulation since the only data available was an integrated relative displacement of the top floor. A better approximation was done for the second 0.5g simulation since the absolute accelerations were measured at each level of the structure. 51 Experimental Study 3.3.2.2 Moment-Rotation Response The high nonlinearity of the connection response and the difficulty to measure stresses within wooden members considerably limited the possibility of deterrnining precisely the bending moments within the structure and the connections. The bending moments were approximated assuming equal moment and rotation in all the connections for a given level at each time step, with a 20% difference between the two levels as observed from the measured rotations during the test. The lateral forces were found by multiplying the acceleration and mass at each storey (Appendix A). The fact that the proportion of the rotation between levels is not linearly correlated to the proportion of moments due to the nonlinearity of the curves was neglected in the calculation. The hysteresis curves that were developed through cyclic testing were not used to ascertain the stiffness of the connection since it would eliminate the possibility of comparing the dynamic results with these cyclic test results. The moment-rotation curve for the first two 0.15g simulations showed that the response of the connections stayed mostly in the elastic range, with a maximum rotation of 0.007 radians (Figure 3.18a). No major damage was caused to the connections during these tests and, as observed earlier, the displacement time histories of the top floor were almost identical for these two simulations (Figure 3.16). The LVDTs measurements for these tests were very small, which led to relatively imprecise measured rotations for the mid connection and useless results for the top connection. For the first test at 0.5g, the rotation of the connection had to be calculated from the acceleration record at the top floor due to the data acquisition problems described earlier. This approximate calculation led to reasonable results for the maximum rotation, yet the rotation time history was very imprecise and resulted in an irregular shape of the moment-rotation curve (Figure 3.18c). The pinched hysteretic behaviour 52 Experimental Study defined through cyclic testing could not be identified from this dynamic moment-rotation graph which was determined by integration. The subsequent 0.5g simulations had moment-rotation curves very similar to each other. Whereas the rotations were properly measured for the third 0.5g simulation, no data were available for the second 0.5g simulation and the rotations were found through integration of the floor accelerations. Despite some anomalies in this later curve, a comparison with results from the third test clearly shows a similarity (Figure 3.18d,f). The maximum rotation was the same for these two tests and slightly higher than for the first one. The loops were also more pinched due to the damage already inflicted to the connection by the first 0.5g test. It was concluded that each connection had to undergo a larger rotation during the second 0.5g test to absorb the same amount of energy which was achieved through further crushing of the wood and therefore increasing displacements. For the subsequent excitations, however, the structure reached an equilibrium where the amplitude of displacement allowed for enough dissipation of energy through the bending of the steel dowel and no further damage was observed in the connections. The fifth test had a maximum base acceleration of 0.15g and occurred after two O.Sg simulations. The reduced stiffness of the connections due to damaged wood caused a response significantly larger than for the first 0.15g test (Figure 3.18e). The comparison between the moment-rotation curve found through these dynamic tests and cyclic testing was greatly inhibited by the monitoring problems during the first two shake table tests at maximum amplitude. Uncertainties were also introduced due to the approximate calculations of the connection bending moment time histories. Nevertheless, the data related to the third and forth 0.5g dynamic excitations clearly shows the same inner loops as found during the cyclic tests and supports the 53 Experimental Study assumption that the cyclic moment-rotation curves are applicable for dynamic analytical simulations. Even for the tests with monitoring problems, the first test at 0.5g clearly produced a fatter curve than the following tests, despite the undefined hyteresis loops. 3.3.2.3 Observed Damage After the set of shake table tests, an inspection of the connections revealed a slight change in the position of the dowels (Figure 3.19). These variations could be observed by comparing with the original connection but would probably not have been detected by a post earthquake inspection of a building. The dissection of the connections after testing showed very little damage of the wood surrounding the dowels (Figure 3.20a). The density of the wood around the dowel seemed slightly increased, but no gaps could be observed. The level of deformation experienced in the connections during these dynamic tests was much smaller than the total ductile capacity of the connection developed through cyclic testing. Nevertheless, from the nonlinear response monitored during the dynamic test, more severe observable damage was expected in the connections (Figure 3.20b). One possible explanation for the small damage could be that the high loading rate during dynamic test leads to a higher resistance of the connections with less damage than for the much slower cyclic tests. More testing would be needed to confirm this hypothesis. Even if more ductility was available in the connection, this level of movement seemed to be the maximum tolerable to satisfy the serviceability limit states. 3.3.3 Natural Frequencies of the Structure Even though a complete modal analysis using the frequency response function of the shake table test results could not characterise the highly nonlinear response of the tested frame, the natural frequencies at 54 Experimental Study different stages of the tests were used to identify the level of damage. The first natural frequency between individual simulations was found by ambient vibration testing. The initial natural frequency of 3.0 Hz was not affected by the 0.15g simulations, but decreased to 2.8 Hz and 2.7 Hz after the first and the second O.Sg simulations, and stayed unchanged at 2.7 Hz for the following 0.5g simulations (table 3.2a). This is consistent with the displacement response of the structure during the different simulations. A larger reduction of the frequency was expected considering the highly nonlinear response of the frame to the high amplitude excitations. A possible explanation could be that the friction resistance in the connections caused an artificially high initial stiffness in the very low displacement range that is being measured in the ambient vibration tests. On the other hand, the wood damage observed within the open connections was also smaller than expected (see section 3.3.2.3). An attempt was made to gain more information about the natural frequency of the damaged structure by analysing the frequency response functions from shake table tests using windowing techniques. No satisfactory results were obtained as no clear peak could be identified in the calculated frequency response function that would represent a fundamental frequency. A more sophisticated analysis of the shake table test results might be appropriate to obtain more clarity on the natural frequencies of the frame, undamaged and damaged, as well as the accuracy of ambient vibration test results for this type of applications. 3.3.4 Evaluation of the Measuring System 3.3.4.1 Measuring System for Connections The measurement of the connection rotation using the triangulated set of LVDTs relied on a complex data reduction algorithm with the result that the precision of the calculated displacements were very sensitive to the precision of all the individual measurements. Prior to the experimental testing, an EXCEL spreadsheet was used to solve the triangulated system of displacement values investigating the accuracy of the different 55 Experimental Study logarithms with the aim to minimise the effect of measurement errors on the final results. An AutoCad drawing was used to simulate a relative displacement of the connection and the corresponding response of the LVDTs, with known errors which were introduced to determine their effect: Later, a FORTRAN program was created using the chosen algorithm to efficient processing of the displacements and rotations of the connections for large amounts of data (Appendix B). The precision of the final response relative to the precision of individual variables was calculated with a Mathcad worksheet using partial derivatives of the equations relative to each variable (Appendix B). The procedure to judge the best algorithm to solve the system of LVDTs would have been greatly simplified if the Mathcad worksheet was used earlier. In order to ascertain the precision of individual LVDT measurements, the amplitude of the noise at the beginning of the shake table test recordings was measured, using three different simulations for each instrument to verify its invariability. For the mid connection, the three LVDTs had a noise level between TO.05 mm and TO.07 mm, which led to a precision on the final results of 0.3 mm for vertical motion (3% of maximum displacement), 0.3 mm for horizontal motion (400% of maximum displacement), and 0.0008 rad for the rotation beam-column (3% of the maximum rotation). A higher noise level of T0.25 mm was observed for the three LVDTs of the top connection, which led to a precision of 0.9 mm for vertical motion (10% of maximum displacement), also 0.9 mm for horizontal motion (1500% of niaximum displacement), and 0.003 rad for the rotation beam-column (11% of the maximum rotation). The rotation measured between the plates and each member had a precision of 3.5% for the mid connection measurement. For the top connection, the precision of the measured rotation was 10% for the beam-plate rotation and 15% for the plate-column rotation. The most important information regarding the connections was related to the relative rotation, and the redundancy of this measurement enabled the verification of these results. For the middle connection, it 56 Experimental Study was possible to establish a good correlation between the three relative rotations: beam-column, column-plate and plate-beam. For the top connection, the poor precision of the LVDTs between beam and column, and a defective LVDT on the plate-beam arrangement limited the accuracy of these results, especially for small rotations. For each simulation, the beam-column rotation measured through the triangles was compared with the addition of the beam-plate rotation and the plate-column rotation. When discrepancies occurred, this latest calculation was used because these measurements were less sensitive to imprecision. The third 0.5g simulation had sufficient displacements to obtain a good precision of the results and a good correlation was found between the two different ways of measurement. It is interesting to notice that, except for a few cases, no major error was found in the time histories of the rotations since they followed each other, as well as the structure displacement time histories, very well. The measurement of the relative horizontal and vertical motion within the connection was, however, too small to be captured with the measurement system used. 3.3.4.2 Moment in Connections The approximation of the bending moments was important in the analysis of the connections, yet the technique used for this calculation was very approximate and could not be verified. 3.3.4.3 Integration of Acceleration Time Histories The problems encountered during the monitoring of the first two 0.5g simulations considerably impeded the comparison of the connection behaviour between the cyclic tests and the dynamic simulations. Using the results of the third 0.5g test, as well as results from cyclic testing, only an approximation of the actual behaviour of these connections could be done. As far as the prediction of the displacements, which have been obtained from integrating acceleration data, is concerned, a few comments: By using the second 0.15g simulation, where both integration of the acceleration and measured displacement were compared, the integrated absolute displacement time history was close to the measured displacements at the top floor 57 Experimental Study (Figure 3.15). The integration errors are enhanced, however, in calculating the relative displacements, which was done by subtracting the table displacement from the absolute displacement of the top floor. The maximum values of the relative displacement have a discrepancy of 60% between measured values and integrated values, which is not acceptable. 3.3.4.4 Camcorder The video recording of the experiment was being used to gain a visual appreciation of the amplitude of the excitation, as well as the response of the frame during the tests. Close-up recording of one specific mid-storey connection showed the large rotation experienced. Although it was originally intended to use digitised video recordings for displacement measurements of the connection, it was found that the accuracy was inadequate. 3.3.4.5 Damage in the Connections For now, only visual observations of the cut-open connections were used to assess the physical damage in the connections. Other techniques to determine the wood density could be investigated to determine with more precision the real damage within these connections. 3.4 Concluding Remarks To date, the experimental section of this global project has included various testing programs. Monotonic tests were conducted to assess the basic material characteristics used in the calibration of a finite element model. Cyclic tests on different configurations of connection were undertaken to ensure ductile behaviour by checking design parameters such as the diameter of the dowels, the cmshing capacity of the wood, the adequacy of end and edge distances, the dowel spacing and the required gap between beam and column. The hysteresis loops defined through cyclic testing on eight pin double connections using 12.7 mm mild 58 Experimental Study steel dowels served in the comparison between cyclic and dynamic behaviours, the verification of a finite element model, and the analytical simulation of the frame response using the DRAIN-2DX program. Shake table tests on a two-storey Parallam® frame with similar eight pin connections were an important part of the study and helped to define the dynamic response of semi-rigid timber frames. The construction of the tested frame needed special attention to permit the assembly of the members with tightly fitted connections. The acceleration record of the Landers earthquake measured at Joshua Tree station was used for the test. It was factored at two specific amplitude levels: a low amplitude simulation with a maximum acceleration of 0.15g and a high amplitude simulation with a maximum acceleration of 0.5g. Analyses with SAP90 and DRAIN-2DX served to approximate the response of the structure prior to testing. During the shake table test, the behaviour of two typical connections was monitored in detail in addition to data pertaining to the overall response of the structure. The response of the frame was also recorded by camcorder during the experimental simulations. The analysis of the experimental data recorded during the shake table test led to several conclusions. The response of the frame in the in-plane longitudinal direction was mainly controlled by the response of the connections, and no significant out-of-plane or vertical motions were recorded during the test. For the first 0.1 Sg simulations, the connection response remained in the linear range and no observable damage occurred. A highly nonlinear response of the connections, however, caused large relative displacements in the structure during the larger excitations with a maximum acceleration of 0.5g. By testing the structure with a series of high amplitude simulations, it was observed that the response of the connections, and consequently of the frame, increased by about 20% for the second test, yet reached an equUibrium point at which subsequent excitations at the same amplitude created no further damage. The damage caused in the 59 Experimental Study connection by these high amplitude excitations, however, significantly affected the response to a subsequent smaller excitation. Even though the moment rotation response of the connections under dynamic loading seemed to exhibit the same hysteresis behaviour as under cyclic loading, no definite conclusion can be made about the applicability of cyclic hysteresis curves for dynamic ^analytical simulations. Unfortunately, instrumentation problems during the first high level simulations limited the capability to provide accurate time histories for the connection relative rotations. Furthermore, the bending moments in the connections could only be approximated for the dynamic tests because of the influence of member curvatures and the nonlinearity of the connection responses. The determination of residual damage using changes in the fundamental natural frequency, which were measured by ambient vibration testing between each test, raised questions regarding the applicability of ambient vibration testing for this type of structure. The surprisingly small observable damage on cut-open connections after the shake table test would also require more investigation regarding the comparison of residual damage between dynamic testing and slower cyclic testing, which could be influenced by wood dynamic properties. A certain level of imprecision was inherent in the analysis of the connection, mainly due to unexpected problems during the recording of some tests, but also because of a very complex measuring system. The importance of redundancy in significant measurements can only be emphasised by the results from this study. 