UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Verification of an analytical hysteresis model for dowel-type timber connections using shake table tests Wong, Ernie Y. 1999

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2000-0305.pdf [ 9.53MB ]
Metadata
JSON: 831-1.0050148.json
JSON-LD: 831-1.0050148-ld.json
RDF/XML (Pretty): 831-1.0050148-rdf.xml
RDF/JSON: 831-1.0050148-rdf.json
Turtle: 831-1.0050148-turtle.txt
N-Triples: 831-1.0050148-rdf-ntriples.txt
Original Record: 831-1.0050148-source.json
Full Text
831-1.0050148-fulltext.txt
Citation
831-1.0050148.ris

Full Text

VERIFICATION OF AN ANALYTICAL HYSTERESIS M O D E L FOR DOWEL-TYPE TIMBER CONNECTIONS USING SHAKE TABLE TESTS  by ERNIE Y. WONG B.A.Sc, University of British Columbia, 1997  A THESIS SUBMITTED IN PARTIAL FULFDLLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCD2NCE 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 November 1999. ©Ernie Y. Wong, 1999.  In  presenting  this thesis  in partial fulfilment  of  the  requirements  for  an  advanced  degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for copying  of  department  this thesis for scholarly purposes or  by  his  or  her  may be granted  representatives.  It  is  by the head of  understood  that  publication of this thesis for financial gain shall not be allowed without permission.  Department of  C\  vlu  The University of British Columbia Vancouver, Canada  DE-6 (2/88)  1  <U&?Bie.r  fJ'&j  extensive  copying  my or  my written  ABSTRACT The purpose of this study was to verify the analytical results of a non-linear finiteelement model for predicting the seismic response of ductile dowel type momentresisting connections. The analytical model, F R A M E , is based on each dowel modelled as an elasto-plastic beam in a non-linear medium, thus allowing the formation of gaps between the beam and the medium.  The model calculates the response using basic  stress/strain information of the connector and the surrounding medium. The advantage of using basic material properties is that the model can automatically adjust to any connection configuration and input history. To validate the results of the analytical model, four test specimens were built and tested on the U B C Earthquake Engineering Shake Table.  The objective was to assess the  analytical model, to see i f its results could accurately simulate the response of connections subjected to different earthquakes and with different configurations.  The  specimens were made from parallel strand lumber (Parallam®), connected to the shake table by steel plates and tight-fitting mild steel dowels. A mass was placed on top of the column to simulate building loads. A Northridge(1994) and a Kobe(1995) earthquake record were used in this experiment. The relative displacements at the top and bottom of the column were obtained from the tests and used to verify the results from the analytical model. The basic material properties used in the analytical model were obtained through wood bearing tests on the Parallam® and tensile yield tests on the steel dowels. The comparison of the experimental results with the analytical predictions from F R A M E showed that F R A M E can accurately model the seismic response of ductile dowel-type timber connections. Of particular importance was the fact that F R A M E is able to adapt to different connection configurations and different excitation signals.  Ill •  TABLE OF CONTENTS ABSTRACT  ii  T A B L E OF CONTENTS  iv  LIST OF TABLES  vii  LIST OF FIGURES  viii  ACKNOWLEDGEMENTS  x  I. INTRODUCTION  1  1.1  Overview  1  1.2 Aims and Objectives 1.3 Scope 1.4 Outline of Thesis  2 3 3  IL BACKGROUND  5  2.1 Behaviour of Wood Under Applied Stresses 2.1.1 Parallel Strand Lumber 2.2 Seismic Performance of Timber Structures 2.2.1 Timber Connections 2.2.2 Ductility 2.2.3 Damping 2.2.4 Hysteresis 2.3 Analytical Models for Hysteretic Response 2.3.1 The Ceccotti-Vignoli Model 2.3.2 The Bouc-Wen-Baber-Noori Model 2.3.3 The Chui-Ni-Jiang Model 2.3.4 The Foschi Model 2.4 Summary III. ANALYTICAL MODEL  5 6 7 8 9 9 10 11 12 12 15 16 16 17  3.1 Introduction 3.2 The Dynamics Problem 3.3 Connector Analysis 3.3.1 The Finite Element Model 3.3.1.1 Modelling of the Steel 3.3.1.2 Modelling of the Wood Medium 3.3.2 Virtual Work for Connectors 3.3.3 Virtual Work for the Wood Medium -iv-  17 18 19 19 20 20 24 25  3.4 Connection Analysis 3.4.1 Description of Analysis for the Experimental Model 3.4.2 Virtual Work for the Connection 3.4.3 Numerical Integration 3.5 Model Inputs and Outputs 3.6 Concluding Remarks IV. TEST SPECIMENS AND ANALYTICAL PREDICTIONS 4.1 Description of Connection 4.2 Modelling of the Test Specimens 4.2.1 Masses 4.2.2 Boundary Conditions 4.2.3 Damping of the Connection 4.3 Simulation Results 4.3.1 Natural Frequency 4.3.1.1 3-dowel configuration 4.3.1.2 6-dowel configuration 4.3.2 Model Response 4.3.2.1 Summary 4.3.2.2 Final Model Results 4.4 Concluding Remarks V. EXPERIMENTAL TESTING  26 28 30 32 33. 33 35 35 37 37 38 40 41 41 .42 43 43 44 44 47 49  5.1 Mono tonic Tests 5.1.1 Dowel Yield Strength 5.1.2 Embedment Properties 5.2 Shake Table Tests 5.2.1 Design of Connection 5.2.2 Construction of Connection 5.2.3 Experimental Procedure 5.2.3.1 Choice of Excitation Record 5.2.3.2 Instrumentation and Monitoring 5.2.3.3 Testing Sequence 5.3 Impact Tests 5.3.1 Results 5.4 Shake Table Tests 5.4.1 Displacement Results 5.5 Concluding Remarks  49 49 49 50 50 52 58 58 61 62 63 63 64 64 68  VI. COMPARISON OF RESULTS  70  6.1 Overview 6.2 Natural Frequency  70 70 -v-  6.3 Comparison of Displacements 6.3.1 Configuration #1: 6-Dowel Connection with Landers Record 6.3.2 Configuration #2: 3-Dowel Connection with Kobe Record 6.3.3 Configuration #3: 3-Dowel Connection with Landers Record 6.3.4 Configuration #4: 6-Dowel Connection with Kobe Record 6.4 Correlation of Results 6.5 Power Spectral Density and Coherence of Results 6.6 Hysteresis Results for A Connector 6.7 Concluding Remarks VII. CONCLUSIONS AND FUTURE CONSIDERATIONS 7.1 Conclusions 7.1.1 Analytical Predictions 7.1.2 Experimental Results 7.1.3 Comparison of Results 7.2 Future Considerations  71 71 72 73 74 75 78 84 87 88 88 88 89 90 90  REFERENCES  92  APPENDIX A  95  A l . Shape Functions for beam elements A2. Input data files for F R A M E  96 98  A3. Output data files from F R A M E  100  APPENDIX B  104  B l . Calculation of a coefficient for damping  105  B2. Parallam® Data sheet  106  APPENDIX C  107  C I . Material bearing curves for steel dowels in Parallam®  108  C2. C3. C4. C5. C6.  110 112 116 123 124  Data sheets for accelerometers and stringpots Data sheets for shake table channels MathC A D worksheets for spectral analysis of results Layout of shake table bolt hole locations Calculation of Hysteresis Plots from Experimental Results  -vi-  LIST OF TABLES  Chapter 4 Table 4.1 - Summary of final F R A M E simulations  45  Chapter 5 Table 5.1 - Bearing parameters for 12.7mm dowels in Parallam® Table 5.2 - Summary of tests  50 63  -vii -  LIST OF FIGURES Chapter 1  Figure 1.1- Portal Frame Connections  2  Chapter 2  Figure Figure Figure Figure Figure  2.1 2.2 2.3 2.4 2.5  - Stress-Strain Relationship in Wood - Hysteresis diagram for typical timber connection - Slope Parameters for DRAIN-2DX Connection Models - Hysteretic SDOF System - Assemblage of Finite Element for Nailed Wood Joint  6 11 13 14 15  - Beam element for F R A M E model and the DOF's - Coordinate transformation for each beam element - Stress-Strain Relationship for Steel Dowel - Parameters for Load-Deformation for Wood Foundation - Typical Loading and Unloading Scheme in F R A M E - Load-Displacement relationship for wood foundation - Steel Dowels inserted in Parallam® member - Free-Body Diagram of Connection Model - Actual Connection Specimen  19 20 21 22 23 24 27 28 29  Chapter 3  Figure Figure Figure Figure Figure Figure Figure Figure Figure  3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9  Chapter 4  Figure 4.1 - Schematic and real connection detail 36 Figure 4.2 - Connection Configurations 36 Figure 4.3 - Test Specimen ready for testing 37 Figure 4.4 - Model of Experimental Mass 38 Figure 4.5 - Modeling the movement of the connection 39 Figure 4.6 - Coordinate axes for connector model 39 Figure 4.7 - Acceleration pulse for calculation of natural frequency of connection .42 Figure 4.8 - Free vibration response of 3-dowel configuration 42 Figure 4.9 - Free vibration response of 6-dowel connection 43 Figure 4.10 - Weight of Mass vs. Horizontal Displacement for connection configurations . . . . 44 Figure 4.11- Displacement Results from F R A M E simulations 45 Chapter 5  Figure Figure Figure Figure Figure Figure Figure Figure Figure  5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9  - Yield Strength of 12.7 mm Diameter Steel Dowels - Load and Dowel Orientation in Parallam® - Loading Direction of test specimens - Steel plate hole pattern - Parallam® hole pattern - Elevation of test connection - Test connection detail - Plan of connection geometry - Landers Earthquake record used in tests  -viii -  50 51 51 53 54 55 56 57 59  Figure Figure Figure Figure Figure Figure Figure Figure  5.10 - Kobe Earthquake record used in tests 5.11 - Spectral Analysis of Landers earthquake record ... • 5.12 - Spectral Analysis of Kobe earthquake record 5.13 - Spectral Analysis of impact tests on specimens 5.14 - Response of specimen #1 to Landers earthquake 5.15 - Response of specimen #2 to Kobe earthquake 5.16 - Response of specimen #3 to Landers earthquake . . 5.17 - Response of specimen #4 to Kobe earthquake  Chapter 6 Figure 6.1 - Comparison of Displacement Results for Configuration #1 Figure 6.2 - Comparison of Displacement Results for Configuration #2 Figure 6.3 - Comparison of Displacement Results for Configuration #3 Figure 6.4 - Comparison of Displacement Results for Configuration #4 Figure 6.5 - Correlation of Results for each configuration Figure 6.6 - Power Spectral Density Comparison of Results for Configuration #1 Figure 6.7 - Coherence of Displacement Results for Configuration #1 Figure 6.8 - Power Spectral Density Comparison of Results for Configuration #2 Figure 6.9 - Coherence of Displacement Results for Configuration #2 Figure 6.10 - Power Spectral Density Comparison of Results for Configuration #3 Figure 6.11 - Coherence of Displacement Results for Configuration #3 Figure 6.12 - Power Spectral Density Comparison of Results for Configuration #4 Figure 6.13 - Coherence of Displacement Results for Configuration #4 Figure 6.14 - Comparison of Hysteresis for Configuration #1 Figure 6.15 - Comparison of Hysteresis for Configuration #2 Figure 6.16 - Comparison of Hysteresis for Configuration #3 Figure 6.17 - Comparison of Hysteresis for Configuration #4  - ix -  59 60 60 65 66 67 67 68  72 73 74 74 76 79 79 80 81 82 82 83 83 85 85 86 86  ACKNOWLEDGEMENTS I would like to acknowledge the contribution of those who assisted me in completing this research project.  First of all, I am very grateful to my thesis supervisor, Dr. Ricardo O. Foschi, for his constant encouragement, guidance, and financial support during my years at U B C . With his unfailing support, I have strengthened my understanding of structural engineering and dynamics.  I also acknowledge the contribution of Dr. Helmut G.L. Prion, for his  technical support during this study and for expanding my interest in the area of timber structures.  They were both inexhaustible sources of knowledge for me and were  invaluable in helping me to the successful completion of this work. To them both, I give my thanks.  During this project, several people were integral in the successful completion of the shake table tests. These include Howard Nichol, in the shake table lab, and Doug Smith and Harold Schrempp in the Civil Engineering machine shop. I would also like to thank them for I greatly appreciated their help during this project.  I also gratefully acknowledge the financial support by Forest Renewal B C , administered by the Science Council of B C , for the duration of this project.  And last but not least to my colleagues, Felix, Hong, Nii, Tomas, Riyahd, and David for their support, their knowledge, and their humour.  Introduction  Chapter 1 INTRODUCTION 1.1 Overview Designers of timber structures have historically had little guidance in modelling momentresisting connections for seismic loads. Over the years, several analytical models have been created to try to capture the non-linear response of a timber connection under reversed cyclic loading, also known as a hysteretic response. Existing empirical hysteresis models require the calibration of several parameters to experimental results for each different connection type and input history. This is neither efficient nor economical for the analysis of numerous connection types and input histories. Ideally, an analytical model that uses only fundamental material properties to computationally produce realistic hysteresis plots would be very beneficial.  Timber structures have traditionally performed well during earthquakes, but they are by no means immune to their effects.  Recent earthquakes in California, Japan and New  Zealand have caused the failure of numerous timber structures resulting in heavy casualties and severe economic losses. In the case of the Kobe (Japan) earthquake (1995) many traditional timber structures with heavy roofs and little or no lateral strength were destroyed. Similar collapses have occurred in the Coalinga (1983) and Northridge (1994) (California) earthquakes because of the lack of proper lateral-load-resisting systems (LLRS) and inadequate detailing of the foundation and structural framing connections. The four most common types of L L R S in timber structures are shear walls, diagonal bracing, cantilever columns, and moment-resisting frames. The present trend in North American low-rise timber construction favours the use of shear walls and diaphragms while for larger timber structures, braced and moment-resisting frames are used.  