60 Experimental Study SIMULATION units 1 s 0.15 g 2 0.15 g 3 0.5 g 4 0.5 g 5 0.15 g 6 0.5 g 7 0.5 g Max. Table Acceleration (9) .16 .16 .46 .45 .15 .46 .46 Max. Top Acceleration (g) .20 .22 •75 .93 .36 .87 .86 Total Relative Top Displac. (mm) 18.1 19.7 192.3 220.7 83.4 217.5 224.5 Total Connection Rotation (*103rad) 3.5 3.9 39.3 46.2 16.0 46.1 47 1 s t Natural Frequency (Ambient Vibration Testing) (Hz) 3.0 3.0 2.8 2.7 2.7 2.7 ~ SIMULATION units 1 s 0.15 g 2 0.15 g 3 0.5 g 4 0.5 g 5 0.15 q 6 0.5 g 7 Max. Table Acceleration (g) .16 .16 .46 .45 .15 .46 .46 Max. Mid Acceleration (g) .17 .17 2 56 .26 .54 .58 Max. Top Acceleration (g) .20 .22 .75 .93 .36 87 .86 Max. Base Shear Force (kN) 2 8.3 2 34.2 8.9 33.8 33.8 SIMULATION units 1 u 0.15 g 2 0.15 g 3 0.5 g 4 0.5 g 5 0.15 g 6 0.5 g 7 0.5 g Max. Table Displacement (mm) 18.7 20.8 65.6 68.7 20.9 68.8 69.0 Max. Mid Displacement (mm) 21.7 25.7 2 90.5 32.7 100.1 99.9 Max. Top Displacement (mm) 25.1 30.0 107.6 129.0 48.5 145.6 145.6 Max. Relative Mid Displac. (mm) -5.38 +5.86 -5.39 +6.29 7 -61.7 +63.4 -22.7 +23.8 -58.0 +61.6 -61.8 +62.2 Total Relative Mid Displac. (mm) 11.24 11.68 2 125.1 46.5 119.6 124.0 Max. Relative Top Displac. (mm) -8.42 +9.70 -9.62 +10.0 -94.8 +97.5 -106.1 +114.6 -40.3 +43.1 -106.2 111.3 -110.7 +113.8 Total Relative Top Displac. (mm) 18.12 19.66 192.3 220.7 83.4 217.5 224.5 SIMULATION units 1 u 0.15 g 2 0.15 g 3 0.5 g 4 0.5 g 5 0.15 g 6 0.5 g 7 0.5 g Mid Connection, b-c Rotation OMO^rad) 3.5 3.9 j . 15.0 33 32 Mid Connection, bp+pc Rotation (*10"3rad) 2.3 2 2 16.0 46 47 Mid relative Displacement Height (1375 mm) (*103 rad) 4.3 4.6 2 46.1 17.3 45 45 Top Connection, b-c Rotation (*10"3rad) _ 1 2 2 _ i 28 20 Top Connection, cp+pb Rotation (*10"3rad) _ 1 2 2 14.0 34 33 Top relative Displacement Height (2695 mm) (*103rad) 3.6 3.7 36.0 42.5 16.0 41 42 2 ' Not available due to data acquisition problems during the simulation 3 - Partial Results only available Table 3.2 - Shake Table Test Results 61 Experimental Study .3 Mi CO 600.00 - r 500.00 4-400.00 4-300.00 200.00 4-100.00 Fu = 490 MPa 0.00 0.00 0.05 0.10 0.15 0.20 STRAIN (mm/mm) 0.25 0.30 0.35 Figure 3.1 Typical stress-strain curve for steel dowel material -r Load Time (b) Loading Cycles Load (a) Test Set-Up Displacement (c) Load-Displacement Results Figure 3.2 - Single dowel connection cyclic testing 62 Experimental Study F O U R P I N C O N N E C T I O N N o o o o Figure 3.3 Cyclic tests on two- and four-dowel connections 63 Experimental Study Experimental Study w — 2790 mm Figure 3.7 - Schematic drawing of the test frame Figure 3.8 - Steel bracket for concrete masses 65 Experimental Study Experimental Study Acceleration (g) Acceleration (g) 6 O O 6 O O !u b 4* b o o o o o o ON o o Experimental Study El Centro Record with a Maximum Acceleration of 0.5g SrQQ-I I I I I I I I I I I I I I I I | -0. o 04 -0.02 l l l I I I l l l l l I I 0. b4 -4.00 + 6.00 Rotation (rad) Joshua Tree Record with a Maximum Acceleration of O.Sg •6.00 ! Rotation (rad) Figure 3.13 - Moment-rotation response for the connection as simulated by DRALN-2DX 69 Experimental Study Relative Displacement (mm) Relative Displacement (mm) Experimental Study 8.0 £ 0.0 <o E o -8.0 -0.06 0.00 Rotation (Rad) (a) Test 1 -1s t 0.15g 8.00 0.00 -8.00 8.0 -0.06 0.00 (c) Test 3 - 1st0.5g 0.0 -8.0 0.06 0.06 8.0 0.0 -8.0 y - i ; | r --0.06 0.00 0.06 (b) T e s t 2 - 2 n d 0.15g -0.06 0.00 0.06 (d) Test 4 - 2nd 0.5g -8.0 -0.06 0.00 0.06 -0.06 0.00 0.06 (e) Test 5 - 3 r d 0.15g (f) Test 6 - 3rd 0.5g Figure 3.18 - Moment-rotation hyteresis loops of the monitored mid-connection 74 Experimental Study (a) South-West mid connection (b) North-West mid connection Figure 3.19 - Connections after shake table tests 75 Experimental Study (a) after cyclic tests (b) after shake table tests Figure 3.20 - Observable damage on cut-open connections 76 Development of an Analytical Model Chapter 4 DEVELOPMENT OF AN ANALYTICAL MODEL 4.1 Introduction The dynamic behaviour of structures when subjected to earthquake loads can be simulated using different analytical models. Computer programs, such as ANSYS, ABAQUS, CANNY, DRATN-2DX, and RUAUMOKO, are available to perform dynamic analyses considering nonlinear material properties and second order structural effects. Most of these programs have been developed to simulate the response of steel or concrete structures, without attention to the specific characteristics of timber structures. One exception is DRAIN-2DX, which also incorporates a subroutine specially developed to evaluate the dynamic response of nonlinear timber structures. In this model, the nonlinearity is essentially concentrated at the connection elements, and is simulated using a force displacement curve derived from cyclic testing. Other curve fitting models have also been developed to include the nonlinear response of timber connections in structural dynamic analysis (section 2.4.2.2). At this point, however, this subroutine is the only one that has been included in a commercial package. The complexity of the analysis and the number of parameters involved in these different simulations often determine their precision. All the models developed until now require testing on full size connections prior to the analysis in order to determine the moment-rotation curve under cycling loading for this particular size and arrangement of connection. This is a major impediment to the typical designer who has no access to a testing facility. To reduce the number of costly tests, it was decided, in the present study, to work on the development of a computer program that could simulate the behaviour of timber structures with ductile connectors based on the basic properties of the different materials. 77 Development of an Analytical Model The first step was to model the behaviour of a single connector, for which the analogy of an elasto-plastic beam supported by a nonlinear foundation was used. An analytical program originally developed by Foschi (1995) and adapted for pile foundations by Khan (1995) was chosen for the present work. Two types of connectors were analysed: very small diameter connectors, such as nails, as well as medium size connectors, such as dowels and small bolts. The diameter of the connector had to stay within a limit to ensure ductile behaviour and avoid brittle fracture of the wood around the connectors. Nail connectors were mostly used for shear wall applications, and specific types of behaviour such as withdrawal, extreme yielding, and tear-out of the sheathing were considered. These specific features were not included for the dowel connections that were dealt with in this thesis. A computer program was adapted to simulate the connector response when supported by different layers of wood or steel along its length. The major objective of this project was to develop an analysis to enable the modelling of a semi-rigid multiple-dowel connection between a post and a beam in a moment resisting frame for a timber structure. This connection model had to include the behaviour of each dowel in the simulation of the connection response. An additional objective was to apply the analysis to a moment resisting frame for static and dynamic loading using the subroutine for each connection to determine the response of the frame. Only the basic material properties of each dowel within the frame were to be used. The theoretical background was developed for this model, and full implementation in the frame analysis program, followed by verification, are still to be done in a future project. 4.2 Analysis of a Connector The following is a summary of the model development in Foschi (1995) and Khan (1995). An analysis of the interaction between the connector and the surrounding wood was undertaken using a finite element 78 Development of an Analytical Model model to determine the force displacement relationship from basic material properties. The dowel was analysed as an elasto-plastic beam supported by a nonlinear wood foundation. The stress strain curve of the steel used for the dowel, as well as the bearing capacity of the wood were defined through experimental testing. 4.2.1 The finite element model A finite element model of the connector was developed by dividing its length into a given number of elements. Five degrees of freedom (DOFs) were defined for each node between the elements: displacement, slope and curvature in the direction perpendicular to the beam (w, w', w" in the y direction), as well as displacement and slope in the direction parallel to the beam (u, u' in the x direction). T Wi , Wj', W i " f Wj , Wj', Wj" u;. u;' {a} = { W i , wj', Wi" , U i , u;', W j , Wj', Wj", U j , u/} , 1 Figure 4.1 A finite element and its degrees of freedom (DOFs) These DOFs at the nodes were chosen to enable the consideration of bending deflection and axial displacement in the deformed shape of each element. The multiplication Of these DOFs with assigned shape functions defined the displacements at any point along the length of each element. As an example, the displacement w along one element could be characterised by an equation found multiplying the ten shape functions {Mo} by the values of the ten DOFs defined for this element {a}. The different shape functions used for this analysis are presented in Appendix C. This plane analysis assumed that the beam was only bending in one direction. The variation in the width of the beam section along the y axis was however considered in the calculation of strain within the beam elements. 79 Development of an Analytical Model To achieve nondimensional equations in the analysis, the axes were redefined within each element. The t, axis replaced the x axis, and varied from -1 to 1 along the length of the element, whereas the TJ axis, replacing the y axis, varied from -1 to 1 across the section of the element in the orientation of the applied force. -1 1 Figure 4.2 Coordinate transformation for each beam element Equations were defined to enable the coordinate transformation between the x, y plane and the 4,T| plane. x = x + \ 5JC = - - 8 C ; 2 d * 8 y = — *8rj (4.2) (4.3) (4.4) In the analysis of a beam composed of several elements, the continuity of the deformed shape is guaranteed by the DOFs at a node which are equal for the two adjacent elements. The chosen DOFs ensure continuity in displacement and slope in the y direction and displacement in the x direction. Since each node has five DOFs, the overall beam model has n*5 DOFs. A reduced number of DOFs at each node would have lead to simpler shape functions yet more elements would have been needed to procure the required precision for the analysis. 80 Development of an Analytical Model A Wi , Wi', Wi" w 2 , w 2 ' , w 2 " 4 W 3 , W 3 ' , W 3 H U2 ^ »2 U3 ^ "3 element 1 element 2 node 1 node 2 node 3 {A} = { W i , W i ' , W i " , U i , U l ' , w 2 , w2', w2", u 2 , u2', w 3 , w3', w3", u 3 , u3'} Figure 4.3 Degrees of freedom of a two element model 4.2.1.1 Modelling of the steel Tensile and bending tests were done on dowels to determine the stress-strain relationship of the steel (Section 3.1.1). A bilinear approximation of the experimental curve was used in this analysis. i o k 1 e, / 1 Figure 4.4 Stress-strain relationship of steel dowel Mathematically, this relationship can be expressed as follows. Given a prior state (ob, £o ) then o = oe +E*(e-e0 ), and— = E do but if o > o „ then a = a „ , and — = 0 y y de or if a <-a „ then a = -a „ , and— = 0 ' ' de (4.5) In the analysis of the finite element model, the strain pattern is defined throughout each element to enable the detemiination of the associated stresses. Three main causes of strain were included in this analysis: 81 Development of an Analytical Model the axial deformation, dx 3u (4.6) the bending deformation, y \ \ \ 1 (4.7) and the additional axial deformation due to bending. 3x k=_ s OW 3x / < — 3x' —t ^3w 1 f ^ 2 y3 (4.8) The total strain at any point along the element was defined by the addition of these three components: £ v =u'-yw"+-*(w')2 . 2 (4.9) 4.2.1.2 Modelling of the Wood Foundation The wood foundation was modelled as a series of nonlinear springs along the length of the beam. Testing was undertaken to evaluate the force displacement relationship of the Parallam® material for various configurations. Initial embedment tests did not include the effect of a continuous wood interface around the dowel, and their results were not judged useful in the determination of the parameters of the wood foundation. The technique used instead was to calibrate the parameters in the finite element program to achieve an analytical force displacement curve similar to the test results on single dowel connections. 82 Development of an Analytical Model Various sets of parameters for different orientations of dowel and load were found using this method (Section 3.1.2). A five parameter curve was chosen to approximate the response of the wood foundation (Figure 4.5). P t Q o ; - k *™ A = |wl Figure 4.5 Parameters of the load-deformation law of the wood foundation During this simulation, it was important to keep track of the maximum previous displacement in both directions, since the foundation does not apply any pressure unless the displacement is larger than the gap already formed in the wood. In addition, this absolute value and the sign of the displacement had to be recorded separately since the pressure of the foundation is only applied when the absolute value of the displacement is increasing (Figure 4.6). "> 1 7? 1st cycle -*—2oi cycle • W Figure 4.6 Load displacement relationship for wood foundation 83 Development of an Analytical Model Mathematically, this relationship can be expressed as follows. ( i fA<A m a x , thenP = (e 0+G 1*A)' 1-e i fA>A m a x , then P = +Q2 • ( A - A ^ ) where Pmax=(Q0 +Ql**max)* (4.10) C o 1-e K J 4.2.1.3 Virtual Work Equilibrium Using the force displacement relationship for both materials, the steel dowel and the wood foundation, the Principle of Virtual Work was used to determine the deformed shape of the dowel for an associated applied force. Work is defined as the product of the force with the effective displacement of its application point. In the Virtual Work method, an equilibrium of energy is achieved by considering the work done externally, as well as the internal strain within the material (Popov, 1990). The Virtual Displacement method was used to determine the work done for each element in the finite element model. According to this method, a very small variation is applied to the displacements at the 10 DOFs for a given element, called the vector {So}. Next, the resulting forces in the system are calculated, which are, for this application, internal stress, pressure of the wood foundation and applied load. A balance of internal and external work is achieved to ascertain the response of the system. Internal Strain Energy To calculate the internal strain energy, one can use the strain equation already defined as sy =u'-yw"+^*(w')2, (4.9) and describe the variables u, w' and w" along the beam element as functions of the DOFs at the nodes utilising the defined shape function, and the cj-n axis of coordinates. 