Moment-resisting connections in frames are fairly common in New Zealand. Examples are shown in Figure 1.1 (Buchanan, 1989). The full strength of glulam (glue-laminated lumber) can be developed in such connections but the failure mode is very brittle. Attempts to create a more ductile connection by reducing the number of nails result in  l  Introduction  stress concentrations at highly loaded nails causing unfavourable failures of the glulam beams. (Buchanan & Dean, 1988) In earthquake design for structures, it is important to have ductile connections during earthquakes because a structure is then able to undergo large displacements without a significant loss in strength. This is achieved through the energy absorption capabilities of a ductile connection, which in turn results in more desirable failure modes. One type of connection that has been shown to be very ductile is the steel dowel connection shown in Figure 1. I f (Ceccotti and Vignoli, 1988).  Figure 1.1 - Portal Frame Connections  Connections are by far the most critical and difficult components to design.  This is  especially so in the case of timber structures because timber typically fails in a brittle manner in shear and tension perpendicular to the fibers. Dowel-type connectors are semirigid and have a relatively small diameter shaft, such as in pins, nails, and bolts.  One  type of dowel connection consists of steel plates and tight-fitting ductile mild steel pins embedded in wood. In Europe, the use of timber as a primary structural material has been popular for decades now, and the use of dowel type connections in large timber structures is not uncommon.  Although the response under static loading is well  2  Introduction  understood, the seismic behaviour of a dowel-type connection is presently unclear and difficult to model accurately. The difficulty in modelling arises from the combination of the crushing of wood around the dowels and the yielding of the dowels during repeated loading. Presently, a non-linear finite-element model (Foschi, 1999) is available for the analysis of dowel connectors in timber connections.  1.2 Aims and Objectives This study is part of an on-going research project by the U B C Civil Engineering and Wood Science departments and funded by Forest Renewal B.C. The overall aim of this project is the understanding of the 3-D, dynamic, non-linear, structural behaviour of timber structures and their reliability under earthquake conditions.  Several specific  objectives are envisioned. These include modelling:  •  the behaviour of timber shear walls  •  the behaviour of dowel-type connections  •  and implementing reliability analysis for earthquake damage of timber structures  An analysis program for dowel connections, F R A M E , has been developed (Foschi, 1999). The study in this thesis is aimed at verifying and validating the dowel connector model F R A M E using experimental results obtained from tests of dowel connections under earthquake loading. This was achieved by predicting the behaviour of connections with different configurations and under different earthquakes, and comparing the results with those from experimental shake table tests.  1.3 Scope One specific type of connection was investigated for the purpose of this thesis. Mild steel dowels and Parallam® were used because of the availability of data on their material properties from previous monotonic tests. The connection comprised of two steel side plates connected to the Parallam® member by a circular arrangement of tight-fitting mild steel dowels. The analytical model was able to accommodate a variety of configurations  3  Introduction  and different connector types, but only 3-dowel and 6-dowel configurations were used for this study.  1.4 Outline of Thesis  The work presented in this thesis is organised in the following manner:  •  Chapter 2 presents background information on wood, timber structures, structural timber connections, and a review on analytical hysteresis modelling of wood joints and dowel type connections.  •  Chapter 3 introduces the analytical model, including the finite element model for the connector, modelling of the wood , the dynamic problem and the input and output of FRAME.  •  Chapter 4 discusses the description and modelling of the connections tested, the assumptions and the analytical predictions from the model.  •  Chapter 5 discusses the experimental set-up and procedures for the shake table testing. In addition, the results from the tests are in this chapter.  •  Chapter 6 gives a comparison of the experimental and analytical results.  •  Chapter 7 summarises the conclusions of this study and offers suggestions for future research.  4  Background  Chapter 2 BACKGROUND 2.1 Behaviour of Wood Under Applied Stresses W o o d is similar to all building materials in that it has unique properties, some o f which are favourable and some o f which are not. For example, steel and concrete are extremely important structural materials in construction today even though steel most often needs to be protected from oxidation, while concrete is always subject to deterioration from freeze-thaw cycles. O n the other hand, timber structures have relatively low masses compared to their load carrying capacity because timber is such a lightweight yet strong material. One o f the advantages o f this in seismic design is that the load carrying capacity relative to its weight causes only correspondingly low forces o f inertia due to accelerations caused by earthquakes.  It is its unfavourable behaviour under  different loading conditions that make timber difficult to design with. Timber typically behaves in a linear-elastic manner under flexural and tensile stresses. The failure mode under these loads is brittle and undesirable, while under compressive stresses, timber behaves in an elasto-plastic fashion. In this case, ductile failure modes can be observed. The typical stressstrain behaviour o f wood is shown in Figure 2.1.  O f particular importance is the extremely low  resistance o f wood in the direction perpendicular to grain.  This usually causes problems in  connections where perpendicular to grain stresses are unavoidable. This can lead to failures that occur very suddenly and at a low load level. The ability to identify potential failures o f this type is very important in the seismic design o f timber structures.  5  Background  Figure 2.1 - Stress-Strain Relationship in Wood (Buchanan & Dean, 1988)  Historically, wood has been a widely used construction material due to its availability, lightweight, and ease of construction. Even today, most residential and low-rise commercial construction in North America use wood as its primary structural material. For larger projects, however, the near exclusive use of concrete and steel is due to a better understanding of their structural behaviour. For large timber structures, larger timber members are required. The need for larger and reliable timber structural elements has prompted the development of innovative-engineered wood products such as structural composite lumbers (SCL). One such product is Parallel Strand Lumber. 2.1.1 Parallel Strand Lumber Parallel strand lumber, also known as Parallam®, is a SCL product that is manufactured by gluing parallel strands of wood together. Each strand is typically 15 mm wide and 300 mm long. The strands are coated in resin and oriented parallel to the direction of the member axis, pressed to about one third of their original volume, and then cured via microwave application. Structural composite lumbers are engineered products that typically have three times the bending strength of a similar sized piece of lumber and are approximately thirty percent stiffer (Canadian Wood Council, 1995). This is because SCL's are free of knots and defects, and are produced at a low moisture content (-11%). The finished product has much less variability in material properties than sawn lumbers. In Canada, Parallam is made from Douglas Fir, while in the United States, Southern Pine is used. Member lengths are usually limited to 20 meters for ease of transportation but, since the production  6  Background process involves gluing wood strands together, larger lengths are possible. Parallam® can be used for a variety of structural applications. These include beam and column members in post and beam construction, headers and lintels for light framing construction, and intermediate and large members in commercial building construction. In order to fully utilise timber as a primary structural material in all types of construction, understanding of the structural behaviour of wood is required. 2.2 Seismic Performance of Timber Structures  The fortuitous seismic performance of timber structures have mainly been based on residential housing and other low-rise structures with soft connections and redundant lateral-load-resisting systems (Buchanan, 1984). Failures are usually due to poor foundation connections, no sub-floor bracing, and unsymmetrical layout or lack of lateral-load-resisting elements leading to torsional response (Cooney, 1979). Examples from the Coalinga (1983) earthquake showed that failures in older residential buildings were due to: 1) sliding of the structures off of their foundations because of inadequate connections between house and framing, and 2) extensive roof damage due to a lack of roof diaphragms, inadequate connections between roof segments and between roof and walls (Rihal, 1983). In medium and large-sized multi-storey timber structures, the seismic performance is not as well understood. This is because timber structures typically utilise very different lateralload-resisting systems (LLRS) for small and large-sized timber construction.  The most common  L L R S in low-rise wood structures are shearwalls and diaphragms whereas larger timber structures favour frame systems. The seismic designs of structures usually involve one type of L L R S , sometimes the combination of two.  One system that is often used for medium to large-sized timber structures is the moment-  resisting frame. This type of system involves a post and beam-type structure where the lateral loads are almost exclusively resisted in the column-beam connections.  Moment resisting frames can  usually be designed for elastic response because they are characteristically flexible (Buchanan, 1989.) When ductility is required, however, it must be provided in the connections between the timber members. This is usually achieved via semi-rigid connections with mechanical fasteners as in Figure 1.1. Since the load deformation relationship of timber connections with fasteners such as nails, screws or dowels show a distinct elasto-plastic behaviour, a favourable ductile behaviour  7  Background  even under the stress of an earthquake can be expected with correctly designed timber connections. (Blass, 1989). 2.2.1 Timber Connections  Timber connections are the most important part of design in timber structures. The behaviour of wood dictates that bending and tension failures are brittle while compression failures are typically ductile. Knowing this, it is critical to design connections to limit the bending and tension forces in particular members while taking advantage of the ductile behaviour of wood in compression. In cases where bending and tension may become critical in some members, the connections must be designed to be weaker and ductile so that brittle failures can be avoided in those members.  The  development of full strength in timber members can be achieved through a variety of timber connections. These include ones using connectors such as nails, bolts, and dowels. Mechanically fastened timber joints have traditionally been characterised under monotonic loading conditions, but recent concerns on the behaviour of structures during earthquakes have led to the need to account for the behaviour of timber joints under reversed cyclic loading conditions. The structural response is generally controlled by the nonlinear characteristics of wood in compression and the elasto-plastic characteristics of the dowel. Connections using nails loaded in shear are considered ductile and capable of resisting many reversals of cyclic loading (Dean, Deam, Buchanan, 1989). Under normal cyclic loading, bending of the nails and local crushing of the wood around the nails can cause slackness in the connection, which causes 'pinching' in their hysteresis loops.  This, however, is acceptable because large  displacements can be achieved without failure resulting in favourable ductile behaviour. Problems arise when the nails either withdraw prematurely or they are too large and cause serious crushing in the surrounding wood, subsequently reducing the ductility of the system. Small diameter bolts and dowels behave in a similar manner to nails. Bolts that have a diameter of 12mm or less are capable of bending without instigating fracture in the timber, therefore producing a ductile connection (Wood & Cooney, 1976). The use of slender dowel fasteners with large spacings between them has been observed to result in yielding and allow large amounts of energy absorption. Larger diameter dowels and/or close spacing of the dowels tend to behave in a brittle manner before large displacements can be achieved (Blass, 1994). Ceccotti and Vignoli (1988) tested semi-rigid joints under seismic loading to observe their effects on timber structures.  8  Ductile behaviour was  Background  generally observed in the dowel connections that were tested because large deformations were achieved prior to failure. 2.2.2 Ductility The seismic response of structures is mostly influenced by the ductility of the structural members (Blass, 1989). Ductility is the energy dissipating capacity a structure has after it has exceeded its elastic range of displacement.  Beyond its elastic range, plastic deformations take place before  failure of the structure occurs.  The advantage of designing for ductile structures is that large  deformation capacities can be achieved without inducing excessive forces.  This can lead to large  savings because modern building codes allow for the design for lower lateral forces i f there is sufficient ductility in the structure.  The main source of ductility in timber structures is in the  mechanically fastened joints, especially those with dowel-type fasteners such as pins, nails, and rivets (Chui, N i and Jiang, 1998). Since ductile failure modes are favoured, a capacity design approach is adapted for use in the design of timber structures. Capacity design in timber structures is a relatively new technique because the knowledge of the seismic response of different types of timber connections is still unclear. Many factors contribute to the ductility of a connection. One is the amount of damping it has. 2.2.3 Damping Damping is a very important structural property in the analysis of the seismic behaviour of structures. Damping represents the energy absorption capacity in the system. This includes all of the mechanisms in which the mechanical energy (potential or kinetic) is transformed into other types of energy within a system. One of the advantages of damping is that it reduces dynamic amplification, especially in resonance conditions, which is often the critical one. In an analytical model, the chosen structural stiffness model greatly influences the damping model that is achieved. Linearly elastic stiffness models have the damping usually modelled in terms of a viscous damping coefficient C (Nm s), which represents a damping force proportional to the velocity of the system. _1  Idealised elastic-plastic stiffness models inherently have damping in the system through the hysteretic effect from yielding and/or crushing of the wood (Ohlsson, 1994).  9  Background The damping capacity of a system has the ability to change according to the load type, load magnitude, and amount of damage already present in the system. In the case of nailed joints acting in shear under cyclic loading, Ohlsson (1994) made the following conclusions: •  damping capacity will initially be high but will rapidly decrease under constant amplitude motion  •  during the period of decreasing amplitudes, damping as well as stiffness may be very low  •  during the first cycle of increasing amplitudes, the damping capacity may remain at a reasonably high level  The first conclusion illustrates that undamaged connection material will perform best during the first few cycles of motion and worsen as the motion continues because of induced damage after each successive cycle. The second describes the periods during loading when, for example, a connector has undergone repeated cyclic loading and has crushed a fair amount of material on either side. A hole or gap is formed around the connector whereby the material is unable to produce a resistive bearing force because it is no longer in contact with the connector. The third can be seen as a combination of the first two. Initially, there is no prior damage in the connection and since the amplitudes are increasing, the connector will be in constant contact with the surrounding wood, thus enabling the wood to share in the resistance of the loads. The damping capacity will inevitably drop after this first period of increasing amplitude and decrease the amount of energy absorption in the connection.  This final conclusion is very important when considering  earthquakes of long duration.  2.2.4 Hysteresis  Similar to reinforced concrete and steel structures, timber structures can form "pinched" hysteresis when subjected to reversed cyclic loading (Figure 2.2). The "pinching" is a result of the formation of gaps between the connector and the supporting medium. The gaps are formed from the crushing of wood around the connector. Hysteresis loops are the load-displacement curves for a system that is subjected to reversed cyclic loading. The area within one loop is the amount of energy that is dissipated through one loading cycle, which in turn is a good indication of how ductile the connection is. Consequently, the hysteretic response of a connection is critical in the dynamic response of structures and must be properly represented in any type of analytical model.  10  A  Background  common observation from cyclic tests of typical L L R S ' s is that the hysteresis trace of a wood system is generally governed by the hysteretic charateristics of its primary connection (Foliente, 1995).  Thus, we only need to characterise the hysteretic behaviour of wood connections to  characterise the behaviour of wood structures and structural systems.  1 1  M  M  4 > 1  —  -0.05  i  0  0.05  —  0.1  Rotation (rad)  Figure 2.2 - Hysteresis diagram for typical timber connection (Prion & Foschi, 1994) 2.3 Analytical Models for Hysteretic Response  In the past, analytical models involved the use of test data for their calibration, in an attempt to reproduce the hysteresis loops of a connection.  In these models, however, the behaviour of  individual connectors within the connection is usually not considered. The consequence is that a separate test must be conducted for every connection with a different connector number or configuration. The analytical model must be custom-fit to each connection individually. This can be a very expensive and time-consuming task for modelling a structure with many different connector configurations.  Since the accuracy achieved in the dynamic analysis of a timber  structure depends primarily on the connection analysis, the ability to accurately model a ductile timber connection is very important. The nonlinearity resulting from ductile connections makes its modelling difficult. In this study, dowel connections were used to verify the results of an analytical model, so models for other types of fasteners will not be discussed here. The most recent hysteretic models for dowel type connections include:  the Ceccotti and Vignoli (1989) subroutine for  DRAIN-2DX to simulate the moment-rotation hysteresis curves of tested connections, the model  ll  Background Wayne Stewart degrading stiffness hysteresis model (Stewart, 1987). It is specifically aimed at modeling nailed wooden shear walls and is one of the hysteresis models available in the Ruaumoko Inelastic Dynamic Analysis program (Carr, 1998.)  2.3.1 The Ceccotti-VignoliModel  This model was developed in 1989 at the University of Florence to enable the simulation of connections with nonlinear metal fasteners connecting wood members in DRAIN-2DX.  The  skeleton of a typical moment-rotation curve of a connection was reproduced by this subroutine while keeping track of the maximum previous rotation, to account for the deterioration in the wood using different loading and unloading slopes.  Parameters for the skeleton curves were found  through the moment-rotation curves obtained from cyclic tests of the connection (Figure 2.3). A four-slope model was first developed in 1989. The outside envelope was defined by two loading slopes, K i and K . The unloading slope was equal to the initial loading slope K i . A return slope Ke 2  was defined and an inner slope K 4 , was used to model the pinched loops of subsequent cycles. A six-slope model was later developed in 1991, which included a third loading slope, K , and the 3  option of an unloading slope, K , which is different from K i . 5  The accuracy of these curves was limited to certain amplitude rotations because the model was only capable of producing a limited number of slopes. Typically, the slope parameters were chosen to best represent the maximum outside loops for a given connection, given that the smaller amplitude loops did not significantly affect the overall shape of the response curve. This model is currently one of the few available models that are incorporated into a nonlinear, commercial structural analysis program and geared specifically for ductile timber connections.  One of the major  disadvantages of this model is the requirement for cyclic testing on full-scale connections to obtain the parameters necessary to model their hysteretic response. This must be done for each different connection consisting of different materials and/or configurations.  2.3.2 The Bouc- Wen-Baber-Noori Model  This model is the culmination of work originally started in 1967 by Bouc, who suggested a versatile, smoothly varying hysteresis model for a single degree of freedom (SDOF) mechanical system under forced vibration. In 1980, Wen generalised Bouc's hysteretic constitutive law and  12  Background  Moment  Hysteresis Curve measured from Cyclic Testing  DRAIN-2DX - Four-slope Model  -0.06  0.00  0.06  Figure 2.3 - Slope Parameters for DRAIN-2DX Connection Models (Frenette, 1997)  13  Background  developed an approximate solution for random vibration analysis based on the mechanical model of Figure 2.4. This was extended to a multiple-degree-of-freedom system (MDOF) and modified to admit stiffness and/or strength degradation as a function of hysteretic energy dissipation in 1981. This model was then further modified by Baber and Noori (1986) to incorporate pinching. The final model, known as the Bouc-Wen-Baber-Noori model, is a single-element pinching model which used an equivalent linearization solution to solve for the response to random vibrations (Foliente, 1995).  The model is based on a first order differential equation in time for the hysteretic component of the force. Different types of hysteresis shapes can be accommodated through the use of 13 model parameters. These parameters are used in the calibration of the model to finite element output. Fitting this model is generally a difficult task, and requires supplementing the optimisation algorithm with trial and error guided by judgement (Foschi, 1998). Ft  (c) Figure 2.4 - Hysteretic SDOF System: (a) Schematic Model; (b) Non-damping Linear Restoring-Force Component; (c) Hysteretic Restoring-Force Component (Foliente, 1995)  14  Background  2.3.3 The Chui-Ni- Jiang Model  This is a nonlinear finite-element model that predicts the load-slip response of a single-shear, nailed wood joint under reversed cyclic loading.  The nail is modelled using a finite beam-element  approximation that incorporates the effects of large deformation and the hysteretic nature of the stress-strain behaviour of the material. The effects of shear deformation in the nail and friction between nail and wood are also incorporated into the model. Three types of elements are used to model the nail, wood, and frictional interface between the nail and wood. The nail uses a beam element while the embedment characteristics of the wood are characterised by spring elements, and the interface is modelled using linkage elements (Figure 2.5). He;td-side member  Point-side member  Movable base-line "8" p  Figure 2.5 - Assemblage of Finite Element for Nailed Wood Joint (Chui et al., 1998)  Tests were used to extract parameters of the real stress-strain characteristics of a nail for use with the Filippou-Bertero-Popov (1983) model for representing the stress-strain relationship of materials. To model the wood medium, tests were also performed to obtain the load-embedment properties of the wood (Chui and N i , 1997). The linkage element to model the friction at the woodconnector interface was accounted for by relating the frictional force to the slippage between wood and fastener, and the normal pressure at the contact of the two surfaces. To solve the nonlinear  15  Background problem, a Newton-Raphson iteration method is used.  The load-displacement response of  connections using other dowel-type fasteners can also be analysed with this model when minor adjustments are made. Unfortunately, there is no indication that the model is able to predict the amount of damage that occurs in the connection. Because of this, gaps due to the wood crushing around the connectors are not accounted for in the analysis.  2.3.4 The Foschi Model  The analytical model, F R A M E , developed by Foschi (1999) is based on a previous model (Foschi, 1974) for nails in a wood foundation and has now been adapted to dowel-type connectors. This model uses only fundamental material properties to analytically model a connection, which disposes of the need to perform full-scale tests on connections to obtain their hysteretic response. This model is discussed in detail in the following chapter.  2.4 Summary  The seismic response of timber structures is strongly influenced by the combined response characteristics of the timber and the connections. Connections must be designed to prevent sudden brittle failures, either in the connections or in the timber. Careful detailing of connections is essential to ensure that the intended behaviour occurs, as both strength and ductility strongly influence connection performance. The brittle failure modes of timber are one of the reasons that make the seismic design of timber structures difficult. Ductility plays a major role in the behaviour of a timber connection. The amount of ductility a connection is designed for greatly affects the overall response of a timber structure.  In moment-resisting connections, tests have shown that  connectors such as nails, bolts and dowels can exhibit ductile behaviour. The amount of ductility in a connection is influenced by the amount of damping present. Damping is the component in the dynamic analysis of a system that accounts for its energy absorption capacity.  In timber  connections, damping can be attributed to the yielding of the connectors and/or the crushing of the wood surrounding the connectors.  This crushing of wood causes gaps to form around the  connectors, producing pinched hysteresis loops. Hysteresis plots represent the amount of energy absorbed from each cycle in a system. Many hysteresis models for dowel-type connections are available today but most require the testing of full-sized connections to obtain parameters to fit the curves to experimental results.  16  Analytical Model  Chapter 3 ANALYTICAL MODEL 3.1 Introduction The seismic response of structures can be simulated using computer programs such as ANSYS, CANNY, DRAIN-2DX, and SAP2000.  While all four are capable of  performing nonlinear analyses, some programs are limited to using only the available nonlinear link elements within each program to model local structural nonlinearities such as gaps, dampers, and isolators. Presently, DRAEN-2DX is the only commercial package that contains a subroutine that allows for consideration of nonlinear connections, specifically tailored for timber structures.  It uses a model for the force-displacement  curve or hysteresis loop, which can be calibrated to experimental results. Unfortunately, the model requires results from full-scale testing to determine the moment-rotation curve for each different size, configuration, and composition of a connection. To avoid the need to resort to full-scale tests, an analytical model that can simulate the dynamic behaviour of ductile timber connections, using only basic material properties was developed.  