84 Development of an Analytical Model e , = N Ta - — T | • M j a. +—a T A f , M \ a (4.11) where: Nu Mu M2 are vectors of the shape functions d . L — T| is the position on the y axis a is the displacement vector for the given element. A very small variation of strain at any point in the steel dowel can be written as: Se = Nj Sa-yM T Sa + Sa T MjMj a (4.12) The internal strain energy for a given steel element is calculated by multiplying the stress by the very small strain at the point, ^dem =o(e)Se (4.13) and then integrating over the volume of the given, element. = J o ( e ) 8 e vf r ; (4.14) = I o ( e ) 8 a r [AT, - y M 2 + M 1 M , r « ] vol The volume of one finite element is defined in the cj- n coordinate system as: vol = A d b = J ' | * * 1 J* ^<^*b(r,) (4.15) The internal strain energy of a beam element due to a virtual deformation {Sa} can then be written as: = 8 a T ^ - \ 1 f ' a ( e ) Nt - ^-M2 +M1M]a b(r\)dcl A, (4.16) 4 J-iJ-i |_ 2 J The width along the depth of the section b(T|) is derived for different cross-sections as follows, 85 Development of an Analytical Model for square sections: 4 t 7 h and for circular sections: b(Tl) = b 2 ; * ( 7 ) 2 = d 2 * ( l - T j 2 ) (4.17) (4.18) Virtual Work due to the Wood Foundation The resistance of the foundation also provides work during deformation of the beam. The force involved at any point along the element depends on the displacement w at this point. As mentioned earlier, the intensity, or absolute value, of the displacement w and its direction of loading, or sign, had to be denned separately. The intensity of the displacement was defined as: (4.19) and the equation for very small variation in the intensity of the displacement w was: S\w\ = Syjw2 w 8 w (4.20) 86 Development of an Analytical Model The displacement w at any point along the element was defined in terms of the DOFs at each end of the element and the defined shape functions. A very small variation of this displacement, as well as its absolute value, were also described in that way: 5w = MT 5 a S\w\ = —MT Sa (4.21) IH where is the sign of w \w\ Considering that the external work for a very small variation in the displacement w at a specific point along the beam element is calculated as: The integration of the work produced by the nonlinear foundation along the beam element in response to the virtual deformation {5a} can be expressed in terms of cj and rj as: "Vfoun^n =-8aT±\ip(\w\)f-M08ta (4.23) 2 J - i |w| External work due to the Applied Forces The vector of applied force {R} acting on the beam element also generates work on a given element deformed by the virtual displacement {So}. Va={R}T{Sa] ( 4 2 4 ) 87 Development of an Analytical Model Virtual Work Equation for a beam element The Virtual Work Equation is established by summing all the energy components for a closed system. In this case, three components needed to be included: the internal strain energy, the work due to the reaction of the wood foundation, and the work due to the externally applied force. 8 a 0 = ^ steel foundation ext +8a T A T ' P(\w\)^-M0d^ 2J-i | w | -5a T R Nj -—M2 +MjMj a 2 (4.25) To achieve equilibrium, the total energy balance of the system 0 must equal zero. 9 = ^ - J / J / a ( e ) N 1-^jM2+M1MTla 2J-I \w\ (4.26) Since the nonlinearity of the materials was included in the analysis, an iterative process was needed to solve the energy balance equation and determine the vector of displacement {a} that would achieve equilibrium. Using Newton-Raphson, and a given value of t9* corresponding to a vector a*, one could write e = e*+[ve](a-a*) = o from which: a = a*-[v8] -1 0* (4.27) Evaluation of the Gradient through Partial Derivation Each term of the gradient V9y needed in the iterative process is defined by the partial derivative of the term 9| by the variable Aj: 88 Development of an Analytical Model 50 2 J - i IX BP(H) Ad r • f Mi2.J J V , - M M , +MIM[ :\Hn>** 4 J - l J - l O f l ; I 2 Ad N1-—M2+Ml 2 M\a (4 .1) dA, b( r\)dr]a% da To define this equation, the partial derivation of its components, such as the strain e(£,A) and the intensity of the displacement M(£) needed to be defined. The partial derivative of the stress in the steel dowel was written as: dcs(e) da(e) de da j de daj (4 .29) usmg: 8 = NT a - - r i • Ml a + \a  T M , M [ a 2 (4 .11) de da : = ^ l i - i f I . M f i + ^ [ M 1 M f ] / k « t (4 .30) and the partial derivative of the wood reaction was: dP(\w\) dP(\w\) d\w\ daj d\w\ da j d\ w\ w dw w _, = -— = -—--M da j w da j 07 (4.31) Using these previous definitions, each terms of the gradient became: 89 Development of an Analytical Model ve (4.32) Numerical Integration The functions 9t and VQU had to be evaluated by integration within each element. A numerical integration using Gauss quadrature was used to approximate these values. Broadly, the method consist of choosing some discrete points within the range of integration, evaluating numerically the function at these points, and adding these values, which were multiplied by a specific weight defined through Gauss weighting function. For example, to evaluate a 3 Gauss point integration on a domain running from -1 to £,= 1, the function that needs to be integrated is evaluated at -0.7746, 0, and 4= 0.7746. These three values are weighted by the factors 0.5556, 0.8889 and 0.5556 respectively, and these weighted solutions are added to approximate the solution of the integration over the chosen domain. Increasing the number of points used in the numerical integration would increase the precision of the approximate result. Virtual Work equation for the entire dowel To evaluate the energy balance for the whole beam ensuring continuity of the deflected shaped between adjacent elements, a global equation including all the nodes of the analysed beam was required. Since each node had 5 DOFs, the size of the global vector of displacement {A} was five times the number of nodes. A transformation matrix [L] was needed to determine the position of the 10 DOFs of each element in the global displacement vector for the beam. The vector {9} and its gradient [V9] were calculated for 90 Development of an Analytical Model each beam element, and added using the transformation matrix [L] for each element to form a global equation. Nelem {e}<^ = Z[L*Ke}* Nelem SZL <4-3> * = i The energy equilibrium was used at each load step to determine the displacement vector of the dowel resulting from a given applied force. For each load step, the convergence of the iterative process was reached when the amplitude of the error Oi was smaller than a given portion of the initial error 9 0 . NDOF , if —— < TOL then converged (4.34) NDOF & z el » = i The details of the computer prograrnming is available in Appendix C. 4.2.2 Possible Improvement This model analysed the deflection of the dowel in one plane, and it does not allow for change in the orientation of the applied force. A three-dimensional analysis that would include an evaluation of the gap in the foundation in different orientations has several problems due to the difficulty to evaluate the influence of the gap in the first orientation on the response of the dowel to a loading in a different angle. If the angle of loading is slightly different, the dowel might just follow the first orientation since the gap already exist in the foundation. If the orientation of loading varies by a large angle, another gap might be formed but near the zero displacement position, the damaged wood might influence the force-displacement relationship of the foundation. The importance of considering the dowel in a three- dimensional model can be assessed by the probability of having different orientation of loading in the actual case studied. In this 9 1 Development of an Analytical Model study, the orientation of loading on a given dowel was estimated to stay relatively constant and a plane analysis was judged to be sufficient. 4.3 Analysis of a Connection In this study, the dowels were used with steel plates to form semi-rigid connections between wood members, as shown in Figure 3.4. This analysis considered a single connection as being the connection between a steel plate and a wood member. The displacement of each individual dowel included in this single connection was derived from the relative displacement between the plate and the member. A three-DOF node was set on both the wood member and the steel plate, and the combination of these six DOFs was used to describe the relative movement of each dowel. A connection analysis was developed to transform the force-displacement relationship for each connector described previously in a force-displacement relationship for the single connection using these 6 DOFs. The details of the computer programming is available in Appendix C. 4.3.1 Description of the model The steel plates, as well as the wood member around the connection, were considered as rigid bodies. In a plane analysis, the displacement of any point on a rigid body can then be described as a function of the vertical, horizontal and rotational displacement of another known point on that same body. y Vp, y Vm, Vp Pi * Up, ymi Vm Mi •» Umi xp > X xm > x (a) Steel plate (b) Wood member Figure 4.7 DOFs of the steel plate and wood member 92 Development of an Analytical Model TJPi =Up-0p(yp! -yp) Vp, = Vp + 0p(xp, -xp) Um j = Um - 0m(ym\ - ym) Vm j =Vm + Qm(xm1 -xm) (4.35a) (4.35b) (4.35c) (4.35d) The relative displacement between the plate and the member at this given point 1 was defined from the motions of the two nodes representing the plate and the member. A U t =Um, -Up t = Um-Up-0m(ym1 -ym)+0p(yp1 - yp) A F ^ V m j - V p , = Vm - Vp + 0m (xm t - xm) - 0p(xp t + xp) (4.36a) (4.36b) Since the displacement considered was small, the x and y distance between the two points on the same body were assumed to be constant. A system of equations was assembled in vector format: Atf, ={Qu1}T {8c} ={Gv 1} r{8c} (4.37a) (4.37b) Where: {8c} = Vm Qm Up Vp Qp] 1 0 yml -ym -1 0 ypi -yp j 0 1 xm1 - xm [-(xpv-xp)\ The vectors {Qu} and {Qv} were defined for a given dowel n within the connection, and the above equation was generalised as: AU„={Qun}T -{8c} = {8c}r {Qun} (4.38) 93 Development of an Analytical Model The force developed in the dowel subjected to this relative displacement was calculated using the analysis of a connector described in section 4.2. The internal work produced by the combination of these forces and displacements can be expressed as follows for each connector. 8W„ =Fu„ 5AUn +Fv„ 8AV„ = 5{8c}T.[Fu„{Qun} + Fvn{Qyn}] ( 4 3 9 ) The angle between the force and the relative displacement influenced the sign of the work done. Considering the signs of the forces and displacements that are provided by the analysis in different situations, it was concluded that the work is always the product of the force and the relative displacement, both of which keep their original sign. The internal work for the connection was calculated by adding the work done by all the individual connectors, while the external work was evaluated from the external force applied on the entire connection. W-m =N,Ts{Sc}T \Fun{Qutt} + Fv„{Qyn}) (4.40) n=\ S^«* = { * } r •5{8c} = 8{oc} r {R} (4.41) A Virtual Work Equilibrium equation could be defined for a given connection = 8{6c}r f T«{8c}r [Fu. {Qu, } + Fv. { f i r . } ] - { * } ! <4 4 2> ^ »=1 ' and the Newton-Raphson technique, described in section 4.2.1.3, could be used to solve the connection response to a given applied force. This analysis of the connection was however a transitional subroutine to include the response of each dowel in a frame analysis model, and the complete solution of the equilibrium equation was done using the frame analysis program described in the section 4.4. 94 Development of an Analytical Model 4.3.2 Comparison with Test Response The simulation of single and double connection responses were compared with cyclic test data, using the program developed for the entire frame to solve the system of equations. Results are available in section 4.4.2. 4.3.3 Possible improvement Possible changes in the orientation of loading for the individual connectors was a concern in the global analysis of the connection. As described in section 4.2.2, the plane analysis of the connector does not account for any changes in the orientation of loading. For this reason, it was chosen to fix the first orientation of loading as being the only orientation for the given dowel during the analysis. Thereafter, the future displacement will be the positive or negative from the sign of the dot product between the new vector of displacement and the original displacement, and the wood properties in the orientation of loading was calculated using rlankinson's formula. Even if a connector model that allows different orientations of loading was available, its use would largely increase the numbers of parameters that have to be stored for each dowel of the connections, which is already a concern for large frame analysis. 4.4 Frame Analysis The aim of this global project was to develop a planar frame analysis program able to determine the dynamic response of a timber structure, using semi-rigid dowel type connection. The main concerns regarding the modelling of the connection were mostly dealt with in the previous sections of this chapter. The remaining work regards the assembling of all the linear and nonlinear parts of a frame. The static planar analysis will first be developed, followed by a dynamic analysis. The details of the computer progiarnming is available in Appendix C. 95 Development of an Analytical Model 4.4.1 Static Analysis - PSA-WoodFrame The general plane static analysis program (PSA) developed by Foschi (1993) was used as a base for the development of a static analysis program for a wood structure with dowel type connections (PSA-WoodFrame). The program PSA was developed for a linear structure, and its analysis included the calculation of the stiffness matrix and external force vector, as well as the determination of the displacement of the structure through the solution of the following system of equations. The program PSA included the analysis of fixed, pinned or spring supports and connections, concentrated loads and moments, as well as distributed loads, which will be lumped at the nodes during the analysis. The nonlinear response of the dowel type connection was added to this static analysis program to develop the program PSA-WoodFrame. The evaluation of the P-Delta effect was available in the program PSA but, at this point, was not included in the new program PSA-WoodFrame. The system of equations solved in PSA-WoodFrame included a nonlinear term to consider the response of the dowel type connections derived previously. The inclusion of the new nonlinear term complicated the analysis significandy since the solution of the new system of equations now required an iterative process. The Newton-Raphson method, already used in the analysis of the connector, was also used in this analysis. or (4.43) (4.44) {e} = [K].