The dowel connector program developed by Foschi (1999), called FRAME, is a nonlinear, finite element model that can simulate the dynamic response of ductile timber connections using only basic material properties as input. The basis for the program was the response of an elasto-plastic beam supported in a nonlinear medium that acted only in compression. This allowed for the formation of gaps between the beam and the medium to account for crushing of the medium or yielding of the beam. The original analysis included small diameter connectors such as nails, and medium-sized connectors such as dowels and small bolts. Diameter sizes were limited to ensure that brittle failures of the medium around the connectors were avoided and ductile responses would be observed. Nail connectors are dominantly used in shear wall applications where issues such as friction, withdrawal, sheathing tear-out and large displacement theory need to be considered. This study will focus mainly on dowel connectors, and those issues are not discussed in the context of this thesis.  17  Analytical Model  3.2 The Dynamics Problem The dynamic analysis of timber frames with dowel type connections is primarily governed by the ductility of the connections.  This involves some nonlinearity in the  connection from the elasto-plastic properties of the dowels and the nonlinearity in the wood medium. The equation of motion for a linear system is of the form:  MX  + CX + KX = LX  {3.1}  g  where: M is the mass matrix,  X is the acceleration vector,  C is the damping matrix, X is the velocity vector, K is the stiffness matrix, X is the displacement vector, 1  L is the vector associated with the inertia produced by the ground accelerations sadX is the input ground acceleration vector. g  Equation {3.1}  must be modified by adding the nonlinear terms accounting for the  energy dissipation in the dowel connections. Equation {3.1} then becomes:  N  AdX+OC+KX+YjFiQ where: F is the t  =LX  {3.2}  g  dowel force  Qi is a vector associated with the DOF's of each F; and N is the number of dowels. The mass matrix for the connection models was assembled according to the magnitude of the mass placed on each connection, the distance from the connection centroid to the mass centroid, and the length of the mass. The connection columns were assumed to be massless while a concentrated load was placed on top of each connection.  The mass  length was required to account for the rotational moment of inertia of the mass when the mass is distributed over a large area (see section 4.2.1).  The stiffness matrix was  assembled using the basic material properties of the connection while the damping matrix was determined by considering an equivalent viscous damping matrix. damping was already inherent in the model through the restoring force vector.  18  Hysteretic  Analytical Model  3.3 Connector Analysis The dowel connector was modelled as an elasto-plastic beam supported in a nonlinear wood foundation (Foschi, 1998.). A brief description of this model is included here for completeness.  The stress-strain curve of the steel used for the dowel, as well as the  bearing capacities of steel dowels in wood were defined through routine experimental testing.  The model can automatically adjust to any input history, and develops the  hysteresis loops and displacements for a connection.  3.3.1 The Finite Element Model for a Connection The finite element model of the connector was developed by dividing its length into a given number of elements. Three degrees of freedom (DOFs) were defined for each node between the elements: displacement and slope in the direction perpendicular to the beam (w, w' in the j-direction), as well as the axial displacement in the beam (« in the JCdirection).  The DOFs at the nodes were chosen to enable consideration of bending deflections and axial displacements in the deformed shape of each element. The DOFs were used in shape functions defining the displacements at any point along the length of each element. As an example, the displacement w along one element was characterised by a cubic polynomial in terms of four constants: wi, Wj, wt, w/, as in Figure 3.1. The analysis assumes that the beam is bending only in one direction. The shape functions used for this analysis are presented in Appendix A. Element degrees of freedom:  {a} = { m, W i , W'I, U j , W j , Wi, Wi'  w'j }  Wj, Wj'  Ui  Figure 3.1 - Beam element for FRAME model and the DOFs  19  Analytical Model  To achieve nondimensional equations in the analysis, the axes were redefined within each element. The jc-axis was replaced by a £-axis and varied from -1 to 1 along the length of the member, while the .y-axis was replaced by a r|-axis and varied from -1 to 1 across the section of the element in the orientation of the applied force.  This coordinate  transformation was also to accommodate the need for numerical integration methods to calculate the integrals in the equilibrium equations. In particular, a Gauss quadrature was used and is discussed later. .  y A 1 *  f  f  * ^ ^ ^ /  W .W .V .V .V .W .W . AV IT r '1& I<,<'•;'""<<}" ;  If  , ,  f  ****** *'  *'**•'  Figure 3.2 - Coordinate transformation for each beam element  In the analysis of a beam composed of several elements, the function and slope continuity of the deformed shape is guaranteed by the DOFs at a node, that are equal for the two adjacent elements. The overall beam model has (n+l)*3 DOFs where n is the number of elements.  3.3.1.1 Modelling of the Steel  The steel connectors were modelled using a bilinear approximation of its real stress-strain curve. Tensile tests were performed on numerous steel dowels to determine the yield strength for the batch of steel dowels used in this study (Section 5.1). Figure 3.3 shows the hysteretic stress-strain relationship of a steel dowel.  20  Analytical Model  | "Oy  Figure 3.3 - Stress-Strain Relationship for Steel Dowel  In F R A M E , i f the previous state of stress in the element is {o , s 0  0  }, the stress  corresponding to a new strain s can be calculated by the following mathematical relationships: F(e) = ao + E(s - s ) D  but if, | F(s) | < a = > rj(s) = F(s) and — = E ds y  {3.3}  or if, | F(s) | > a ==> c *F(e)/| F(s) |, and — = 0.0 ds y  y  This produces the elasto-perfectly plastic hysteresis loop shown in Figure 3.3. The strain in each element needs to be defined in order for the associated stresses to be determined. The three main sources of strain that were accounted for in this model were axial strains due to «, bending strains due to w, and axial strains due to w.  These three strain  components were added together to define the strain at any point within an element:  z = u'-yw" + (wf/2. y  {3.4}  3.3.1.2 Modelling of the Wood Medium The wood medium was modelled as a foundation consisting of nonlinear continuous springs. Monotonic tests on different orientations of 12.7 mm steel dowels driven into Parallam® were performed. From these results, a five-parameter curve was created to  21  Analytical Model  approximate the response of the wood medium in compression for the different directions of loading (Figure 3.4).  Qo  Q 2 : variable for decay / K: initial stiffness  A  i  A  m a x  : max  displacement Figure 3.4 - Parameters for Load - Deformation Relationship for Wood Foundation  The mathematical relationship for this curve is expressed as: if A < A  m a x  , then p = (Q„ + Qi*A)*(l - e ^ o )  if A > A  m a x  , thenp = / w *  where P  m a x  6  4( A  e  = (Q + Qi*A 0  ~  m a x  )*(l  and Q = /«(Q )/(((Q - l ) * A 4  2  &%  A n , a x ) 2  3  -e* K  A  /Q  max  )  0  ) ), 2  max  with Q 2 and Q 3 defined as follows: p = Q2*p  max  at D = Q * D 3  m a x  ( Q < L 0 , Q >1.0) 2  3  It is also important for the model to define the unloading path and to remember the gap developed so far. To illustrate a loading and unloading scenario, see Figure 3.5. Assume that at a point the member is pushing against the wood medium and develops a reaction of p(w) as a function of the embedment depth, w. Let's assume that the member reaches  22  Analytical Model  point A before it begins to unload. Point A corresponds to a load p and a displacement 0  wo, which is associated with a displacement Do, by the expression: {3.6}  Do = wo - po/K,  which defines the unloading path from A —» Do. This path is the unloading curve for the model which assumes that unloading will have the same slope as the initial loading curve. When the member reaches a point of p = 0, the corresponding D , or permanent gap 0  formed in the dowel hole, will be saved in memory so that when loading begins again, the reaction will be p = 0 until the previous Do has been reached. At this point, the loading curve will follow the slope of K starting at Do until the previous wo, and then beyond this point it will follow the original curve. Since gaps will be formed to the right and left of the member during seismic loading, values for D  + 0  and D ~ are kept in order to account 0  for the formation of gaps on either side of the member.  Along the lengths of the  members the values of Do and Do" are functions of x, resulting in different gap sizes at +  different locations on the member. p(w)  Po  Do  wo  Figure 3.5 - Typical Loading and Unloading Scheme in FRAME  The following algorithm defines the value of p(w) for each new w: if (w < D ) -> p = 0 0  if (w > D ) -> p = min of {pi = K(w-Do ), P2 = p(w)} 0  if ( p = p2 ) > update Do: Do = w - p/K if (p = 0) or (p = pi) -> D unchanged 0  23  {3.7}  Analytical Model  This algorithm is identical for the right and left side of the member (Figure 3.6). The sum of the corresponding | D | and |Do"| equals the total gap size at any particular point, +  0  at a given time.  Figure 3.6 - Load displacement relationship for wood foundation  3.3.2 Virtual Work for Connectors The Principle of Virtual Work was used to determine the deformed shape of the connectors when subjected to dynamic loading. The Principle states that the external work done on a system (ie. by the loads) must be equal to the internal work in the system (ie. by the stresses) for any virtual, admissible perturbation of the true deformation corresponding to the solution. Thus, to determine the work done in each element of the model, 'virtual strains', {Ss}, were obtained, using Equation {3.4}, in terms of virtual displacements {du} and {Sw} . The internal work was then calculated by,  VJ/elem = Ivol  a(s)5(g)  {3.8}  where vol is the volume of the element. First, the degrees of freedom u(x), w(x), and w'(x) were defined in terms of the vector shape functions, No(£), M ( £ ) and Mi(£) and 0  the degrees of freedom vector {a}. Since w/ = (dw/dx)i, w" = (d w/dx )i let: 2  24  2  Analytical Model  u(x) = N o a T  w(x)=M a  {3.10}  T  0  w'(x)=Mi a T  w"(x) = M a T  2  where, a = {ui, wi, w'i, Uj, Wj,w'j). Now, the strain in {3.4} can be written as:  e = (N  T x  - yM )a + ^a M M a T  T  T  2  l  l  { 3 1 1 }  and the virtual strain is obtained from:  Se = Sa (TYj - yM ) T  +  2  8a M M?a T  {3.12}  x  Therefore the internal work in an element due to a virtual deformation {da} is:  Velern = J  ^  ^  ^  +  a]  vol  3.3.3 Virtual Work for the Wood Medium The force from the medium varies depending on the displacement w along any point of an element. A small 'virtual displacement', Sw, produces an external virtual work, w  e x t  =  ~ \  Jo  P(\ \)ri^ w  w  taking into account the forces acting from either side of the  w  connector.  In terms of the shape functions of {3.10}, the virtual displacement becomes:  Sw = M Sa T  {3.14}  0  The work, Waa, on an element from the wood medium can now be added to the internal work from Equation {3.13}, as  25  Analytical Model  {3.15}  medium  The total vector \\J will then be:  WTotal ~ where  VJ/ t ex  ext Welem Wmedium W, +  {3.16}  is the work of the applied forces {R} acting on the virtual displacements. The  total vector l|/ otai for the element is then assembled into a global vector {\|/}. The t  solution of the problem requires finding {a} so that {\|/} = 0. The global solution vector a is found using the Newton-Raphson iteration scheme: {3.17}  a = a* + [ V y * ] ^ * ) 1  where the V\|/* is the tangent stiffness matrix associated with the vector a* 3.4 Connection Analysis In this study an inverted pendulum was used for analysis and testing, as shown in Figure 3.9. Steel dowels were used with thick steel side plates, as shown in Figure 3.7, to form a semi-rigid wood connection.  This analysis considered the degrees of freedom of the  connection (U, Fand 0) corresponding to the centroid of the connector group. The DOFs were used to describe the relative motion between the connection and the steel side plates, which were assumed to be rigid and part of the ground. Each connector was individually defined by the DOFs according to their geometry, thus a force-displacement relationship for each connector was developed.  The connection response was an  accumulation of the individual connector responses whereby a total force-displacement relationship for the connection using the three DOFs was obtained.  26  Analytical Model  Analytical Model  3.4.1 Description of Analysis for the Experimental Model In this study, the model consisted of steel side plates, a wood member and steel dowel connectors. Figures 3.9 shows the real connection specimen ready for testing. The steel plates and the wood member were both assumed to be rigid while the connectors were flexible and responded to the input history according to their hysteretic behaviour. Figure 3.8 shows a free-body diagram of the model with the inertia forces for the mass, M, and the connector forces, F. The distance e, is the distance from the centroid of the connector group to the centroid of the applied mass, M.  M(U-0e  + aJ  U-0e  iv tfl  —w  wood member (rigid)  A  U  0  ~>  Wi  connector displacements:  1  Fj  connector force: F Connector degrees of freedom: X = {U,V,9) T  ground accelerations  Figure 3.8 - Free-body Diagram of Connection Model  28  s  Analytical Model  The relative displacements (slips) of each connector were defined in terms of the U, V, and 6 degrees of freedom. Thus using the local coordinates cji, and rji shown in Fig. 3.8.  