{x}+{T Fconn}-{R} (4.45) 96 Development of an Analytical Model For each load step, two conditions were needed to reach the convergence of the iterative process: the amplitude of the error Oi was smaller than a given portion of the initial error 9 0, and the displacement difference with the previous iteration was smaller than a given portion of the initial displacement error. NDOF NDOF if — < TOL, and — <, TOLX then converged (4.46) NDOF ' NDOF ° £ e ; 2 ( * w - x 0 ) 2 «=1 i=l 4.4.1.1 Problems in developingPSA-WoodFrame 1. Special attention was needed in the transformation of forces and displacement between the connection coordinates (u-v) which are parallel and perpendicular to the wood member and the global coordinates of the frame (x-y). A transformation matrix [A] was calculated for each connection and multiplied with the transformation vectors{Qu} and{Qv} already used in the connection subroutine. (4.47) Angle of a wood member 2. An extensive study was undertaken to determine the force-displacement slope for each connector used in the calculation of the gradient. A problem was identified related to the use of numerical integration to approximate the solution for the forces and the slopes calculated for each dowel by the connector subroutine (HYST). Since the slope was calculated using two adjacent points, a small error in the 97 Development of an Analytical Model calculation of these points could create a significant error in the calculation of the slope when the two points were closely spaced. This error in the evaluation of the slopes could lead to a diverging solution process when using the Newton-Raphson method. Different solutions have been investigated to solve this problem. The first proposed solution was to directly use the DOFs from each element of each dowel in the system of equations for the frame. This solution led to a very large system of equations since for a typical case with 10 DOFs per element, 5 elements per dowel, 4 dowels per connection, and 8 connections for a simple two storey frame, the vector of displacement {X} was enlarged by 1600 entries, which was considered unacceptable. The second proposed solution was to calculate the slope for each connector using a three point parabolic equation. This method solved the problem in some cases but was still unstable in other cases. The third proposed solution consisted in always using the initial slope calculated for each dowel, which is the largest possible slope in this analysis. This solution did not work since along the analysis the different connectors are deformed at different levels, and using the initial slopes for all the dowel implies that the variation of the force-displacement would be the same for all the dowels within the frame. This was not true since they had different slopes. The final solution included different steps. The initial slope calculated at the first step was used for the first iteration of all the following step. For each following iteration, a new slope was calculated using the force and displacement of the present and the last iterations, yet if this slope was negative or higher than the initial slope, the slope calculated at the previous iteration was used. 3. The total amount of memory needed in the program was relatively large since several parameters were needed for each Gaussian point of each element of each dowel of each connection. An attempt was made to reduce the total number of variables by recalculating some parameters at each iteration. A balance point had to be reached, however, since the recalculation of parameters increased the computing time, which was already very long. 98 Development of an Analytical Model 4. A convergence problem occurred during the analysis; it was due to the determination of the slope, as describe above, or to a loop in the iteration process. To overcome this problem, a "gate" was established to check the convergence process. If the error changed sign, yet it was of the same amplitude, the iterative process continued as usual, but the program did not update the dowel parameters since the displacement found at this iteration was beyond the aimed displacement. If the error changed sign and was larger than the previous error, the iterative process used the initial slope without updating the dowel parameters. If the problem was not solved in three iterations, the step size was split in two and the process restarted at the beginning. Improvement of this solution method could be considered, yet it was solving the problem for the present application. 4.4.2 Hysteresis Loop of a Double Connection: Model Prediction vs. Experiments The program PSA-WoodFrame was used to simulate the static response of a double connection that had been tested previously under cyclic testing. The moment-rotation hysteresis loops of the two responses are compared in Figure 4.8. This comparison was used to verify the correctness of the program and the stability of the iterative process, since the connector subroutine had already been verified prior to this analysis. In general, the moment rotation curve that was produced analytically with PSA-WoodFrame followed the loops described by cyclic testing very well. The return slope was slightly lower than the slope described through cyclic testing, which reduced the energy dissipated during a cycle. The approximation technique to define the wood foundation properties at various angles to the wood fibres using Hankinson's formula gave good results in this particular case. A localised error can be observed from the first loop at high amplitude of rotation in the positive range, which seems to follow the outside envelope even in the low rotation range when previous cycles had already damaged the foundation. The cause of this problem needs to be investigated. The stability of the iterative process, which was a problem in this area of the curve, 99 Development of an Analytical Model may be the cause for this irregularity. The same error is not found for the equivalent curve in the negative range of rotation. Moment (kNm) PSA-WoodFrame Results Compared to Cycl ic Test Results Double Connect ion -0.08 -0.04 0.00 0.04 0.08 Rotation (rad) Figure 4.8 - PSA-WoodFrame results compared to cyclic test results for a 8 dowel connection 100 Development of an Analytical Model 4.4.3 Dynamic Analysis - DPSA-WoodFrame A dynamic analysis incorporates the inertia and the damping capacity of the structure in addition to the stiffness already considered in the static equation of equilibrium. The equation of motion for a linear system is defined as: [M\ X +[C]- X +[K\{X} = [M\ Y where: f M l is the mass matrix , X the acceleration vector (4-48) [C] the damping matrix, X the velocity vector [JT] the stiffness matrix, { X } the displacement vector and { Y } the input base acceleration vector The dynamic response of timber frames with dowel type connections is mainly governed by the ductility of the connections, which provides a means to dissipate the energy from the earthquake. As for the static analysis, the nonlinear response of the connections should be included in the dynamic analysis using a nonlinear force in the equation of motion of the structure. [M]- X +[C\ X +[K\{X} + ( £ j F c o n n } = [M\ Y (4.49) where: is me nonlinear force vector This equation was used in the present analysis to determine the dynamic response of the nonlinear frame studied. The mass matrix was created by considering the different masses acting on the structure, while the stiffness matrix, as for static analysis, was calculated with the material properties of the members. The damping matrix had a limited importance since most of the energy dissipation was caused by hyteresis damping, which was already considered by the nonlinearity of the dowel type connection. Nevertheless, an equivalent viscous damping matrix was added to include other types of damping within the structure. The solution of this dynamic equation of motion required the use of an iterative process since, as for the static analysis, it included nonlinear terms. In addition, a linear acceleration method was used to solve a problem caused by the dependency between acceleration and velocity and the displacement vectors. The 101 Development of an Analytical Model Newton-Raphson method was used in the iterative process to solve this system of equation, the equations for this method can be found in the development of the static analysis, section 4.4.1. This new system of equations needed to be included in PSA-WoodFrame to enable the analysis of structures under dynamic loading. At this point, the theory to implement these changes has been developed, but the programming is not completed. 4.5 Concluding Remarks Four steps were undertaken to enable the simulation of the dynamic behaviour of a timber frame using dowel type connections, based on fundamental material properties. 1. A finite element model simulating the behaviour of a ductile beam supported by a nonlinear foundation developed by Foschi (1995) was adapted to analyse the response of each connector. The solution of the model includes a description of the material properties for the steel and the wood, a virtual work equilibrium, and a Newton-Raphson solution for a nonlinear system of equations. This model was calibrated using the response of a single dowel connection subjected to cyclic test. 2. The development of a subroutine to relate the forces and displacements of the individual dowels to the 6 DOFs of each moment-resisting connection between a wood member and a steel plate, considering local and global coordinates. 3. The incorporation of the forces calculated for each connection in the static analysis of a wood frame. The addition of these nonlinear terms in the equilibrium equation for the frame required an iterative process using the Newton-Raphson method to solve the system of equations. Analytical results from this 102 Development of an Analytical Model static analysis program are in good agreement with experimental results from cyclic testing for a eight-dowel connection model. 4. The inclusion of the inertia and damping of the frame to perform a time step dynamic analysis using the equation of motion. In this analysis, in addition to an iterative process needed due to the nonlinear terms brought by the connection behaviour, a linear acceleration approximation is used to solve the system of equations considering acceleration, velocity and displacement. The development of me finite element model of a connector was done prior to this specific research. The part of this global study done by the author included the development of a subroutine to consider the different individual forces within each connection and its implementation in a static frame analysis program. The next step would be to implement the theory for dynamic analysis in the static analysis program PSA-WoodFrame to developed DPSA-WoodFrame; a program capable of performing a time history analysis and predict the response of planar frame to dynamic loading. 103 Analytical Prediction Chapter 5 ANALYTICAL PREDICTION In an attempt to simulate the hysteretic behaviour of dowel type connections in a frame analysis program, different approaches were explored. This study examined three distinct models: a linear analysis program and two nonlinear programs, both of which perform a step by step time history analysis. The program SAP90 (Wilson and Habibullah, 1992) can perform a linear dynamic analysis in three dimensions. It determines the time history response of the structure to a stochastic excitation by integrating a given number of modal responses. The program DRAIN-2DX performs a nonlinear dynamic analysis of planar structures using the equilibrium method and applying finite element techniques (Prakash et al., 1993). It contains a subroutine, developed at the University of Florence, that simulates dowel type connections with nonlinear elements using parameters from a typical moment-rotation curve of the connection (Ceccotti and Vignoli, 1989). DPSA-WoodFrame is a dynamic analysis program that was partially developed by the author as part of this project. It also uses the equilibrium method for planar structures, but simulates the dowel connections using basic material properties such as wood embedment strength and steel yielding criteria. The forces and displacements are determined for each dowel within the connection. The theoretical background of this program is described in chapter 4. Its development is not completed yet and only partial results can be presented in this comparison. 104 Analytical Prediction The performance of the different analytical models will be evaluated considering the level of information required as input data, as well as the quality of the analytical prediction. The analytical responses will be compared with the results measured during the shake table tests described in chapter 3. The geometric and strength properties of the test-frame were used in the different analyses to enable this comparison. 5. / Modelling of the test-frame 5.1.1 Wood members The timber members were modelled as linear elastic elements in all the models used to simulate the behaviour of the frame. The stiffness and strength properties used were determined by the Parallam® producer from static testing (Appendix D). The Parallam® properties were increased by 10% to account for the dynamic effect. Moreover, the strength properties were divided by 0.90 to elirninate the design resistance factor § already included in these available values. This factor, which is included in limit states design procedures to ensure the reliability of the structure, was not needed in this analysis since the goal was not to design a structure but to predict the behaviour of the frame with maximum accuracy. The size of the sections were measured on site with a measuring tape. The accuracy of these measurements was within 1mm. The sections of the members were 177 x 177mm for the columns, and 177 x 300mm for the beams. A density of 6.5 kN/m3 was used to calculate the weight of the frame. The maximum bending moment and axial strength were calculated assuming a linear distribution of the stresses within the section. Several simulations with DRAIN-2DX were undertaken to verify the influence of the different properties of the wood members on the accuracy of the analytical results. The strength capacity of the members had no effect on the modelling as long as the values stayed within a reasonable range, since the connections were much weaker than these members. The modulus of elasticity was the 105 Analytical Prediction only wood property that affected the response of the frame, because the curvature of the members depended on the stiffness ratio between the members and the connections. For a four-slope modelling of the connections (see section 5.1.4), it was observed that varying the modulus of elasticity of the members mostly affected the inner loops but had no effect on the envelope curve (Figure 5.2). This phenomenon can be explained by a change in stiffness ratio between the connections and the members. For small amplitudes of rotation, the stiffness of the members is close to the stiffness of the connections and has an effect on the results. For larger rotations, the stiffness of the connection is much lower and becomes the major factor controlling the response of the frame. 5.1.2 Base supports and lateral bracing The base supports of the frame allowed free rotation in the in-plane longitudinal direction, and pinned supports were modelled in the different analyses. In the three dimensional linear analysis using SAP90, the out-of-plane lateral bracing system was modelled as a set of linear members, pin-connected to the wood members. For the other analyses in two dimensions, only one of the two timber frames was modelled and no out-of-plane displacement was considered. 5.1.3 Masses and static loads The concrete blocks had two major effects on the response of the structure. Firstly, their weight produced initial forces in the members and connections; secondly, their mass induced inertial forces when acted upon by the horizontal accelerations during the excitation. In the three-dimensional analysis with SAP90, the concrete blocks were simulated by shell elements. These elements were positioned on the beam and had been given values for weight and for mass. 106 Analytical Prediction In the DRAIN-2DX analysis, the initial forces in the members and connections were estimated beforehand and included in the input data, in order to account for the weight of the concrete blocks. The initial bending moments in the connections were approximated for rigid connections due to the relatively high rigidity of the connections at initial displacement. The total mass of the wooden frame, including the concrete block, was also calculated and lumped at the connection. The mass of the frame was only included as a translational mass since the calculation to include the rotational component would be extremely complex, while its effect was deemed to be very small. In a DPSA-WoodFrame analysis, the distributed load, as well as the distributed added mass, should be included in the input file, but no calculation of initial moments in the connections was needed. 