Figure 3.9- Actual Connection Specimen  From geometry (assuming rigidity), the approximate connector displacements in the JC and y coordinates are, respectively, Axj = U - Grii  and  Ayj = V + 0^  {3.18}  or, in matrix form,  Ax, =(1.0 0.0 {3.19}  Av,=(0.0  1.0 £ ) *  Using {3.19}, the principal of virtual work is applied to the free-body diagram in Figure 3.8 to formulate the equations required for the solution.  29  Analytical Model  3.4.2 Virtual Work Equilibrium for the Connection  The Principle of Virtual Work is also used to obtain the equations for the overall response of the connection when subjected to dynamic loading. The virtual displacements at the top of the specimen are, SU-eS0  = (l.O 0.0 -e){dx}=Q(Sx {3.20}  5V = {0.0 1.0 0.6){8x}=Q Sx r  2  where (1.0 0.0 -e) = Q i  and (0.0 1.0 0.0) = Q .  T  2  T  Thus the Principle of Virtual Work is expressed as:  -M(U-fe+aJ*(SU-eS9)-M(V+a )*SV-J]F y  *S\ = 0  {3-21}  t  i  where 8Aj is the virtual displacement associated with each connector force.  The nonlinear terms in {3.21} associated with the forces Fi, require an iterative process to solve this dynamic problem at each time step. A constant average acceleration method (Newmark's Method) was used to formulate the integration over time. The NewtonRaphson iteration process was used to solve the nonlinear problem at each time step. Equation {3.21}, gives  M(Q Q x T  x  + QiQiY*  + Z(F {Q } XI  +  XI  '"=1  =  F {Q }) y  yi  {3.22}  -Ma Q -Ma Q x  l  y  2  Using the Newmark's average acceleration method to solve nonlinear systems, the acceleration, velocity and displacement vectors are related by the following expressions:  30  Analytical Model  x = x  ( x  +  k  + x  k  k + l  )  {3.23} =  x - x  k  k  X  k  + x  +  X  t  k  \  k  +  X  .  k+l  ~  X  x  k  At  +1  2  2  At  ) —  2  At  x  At  k + l  + ——  2  k  *  ^  + —-  k  k  k\  1  k  x  x  +  2  = k + At  +  k  + ( ^- - +  .  k\  x  X  2  x  k  2  2  + x At + x k  At 2 A  .. k  Inserting equations {3.23} into {3.22} gives: 4Mi  {3.24}  =-MqQ -MqQ 4 - ^ (QQ +Q Q % +  +Q Q K  T  2  where x ,x ,x 0  0  0  2  2  2  2  correspond to the previous time step.  Rearranging {3.24} gives:  f r f e s ' +&<2/)* + l ( ^  Q +F Q )+Ma Xi  yi  y  x  ft M a ^ & +  {3.25}  where £?..25£  is known as the 'ow/ o/ balance force' -  the tangent stiffness matrix,  or the 'residual force vector'.  V^F can be evaluated 31  From  by calculating the gradient of  Analytical Model  *P. Evaluation of the gradient of *P involved the partial derivative of each term in the *¥ vector  with respect to the variable Xj. Therefore:  V\|/ij = 9l|/i/3Xj  {3.26}  The tangent stiffness matrix is thus expressed by:  ™~gm +ae/]+Z{^)E?„a;]+Z(5 )iB e/] T  L  i 3  a  k  -  2 7 }  yk  x  Using the Newton-Raphson iteration, and given *F and V^F corresponding to a vector x , an updated x can be found by setting  ¥ = ¥* + (V ¥)*  x  from which:  is obtained.  x  X  = 0  .(x-x)  X = X* - ( V^F )  _1  F*  X  {3.29}  The process is repeated with x taking the place of x*. Convergence is  satisfied when the current norm of the vector x changes less than a tolerance times the previous norm, and when similar small changes occur for the residual force vector.  3.4.3 Numerical Integration The solution of the Virtual Work equations require the use of a numerical integration scheme. F R A M E used a Gauss quadrature to perform the necessary integrations. This method entailes the calculation of the functions at discrete points (known as Gaussian points) within the ranges of integration and multiplying them with specified weighting functions defined by the integration scheme.  To facilitate the use of the Gaussian  integration scheme, the limits of integration have to be converted to the range of -1 to 1. As discussed before, this is achieved by doing a coordinate transformation of the jc-axis to a £,-axis that has a range o f - 1 to 1, and the j-axis to a 77-axis that has a range from -1  32  Analytical Model  to 1. The values of the function at discrete points times their respective weights are summed to get an approximate value for the integral. Increasing the number of Gaussian points used in the numerical integration increases the precision of the approximate solution.  3.5 Model Inputs and Outputs F R A M E requires various input values to carry out the solution of forces and displacements for a connection. The two main input components are the connection geometry data file, and the connection properties data file. The geometry file contains the mass, eccentricity, location of connectors, and acceleration records. The properties data file contain the values for the connector and wood medium material properties, the boundary conditions for each connector, and the convergence tolerances (Appendix A). Currently, F R A M E is configured to produce output files for the rotation of the connection, 0, the vertical displacement of the connection, V , and the horizontal displacement at the mass location, U , at every acceleration step. Also, hysteresis loops for one connector in one direction(eg. parallel to grain) are determined by F R A M E and written to an output file. Samples of each data file are shown in Appendix A.  3.6 Concluding Remarks The development of F R A M E (Foschi, 1999) began in 1995 based on the behaviour of a ductile beam supported by a nonlinear foundation using a finite-element approach. The purpose was to create a program able to simulate the dynamic behaviour of dowel type connections based on fundamental material properties. The main components of this program included: the modelling of the steel connectors using finite elements, the modelling of the wood medium as a system of springs with specified force-displacement relationships, the inclusion of the inertia and damping components in the model to perform a time step dynamic analysis using the Newmark method to relate the accelerations, velocities and displacements, and the Newton-Raphson iteration scheme to obtain the solution. The intention was to implement this analytical model as a subroutine into a full 3-D, nonlinear, dynamic structural analysis program for timber structures, which is currently under development.  33  Analytical Model  Possible improvements that can be made to the model are: including the effect of some initial confining pressure on the connector by the surrounding medium given that the predrilled holes are smaller than the diameter of the connector, the effect of friction between the connector-medium interface given that the connectors are fitted snug into the predrilled holes, and better modelling of the gap formation. F R A M E currently models the gaps formed in the x and y directions independently and finds the corresponding forces generated by the medium in each direction. This is an approximation to what actually occurs during the formation of gaps.  The gaps formed in the model are  calculated along each of the axes but in reality the gaps form an oval-like surface around the connector and vary along the entire length. The correct model performs only a planar analysis of a connection but a 3-D model will be required for implementation into the 3D structural analysis program for timber structures mentioned before.  34  Test Specimens and Analytical Predictions  Chapter 4 TEST SPECIMENS AND ANALYTICAL PREDICTIONS This study examines FRAME'S ability to accurately simulate the dynamic response of dowel type connections by comparing its results to experimental data obtained from the shake table tests described in Chapter 5.  As mentioned in Chapter 3, F R A M E is a  computer subroutine that can calculate the response of dowel connections to seismic loading using basic material properties such as steel yield and wood embedment strengths. Eventually, it will be implemented into a 3-D, non-linear, dynamic analysis program as a subroutine for dowel-type ductile timber connections. The F R A M E model was first used to help design the test, that is, to determine the dowel configurations and an appropriate mass, and also to choose appropriate earthquake records for the shake table tests, so that a desired response within the physical limits of the shake table could be achieved. The lateral displacements at the top of the column were measured during the tests and used to compare with results from F R A M E . Hammer tests were also performed on the test specimens to obtain the natural frequencies of the specimens and to compare their values to the results from F R A M E . 4.1 Description of Connection  The specimens used in the shake table tests consisted of a Parallam® member connected to the shake table via 25.4 mm (1") thick steel base plates, 19.05 mm (%") thick angle plates, and 12.7 mm (1/2") mild steel dowels. Figures 4.1a & lb show the schematic and real connections respectively. The Parallam® member dimensions are 178 x 356 x 800 mm (7" x 14" x 31.5"). The dowels were arranged in a circular configuration with a radius of 105 mm. The specimens had a total of 6 holes that could accommodate a variety of connection configurations. A 6-dowel and a 3-dowel configuration was used for the shake table tests (Figure 4.2). The dowels were approximately 220 mm in length, of hot-rolled G300 steel. A n information data sheet for Parallam® can be found in Appendix B. A mass of 907.2 kg (2000 lbs) was used to simulate building loads on the connection, using two 63.5 mm (2.5") thick steel plates with dimensions of 1500 x 603 mm (59" x 23.75"). The plates were connected to the column via an inverted C200 x 17  35  Test Specimens and Analytical Predictions  steel channel section.  The channel section was securely fastened to the Parallam®  section with long lag screws.  Figure 4.3 shows a complete test specimen ready for  testing.  Figure 4.1a and 4.1b - Schematic and Actual Connection  Figure 4.2a - 6-dowel configuration  Figure 4.2b - 3-dowel configuration  36  Test Specimens and Analytical Predictions  Figure 4.3 - Test specimen ready for testing  4.2 Modelling of the Test Specimens  The test specimens were all modelled as single-degree-of-freedom systems with rigid columns and a mass placed over the centre of the column. The connection between the shake table and the test specimen was assumed to be rigid, as was the connection between the mass and the test specimen. These connections were constructed accordingly, so that they would be stronger than the dowel connection and not have to be considered as independent connections in the analytical model. The Parallam® behaviour was modelled as a non-linear foundation, using bearing values obtained from available monotonic tests on steel dowels on Parallam. The steel dowels were modelled as elasto-plastic elements, using E and a values obtained from monotonic tension tests. y  4.2.1 Masses  The mass in the analytical model is assumed to behave as a point mass acting through the centroid of the connection. The masses in the test, however, were large steel plates that overhung the column on all 4 sides (Figure 4.4). Even though the mass was placed so that there was no eccentricity, the distribution of mass over such a large area must be accounted for. Since a point mass will not accurately model the overhang effects of the  37  Test Specimens and Analytical Predictions  steel plates, a rotational moment of inertia term needed to be incorporated into the model's mass matrix.  model specimen  Figure 4.4 - Model of Experimental Mass  4.2.2 Boundary Conditions The boundary conditions for the attachment of the dowels at the steel side plates were difficult to model. It was unclear as to whether the dowels could be assumed fixed or pinned at the ends.  Figure 4.5 shows a plan view of the cross-section of a dowel  connector inside the connection. The wood medium was assumed to move as a single layer relative to the steel side plates that were assumed to be rigid.  38  Test Specimens and Analytical Predictions  symmetry  a steel side plates (assumed rigid)  Parallam®  Figure 4.5 - Modelling the movement of the connection  Due to symmetry of the connection geometry, the analytical model was executed using only half of the connector (Figure 4.5) with the corresponding calculated loads having to be multiplied by two. If the co-ordinate axes are defined as shown, symmetry of the dowel resulted in boundary conditions u = 0 and w' = 0 at node b and w = 0 and possibly w' - 0 at node a. The w' = 0 boundary condition at node a was debated because the bearing properties of 12.7 mm diameter steel dowels on steel were unknown, so the rotational rigidity of a dowel end in the side plate was unknown. It was first speculated that the dowels should be modelled so they could rotate in the plane of the plates but upon further consideration the ends were modelled as fixed (w = 0, w' = 0) but able to slide freely in and out of the hole (u * 0). This was because the dowels were tight fitting as they had to be hammered into the holes in the steel plates. center of cross-section  u  Figure 4.6 - Coordinate axes for connector model  39  Test Specimens and Analytical Predictions  To better ensure that out-of-axis rotation of the dowels did not occur in the plate holes, 19 mm (3/4") thick steel side plates were used. A plate thickness larger than the dowel diameter was assumed to be sufficiently rigid compared to the dowels.  4.2.3  Damping of the Connection  Hysteretic damping of the system was inherent in the analytical model, due to the elastoplastic assumptions of the connector and the non-linearity of the wood behaviour. In addition, viscous damping was introduced and values were estimated for each dowel configuration according to their calculated natural frequency.  These values were  confirmed from the values obtained from the free vibration tests of each specimen (Appendix B)  The differential equation that governs free vibration of a single-degree-of-freedom (SDOF) system with damping is: mii Dividing by m gives:  + cu + ku = 0  .. u +  c  , u + ku  {4.1}  . = 0  777  Using:  fk~ co = - / — Vm  C  and  — = 2.qco m  Equation {4.1} becomes:  u  +  2%co  u  +  co u 2  =  0  fr * 2  where £ is the damping ratio, m is the mass, c is the damping constant, k is the stiffness, and co is the natural frequency of the system. It was assumed that the damping ratio £ = 0.05 (5 %). This ratio was used to obtain the constant c = 2£<am.  