5.1.4 Connections Dowel-type connections are designed to be key components in the resistance of lateral loading for moment resisting timber frames, and their moment-rotation characteristics are of major importance. The moment-rotation curve of such a connection under cyclic loading demonstrates a hysteretic behaviour where energy is dissipated through yielding of the steel dowels and crushing of the wood. The simulation of this hysteresis response is complicated by the pinched curve around the zero displacement position due to permanent damage of the wood around each fastener (Figure3.6). The SAP90 program did not take into account nonlinear elements in its analysis. The connections were modelled as small flexible beam elements between beams and columns. The flexibility of these elements was estimated using the initial slope of the moment-rotation curve determined from cyclic tests of a typical connection. The modulus of elasticity of these elements was kept the same as for the beam members, and the moment of inertia of the section was adjusted, considering the length of the element, to achieve a 107 Analytical Prediction representative rotational stiffness for the connection (Appendix D). Particular attention was paid to assure that the shell element simulating the mass was connected to the node between this small element and the beam, instead of the corner node between this element and the column. The DRAIN-2DX analysis included a subroutine developed at the University of Florence to enable the simulation of nonlinear metallic fasteners connecting wooden members (Cecotti and Vignoli, 1989). This subroutine reproduced the skeleton of a typical moment-rotation curve of the connection, keeping track of the maximum previous rotation for each connection and using different slopes to take into account the deterioration of the wood. Parameters were defined using a moment rotation curve developed from a cyclic test on a connection (Figure 5.1). In the four-slope model, the outside envelope was defined by two loading slopes, K i and K 2 , an unloading slope equal to the initial loading slope K u a return slope K e , and a force associated with zero displacement, F0. The model also included an inner slope, K * , to simulate the pinched loop for subsequent cycles. In addition to these properties, the six-slope model included a third slope, K 3 , in the definition of the outside envelope, as well as the possibility of using an unloading slope, K s , different from the initial loading slope, Kv Since these skeleton models simulated only a limited number of slopes, the parameters were defined to achieve a maximum of precision for a certain amplitude of rotation. For each analysis, a first simulation with approximate parameters was done to determine the maximum amplitude of rotation for a given excitation. This was needed to select the slope parameters that best represent the maximum outside loops used for this analysis. Smaller amplitude loops are also present in an earthquake excitation, but it was found that the lack of accuracy for the smaller cycles did not significantly affect the overall shape of the response curve. The curve parameters used for this study were determined from the moment-rotation curve defined by cyclic testing of an eight-pin connection. Analytical models with four- and six-slope skeletons were available; both results were compared with experimental response. 108 Analytical Prediction In DPSA-WoodFrame, the forces and displacements were analysed for each dowel within a connection through a time-step procedure. For this reason, the analysis required the embedment properties of the wood in the directions parallel and perpendicular to grain, the yield criteria of the steel, as well as geometrical properties of the dowel and the connection (figure 5.3). Experimental tests were needed to determine the material properties, but, unlike for DRAIN-2DX, no cyclic testing on a full connection was required. This analysis simulated the damage imparted to a connection by keeping track of the gap in the wood surrounding each dowel- The embedment properties of the wood at an angle to the wood fibres were calculated from the perpendicular and parallel to grain properties using Hankinson's formula. 5.2 Earthquake excitation 5.2.1 Input acceleration record A factored version of the Landers earthquake from Joshua Tree station in the East-West direction was reproduced during the shake table test. During the first test with maximum acceleration of 0.5g, the shake table acceleration was monitored, and this record was used as input excitation for the different analytical simulations. For SAP90, the record was a time history with 0.02 second time increments for a total time of 70 seconds. Smaller time increments could not be used due to the limited memory capacity of the version of SAP90 used for this analysis. For DRATN-2DX, the same acceleration record was used with a time increment of 0.0039 second since it was able to handle larger data files. The acceleration record for DRAIN-2DX was, however, cut to 50 seconds to limit the number of steps in the analysis to 10 000 using a time increment for the analysis of 0.005 seconds. This limitation later proved to be unnecessary. 109 Analytical Prediction 5.2.2 Consecutive excitations The damage caused in the connection during one test affected the response of the frame to subsequent excitations, and it was important to include this residual damage in the analysis. The SAP90 analysis reproduced only linear connections and did not take into account any damage in the connections. The DRAIN-2DX program could simulate consecutive excitations within one analysis, accumulating the damage from one simulation to the next. The number of consecutive simulations was limited, however, since the simulated responses of the frame were recorded in the same file. To limit the length of the output file to 40 000 entries, a maximum sequence of three consecutive simulations was analysed. The sequence of simulations studied was 0.15g, O.Sg and 0.5g. Minimum damage was caused to the connection by the 0.15g simulation. It was, however, important to analyse two simulations at 0.5g since the damage caused by the first one highly influenced the response to the second excitation. Since it was not possible to use different acceleration records for the different simulations, the 0.1 Sg simulation was not the actual acceleration of the shake table during that test but 30% of the 0.5g record. Therefore, the comparison of the response from this simulation to the shake table test results was possible for the extreme values, but not for the whole time history. 5.3 Damping SAP90 performed a linear analysis utilising the mode superposition method to determine the response of the structure. Each mode could have a different damping ratio. In this case, a 5% damping ratio was chosen for all the modes since it was a medium value often used and no other information regarding damping allowed for a better approximation. For the DRA1N-2DX analysis, a viscous damping ratio of 5% was also recommended by Ceccotti et al. (1994). For large deformations, the hysteretic damping was automatically included in the analysis by the 110 Analytical Prediction moment-rotation curve of the connections, and the importance of viscous damping was limited. For small deformations, the skeleton model of the connection stayed in the linear range and the analysis did not include the hysteresis damping found in reality. Therefore, the addition of sufficient viscous damping to overcome this lack was important and it had a major effect on the response of the frame. The viscous damping coefficient, which was included in the input file, was calculated using the damping ratio and the natural period of the frame, measured from Ambient Vibration Testing. In DPSA-WoodFrame, hysteretic damping due to the connection behaviour was automatically included in the analysis through the hysteresis loops. An additional equivalent viscous damping component could also be added by specifying the two Rayleigh factors in the input file. 5.4 Analytical results To enable a comparison with the shake table test results, parameters from the test-frame were used in the analyses, and the actual shake table accelerations were entered as ground accelerations. Analytical simulations were performed for a low amplitude simulation (0.15 g), as well as for the first and the second simulations at high amplitude (0.5 g) to observe the ability of the model to account for previous damage sustained by the structure. Only SAP90 and DRAIN-2DX were used for this comparison, since DPSA-WoodFrame was still in the development phase. 5.4.1 Linear analysis (SAP 90) The SAP90 program simulated the linear response of a frame using a time history analysis. This analysis had a limited application since it did not include the nonlinearity or the increasing damage within the connections. SAP90 was first used to approximate the response of the frame during its design. Following the shake table test, the comparison of the SAP90 analytical results with the actual response of the frame allowed the assessment of the degree of error resulting from a linear analysis. I l l Analytical Prediction 5.4.1.1 Determination of results The program SAPTIME was used to create time history files for the displacement of given joints, as well as total base shear for the structure. A problem was encountered in determining the dynamic forces in the different elements. The solution files of SAP90 such as .F3F and .FEF included the results from the static analysis only, and SAPTIME had a memory problem in plotting the time histories of element forces. Since the solution provided by SAP90 was very far from the actual behaviour of the test-frame, bending moments in the connections were approximated using the horizontal acceleration of these nodes. The exact solution of the base shear was judged to be sufficient to evaluate the performance of the SAP90 model regarding its prediction of the forces within the structure. J. 4.1.2 Comparison of results For the simulations at 0.15g, the linear analysis overestimated the displacements and the forces when compared to the shake table test. For the larger amplitude simulation at 0.5g, the calculated relative displacements of the top floor were too small with large associated forces (table 5.1, Figure 5.6). This observation supports the present seismic design philosophy, which promotes ductility and larger deformation in the structure to dissipate the energy and reduce the intensity of the forces involved. The first natural frequency of the frame was estimated by SAP90 using a modal analysis at 1.9 Hz, a value that was much lower than the measured 3.0 Hz from the ambient vibration test on the undamaged structure. It is assumed that the friction between the different components of the connections caused this increased stiffness at the ambient vibration level, which would disappear rapidly with larger excitations. 5.4.2 Nonlinear analysis with DRAIN-2DX In preparation for the shake table tests, DRAIN-2DX was used to simulate the response of the frame, subjected to different earthquakes, and to select a suitable record that would produce a highly nonlinear 112 Analytical Prediction response of the connection. An earlier version of DRAIN-2DX (version 1.03, December 1992) with a four-slope model was used. After the test, a more refined analysis was performed. A new version of the program (version 1.10, November 1993) was available, where a six-slope model could also be used for the simulation of the connections. The model was also refined by adding the mass of the frame to the concrete masses in the input file. The parameters of the connection model were adjusted for the exact range of rotation found during the first simulation. It was possible to use consecutive simulations to analyse the effect of cumulative damage on the connection. 5.4.2.1 Comparison of moment-rotation curves The moment-rotation curves simulated by DRAIN-2DX were compared with the curve obtained from previous cyclic testing, since the results from the first shake table test at O.Sg did not include any measurement of the connection rotations. The main goal of this comparison was to determine if the parameters used for the simulation of the connection were accurate. Firstly, the maximum range of rotation for a particular amplitude of excitation was deterrriined, and the curve parameters were redefined with more precision (Figure 5.1). Secondly, different amplitudes of excitation were used to verify that the simulated connection followed the typical curve. As expected, the six-slope model followed the outside curve with more accuracy (Figure 5.4). The value F 0, which deterrnines the residual force for a zero displacement, varied for the tested curve, but had to be defined as a single value for the simulation. Even though a median value was chosen, a significant error could be expected from the simulation. As discussed in section 5.1.4, the parameters were chosen to best reflect the connection behaviour at the largest rotation of 0.04 radian. With these slope parameters, the curve simulating the first excitation at 0.15g stayed strictly linear. The six-slope model was used to try to induce some hyteretic damping by having the unloading slope K 5 larger than the loading slope Ki, which, however, did not produce better results. 1 1 3 Analytical Prediction The program DRAIN-2DX was also capable to simulate the behaviour of a frame subjected to a series of consecutive excitations. The general shape of the simulated moment-rotation curve for the second excitation at 0.5 g was compared with the curve found through shake table test. In both cases, the pinched envelope due to the degradation of the connection is evident (Figure 5.5). 5.4.2.2 Comparison of displacements and rotations The top floor displacement time history was the main variable chosen to compare results between analytical simulations and shake table tests. For the 0.15g simulation, the analytical response of the frame was larger than the experimental results for the whole duration of the excitation (Figure 5.6(a)). This error was mainly due to the lack of hyteretic damping at this level of rotation, since the curve parameters were chosen for the maximum rotation of the connection of 0.04 radians. The error was smaller for the six-slope model than for the four-slope model since it was possible to reproduce the outside curve with more accuracy in the six-slope model. For the first simulation at 0.5g, the six-slope model seemed to reproduce the response with more accuracy, both for the first cycles with large displacement, and for the smaller displacement cycles (Figure 5.6(b)). This could be attributed to a better fit Of the outside envelope and initial slope. Unfortunately, the top displacement of the frame was not measured during this shake table test, and it had to be determined by double integration of the acceleration record. The inaccuracies created by this procedure limited the ability to compare the time histories of the relative displacements. The extreme values of displacement were the only values that were compared with confidence since they could be verified relative to the subsequent excitation. 114 Analytical Prediction For the second simulation at 0.5 g, the connections were already damaged, and the model followed the inner loops which were identical for both models. Therefore, the analytical response was similar for both models (Figure 5.6 (c)). In general, this analysis provided a good response, except for a drift which was caused by accumulation of residual displacement during the three analytical simulations. The drift reached its maximum displacement for this last simulation. For the rotation of the connection, the observations for the 0.15g simulation and the second 0.5g simulation are similar to those for the relative displacements of the top floor (Figure 5.8). This similarity was expected since the relative displacement of the structure was mainly caused by the rotation of the connections. No comparison was possible for the first simulation at 0.5g since no connection rotations were recorded during this shake table test (see Chapter 3). 5.4.2.3 Comparison of top acceleration The shake table accelerations were subtracted from the accelerations measured at the top floor during the tests to permit the comparison with the relative acceleration record generated by the DRAIN-2DX analyses. The choice of the node at which the analytical record was measured had no importance since a translational slaving was imposed on the nodes at each side of the two connections, so that they would have the same horizontal or vertical displacements. For the 0.15g simulation, the four-slope model overestimated and the six-slope model underestimated the maximum response (Figure 5.7(a)). For both models, the shape of the acceleration time history did not follow the measured acceleration. For the large amplitude simulation, both models gave very good results compared with experimental testing (Figure 5.7(b)). In terms of maximum acceleration, the two models had error within ±20% for the 0.15g excitation and an error of 10% for the two excitations at 0.5g. 115 Analytical Prediction 5.4.2.4 Comparison of forces The base shear was the only force measured during the shake table tests. The analytical maximum base shear was calculated for each simulation by adding the shear force at the end of each column. This information was available in a summary produced in the DRAIN-2DX output files before the time histories for each simulation. The analytical simulation at 0.15g produced excessive forces and displacements. The next two simulations at 0.5g resulted in maximum forces that correlated with the measured response from the shake table testing (Table 5.1). The maximum moments in the connections were also compared (Table 5.1). These moments were approximated for the shake table tests using the floor accelerations. As observed for the base shear, the analytical moments were too large at low amplitudes, and correlated with the measured response for the large amplitude excitations. 5.4.2.5 Possibilities of improvement During this investigation, several parameter adjustments were tried to study the behaviour of the DRAIN-2DX analytical models. Suggestions are made regarding other factors that could be varied to improve the analytical results. • The residual displacement accumulated during consecutive analytical simulations were not part of the actual behaviour of the frame. It could be the result of the shortened acceleration records that only included the important part of the excitation. If needed, a linear displacement shift can be applied in the analytical process. 116 Analytical Prediction • The six-slope model could be used to generate some hysteretic damping at small rotations by using an unloading slope larger than the initial slope. The small loops created by this procedure would dissipate energy and represent more accurately the response of the frame to the 0.15g simulations. This was tried, but without success at this point. • The use of the actual record of the shake table acceleration at 0.15g would also improve the analytical response of this simulation. An analysis with only one simulation at 0.15g using the actual record and adjusting the curve parameters to this smaller range of rotation would allow to observe its influence on the simulated response. • To simulate the third test at 0.15g, which occurred after the larger excitations, a second acceleration record will need to be included in the DRAIN-2DX analysis. According to the program manual, it is possible to simulate consecutively two different acceleration records, but no success was achieved and further investigations are needed in this area. • The new version of the program using a six-slope model, allows for the return slope K6 to be either constant or variable (Figure 5.1). To have a varying slope, the parameter Kg has to be set to -1. During this investigation, the program did not complete the analysis of the model with a constant slope Kg. An incomplete loop formed by the different slopes was probably the cause of this error. Further investigation is needed to determine which value of Kg would be accepted by the analysis. The subroutine was recently improved to overcome a problem regarding the Kg slope, and it could be investigated to see if it solves this problem. 117 Analytical Prediction 5.5 Concluding Remarks By comparing the different analytical results with shake table testing for the same structure, several conclusions could be drawn. Programs such as SAP90 are available to perform a three-dimensional dynamic linear analysis, and the information required to simulate the connections is limited to an approximate stiffness value for an equivalent spring. A linear response can provide good results for very small excitation levels, yet, for larger excitation, the linear model greatly overestimates forces and underestimates displacements. The DRAIN-2DX program performs a dynamic analysis considering only two dimensions, but it can simulate the specific nonlinearity of dowel type connections. To model the behaviour of these connections in the analysis, a typical moment-rotation hysteresis curve, which is defined from a cyclic test on a similar connection with the same geometry, is needed. The different connection parameters are adjusted on the hysteresis curve for the expected range of rotation to optimise the accuracy of the response. Once the parameters are adjusted, a good simulation of the dynamic response of the frame can be obtained. For the simulations performed in this study, the adjustment of the curve parameters was done using the range of rotation for the 0.5g simulations. For this reason, the response of the frame to the small amplitude excitation at 0.15g was overestimated. The response to the 0.5g simulations was generally close to the experimental behaviour. For the first simulation at 0.5g, the. six-slope model was more accurate than the four-slope model, yet for the second simulation at 0.5g the two responses were similar. An analysis with DPSA-WoodFrame would only require material properties to simulate the dynamic behaviour of a frame using dowel type connection. More work needs to be done on this program, however, before any dynamic results can be compared. 118 Analytical Prediction Maximum Maximum Shake Linear DRAIN-2DX Analysis Table Values Ground units Test Analysis 4 Slope 6 Slope Model Acceleratio n SAP90 Model Total Relative Displacement • 15 g mm 20 33 48 29 1st 0.5g mm 1921 109 240 221 "O^g m 217 - 238 209 Relative Acceleration Top Floor .15 g m/s2 -2.9 to 2.9 -2.8 to 2.4 -3.5 to 3.2 -2.3 to 2.3 1st 0.5g m/s2 -9.3 to 8.8 -11.1 to 9.3 -10.4 to 10.0 -10.3 to 9.9 ""O g^ m/s2 -9.9 to 9.3 - -10.9 to 10.1 -10.3 to 11.1 Total Base Shear • 15g kN 8 11 11 8 1st 0.5g kN * 33 26 25 ""O g^ N 34 - 26 25 Rotation in Connections .15 g 103rad 3.9 4.8 8.0 3.9 1st 0.5g l(r3rad 39.31 19.2 43.0 39.0 ""CSg 10"3rad 46.1 - 46.1 42.1 Moment in Connections .15 g kN.m l.l 1 * 3.8 2.3 1st 0.5g kN.m 6.51 * 6.8 6.9 -"O^g N.m 7.11 * 6.9 6.9 * No results due to memory limitation 1 Estimated using measured accelerations Table 5.1 Comparison of Maximum Values of the Frame Response 119 Analytical Prediction Moment Hysteresis Curve measured from Cyclic Testing -0.06 0.00 0.06 Figure 5.1 - Slope parameters for DRALN-2DX connection models 120 Analytical Prediction P(W) TQ2 S T E E L WOOD Figure 5.3 - Material properties needed in DPSA input file 121 Analytical Prediction Moment (kN.m) 6 . 0 0 — I 0 . 0 0 - 6 . 0 0 — DRAIN-2DX - 4 Slope Model i i / DRAIN-2DX Cyclic Test 6 . 0 0 — \ 0 . 0 0 — \ - 6 . 0 0 DRAIN-2DX - 6 Slope Model DRAIN-2DX Cyclic Test - 0 . 0 6 0 . 0 0 Rotation (Rad) 0 0 6 Figure 5.4 - Moment-rotation curve from DRAIN-2DX compared to cyclic test results 122 Analytical Prediction Figure 5.5 - Moment-rotation curve from DRAIN-2DX compared to shake table test results 123 Analytical Prediction 20.00 — 1 20.00 — , 20.00 20.00 20.00 20.00 0.00 Shake Table Tes t 20.00 DRAIN-2DX Analys is - 6 Slope Model Figure 5.6(a) - Relative displacement of the top floor - 0.15g simulation 124 Analytical Prediction Shake Table Test 0.00 -100.00 100.00 H -100.00 H SAP90 Analysis DRAIN-2DX Analysis - 4 Slope Model 4 0 0 0 Time (sec) 5 0 0 0 Figure 5.6(b) - Relative displacement of the top floor - first 0.5g simulation 125 Analytical Prediction Shake Table Test 100.00 (mm) 0.00 -100.00 SAP90 Analysis DRAIN-2DX Analysis - 4 Slope Model 0.00 10,00 Time (sec) 50.00 Figure 5.6(c) - Relative displacement of the top floor - second 0.5g simulation 126 Analytical Prediction Shake Table Test SAP90 Analysis DRAIN-2DX Analysis - 4 Slope Model 0.00 30.00 4 0 0 0 T i m e (sec) 5 0 0 0 Figure 5.7(a) - Relative acceleration of the top floor - 0.15g simulation 127 Analytical Prediction 10 —, 0.00 10.00 20.00 30.00 4 0 0 0 T i m e (sec) 5 0 0 0 Figure 5.7(b) - Relative acceleration of the top floor - first 0.5g simulation 128 Analytical Prediction 10.00 i 1—n—r 20.00 30.00 40.00 Time (sec) Figure 5.7(c) - Relative acceleration of the top floor - second 0.5g simulation 129 Analytical Prediction Shake Table Test SAP90 Analysis 0 —\ Not Available DRAIN-2DX Analysis - 4 Slope Model 0.00 10.00 20.00 30.00 40.00 T . 50.0 Time (sec) Figure 5.8(a) - Rotation within the mid connection - 0.15g simulation 130 Analytical Prediction 0.04 (Rad) Shake Table Test 0 —\ Not Available -0.04 0.04 SAP90 Analysis 0.00 Not Available DRAIN-2DX Analysis - 4 Slope Model 0.00 T ime (sec) 50.0 Figure 5.8(b) - Rotation within the mid connection - first 0.5g simulation 131 Analytical Prediction Shake Table Test o H SAP90 Analysis Not Available DRAIN-2DX Analysis - 4 Slope Model DRAIN-2DX Analysis - 6 Slope Model Not Available i | i | 10.00 20.00 4 0 0 0 Time (sec) 5 0 0 0.00 .  .  30.00 Figure 5.8(c) - Rotation within the mid connection - second 0.5g simulation 132 Conclusions and Further Studies Chapter 6 C O N C L U S I O N S A N D F U R T H E R STUDIES The dynamic behaviour of dowel type connection and its influence on the seismic response of a two storey timber frame were investigated. This study included three specific parts: the experimental testing including cyclic test on specific dowel connections and a shake table test of a two-storey Parallam® frame; the development of a dynamic analysis program for timber frames including a finite element model based on material properties to simulate the specific behaviour of individual dowels; and the comparison of analytical results from different commercial dynamic analysis programs with the response of the frame monitored during the shake table test. 6.1 Conclusions 6.1.1 Experimental Testing Cyclic tests on different configurations of connections using a centre steel plate and tight fitting steel dowels showed good ductility and a satisfactory response to several loading cycles, with limited strength degradation and acceptable pinching of the hysteresis loops. The cyclic response of these connections was used to verify the finite element model of connections based on basic material properties, as well as to calibrate the connection stiffness and hysteresis curve for the commercial frame analysis programs. The main experimental part of this study was a shake table simulation of a two-storey frame using Parallam8 members and eight dowel double connections with hidden steel plate. The main conclusions from the analysis of the data provided by the momtoring of this test are: • The in-plane longitudinal motion of the frame was mainly controlled by the stiffness of the semi-rigid dowel connections, while the vertical or out-of-plane motions recorded were negligible. 133 Conclusions and Further Studies • The moment-curvature relation of the connection under dynamic loading showed the same hysteresis behaviour as it did under cyclic loading, with a good elastic response and relatively high stiffness under small excitation, and a highly nonlinear response under high amplitude excitation. • The displacement response of the frame increased by about 20% for the second high amplitude simulation, yet it reached an equilibrium point at which consecutive excitations at the same amplitude created no further damage. • The displacement response of the frame to the low amplitude excitation increased by 400% for the frame with damaged connections compared to the response of the undamaged frame to the same excitation. 6.1.2 Development of an Analytical Model The development of the frame analysis program was done in several steps. Prior to this study, a finite element program had been developed by Foschi (1995) to simulate the behaviour of a mild steel dowel supported by a wood foundation, based on fundamental material properties. This model was implemented in a connection analysis program, during the present study, and proved to provide good results compared to cyclic test data on the eight pin double connection studied. The development of a dynamic analysis program to simulate the response of a timber frame was outlined. The analysis of an entire frame using this technique involved a large number of variables resulting in excessive computing time, mainly due to the iterative process. Problems related to the iterative process still need to be resolved to ensure convergence for all cases. 6.1.3 Analytical Prediction Analytical results from two commercial computer programs were compared with the response of the timber frame to the shake table excitations. The behaviour of the frame was initially analysed with a linear model using SAP90. This program did not consider the hysteresis damping in the connection, and resulted 134 Conclusions and Further Studies in larger forces and smaller displacements than the shake table results. A non-linear analysis of the frame was also performed by DRAIN-2DX. The program executed a time history analysis of the frame using non-linear elements to simulate the connection. The parameters for these nonlinear elements were calibrated to follow the hysteresis curve developed through cyclic testing of a similar connection. This analysis overestimated the response of the system for small excitations, but provided a good prediction of forces and relative displacements for high amplitude ground acceleration records. This model was able to consider damage caused in the connection by previous loading. In conclusion, the dynamic response of a timber frame with semi-rigid dowel connections was found to be highly controlled by the response of the connections. The connections of the frame formed with eight tight fitting mild steel dowels and a hidden steel plate provided good ductile behaviour through-out a series of consecutive earthquakes excitations. It was evident, however, that the analytical simulation of semi-rigid timber frames requires a good understanding of the connection behaviour. For small excitations that remain in the elastic range of the connection, a linear analysis program was found to provide satisfactory results, but a nonlinear analysis simulating the hysteresis curves of the connections is required for any level of excitation surpassing this range. Special attention needs to be paid to the selection of parameters for empirical hysteresis models to assure that the assumed response accurately reflects excitation levels as experienced in an earthquake. 6.2 Further Studies Considering the importance of semi-rigid connections in seismic design of timber structures, as well as the promising results obtained from this study regarding the dynamic behaviour of the tested frame and the analytical prediction of its response, several research topics are suggested for further study. Thereby, the body of knowledge regarding semi-rigid frame for timber constructions can be expanded, and permit the 135 Conclusions and Further Studies ultimate goal to be reached, namely the accurate prediction of the response of timber structures under earthquake excitation. Future research topics recommended are: • Verification of the analytical program PSA-WoodFrame with test results for different configurations of the connection, using several steel plates and more dowels. • Verification of the static analysis program PSA-WoodFrame for a multi-degree of freedom frame with more than one double connection, with special attention paid to problems related to the stability of the iterative process. • Development of the dynamic analysis program DPSA-WoodFrame using the theoretical background presented in this study, and investigating problems related to the stability of the iterative process. • Further analysis with DRAIN-2DX including a modification of the different curve parameters to enable a better simulation for lower excitation levels. • Investigation of the precision of a DRAIN-2DX analysis, using curve parameters determined from hysteresis loops simulated by the finite element model PSA-WoodFrame for a specific connection configuration. • Development of different monitoring systems to measure the response of the connection during shake table tests of a frame that will be more accurate and limit the possibilities of faulty readings • A study on the dynamic response of composite systems including shear walls and semi-rigid frames around large openings, considering the behaviour of a multi-storey building with the possibilities of soft storey effect as well as torsion due to differential stiffness of the different lateral load resisting systems. • Investigation regarding the reliability index of frame with dowel type connections using the reliability of the different basic material properties. • Exploring the accuracy of ambient vibration tests on semi-rigid timber frames, analysing the natural frequency degradation and the possibility of using the concept of a damage index on such frames. • Research on non-destructive techniques to assess the damage of dowel type connections and to approximate its response to subsequent excitations. 136 References R E F E R E N C E S Aasheim E. 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(1992) Reliability of Bolted Wood Connections, Journal of Structural Engineering, ASCE, Vol. 118, No. 12, 3362-3375 143 APPENDIX A 144 • top M o p 2 . 6 ^ 7 ) V^AlkAfW, \afl.Tv, CY.rm— 0-S\ A _ W L <Li-0 ^ ^..111,1^ JtC cL^ -0_. "T, CJ 2 - ~1 o 9, -\ H-Up / V ^ b -» -1,62- -U /o,o<| O z a, 8> wo" 3 ^oA 3 5~-0, i 5s 2 3,S Q, >6 • so S-O, iS^ C / w i 'A ) a 5 - e ^ 6 - O, 5"< . . a . 4 1.6 &z -<\\ • 16 s <1 S> to'1 6 t o p : e33 - e > 7 - o. S"< <3L_ - ^ 2 , 2 < 2 v o - 3 © 3 = 3 1 ' / o - 3 CcfL X 1 I 4 ? "Si Q-tj <<«« Carv-uja.cTU.o-y , I > A J \ A . "Lkfl- M e l d ^ Sbar*-. j ^ a i t c d>x -<UJ 6 148 *2 {flvr.i H 1^ 9 APPENDIX B 150 Monitoring of the Connections To monitor the relative displacement within a connection, seven LVDTs were used, as described in section 3.2.2.3. The three relative rotations: column-plate, plate-beam and column-beam, were measured in addition to the relative horizontal and vertical displacement between the beam and the column. A schematic drawing of the connection served to show the different measurement needed in the trigonometric transformation between LVDT measurement and displacement values. • . . 