To obtain the natural frequency co of the connection model, a triangular acceleration pulse was used as an input record in FPvAME. The calculated connection response to this 40  Test Specimens and Analytical Predictions  pulse, and the free vibration that followed, was used to estimate the period T as the time between peaks in the free vibration portion of the response. Using the expression: <D =  2n/T  where T is the period, the system's natural frequency, co was calculated. Assuming a damping ratio of £ = 5% (typical for wood structures), a value a = 2^a> was calculated and used in the viscous damping parameter c, for the model.  See Appendix B for  calculations for a for dowel configurations with different masses.  4.3 Simulation Results The analytical results from F R A M E were compared to results obtained from the shake table tests. F R A M E was used to provide results for the fundamental natural frequency for each configuration as well as the horizontal displacement response at the top of each test specimen. Prior to the shake table testing, F R A M E was first used to estimate the response of different connection configurations with different masses, so that responses within the physical limitations of the shake table could be obtained. Two connection configurations with identical masses were chosen from these results to be constructed for the shake table tests.  After the shake table tests were performed, the real table  acceleration records were input back into F R A M E so that the final model simulations could be performed and their results compared to the experimental results.  4.3.1 Natural Frequency The natural frequency of the connection test models was estimated by using F R A M E with a triangular pulse with a post-pulse tail of zeros as an input acceleration (Figure 4.7). The results gave the free vibration response of the model system whereby a fundamental period could be determined and a natural frequency for the system was calculated. The simulation was performed for a 2000 lb (907.2 kg), 3-dowel configuration and a 2000 lb, 6-dowel configuration.  41  Test Specimens and Analytical Predictions  600 n  -100 J time(s)  Figure 4.7 - Acceleration pulse for calculation of natural frequency of connection  4.3.1.1 3-Dowel Configuration  The response from F R A M E to the acceleration pulse for the 3-dowel system is shown in Figure 4.8. A fundamental frequency of 5.13 H z was obtained. This value was used to calculate the viscous damping coefficient for this system which was then used in the comparison with the results from the experimental tests.  Figure 4.8 - Free vibration response of 3-dowel configuration  42  Test Specimens and Analytical Predictions  4.3.1.2 6-dowel configuration The analytical response of the 6-dowel configuration to the acceleration pulse is shown in Figure 4.9. A fundamental frequency of 7.14 Ffz was obtained. A coefficient for viscous damping of this system was also calculated using the fundamental frequency as discussed in section 4.1.3. As expected, this system had a higher natural frequency because it had more dowels and thus it was stiffer.  0.15 -,  -0.3 J  time(s) Figure 4.9 - Free vibration response of 6-dowel connection  4.3.2 Model Response The seismic response of the connection models to the Landers(1992) and the Kobe(1995) earthquakes were investigated. F R A M E is capable of returning results for the U , V and 8 displacements for the top of the specimen as well as the hysteresis loops for one chosen connector in a given direction. In this study, the horizontal displacement at the column top and the hysteresis for a chosen connector was compared with the experimental results.  Visual inspection of the other plots (V, and 9) was helpful in verifying the  robustness of the results. 4.3.2.1 Summary 43  Test Specimens and Analytical Predictions  The initial simulations prior to the shake table testing included: •  3, 4, & 6-dowel connection configurations  •  each with 1000, 2000, 3000, 4000, or 5000 lb weights on top  A summary of these initial results is shown in Figure 4.10 below. 140 120 ? d.  100  3-dowel Analysis performed using modified  configuration  Landers record (-historical acceleration amplitude x 2. This w a s done to get a m o r e d u c t i l e resp o nse f ro m t he co nnect io n configurations.)  80 -|  cn  b * a  60 40 20 0 1000  2000  3000 Weight  4000  5000  6000  (lbs)  Figure 4.10 - Weight of Mass vs Horizontal Displacement for different connection configurations  From these results two connection configurations with the same mass were chosen for the construction of test specimens. A 3-dowel and 6-dowel configuration each with a 2000 lb mass was chosen to be used as test specimens for the earthquake shake table tests. This was because their initial displacement results from F R A M E showed that they would not exceed the displacement limits of the shake table, and their displacments were large enough for the displacement transducers to measure accurately.  4.3.2.2 Final Model Results  Having decided on appropriate connection geometry, boundary conditions, viscous damping coefficient values, and mass, the final model configurations were used to calculate the response to the real acceleration records acquired from the shake table tests on the specimens. The reason for using the actual recorded displacements was because  44  Test Specimens and Analytical Predictions  there is always an error between the input record and the record that the shake table can actually produce. To limit the error in this study, the actual table accelerations recorded during the tests were used in the F R A M E simulations. A total of four final simulations were performed. It was decided that two connection configurations, each subjected to two different earthquakes would be adequate to gauge the accuracy of the analytical model.  A summary of the final simulations is shown in Table 4.1.  The horizontal  displacement plots at the mid-mass locations for each simulation are shown in Figures 4.11(a-d).  Simulation No.  No. of dowels  Max Weight of Mass Earthquake Residual Displacement Displacement kg (lbs) Record (mm) (mm) 907.2 (2000) 9.78 1.018 Landers  1  6  2  3  907.2 (2000)  Kobe  18.66  3.98  3  3  907.2 (2000)  Landers  17.95  4.295  4  6  907.2 (2000)  Kobe  7.19  0.208  Table 4.2 - Summary of final FRAME simulations.  Displacement Results for 6-Dowel Landers Configurations  50  time(s) Figure 4.11a - Displacement results from FRAME for configuration #1  45  Test Specimens and Analytical Predictions  Displacement Results for 3-Dowel Kobe Configuration  FRAME results  time(s)  Figure 4.11b - Displacement results from FRAME for configuration #2  Displacement Results for 3-Dowel Landers Configuration 17.95  4.295  F R A M E results time(s)  Figure 4.11c - Displacement results from FRAME for configuration #3  46  Test Specimens and Analytical Predictions  Displacements Results for6-Dowel Kobe Configuration  -6 -8 time(s)  4.4 Concluding Remarks  The use of F R A M E was instrumental in the pre-experiment and post-experiment stages in this study. F R A M E was used first to determine an appropriate connection geometry and mass for the shake table tests, based on the material properties of the connection and experimental limitations.  A six-hole circular configuration was chosen given its  flexibility to accommodate two distinct and very different connection configurations. A mass of 2000 lbs was chosen to satisfy a desired ductile response based on the chosen 3dowel and 6-dowel configurations.  F R A M E was then used to calculate the natural  frequencies of the test models as well as their response to seismic loading. Modifications to the boundary conditions in the analytical model were necessary to accurately model the test specimens.  Also, because the mass that was used in the experiment was  distributed over a large area, overhanging the top of the column, modifications to the mass matrix of the analytical model were necessary to account for the additional rotation of the specimen caused by the overhanging mass.  Hysteretic damping is inherent in  F R A M E but a viscous damping component was also included in the analysis based on the connection stiffness. The real table accelerations during the tests were then used to  47  Test Specimens and Analytical Predictions  simulate the responses of each connection. These results were then compared to the resultsfromthe shake table tests and are discussed in Chapter 5.  Thefreevibration results showed that the connections were reasonably stiff and may not respond well to the chosen earthquakes, which have a high content of low-frequency vibrations. The resultsfromthe earthquake simulations showed that some damage might occur in the real specimens, particularly the 3-dowel configurations. This was evident from the emergence of residual displacements at the end of each response record. Also, the general shape of the responses was similar to the shapes of the input acceleration records, which was a good first indication that FRAME was producing logical results.  48  Experimental Testing  Chapter 5 EXPERIMENTAL TESTING The experimental part of this study consisted of monotonic tests on the steel dowels, impact tests, and shake table simulations of the moment-resisting steel dowel connection. Specific objectives of these tests were: •  to determine the yield strength of the steel dowels  •  to determine the seismic response of different connection configurations to different earthquakes determine the fundamental natural frequency of each connection configuration  •  5.1 Monotonic Tests 5.1.1 Dowel Yield Strength The purpose of the monotonic tension tests on the V2" diameter steel dowels was to get an indication of their batch yield strength. The average length of each dowel for this test was 150 mm. The results of these tests, for four dowels, are shown in Figure 5.1, with an average yield strength, a , of 345 MPa and an elastic modulus, E , of 2 x l 0 y  5  MPa.  5.1.2 Embedment Properties Monotonic testing of 12.7 mm steel dowels embedded in Parallam® was performed in a prior study (Bimha et al., 1996) and the results used to determine the parameter values are shown in Table 5.1. and defined by Equation {3.5}. From these tests, bearing parameters for the following configurations were available: load parallel to grain with dowel parallel to strands, load perpendicular to grain with dowel perpendicular to strands, load parallel to grain with dowel perpendicular to strands, and load perpendicular to grain with dowel parallel to strands.  Figure 5.2 shows the orientation of the materials and  loads. The results from these tests are input as basic material properties in the analytical model.  These values were obtained by cutting a slot into a block of Parallam® and  inserting a steel plate into the slot. A steel dowel is then inserted into a tight hole in the Parallam® and a reinforced hole in the steel plate. The steel plate is then loaded under cyclic tension and compression while an acquisition machine records the load and stroke.  49  Experimental Testing  |Fu " 490 MPa  ioaoo +  .  o.oo4 0.00  1 OAS  f—  1 0.10  0.15  1  h-  1  0.20  025  O30  STRAIN (mirtun)  OSS  Figure 5.1 - Yield Strength of 12.7 mm Diameter Steel Dowels  Orientation  Parallam® Bearing Parameters(dowel dia. = 12.7mm) K (kN/mm )  Qo (kN/mm)  Ql (kN/mm )  Q2  Q3  Dmax (mm)  1: Load Parallel to Grain, Dowel Parallel to Strands  0.650  0.530  0.00735  0.5  6.3  7.5  2: Load Perpendicular to Grain, Dowel Perpendicular to Strands  0.310  0.470  0.0470  0.5  1.942  11.25  3: Load Parallel to Grain, Dowel Perpendicular to Strands  0.690  0.690  0.00938  0.5  3.953  15.0  4: Load Perpendicular to Grain, Dowel Parallel to Strands  0.300  0.210  0.0525  0.5  1.633  15.0  2  2  Table 5.1 - Bearing parameters for 12.7 mm steel dowels in Parallam® (Bimha et al., 1996)  The above bearing parameters correspond to values for equations that produce the backbone curve for the medium. These curves represent the compressive properties and are used in the analytical model, as discussed in Chapter 3. Orientations 2 and 3 were the two configurations that were relevent to this study.  The plots used to determine the  variables Q 2 and Q 3 for orientations 2 and 3 are located in Appendix C.  50  Experimental Testing  j . H I • i •j:/i"ur: :i »i  orientation 1  vi • i •  1  orientation 2  . . . i - ^ . . . . ••••••  orientation 3  ^  •  orientation 4  Figure 5.2 - Load and Dowel Orientation in Parallam®  5.2 Shake Table Tests The purpose of the shake table tests on the Parallam® and steel dowel moment-resisting connections was to compare the results with those obtained from the connection model, F R A M E (Foschi, 1999). These comparisons served as a verification of the accuracy and validity of the model. The connections were all loaded in only the E-W direction (Figure 5.3).  The design and construction of the connections, choice of excitation records,  instrumentation and monitoring, and the experimental procedure, are discussed in the sections to follow.  Figure 5.3 - Loading direction of test specimens  5.2.1 Design of Connection The Parallam® and steel dowel connection was chosen for the investigation of the seismic response of a ductile timber connection and for comparison to predictions from a non-linear finite-element hysteresis program. Data was already available for 12.7 mm 51  Experimental Testing  steel dowel embedment in Parallam®, and these materials were then used in this study. Initially, a square 4-dowel configuration was chosen because its response obtained from F R A M E to the Landers earthquake record showed a non-linear response, either through crushing of the wood or yielding of the steel dowels. However, a 6-dowel circular configuration was finally chosen because it offered more choices in terms of connection geometries. A three, four or six dowel configuration could be obtained with the adopted circular geometry.  5.2.2 Construction of Connection The connection construction began in September, 1998.  The most difficult part of  construction was drilling the connector holes in the Parallam® and the steel side plates because extreme accuracy and consistency was required for assembling and reusing the side plates for all four test specimens. A drilling procedure was devised by the Civil Engineering machine shop where the drilling was eventually successfully completed. Dimensions of the materials used as well as the 6-dowel configuration are shown in Figures 5.4 - 5.7  The holes were drilled so that the dowels' fit was snug. Both the steel and the Parallam® specimens were drilled with the same bit. Expansion of the Parallam® due to heat during the drilling and deformities in the steel dowels amounted to a tight fit between the dowels, Parallam®, and steel side plates. The dowels were inserted into the specimens simply by individual hammering. Once the side plates were attached to the column with the dowels, the specimen was attached to the shake table by thick steel base plates that were bolted directly into the table. The mass was attached to the column with the use of a C200 x 17 steel section. The channel was bolted to the column with four lag screws at the top of the column and two angle plates that were bolted on the east and west sides of the column. Spacers were then put onto the channel and the mass was then placed to bear on spacers. The mass was attached to the channel with a 1" bolt at either end of the channel. A l l of the large steel plates were connected by bolting them with four readyrods. The masses were further secured from falling onto the table during testing by  52  Experimental Testing  strapping them loosely to the two overhead cranes. The testing, dismantling and set-up of each specimen required approximately one full day to accomplish.  STEEL CONNECTOR PLATEO/4*)  0.3050  0.3800 0.3209 0.0127  T  0.2300 0.1391  •0.047$ •0.1000•0.2050•0.2575-  Figure 5.4 - Steel plate hole pattern  53  Experimental Testing  PARALLAM SPECIMEN  0.9140  0.3099  Figure 5.5 - Parallam® hole pattern  54  Experimental Testing  55  Experimental Testing  Experimental Testing  Figure 5.8 - Plan of connection geometry  57  Experimental Testing  5.2.3 Experimental Procedure  The main experimental part of this study consisted of the shake table tests on the steel dowel connections. In order to investigate F R A M E ' S capability to adapt to different connection geometries and different earthquake input records it was decided that two configurations (3-dowel and 6-dowel) be built and tested. Two dowel arrangements were each tested with two different earthquake records. In this way, a complete comparison was possible, verifying the analytical model for variations in geometry and in input excitation..  5.2.3.1 Choice of Excitation Record  The earthquake acceleration input records were chosen from those already available in the shake table's computer system. Records were chosen so that noticeable non-linearity in the connection behaviour could be expected. This was achieved by selecting records that had a main frequency content close to the fundamental frequency of the connections. Simulations performed by F R A M E to investigate the fundamental frequency of the models are discussed in Chapter 4.  Impact tests on the test specimens were also  performed to determine the fundamental frequency of the connections and compare them to the F R A M E results. The frequency content of the earthquake records was obtained from a spectrum analysis. The analysis involves using the fast-Fourier transform (fft) algorithm to convert the acceleration data from time domain to frequency domain. Based on these results, the earthquake records chosen for the shake table tests were the Landers, 1992 (Figure 5.9) and the Kobe, 1995 (Figure 5.10) earthquakes.  The Landers record  from the Joshua Tree Station in California was found to have a main frequency of 1.71 Hz and secondary frequencies between 5 - 6 . 5 Hz. The Kobe earthquake in Japan had a main frequency content between 1.5 - 3.5 Hz with secondary frequencies between 6.5 8.5 Hz. Figure 5.11 and 5.12 shows the results of the analyses.  58  Landers Earthquake Record (200%) 1  i  •0.8  J  time(s) Figure 5.9 - Landers Earthquake record used for shake table tests  Kobe Earthquake Record 1  i  -1.5 J  time(s) Figure 5.10 - Kobe Earthquake record used for shake table tests  59  Experimental Testing  Spectral Analysis of Landers Earthquake Record  250 200  3" '55 c o  1.71  ,2.69  150 100 50  0 10  •50  15  frequency(Hz)  Figure 5.11 - Spectral analysis of Landers earthquake record  Spectral Analysis of Kobe Earthquake Record 250 200  c  100  frequency(Hz)  Figure 5.12 -Spectral analysis of Kobe earthquake record  60  20  Experimental Testing  5.2.3.2 Instrumentation & Monitoring The response of the connection to the seismic loading was recorded with an extensive network of stringpots and accelerometers.  The stringpots are position transducers  specifically used to measure displacements and can be sensitive to approximately 0.05 mm. Accelerometers are devices used to measure accelerations and can be tailored to measure in the x, y, and z directions simultaneously. Data sheets for these devices can be found in Appendix C. In addition, two video cameras were used to observe the specimen response. One camera was focused on monitoring only the connection while the other was used to monitor the overall specimen response.  The longitudinal displacement (E-W) response of each test specimen was the record of most interest. There were three important issues to consider in the monitoring of each test. •  To obtain accurate longitudinal displacement records at the mass centroid so that a comparison could be made with the results from F R A M E .  •  To observe the transverse (N-S) motions of the specimen to investigate i f there was any out-of-plane motion or in-plane rotation of the mass that could corrupt the results from the E-W direction  •  To observe the 'rocking' of the mass to investigate i f the mass was moving independently of the column, which would indicate that the connection between the column and the mass via the channel was inadequate  The displacement response in the E - W direction was recorded using three stringpots. One was attached to the centroid of the transverse cross-section of the connection, one was attached to the top of the column, while the third was attached to the centroidal location of the mass. The actuator measured the displacement of the shake table and its record was available for each test. The combination of these four displacement records allowed the investigation of the relative displacements of the specimen at the stringpot locations. The movement of the test specimens in the transverse direction (N-S) was monitored using two stringpots, one at each end of the mass. A l l stringpots were attached to rigid frames that were independent of the shake table. Two accelerometers were also attached to the top of the mass (channels 25 - 30 on pg. 115, Appendix C), to monitor  61  Experimental Testing  accelerations in all three orthogonal directions (u, v, z). They were used to monitor the free vibrations of the specimens during the impact tests as well as to measure the accelerations during the shake table tests.  The information from the stringpots and accelerometers were all recorded on two independent data acquisition systems. They were the Labview and Global Lab systems. Labview was the system from which the results in the following sections were obtained. The Global Lab system essentially served as a back-up in case data from Labview were not recorded, or suspicious. A visual inspection of the data served as an initial check to see i f any problems were evident. Appendix C (pgs. 113 - 115) contains the data sheets that show the channel numbers that correspond with the output data records.  5.2.3.3 Testing Sequence  Testing of the connections started on Feb 23, 1999 after a full week of preparation of the specimens for installation onto the shake table. Impact tests were performed before and after each shake table test to determine the natural frequencies of the specimens. A l l tests used a total mass of 2000 lbs. A chronological summary of the test dates, connection configurations, and earthquake records used are shown in Table 5.2.  62  Experimental Testing  Test  Configuration  impact(pre) #1  Earthquake Recorded Record Data (excitation level) n/ a  Date  A l l data  February 23,  impact(post) #1  A l l data  1999.  impact(pre) #2  all data  Shake table #1  6 - dowel  Landers (200%)  all data  February 24,  impact(post) #2  all data  1999.  impact(pre) #3  all data  Shake table #2  3- dowel  Kobe (100%)  all data  February 25,  impact(post) #3  all data  1999.  impact(pre) #4  all data  Shake table #3  Shake table #4  3-dowel  6-dowel  Landers (200%)  Kobe (100%)  impact(post) #4  all data  February 26,  all data  1999.  Table 5.2 - Summary of Tests  5.3 Impact Tests Impact tests were performed before and after every shake table test on a specimen to determine the natural frequencies of the specimens. These tests not only served to verify the results from the analytical model but also to investigate i f any damage was evident in the connection after each shake table test. A rubber mallet was used to hit the test specimen at the mass level. Accelerometers that were placed on the mass were then able to measure the free vibrations of the system in all 3 orthogonal directions. These results were then input into a MathCAD worksheet (Appendix C - pgs. 116 - 122) where a spectral analysis was used to find the natural frequencies of the specimens before and after each test.  5.3.1 Results Unfortunately, the impact tests were not performed on test specimen #1 (6-dowel), so a comparison between the natural frequencies for test #1 was not possible. The analysis of  63  Experimental Testing  the pre- and post-test impact tests for specimen #2 (3-dowel) revealed that the connection had undergone some damage because of the significant drop in the fundamental frequency of the system from - 9 Hz to -5.5 Hz (Figure 5.13a). The impact results from test #3 (3-dowel) showed no signs of damage to this connection (Figure 5.13b). These results are suspicious because the displacement record at mid-mass of this specimen showed that there was permanent deformation evident in the connection due to the residual displacement of the specimen at the end of the record. Specimen #4 (6-dowel) showed some signs of damage from the shake table testing, although not nearly as extensive as specimen #2. There was an evident shift of the fundamental frequency from -9.4 Hz down to -7.8 Hz (Figure 5.13c).  5.4 Shake Table Tests The longitudinal displacement response results from the shake table tests were the most important part of this study. In addition to these results, the transverse response results were also obtained.  These were used to investigate the out-of-plane motions of the  specimen so as to determine i f they were significant enough to corrupt the results in the longitudinal direction. Also, displacement results for the top of the column and the midmass locations were carefully monitored to determine i f the mass was rotating independently of the column during the shake table tests. The out-of-plane displacements of each test were in the order of 2-3 mm, and were assumed to be negligible because such small displacements could have been a result of the transverse stringpot measuring the longitudinal displacement of the specimen. The strings were about 2 meters in length.  5.4.1 Displacement Results The first shake table test comprised of a 6-dowel connection with a mass of 2000 lbs subjected to the modified Landers earthquake record. During the test, the mass reached a maximum acceleration of 1.4g while also obtaining a maximum deflection of 8.8 mm (Figure 5.14). Visual observation of the specimen during the test revealed that it responded quite violently to this earthquake.  The second test involved of a 3-dowel  connection with a mass of 2000 lbs subjected to the Kobe earthquake record.  64  A  Experimental Testing  a)  Spectral Analysis of Test Specimen # 2 (3-dowel, 2000 lbs config.) 0.003 8.98  0.0025 0.002  m c  0.0015  • postKobe - preKobe  0.001  5/7  19.92  0.0005 0 10  -0.0005  20  30  40  frequency(Hz)  b)  Spectral Analysis of Test Specimen #3 (3-dowel, 2000 lbs config.) 0.003 6.64  0.0025 0.002 JJ 0.0015  *j  postLanders  17.97  c  preLanders 0.001 H  0.0005 0  10  20  30  40  -0.0005 frequency(Hz)  Spectral Analysis of Test Specimen # 4  c)  0.0025  (6-dowel, 2000 lbs config.) n  0.002 7.8]  0.0015  I  • postKobe  0.001  - preKobe  c 19.92  0.0005  10  20  30  -0.0005 frequency(Hz)  Figure 5.13(a-c) - Spectral Analysis of impact tests on specimens  65  40  Experimental Testing  maximum acceleration of 1.18g at the mass level was reached, as well as a maximum displacement of 22.6 mm. Observation of this specimen also revealed a violent response to this earthquake input. The displacement results for test #2 are shown in Figure 5.15. Test #3 consisted of the second 3-dowel connection with a 2000 lb mass but subjected to the Landers earthquake record. A maximum top acceleration of 1.29g was recorded as well as a top displacement of 20 mm. The displacement results for this test are shown in Figure 5.16. The final test consisted of a 6-dowel connection with a 2000 lb mass and subjected to the Kobe earthquake record. A maximum acceleration of 1.2g was recorded at the mid-mass level and a maximum displacement of 6.8 mm was observed. The longitudinal response of this specimen is shown in Figure 5.17.  There was no significant permanent  damage for the specimen in test #1, in  correspondence with a residual displacement of ~0.5mm. A permanent displacement of 2mm was evident for the specimen in test #2. Permanent damage was apparent in this connection also. There was little or no apparent permanent damage after test #4 because the residual displacements of the connection were virtually zero. A residual displacement of ~4 mm was observed, on the other hand, for test #3.  