11". • i i i • • • • a 151 Using the connection parameters measured prior to testing, as well as the LVDTs measurements, the deformed shape of the connection was calculated. It was assumed that the steel plate and the wood members were infinitely rigid compared to the dowels. (c) Parameters at each time step during the test (Deformed Connection) 152 Calculation of the relative Displacement Undeformed Connection 0 i = cos - l f n2 D - E11 -A 2 A -2 * E * A - 1 ,8l ~ Si s in ( ) # 1 1 - 0 1 - v 0 i = cos - 1 s ( A 2 A ' - C 2 - £ 2 ^ -2* C * £ /*! '= £ * c o s ( # n ) A2 = C * c o s ( # u + e 2 ) - 1 A l ~ A 2 angle = sin ( ) Deformed Connection 0 i = cos - l ( D'2 - E'2 - A2 ^ -2* E'*A # 1 1 = # ' l ~ ^ y 0 1 = cos - 1 f A 2 - C ' 2 - £ . 2 -2 * C ' * E y 4 j = £ ' * C O S ( 0 ' l l ) yf 2 = C ' * c o s ( # ' 1 1 + # ' 2 ) angle ' = s in 1 ( A* - A ) rotation beam - column Aangle = angle - angle' Vertical displacement Ayt = A[ - At &y 2 = A z ~ A 2 Horizontal displacement 5 d = 8 d - Aangle Ax i = D cos 8 d - D ' cos 8 d 153 Connection parameters measured prior to testing Mid-Height (mm) Top measured drawing* measured drawing* A 337 337 A 330 330 D 342 341 D 338 338 C 626 627 C 613 612.7 E 454 527.7 E 458 457.9 gl 20 gl 20 g2 50 g2 48 hi 79 hi 88 h2 81 h2 80 h3 84 h3 82 A 218 219.5 A 202 , 203 vl 14 vl 15 vl+v2 363 . ;. vl+v2 358 vl+v2+v3 583 Vl+v2+v3 562 i2 179 i2 192 jl 13 jl 15 J2 80 J2 80 J3+J2 210 J3 169 beam column beam column 11 99 86 11 102 105 bl 78 36 bl 366-190 36 12 99 86 12 100 206 b2 179 140 b2 265-190 136 13 200 202 13 204 105 b3 78 37 b3 367-190 35 14 200 202 14 203 206 b4 179 138 b4 264-190 137 * The length A ,C ,D and E were calculated from a AutoCad drawing o f the connection using a l l the other basic parameters measured. 155 c C '• TRIGO IS TO SOLVE CONNECTION DISPLACEMENT FROM THE TEST ON THE TIM BER FRAME c (Middle Connection) C -C Written by: Caroline Frenette C C i v i l Engineering Department C University of B r i t i s h Columbia C C June 1996 C c IMPLICIT REAL*8(A-H,O-Z) CHARACTER*40 NAME,NAME1,NAME2,name3 CHARACTER*80 TITLE,TITLE1 DIMENSION DH1(20000),DH2(20000),DV1(20000),DV2(20000) DIMENSION DD(20000),DE(20000),DC(20000) • c C * INPUT c WRITE(*,900) 900. FORMAT(/' ENTER THE TITLE'/) READ(*,1002) TITLE1 WRITE(*,1000) 1000 FORMAT(/' ENTER 1 IF DATA FILE EXISTS, 0 IF IT IS TO BE CREATED'/) READ(*,*) NANS IF (NANS.EQ.1) GO TO 10 WRITE(*,1001) 1001 FORMAT(/' ENTER NAME OF INPUT FILE TO BE CREATED'/) READ(*,1002) NAME 1002 FORMAT(A) OPEN(UNIT=l,FILE=NAME,STATUS='UNKNOWN') WRITE(*,888) 888 FORMAT(/' ENTER PROBLEM TITLE'/) READ(*,1002) TITLE WRITE(1,1002) TITLE 3 WRITE(*,1003) 1003 FORMAT(/' ENTER A'/) READ(*,*) Al WRITE(1,1009) Al 1004 FORMAT(15) 1009 FORMAT(5E15.6) WRITE(*,1006) 1006 FORMAT(/' ENTER D*/) READ(*,*) Dl WRITE(1,1009) Dl WRITE(*,1008) 1008 FORMAT(/' ENTER C'/) READ(*,*) Cl WRITE(1,1009) Cl WRITE(*,1012) 1012 FORMAT(/' ENTER E'/) READ(*,*) E l WRITE(1,1009) El WRITE (*, 1015.) 1015 FORMAT(/' ENTER DELTA'/) READ(*,*) DELTA1 WRITE(1,1009) DELTA1 WRITE(*,1018) 156 1018 FORMAT (/' ENTER Gl AND G2 ' /) READ ( *, * ) G1,G2" . WRITE(1,1009) G1,G2 WRITE(*,1020) • 1020 FORMAT(/' ENTER H2-H3'/) READ(*,*) H2MH3 WRITE (1, 1009.) H2MH3 WRITE ( *•> 1028) 1028 FORMAT(/' ENTER DISTANCE BETWEEN -LVDT VERTICAL'/) READ (*, *) DELTAY ' •' ' • ' WRITE(1,1009) /DELTAY WRITE(*,1032) 1032 FORMAT(/' ENTER DISTANCE BETWEEN LVDT HORIZONTAL'/) READ(*,*) DELTAX ' WRITE(1,1009) DELTAX WRITE(*,1035) 1035 FORMAT(/* ENTER DISTANCE TO CENTER OF THE CONNECTION(X AND Y)'/) READ (*,*) XCON,YCON WRITE(1,1009) XCON,YCON CLOSE(1) GO TO 11 C ' 10 WRITE(*,1021) 1021 FORMAT(/' ENTER THE NAME OF THE INPUT FILE'/) READ(*,1002) NAME OPEN(UNIT=2,FILE=NAME,STATUS='OLD') READ(2,1002) TITLE READ(2,*) Al READ(2,*) Dl READ(2,*) Cl READ(2,*) E l READ(2,*) DELTA1 READ(2,*) G1,G2 . READ(2,*) H2MH3 READ(2,*) DELTAY READ(2,*) DELTAX READ(2,*) XCON,YCON CLOSE(2) C 11 WRITE(*,1040) 1040 FORMAT(/' ENTER NAME OF DATA FILE'/) READ•(*, 1002) ' NAME1 WRITE(*,1045) 1045 FORMAT(/' NUMBER OF STEPS IN DATA FILE'/) READ(*,1004) NSTEP OPEN(UNIT=3,FILE=NAME1,STATUS='OLD') DO 1=1,NSTEP READ (3, *) DH1 (I) ,DH2.(I) , DV1 (I) , DV2 (I) , DD (I) , DE (I) , DC (I) END DO CLOSE(3) WRITE(*,1050) 1050 FORMAT(/' ENTER NAME OF OUTPUT FILE 1 and 2'/) READ(*,1002) NAME2 READ(*,1002) NAME3 OPEN(UNIT=4,FILE=NAME2,STATUS='UNKNOWN') OPEN(UNIT=5,FILE=NAME3,STATUS='UNKNOWN') WRITE(4,1002)TITLE1 C c C 157 C * MAIN PROGRAM, SIZE OF VECTORS C 100 TETA1 = ACOS(((D1**2)-(E1**2)-(A1**2))/(-2*E1*A1)) . DEDAV = ASIN((G2-G1)/Al) DEDAH = ATAN(H2MH3/DELTA1) TETA11 = TETA1-DEDAV TETA2 = ACOS(((DELTA1**2)-(Cl**2)-(El**2))/(-2*C1*E1)) A l l = E1* COS(TETA11) A12 = Cl*COS(TETA11+TETA2) ANG1 = ASIN((A11-A12)/DELTA1) ANGLE1 = ACOS(((El**2)-(Al**2)-(Dl**2))/(-2*A1*D1)) C , : . DO 110 I=1,NSTEP C1P = Cl + DC(I) E1P = E l + -DE (I) DIP = Dl + DD(I) TETA1P = ACOS(((D1P**2)-(E1P**2)-(Al**2))/(-2*E1P*A1)) TETA11P = TETA1P-DEDAV - TETA2P = ACOS( ( (DELTAl* :*2)-(ClP**2)-(ElP**2)-)/(-2*ClP*ElP) ) A11P = E1P'*.COS(TETA11P) A12P'= C1P * COS(TETA11P+TETA2P) • YY1 = A l I P - A l l YY2 = A12P-A12 ANG1P = ASIN((A11P-A12P)/DELTA1) DANG = ANG1P - ANG1 ANGLE IP = ACOS(((ElP**2)-(Al**2)-(DlP**2)-)/(-2*Al*DlP)) DANGLE1 = ANGLEIP-ANGLE1 XX = -(Dl*COS(DEDAH)-D1P*C0S(DEDAH-DANGLE1)) ,XXC = XX + (XCON*SIN(DANG)) + (YCON*(1-COS(DANG))) YYC - YY1 + (YCON * SIN (DANG) ) - (XCON* (1-COS (DANG) ) ). C ANGBP = ATAN((DH1(I)-DH2(I))/DELTAX) ANGPC = ATAN((DV1(I)-DV2(I))/DELTAY), C WRITE(4,1009) XXC,YYC,DANG,ANGBP,ANGPC WRITE(5,1009) xx,yyl,danglel 110 CONTINUE CLOSE(4) CLOSE(5) STOP END 1 5 8 •;ocooo .', -.1 cn co I '/] V] (/) -I -I •'~,nr,uo s CD 9 .1 Pc ^ 3 D a > £ * 'Tka^o— Cix^«_juSoAj^cr«-^ ULXJJN-Q- (Aon*-?-. u A - i - < ^ O _ V^oiLuOixA CX3J^<!,JLS^O^>,*,syy^ 13^ 103 C V / O T L. v' DT \ -0,0 3 * o ( O f l 4 -0,0 3 2. -> otO 1 <\ & - o ( 1 o, 1 -o, 13 o, \ 1 e. -o, i < • - o, 10 | C ,  L V O '  t -o.o0! o (o5" 2. -0,03 -*» 0(G»"2-<\ - 0 , 0 5 ^ 0 , 0 ^ 3 5" -o, a 5" o^3 £ -o , o & •»>o )o5" "? -O t O fe -%»o,o^ ^ -0,0 1*1-* o, £34 1 0 ~o \ 3 ^ °» 1 3 1 1 - o, l o,o°i 13 -o.is' \' 2 -=*- 0 , 0 3 " 8 10 l \ \-z 1 s I 2 + 5^" c /I D 1 6 0 Calculat ion of error produced in the calculat ion of the rotation for beam-plate and plate-column pc and bp are inversed for the mid connect ion, so Iv is the horizontal and lh id the vertical. dl =1.625 d2 := -1.851 lv := 130 rotpc(dl,d2,lv) .-dl - d2 lv d3 - 0.974 d4 := 1.553 lh := 179 rotbp(d3,d4,lh) d3 - d4 lh ddl = .05 dd2 := .04 div := 1 errpc : = errbp : = relerrpc : = relerrbp = ddl rotpc(dl, d2,lv) rotbp(d3,d4,lh) dd3 errpc |rotpc(dl,d2,lv)| errbp |rotbp(d3,d4,lh)| rotpc(dl,d2,lv) = 0.027 errpc = 8.98-10~4 relerrpc = 0.034 ddl dd3 dd3 := .02 dd4 := .05 dlh = 1 dd2 rotpc(dl,d2,lv) dd4 rotbp(d3,d4,lh) dd2 dd4 — rotpc(dl,d2,lv) d lv — rotbp(d3,d4,lh) d lh rotbp(d3,d4,lh) = -0.014 errbp =4.699-10 relerrbp = 0.033 -4 6 Calculat ion of error produced in the calculation of the rotation for beam-plate and plate-column top connect ion dl := 1.625 d2 = -1.851 lv = 192 rotpc(dl,d2,lv) : = dl - d2 W d3 := -0.974 d4 := 1.553 lh = 169 rotbp(d3,d4,lh) d3 - d4 lh ddl := .1 dd2 = .4 div := 1 errpc : = errbp : = relerrpc rel errbp ddl rotpc(dl,d2,lv) -rotbp(d3,d4,lh) dd3 errpc |rotpc(dl,d2,lv)| errbp |rotbp(d3,d4,lh)| rotpc(dl,d2,lv) = 0.018 errpc = 0.003 relerrpc = 0.149 ddl + dd3 + dd3 := .13 dd4 := .1 dlh := 1 — rotpc(dl,d2,lv) dd2 -—rotbp(d3,d4,lh) d d4 rotbp(d3,d4,lh) = -0.015 errbp = 0.001 relerrbp = 0.097 dd2 + dd4 + div rotpc(dl,d2,lv) dlh -rotbp(d3,d4,lh) \(oZ A D - -0,208 c - " o , e z.S 7 ' / o ' ^ / N A ^ i 0 ( t I /A^AA T o p (2 A D - o , ^ 5 • A 6 • IS,OS d -- ± 0 , 2 . 5 " •S •/£>"* . lO,l o,zB .. O , o o 3 IO ,0 0,0C o , 9 (*3 ' J O O O O C ,-> vi vx/iViVi i , •/! tVl Ul >- — * vi ui u) tr ir ar><- — <pjr< A A C C L ^ Vnnan' A •a, kO,OS 1 0 ,05" io , l i o,4 1 o,B i o , < £ • io lop r^ lb i ic r* - b p o,oo I ^ ^ L ^ r o ^ J c n * - hc\ors ^ o r W D T 3 Sheet2 Ivdt m m N 1 1.89 1 8 9 0 " 1 2 4 ? 2 1.241 3 0.95 0.950 4 1.87 1.870 5 1.515 1.515 6 1.722 1.722 7 1.878 1.878 8 3.25 3.250 9 1.525 1.525 10 3.65 3.650 11 3.31 3.310 12 4 .795 4 .795 13 7.51 7 .510 14 7.46 7.460 \bS U v / D T 4 2 1 0.5 0 -0.5 -1 -2 -4 slope MO) OUT slope MO) IN slope MO) slope mm/V V/mm inch/V V/inch 8 2509 1.237 0.6253 0.318 0.1628 0.0092 -0.1443 -0.2987 -0.6054 -1.2143 3.259973 -0.03245 3.249109 -0.03105 3.25779 -0.03266 3.244997 -0.02979 3.269447, -0.02646 3.253509 -0.02974 9 2_512 2.61 1.3062 0.6527 0.3243 -0.0031 -0.3345 -0.6596 -1.3142 -2.62 1.528636 0.006488 1.525605 0.006146 1.530899 0.00275 1.527409 0.004339 1.52948 0.008496 1.526744 0.007229 10 1094 1.1003 0.5514 0.2764 0.1396 0.0021 -0.1343 -0.2717 -0.5451 -1.0893 3.652299 -0.01193 3.648299 -0.00959 3.642323 -0.00778 3.641658 -0.00765 3.664605 -0.00609 3.653898 -0.00811 11 1095 1.2053 0.6032 0.301 0.1493 -0.0022 -0.1533 -0.3046 -0.607 -1.2066 3.313855 0.005486 3.304692 0.006421 3.31306 0.00475 3.303567 0.00674 3.321028 0.010204 3.306691 0.007142 12 2361 0.8363 0.4176 0.2083 0.1026 0,0011 MD.1035 •-0.2082 -0.4144 -0.8348 4.792504 -0.00266 4.80861 -0.0024 4.782004 0.002372 4.792729 0.000806 4.788643 -0.00613 4.814222 -0.00242 13 14 5 color fuH 2510 0.5346 0.5297 2.637 0.2651 0.2618 1.321 0.01304 0.127 0.6594 0.0616 0.0612 0.326 0.0006 -0.001 -0.0007 -0.0684 -0.0681 -0.3333 -0.1334 -0.1327 -0.6597 -0.2665 -0.2706 -1.32 -0.5357 -0.5458 -2.633 7.521509 0.107858 7.706398 0.140873 6.765517 0.316116 6.100069 0.355976 7.47199 -0.00052 7.507232 0.002783 7.466731 0.031941 7.557357 0.024184 7.507072 0.030566 7.58462 0.023626 7.331595 -0.00699 7.424055 0.001781 1.5172706 0.0005563 1.5149805 0.0015799 1.5155249 0.0018431 1.5118262 0.0035456 1.5201456 0.0039409 1.5171247 0.0025429 6 2360 2.328 1.1634 0.5818 0.2896 0.0004 -0.2897 -0.5781 -1.1568 -2.317 1.72274 -0.0041 1.72415 -0.0026 1.71804 0.00077 1.71892 0.00041 1.72629 -0.0012 1.72874 -0.0002 7 2359 2.13 1.063 0.531 0.262 0.0016 -0.271 -0.534 -1.065 -2.13 1.87824 0.002588 1.879177 0.003329 1.877358 0.00277 1.881513 0.001226 1.879014 0.00261 1.87818 0.002298 not g o o d nnt fjnnt* 3.25 1.525 3.65 3.31 4.795 7.51 7.46 1.515 1.722 1 878 0.307692 0.655738 0.273973 0.302115 0.208551 0.133156 0.134048 0.660066 0.58072 0 532481 0.127953 0.060039 0.143701 0.130315 0.18878 0.295669 0.293701 0.0596457 0.0678 0 073937 7.815385 16.65574 6.958904 7.673716 5.297185 3.382157 3.404826 16.765677 14.7503 13 52503 1 2 5 i_vtrr 605-02 605-01 2510 3 1.6048 2.415 1.979 2 1.0591 1.6114 1.321 1 0.5263 0.8071 0.6594 0.5 0.2665 0.4022 0.326 0 0.003 -0.0012 -0.0007 -0.5 -0.2602 -0.4047 -0.3333 -1 -0.5208 -0.807 -0.6597 -2 -1.05 -1.6099 -1.32 -3 -1.59 -2.417 -1.9777 1.885721 1.241497 1.515885 -0.00811 0.000566 0.001011 1.899181 1.241111 1.514981 -0.00648 0.000372 0.00158 1.874123 1.241662 1.513844 0.00322 0.000104 0.002727 1.894362 1.240121 1.511826 -0.00346 - 0.00087 0.003546 1.884567 1.24239? 1.51807 -0.01171 0.001977 0.002929 1.899844 1.243461 1.517125 -0.00677 0.002511 0.002543 slope mm/V 1.89 1.241 1.515 V/mm 0.529101 0.805802 0.660066 inch/V 0.074409 0.048858 0.059646 V/inch 13.43915 20.46737 16.76568 3 4 HP-JO HP-KL 0.000 0 0.0019 -0.0004 0.100 -2.54 -2.68 -1.3764 0.105 -2.667 -2.81 -1.432 0.110 -2.794 -2.924 -1.4845 0.210 -5.334 -5.456 -1.9977 0.97695 2.4009813 0.042688 0.3539147 0.950719 1.8661349 -0.00086 0.0027193 0.95 1.87 1.052632 0.5347594 0.037402 0.073622 26.73684 13.582888 66 CD E to "O o o o ' o CO LL o 15 t_ .a O 1 6 7 oqoqoo rrSjfijtjiuui — UJUJUJ g g 3 v> w co cc tr 2 <jp w cu r\i ry 3 I A o 3 o p r o v e J 2 - V . O L O A V O O - A A _ £»S*aoJi-5 2 , II ^ - — • — •-\ ii. 8 , 3 ! i — - — 3-4, a " ° , I S | £ , 5 " 0 , 4 6 "«3,o. 3 3 , 8 APPENDIX C 169 Degrees of freedom of a beam element: In the direction perpendicular to the beam element: w = A / Q (^)a - Displacement w'=Ml (l)a - Slope w"=Ml (I,) a - Curvature In the direction parallel to the beam element: u = N% (£,)a -Displacement u'=Nl (Z,) a - Slope Shape functions: M 0 (l,cl) = (8-15c: + 10c;3-3S5 )/16 M 0(2, t;) = (5-7^-6^2 + 10*; 3 +V - 3 £ s ) ( A / 3 2 ) M 0(3,4) = (l-ci-2 t; 2 +2^3 + $ 4 - $ s ) ( A 2 /64) Mo(4,^) = 0 M 0(5,^) = 0 M 0 (6,i;) = (8 + 15i;-10c=3 +3<;5)/16 A/ 0 (7,^ ) = (-5-74 + 642 + 10;; 3 - 4 4 -3cl5)(A/32) M0(S,B,)=ll + ^ -2V -2i; 3 + S4 + ^ 5 ) ( A 2 /64) A f 0 ( U ) = 0 A/0(lO,c;) = 0 M , ( l , ^ ) = (-15 + 3042 - 1 5 £ 4 ) ( 2 / 1 6 A ) M,(2,c;) = (-7-12c; + 30c;2 + 41; 3 -15cl4 )/16 M 1(3,4) = (-l-4c; + 6c;2 + 4 £ 3 -5<;4)(A/32) M 1(4,4) = 0 M1(S£) = 0 Mx (6,S) = (l5-30c:2 + 15i;4)(2/16A) A/j (7,c;) = (-7 + 12i; + 30c;2 -4c;3 -15c;4 )/16 M t (8,c;) = (l-4c;-6c;2 +443 + 5Z,4 )(A/32) M,(lO,c;) = 0 170 M2 (1,5) = (60i; - 60£ 3 )(4 /16 A 2 ) M2 ( 2 ,5 ) = (-12 + 60c;+ 12c;2 - 6 0 5 3 ) (2 /16A) M 2 (3 ,5 ) = (-4 + 125 + 125 2 - 2 0 5 3 )/16 M 2 ( 4 , 5 ) = 0 M2 ( 5 ,5 ) = 0 Mi ( 6 , 5 ) = ( - 6 0 5 + 6 0 5 3 )(4/16A 2 ) M2 (7 ,5 ) = ( l 2 + 6 0 5 - 1 2 5 2 - 6 0 5 3 ) (2 /16A) M2 (8,5) = ( -4 -125 + 125 2 + 2 0 5 3 )/16 M2 (9 ,5 ) = 0 M 2 ( 1 0 , 5 ) = 0 J V 0 ( i , 5 ) = o 7 V 0 ( 2 , 5 ) = 0 i V 0 ( 3 , 5 ) = 0 NQ (4,5) = ( 2 - 3 5 + 5 3 ) / 4 7V0 ( 5 , 5 ) = ( l - 5 - S 2 + 5 3 ) ( A / 8 ) N0(1,Z,) = 0 /V 0 (8 ,5) = 0 7V0 ( 9 , 5 ) = (2 + 3 5 - 53 ) / 4 M i o , 5 ) = ( - i - 4 + S 2 + £ 3 ) ( A / 8 ) TV, (1,5) = 0 7V 1 (2 ,5 ) = 0 / V 1 ( 3 , 5 ) = 0 Nt (4,5) = ( -3 + 3 5 2 ) / ( 2 A ) / V 1 ( 5 , 5 ) = ( - l - 2 5 + 3 5 2 )/4 Nl(6,Z,) = 0 Nt (7 ,5 ) = 0 /V,(8,5) = 0 7V1 ( 9 ,5 ) = ( 3 - 3 52 ) / ( 2 A ) A ^ 1 ( l 0 , 5 ) = (-l + 25 + 3 5 2 )/4 171 Computer Programming - HYST.FOR The program HYST.FOR was created to evaluate numerically the response of a given dowel when subjected to a known force or displacement. HYST.FOR 1. Input: - Length and diameter of the dowel - parameters for steel yielding (E, Sy) - Number of wood layers and thickness of each layer - parameters for each wood foundation (QO, Q1, Q2, XK, DMAX, PMAX) - Boundary Conditions - Number of finite elements - Number of Gauss points per element in x and y directions - Tolerance for convergence of the system of equations 2. Set Shape Functions at the different integration points along the length of the element 3. Set initial values for each Gauss point: - foundation: maximum positive and negative prior displacement and actual last force to 0.0, last sign for displacement to 1, and slope force/displacement to XK. - Steel: last stress and last strain to zero, slope stress/strain to E 4. Input known force or displacement at a given node 5. Set a first vector of displacement {Ai} 6. For each Gauss point of each element, using the displacement vector {X0} and the Shape Function, determine: - foundation: the displacement, the force and the slope - Steel: the strain, the stress and the slope 7. Using the Weighting Function determine the energy balance {0} and its gradient V0. ndem AH nx W f Tld r T 1 I / \ +^P(\w\)t 2 i = i .1/ Moi)k +R 172 AH nx »y r _ i 4 i=i i=i 1 Jv A "* MQjMQi ) dw 8. Introduce boundary conditions, and include known force or displacement. 9. Solve for the new displacement vector. {AM-A,).[ve]-*{-0) 10. Check convergence: if converged go to the next time step and go back to 4. NDOF „ I 9] if —— < TOL then converged NDOF _ s #S 11. Update: - foundation: maximum positive and negative prior displacement, actual last force, last sign for displacement, and slope force/displacement. - Steel: last stress, last strain, and slope stress/strain 12. go back to 6. for another iteration with the new {A;+i} Computer Programming - CONNECTG This subroutine was created to include the work done by the forces within each connector in the analysis of a frame. The vector of forces and the matrix of the gradient of this last vector were calculated by the subroutine for a given connection. CONNECTG 1. input: - x, y coordinates for the member node - x, y coordinates for the plate node - Number of dowels - x, y coordinates for the all the dowels 2. Call HYST to determine the initial slope in the two directions parallel and perpendicular to the wood member (KX, KY). 173 3. Calculate {Qu} and {Qv} for all the dowels and multiply by [A], the transformation matrix from local coordinates, which are parallel and perpendicular to the wood member, to global x, y coordinates. Angle of a wood member { XN } global = [A]-{ XN } local x l cos a -sin a -ui sin a cos a v i 1 Or X2 cos a. -si/i a U2 sin a cos a 1 4. Calculate the initial gradient for the connection 5. Set force vector equal to zero 6. Return to main program to calculate the first approximation of displacement {XN} 7. First approximation: The subroutine needs to determine the gradient vector considering the orientation of each dowel for the given vector of displacement for this connection {XN} For each dowel: a) Calculate the vector displacement for this dowel b) Determine the intensity and orientation of this displacement c) Calculate the wood bearing properties in this direction using Hankinson's formula (Note that this orientation is now fixed for this given dowel for the entire analysis) d) Calculate a new gradient using the initial slope in this direction for each dowel. e) Set the parameters for the dowel calculation to initial values Return to the main program to calculate the second approximation of displacement {XN} 8. Other approximations: The subroutine needs to determine the force vector and the gradient vector for the given vector of displacement for this connection {XN} For each dowel: a) Calculate the intensity of the displacement and the direction, (since the orientation is now set to the original orientation) b) Update the parameters for the dowel calculation from the last step or from the last iteration. c) Calculate the force corresponding to this displacement using the subroutine HYST d) Calculate the slope using last iteration. e) If the slope is negative or if it is larger than the initial slope, the slope calculated during the previous iteration is used. f) Add the result from this dowel to the force vector and to the gradient. g) Update the parameters for the dowel calculation for the next iteration or the next step. 174 9. Return to main program to calculate the next approximation of displacement {XN} Computer Programming PSA-WoodFrame The program PSA-WoodFrame was created to evaluate the response of an entire frame to statically applied forces using the subroutine described previously for the determination of the connections responses. PSA-WoodFrame 1. Input: - Number of nodes, coordinates of each nodes - Number of members and member properties - Support conditions - Number of dowel-type connections - Constant concentrated loads on nodes - Constant distributed loads on members 2. Calculate the Stiffness matrix for all the linear elements 3. For each connection, calculate the transformation matrix from 6 DOFs of the connection, {XN}, to all the DOF of the structure{X}. 4. Calculate the constant force vector {R} FOR EACH LOAD STEP: 5. Enter the load for this load step 6. Add the varying load to the constant force vector {R} to determine the force vector for this load step {RR} FOR THE FIRST LOAD STEP: 7. Calculate the initial slopes for all the connections, and calculate an equivalent stiffness matrix assuming a linear system [Ko* 1 = [K] + [£ df/dx for all connection] 8. Introduce boundary condition from support parameters, and approximate linearly the first vector of displacement for the structure. {X<,} = [KOVMRR} 9. Using the subroutine CONNECT, determine the orientation of displacement for each dowel and calculate the initial slopes considering the new wood parameters in this orientation for each dowel for all the connections. Recalculate an equivalent stiffness matrix using this new oriented stiffness. [VGo* 1 = [K] + IX df/dx for all connection] 175 10. Introduce boundary condition from supports parameters, and recalculate a vector of displacement for the structure using the new equivalent stiffness. {x0} = [Veo']-|MRR} FOR ALL OTHER LOAD STEPS: 11. For the first iteration only, use the displacement vector { X|} and connection force vector {XFconn} from last iteration of last step, and use the gradient (V60* ] calculated at the first step, go to 15 12. Calculate Displacement vector {XN} for each dowel-type connection 13. Call subroutine CONNECT to calculate force vector and slope matrix associated with this displacement. 14. Add force vector and slope matrix to the global nonlinear force vector {XFconn} and gradient rnatrixtXdFconn /dX]. 15. Calculate the error on this approximation and the gradient matrix {e1}=IK].{X1} + {ZFconn}-{RR} [V6i] = [K] + [ZdFco,,,, /dX] 16. Approximate a new displacement vector {xM} = {x,} + [ve,]-1* {6,} 17. Check convergence: a) If Z0| / 20o < tolf, converged go to 18 b) If 2{Xi+i -X, }/ Z{Xm -XO }< tolx, converged goto 18 c) If EOi+i / £6| < 0 and P0|+i | < P0| I, go to 12, but do not update dowel parameters in next iteration d) For the first 3 times that £O|+i/£0i < 0 and \EBM | » [Z0, |, go to 12, use initial slopes, and do not update dowel parameters in next iteration e) After the first 3 times that Z0i+1/Z 0, < 0 and (Z0i+i I » P0i I, split the step in two and start again the load step at 6. 18. When converged update dowels parameters for each connect using CONNECT and go to 5 for the next load step Computer Programming DPSA-WoodFrame Even if the following process have not been implemented yet, an outline of the procedure is proposed. DPSA-WoodFrame (Note: In the following description * indicates modification from PSA-WoodFrame) 1. Input: - Number of nodes, coordinates of each nodes - Number of members and member properties 176 - Support conditions - Number of dowel-type connection - Constant concentrated loads on nodes - Constant distributed loads on members *- Concentrated masses on nodes *- Distributed masses on members *- Factors for equivalent viscous damping (Rayleigh damping) *- Parameters a, 8 for the linear acceleration approximation 2. Calculation of Stiffness matrix for all the elements except dowel-type connection *2a. Calculation of Mass and damping matrices *2b. Calculation of the [AA] matrix 3. For each connection, calculation of transformation matrix from 6 DOFs {XN} of the connection to all the DOFs of the structure{X}. 4. Calculate constant static load vector {R} FOR EACH TIME STEP: *5. Enter the acceleration for this time step {Y} *6. Add the dynamic load [M]*{Y} to the force vector {R} to cktermine the force vector for this time step {RR} *6a. Calculate the {B} vector for this time step using the displacement, velocity and acceleration vectors from the previous time step. FOR THE FIRST TIME STEP: *7. Calculate the initial slopes for all the connection, and calculate an equivalent [AA] matrix assuming a linear system [AAo* ] = [AA] + [I df/dx for all connection] *8. Introduce boundary condition from supports parameters, and approximate linearly the first vector of displacement for the structure. {Xo} = [AAo,]1*({RR}+{B}) *9. With CONNECT, determine the orientation for each dowel and calculate the initial slopes considering the new orientation between the load and the wood for each dowel for all the connections. Calculate the initial gradient matrix using this new oriented stiffness for the connectors. [VGo* ] = [AA] + [Z df/dx for all connections] *10. Introduce boundary condition from supports parameters, and recalculate a vector of displacement for the structure using the new equivalent stiffness. {X0} = rve0*]-1*({RR}+{B}) 177 FOR ALL OTHER TIME STEP: 11. For the first iteration, use the previous displacement vector { X ; } and connection force vector {EFconi,} from the last iteration of the last step, and use the initial gradient ( V 0 O " ] , go to 15 12. Calculate Displacement vector for each dowel-type connection 13. Call subroutine CONNECT to calculate force vector and slope matrix associated with this displacement. 14. Add force vector and slope matrix to the global nonlinear force vector {XFconn} and gradient matrix [EdFeo^/dX]. * 15. Calculate the error on this approximation and the gradient matrix {6|}= [AA] * {X|} + {XFconn} - ({RR}+{B}) [Ve1] = [AA] + pdFcooll/dX] 16. Approximate a new displacement vector {x1+1} = {x,} + rye,]1* {e,} 17. Check convergence: a) If 26, / X80 < tolf, converged go to 18 b) If Z{Xm -X,}/ I { X H . ! - X O }< tolz, converged go to 18 c) If £6t+i / XG| < 0 and |ZOi+i | < |Z9,1, go to 12, but do not update dowel parameters in next iteration d) For the first 3 times that L8,+i/Z8, < 0 and [LBM \ » £ 9 , |, go to 12, use initial slopes, and do not update dowel parameters in next iteration e) After the first 3 times that 26,+1/E 6| < 0 and pE6|+i1» fE8| |, split the step in two and start again the load step at 6. *18. Converged: a) Update dowels parameters for each connect using CONNECT b) Calculate the velocity vector for this time step using linear acceleration equation c) Calculate the acceleration vector using the general equation of motion to limit error due to the linear acceleration approximation. d) Go to 5 . for the next time step 178 A P P E N D I X D 179 B.C Bulletins from The Northwest Division T R U S J O I S T MACMILLAN P R O D U C T A P P L I C A T I O N A S S U R A N C E March 30, 1993 = PAA BULLETIN No. 4.1.0 Page 4 of 6 LTMrr STA' MICRO =LAM° LVL AND PARA! STIFFNESS (A Resistance Factor of 0.90Jis Included) ORED STRENGTHS AND •7 1.8EWS MICRO=LAM« LVL 2.0E WS PARALLAM* PSL Shear Modulus of Elasticity G 177,800 197.600 Modulus of Elasticity E 1.8x10' 2.0X10«A). |V. Flexural Stress Fb 4 1 1 0 m Compression Perpendicular to Grain, Parallel to Glue Line FcA 1185°' 1 0 3 0 m Compression Parallel to Grain Fc,, 3890 4585 > Iitf- <'>-1 Horizontal Shear Perpendicular to Glue Line Fv 450 460 (II For a 12 inch depth, d, for others *:uHipiy by ^ ^ [ - j j " * 5 (MICRO=LAM* LVL Only) °' For a 12 inch depth, d, for others multiply by "up X (PARALLAM PSL Only) ^ " ~ ^ c - / *1 fi.u) • -> l.!2x !.!0 P l Fcj. shall not be increased for duration of load (G-52G) 31 ^KJ: -to U i i Ktflicti'a/ vavdbitt't^ . fb^ £<r>v.ruc*I«* u.£c <j px/-6 b c e o , v a U M A s P Q 0^(,<, l t . r 80 A P S o V O O O O O j vo c/i 011-1-"'••SiliujSg ~i 5> 55 Ol CZ 1 cru a . c *— a - £ 3 51 ~T~ Ho si 5 fc-v • • I : 3,3 " O 6 A U * * 4 icy 181 r SAP90 INPUT FILE Model of the Frame - Pinned base, Modeled spring connections, both sides braced S Y S T E M V = 1 0 L = 1 J O I N T S 1 X = 0 Y = 0 Z = 0 2 X = 2 . 7 6 9 Y = 0 Z = 0 3 X=2 .769 Y = 2 . 0 0 0 Z = 0 4 X = 0 Y = 2 . 0 0 0 Z = 0 5 X = 0 Y = 0 Z = 1 . 3 7 5 6 X = 2 . 7 6 9 Y = 0 Z = 1 . 3 7 5 7 X = 2 . 7 6 9 Y = 2 . 0 0 0 Z=1 .375 8 X = 0 Y = 2 . 0 0 0 Z = 1 . 3 7 5 9 X = 0 Y = 0 Z=2 .695 10 X=2 .769 Y = 0 Z = 2 . 6 9 5 11 X=2 .769 Y = 2 . 0 0 0 Z=2 .695 12 X = 0 Y = 2 . 0 0 0 Z=2 .695 13 X = 0 . 1 0 0 ' Y = 0 Z=1 .375 14 X=2 .669 Y = 0 Z = 1.375 15 X = 2 . 6 6 9 Y = 2 . 0 0 0 Z=1 .375 16 X = 0 . 1 0 0 Y = 2 . 0 0 0 Z=1 .375 17 X=0 .100 Y = 0 Z=2 .695 18 X = 2 . 6 6 9 Y = 0 Z=2 .695 19 X = 2 . 6 6 9 Y = 2 . 0 0 0 Z=2 .695 2 0 X=0 .100 Y = 2 . 0 0 0 Z=2 .695 R E S T R A I N T S 1,20,1 R=0,0,0 ,0 ,0 ,0 1,4,1 R=1,1,1 ,0 ,0 ,0 F R A M E N M = 6 Z=-1 1 E=1 .52E10 W = 3 2 0 M = 3 2 . 9 4 S H = R T = 0 . 3 0 0 0 , 0 . 1 7 7 8 : B E A M S 2 E=1 .52E10 W = 2 0 0 M = 2 0 . 9 7 S H = R T = 0 . 1 7 7 8 , 0 . 1 7 7 8 : C O L U M N S 3 E = 2 E 1 1 A = 9 2 7 E - 6 1=0.518E-6.0.518E-6 W = 7 2 M=7.28 : B R A C E S L 7 6 X 7 6 X 6 . 4 4 E=2E11 A = 4 7 3 0 E - 6 l=22.2E-6 ,7 .07E-6 J = 1 9 3 E - 9 W = 3 7 0 M=37 : L A T E R A L B E A M W 150X37 5 E=1 .52E10 W = 3 2 0 M=32 .94 A=0 .054 l=3 .3E-6 ,7 .0E-6 J=1 .822E-6 : M O D E L E D S P R I N G S 6 E=2E11 A = 3 1 . 6 7 E - 6 l=7 .98E-11.7 .98E-11 J=8 .08E-10 M = 0 . 2 4 9 W = 2 . 5 : W I R E B R A C I N G 6 .35mm DIA 1 1 5 M = 2 LP=-2,0 G=7,1,1 ,1 : C O L 9 13 14 M=1 LP=-2,0 11 16 15 M=1 LP=-2,0 13 17 18 M=1 LP=-2,0 15 2 0 19 M=1 LP=-2,0 : B E A M S :10 6 7 M = 4 LP=3,0 :12 5 8 M = 4 LP=3,0 :14 10 11 M = 4 LP=3,0 :16 9 12 M = 4 LP=3,0 : L A T E R A L B E A M S 17 1 8 M = 3 LR =1,1,0,1,1,0 LP=1 ,0 18 4 5 M = 3 LR =1,1,0,1,1,0 LP=1 ,0 19 2 7 M = 3 LR =1,1,0,1,1,0 LP=1 ,0 2 0 3 6 M = 3 LR =1,1,0,1,1,0 LP=1 ,0 183 21 5 12 M = 3 LR=1,1 ,0 ,1 ,1 ,0 LP=1 ,0 22 8 9 M = 3 LR=1,1 ,0 ,1 ,1 ,0 LP=1,0 2 3 6 11 M = 3 LR=1,1 ,0 ,1 ,1 ,0 LP=1 ,0 2 4 7 10 M = 3 LR=1,1 ,0 ,1 ,1 ,0 LP=1 ,0 : B R A C E S 25 5 13 M = 5 LP=-2 ,0 2 6 14 6 27 15 7 2 8 8 16 2 9 9 17 30 18 10 31 19 11 32 12 2 0 : M O D E L E D S P R I N G E L E M E N T S 3 3 1 6 M = 6 LR=1,1,0 ,1 ,1 ,0 LP=1 ,0 3 5 4 7 M = 6 LR=1,1 ,0 ,1 ,1 ,0 37 5 10 M = 6 LR=1,1 ,0 ,1 ,1 ,0 3 9 8 11 M = 6 LR=1,1 ,0 ,1 ,1 ,0 4 0 2 5 M = 6 LR=1,1 ,0 ,1 ,1 ,0 41 6 9 M = 6 LR=1,1 ,0 ,1 ,1 ,0 4 2 3 8 M = 6 LR=1,1 ,0 ,1 ,1 ,0 4 3 7 12 M = 6 LR=1,1,0 ,1 ,1 ,0 : W I R E B R A C I N G , Used in pre l iminary tests S H E L L N M = 2 Z=-1 . 1 E=2e11 U = 0 . 3 W = 3 . 2 3 E 4 M = 3 . 3 E 3 2 E=2e11 U=0.3 W = 3 . 2 3 E 4 M=3 .3E3 1 J Q = 1 3 , 1 4 , 1 6 , 1 5 E T Y P E = 0 M = 1 TH=0.1 ,0 .1 :LP=3 2 J Q = 1 7 , 1 8 , 2 0 , 1 9 E T Y P E = 0 M = 1 TH=0.1 ,0 .1 :LP=3 T I M E H A T Y P E = 0 N S T E P = 3 5 7 0 DT=0 .02 NF=1 D=0 .05 N V = 1 0 NF=1 PRIN=1 N P L = 1 DT=0.02 N A M = 1 0 0 a LC=-1 NF=1 S = 9 . 8 1 e - 3 A N G L E = 0 . 0 A T = 0 . 0 \B<\ oogooo '/) w to yj 01 (/) si§§§ < 1-3 2. C. - I 3 2 0 ^ ^ M -N J O < L _ \ J v U M P £ D M A S S / 2- ^/U5-v*-«0 2 < 2 Kf - M q ^ 8 5 " OA^LAC = \ 3*5" ^ \^\U"6' >lo~u /^JLA »OCU_ = 0 , 4 2 3 4 0,0 15" - O, <J98 s O, <l £ 3 + 0 , 0 & 0 : O , -^8*4.Mg £ - 2 , 0 £ 2 ' / 6 ^ - 0 , 0 6 689 5 l3^9oHR. Fc.// = 4 5 - 0 5 " ^ : ^ 3 1,6 /So »«'» 6» 5 3 g,p H(P<Z| 11 — * 1-2. = 4 1 , 9 M R , 15 = cr^_i = . . r b j _ -38,? '/o6 NJ^<-P Y „ fc,r • A^ 1,2.1 ' / O 6 N / r 0 | q 1 ' / O ^ f N / \*1 2 ) , o , H l r b : 3 ^ , 5 " " = 39 ,5 - M-Po. \ 8 ? - C O N J r O C C L T i O N J C^fo^-p 3) k, r -4 6 "4 ' / O 6 '4 ' 2.1 ' /O k 9 - 4 3 ' / o 4 - o (o» 3 - S T AT Id LO A D S / / / V j K -4 a 8 188 • DRAIN-2DX INPUT FILE ! TELAIO Caroline per drain NEW ! CARICHI: 16.6 KN/2 per piano ! TERREMOTO ORIZZONTALE ! 100A = nome accelerogramma (senza N) ! 12800 = numero valori di accelerazione ! 9810 = accelerazione di picco ! 50.00 = tempo totale ! 0.003906 = delta t *STARTXX ! 0,execute * 1,data checking only CFRAME100 . 0 0 0 0 FRAME100 | ******************************************************* •NODECOORDS 1 XXXXXXXXXXYYYYYYYYYY ! NODI Dl BASE C 1 .0000.0 00000.0 C 2 2619.0 00000.0 ! NODI PILASTRO | LIVELLO C 3 0000.0 1375.0 C 4 2619.0 1375.0 ! NODI PILASTRO II LIVELLO C 7 0000.0 2695.0 C 8 2619.0 2695.0 ! NODI TRAVE I LIVELLO C 5 0000.0 1375.0 C 6 2619.0 1375.0 ! NODI TRAVE II LIVELLO C 9 0000.0 2695.0 C 10 2619.0 2695.0 *RESTRAINTS ! XYR NF S 110 1 S 110 2 | ******************************************************* 'SLAVING ! XYR MASTER SLAVED 5 1 1 0 3 5 5 1 1 0 4 6 S 1 1 0 7 9 S 1 1 0 8 10 | ******************************************************* •MASSES ! XYR MASS[Mg] NF NL S 110 0.4935 5 6 S 110 0.4805 9 10 | ******************************************************* *ELEMENTGROUP ! EVENT PD G R U P P O I 02 1 0 PILASTRI (TYPE 02) GRUPPO 1 !STIF*EXC*YIELD (NUMBER) 1 0 1 ISTIFFNESS TYPE ! YOUNG[MPA] STR HARD AREA[mm] INERTI[mm] Kii Kjj Kij 1 15170 1 31329 81.79e6 4 4 2 IYIELD SURFACE ! CODE My+ My- Pyc Pyt MA PA MB PB 1 2 38.72e6 38.72e6 0.97e6 1.0e6 0.90 0.30 0.90 0.30 IGENERATION !ELE NODEi NODEj INC STIFF YIELDi YIELDj 18^ 1 1 3 1 1 1 1 2 2 4 1 1 1 1 3 3 7 1 1 1 1 4 4 8 1 1 1 1 ! •ELEMENTGROUP ! EVENT PD GRUPP0 2 02 1 0 TRAVE (TYPE 02) GRUPPO 2 !STIF*EXC*YIELD(NUMBER) 1 0 1 (STIFFNESS TYPE ! YOUNG[MPA] STR HARD AREAfmm] INERTI[mm] Kii Kjj Kij 1 15170 1 53100 398.2e6 4 4 2 IYIELD SURFACE ! CODE My+ My- Pyc Pyt MA PA MB PB 1 1 104.87e6 104.87e6 [GENERATION !ELE NODEi NODEj INC STIFF YIELDi YIELDj 1 5 6 1 1 1 1 2 9 10 1 1 1 1 l . *ELEMENTGROUP ! EVENT PD STIFF DAMPING GRUPPO 3 03 0 0 0.001 CONN. ELASTICA ROTAZ.(TYPE 07) GR.3 !NUMBER OF TYPE OF CONN 1 [DEFINE EACH TYPE ! U1 U2 K1 K2 K3 K4 K5 K6 F0 COD 1 .008 .03 464e6 85e6 43e6 145e6 500e6 -1 0.5e6 3 IGENERATIO !ELE NODEi NODEj TYPE 1 3 5 1 2 6 4 1 3 7 9 1 4 8 10 1 | ************************************************ •RESULTS ! SPOSTAMENTO NODO 8 INODDISP 1ST NODE LAST NODE STEP NSD 2218 INODEacc 1ST NODE LAST NODE STEP !NSA 2211 ! CONNESSIONE SEMIRIGIDA DESTRA IN BASSO (LOWER) IELEM GROUP ELEM E 2 3 2 | ******************************************************* *NODALOAD CCCC CARICO DISTRIBUITO 5 00000 -4150 -1.6683E6 5 '• S 00000 -4150 1.6683E6 6 S 00000 -4150 -1.6683E6 9 S 00000 -4150 1.6683E6 10 | ******************************************************* *ACCNREC 100Ainpac1 (1e12.0) 100A 12800 .1 0 2 .0039059 0.0 | ******************************************************* •PARAMETERS ! OUTPUT STATIC - LOAD STEP ! RESULTS ENVELOPE OS 0 100 0 100 ! OUTPUT DYNAMIC ! STEP RESULTS TIME ENVEL OD 0 0 1 0 0 50 *STAT STATIC ANALYSIS N CCCC L0.05 | ****************************************************** *ACCN 30,1ST ITMINC MAX CODE DT 50.00 16000 1 0.003906 IADIR NAME SCALE 1 100A 2943 j ******************************************************* *REST | ******************************************************* *STAT STATIC ANALYSIS N CCCC L0.05 j ******************************************************* *ACCN 100,2nd ITMINC MAX CODE DT 50.00 16000 1 0.003906 IADIR NAME SCALE 1 100A 9811 I * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * : *REST | ******************************************************* *STAT STATIC ANALYSIS N CCCC L0.05 | ******************************************************* *ACCN 100,3rd ITMINC MAX CODE DT 50.00 16000 1 0.003906 IADIR NAME SCALE 1 100A 9812 | ******************************************************* *STOP 

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