Experimental Results for 6-Dowel Landers Configuration  time(s)  Figure 5.14 - Seismic response of specimen #1 to the Landers earthquake  66  Experimental Testing  Figure 5.15 - Seismic response of specimen #2 to the Kobe earthquake  Experimental Result for 3-Dowel Landers Configuration  time(s)  Figure 5.16 - Seismic response of specimen #3 to the Landers earthquake  67  Experimental Testing  Experimental Results for 6-Dowel Kobe Configuration  test results  time(s)  Figure 5.17 - Seismic response of specimen #4 to the Kobe earthquake  5 . 5 Concluding Remarks  The experimental component of this study was part of an on-going project to better understand and model the response of ductile timber connections. Presently, the project has produced results from numerous tests that are available for the calibration and verification of the finite element model.  These tests included:  monotonic tests to  evaluate the basic material properties used in the calibration of the model (Bimha et al., 1996), shake table tests on a two-storey Parallam® frame to investigate the dynamic response of semi-rigid timber frames (Frenette, 1997) and this study, the objective of which was to verify the seismic response predicted by the finite element model through direct shake table testing of different configurations of steel dowel moment-resisting connections under different excitation records. A 3-dowel and a 6-dowel connection configuration was used to help investigate the accuracy of the seismic simulations from the finite element model. Two earthquake  68  Experimental Testing  records were chosen for the tests: 1) the Landers, California earthquake which occurred in 1992 and recorded at the Joshua Tree Station, as well as, 2) the 1995 Kobe, Japan earthquake. The location at which the Kobe earthquake was recorded is unknown. The Landers earthquake was modified (historical amplitude x 2) to accommodate the high natural frequencies of the connections. The modified record had the same duration but its acceleration amplitudes were approximately doubled. A maximum acceleration of 0.87g was reached, which is almost three times the design earthquake that Canada uses for design. F R A M E was used to approximate the response of the connections prior to the shake table testing.  In addition to the longitudinal displacements of each specimen, the transverse or out-ofplane motions were recorded to see that no excessive motions in the transverse direction would corrupt the longitudinal results. Transverse displacements were in the order of 2-3 mm and assumed to have negligible effects on the longitudinal results. Two camcorders were also used to observe the response of the test specimens. One was focused mainly on the connection while the other observed the entire specimen response.  Visual  observations concluded that the 3-dowel connections responded more aggressively to the Kobe earthquake record while the 6-dowel connections responded more violently to the Landers earthquake record.  A qualitative analysis of results from the impact tests showed that the 3-dowel connections suffered considerably more damage than the 6-dowel connections, as expected.  Residual displacements evident in the results confirmed the occurrence of  either the yielding of the steel dowels or the crushing of wood during the tests. Initial results from the impact tests were used to compare with the natural frequency results calculated by F R A M E . The longitudinal displacements of the connections ranged from ~7 mm for test #4 to -23 mm for test #2. Maximum accelerations were within the 1.2 1.4g range for each test.  69  Comparison of Results  Chapter 6 COMPARISON OF RESULTS 6.1 Overview The longitudinal or horizontal displacements at the middle of the applied mass of each test specimen were recorded during the shake table tests and used to compare with the analytical predictions calculated by F R A M E . Impact tests were also performed on the test specimens to obtain the natural frequencies and compared with the natural frequency results obtained from F R A M E .  Direct numerical comparison of the experimental and  analytical fundamental frequencies was a good initial indication of the accuracy of the model. Comparison of the displacement responses was done by comparing the overall shape of the responses, the maximum displacement values, and the residual displacement value achieved. As well, a correlation, power spectral density and coherence analysis of the displacement records were completed on the experimental and analytical results. A l l of these comparisons were used to gauge the ability of F R A M E to simulate the earthquake response of different dowel connection configurations.  6.2 Natural Frequency  Results from F R A M E for the 3-dowel configuration showed a fundamental frequency of 5.13 Hz (Figure 4.8) while the experimental impact tests of the 3-dowel configurations produced results of 8.98 Hz (Figure 5.13a) and 6.64 Hz (Figure 5.13b).  Similarly,  F R A M E results of the 6-dowel configuration gave a fundamental frequency of 7.14 Hz (  (Figure 4.9) while the lone impact test for the 6-dowel configuration resulted in a fundamental frequency of 9.38 Hz (Figure 5.13c). Discrepancy between the analytical and experimental values was likely due to increased friction in the test specimens caused by variability in the fabrication and construction of the test connections.  Increased  friction would result in more stiffness and thus a lower fundamental period or a higher fundamental frequency.  70  Comparison of Results  6.3 Comparison of Displacements The data acquisition system for the shake table recorded the acceleration and displacements at several locations on the specimen as well as the table itself. To obtain the relative displacement results of the connection, the shake table displacements were subtracted from the displacements at the mid-mass location. The actual shake table accelerations recorded for each test were then used as input earthquake records in F R A M E to obtain the analytical predictions for the connection response.  6.3.1 Configuration #1: 6-Dowel Connection with Landers Record The first shake table test consisted of a 6-dowel connection subjected to the modified Landers earthquake record. The test results and the results from F R A M E are shown in Figure 6.1. Initial observation indicates that the two displacement records correspond reasonably well in overall shape, intensity, and the magnitude of the average residual displacement.  The maximum displacement amplitude observed at the first peak was  8.88mm in the test results while results from F R A M E showed a displacement of 7.97mm. Their magnitudes only differ by -10%. Similar results are observed at the second peak where the maximum positive displacements are 9.78mm and 7.11mm for the F R A M E results and test results respectively.  The maximum negative displacements at this  location are very similar, as can be seen from Figure 6.1. Residual displacement of the connection due to permanent deformation of the wood around the connectors and/or the plastic deformation of the steel dowels was evident in the experimental and analytical results. Test results displayed a residual displacement 0.496mm while the results from F R A M E showed a final residual displacement of ~1.018mm. A l l values are shown in Figure 6.1.  71  Comparison of Results  1S  n  Comparsion of Displacements for 6-Dowel Landers Configuration 9.78  10 \  1.018  30  40  50  test results FRAME results -10  J  time(s)  Figure 6.1 - Comparison of Displacement Results for Configuration #1  6.3.2 Configuration #2: 3-Dowel Connection with Kobe Record The next shake table test consisted of a 3-dowel connection subjected to the Kobe earthquake.  Initial observation of the displacement results for this configuration show  that even though the general shapes of each record correspond, the magnitudes of their amplitudes are noticeably different.  A n absolute maximum displacement amplitude of  22.62mm was obtained from the test results while the results from F R A M E showed a maximum absolute displacement amplitude of -18.66mm.  FRAME'S prediction of  response amplitudes during the segment of major shaking (12 seconds - 25 seconds) is smaller. Then the next 15 seconds of the record have displacement amplitudes which are comparable, and the residual displacements are similar in magnitude.  A residual  displacement of 3.98mm is obtained from the predictions from F R A M E while a displacement of 2.29mm occurs in the test specimen. Comparison of the results is shown in Figure 6.2.  72  Comparison of Results  Comparison of Displacements for 3-Dowel Kobe Configuration 22.623  3.98  j^'^SWfW^I  1  30  35  2.288  40  45  - test results FRAME results  Figure 6.2 - Comparison of Displacement Results for Configuration #2  6.3.3 Configuration #3: 3-Dowel Connection with Landers Record The third test specimen consisted o f a 3-dowel connection subjected to the modified Landers earthquake record. The displacement results from the test and from F R A M E are shown in Figure 6.3.  Again, the general shape and magnitude o f the displacement  responses are similar for this configuration.  A maximum absolute displacement o f  17.95mm occurred in the F R A M E prediction while the test results showed a maximum absolute displacement o f 20.05mm. The average residual displacement occurring in the test results was 4.07mm while F R A M E predicted an average residual displacement o f 4.29mm. The maximum displacement values were within 10% o f each other while the residual displacement values differed by no more than ~ 5%.  73  Comparison of Results  Comparison of Displacement for 3-Dowel Landers Configuration  4.071  — test results FRAME results time(s)  Figure 6.3 - Comparison of Displacement Results for Configuration #3  6.3.4 Configuration #4: 6-Dowel Connection with Kobe Record The final test specimen consisted of a 6-dowel connection subjected to the Kobe earthquake record. The plots of the two records in Figure 6.4 show that the results are alike in general shape and magnitude. The maximum absolute displacement predicted by F R A M E was 7.19mm while the test results showed an absolute maximum displacement of 6.84mm. Residual displacements of 0.208mm and 0.161mm were obtained by the F R A M E analysis and test results respectively.  Comparison of Displacements for 6-dowel Kobe Configuration  0.208 0.161 (11/. M i | r W A * 4 * W L M » * » ' 30 40 5t 0  — test results FRAME results  -6.841  time(s)  Figure 6.4 - Comparison of Displacement Results for Configuration #4  74  60  Comparison of Results Figure 6.4 - Comparison of Displacement Results for Configuration #4 6.4 Correlation of Results  A first quantitative method of data comparison is to find the correlation between the two sets of data, one experimental and the other predicted by F R A M E , at each time step. Correlation is a measure of the relationship between two data sets, and the correlation coefficient is the results of taking the covariance of the two data sets divided by the product of their standard deviations.  Finding the correlation of two sets of data is  analogous to finding whether the signals move in phase, that is, whether large values of one set are associated with large values of the other set. Perfect correlation results in a correlation coefficient of 1. A correlation coefficient of 0 means that the two sets of data are unrelated or independent.  Figures 6.13a-d show the correlation plots and their  corresponding correlation coefficients. The more linearly correlated two sets of data are, the more linear the plot and the closer the correlation coefficient is to being unity. For a perfect correlation between the experimental and the predidcted signals, all points should lie on the 45° incline shown. Configurations #1 and #3 show good correlation of results while configuration #4 shows reasonable correlation. The 3-dowel connection configuration subjected to the Kobe earthquake(configuration #2) shows the worst correlation of all of the results, as expected from the dissimilar response shapes shown in Figure 6.2. To analyse the results further, two more methods of analysis are used to compare the two signals.  75  Comparison of Results  25  25  FRAME (mm)  Figure 6.5a - Correlation of Configuration #1  FRAME(mm)  Figure 6.5b - Correlation of Configuration #2  76  Comparison of Results  Figure 6.5c - Correlation of Configuration #3  45° incline line  correlation coefficient = 0.5636  FRAME (mm)  Figure 6.5d - Correlation of Configuration #4  77  Comparison of Results  6.5 Power Spectral Density and Coherence of Results The power spectral density (PSD) and coherence analysis are other methods that can be used to compare different sets of data.  These two methods of analysis compare the  frequency content of the displacement signals by converting the signals from the time domain to the frequency domain. A power spectral density is a function of frequency and can be used to see if the frequency contents of different signals are in agreement. This is true i f frequency peaks match between the two signals. The PSD is obtained directly from the Fourier spectrum of a signal, which represents, for each frequency, the corresponding amplitude. The PSD is defined as the square of the Fourier amplitude, at a given frequency, divided by  2*7i.  The coherence function is a function of frequency as  well, and provides a cross-correlation between two signals. A coefficient is associated with the frequencies of the two signals and varies between -1 and 1.  A coherence  coefficient of 1 indicates perfect in-phase correlation (ie. the two signals are exactly the same), while a coherence coefficient of -1 indicates a perfect 'mirror' image of one signal to the other.  A coherence coefficient of 0 means that there is no correlation  whatsoever between the two signals.  Generally, a coherence coefficient above 0.5  represents good correlation between two signals.  A power spectral density analysis of the results from the shake table test and F R A M E for configuration #1 (6-dowel with Landers earthquake record) is shown in Figure 6.6. The plot shows that the two dominant frequencies in each signal are approximately 3 Hz and 6 Hz. Results from F R A M E showed two dominant frequency peaks occurring at 2.95 Hz and 6.3 Hz.  Similarly, the results from the shake table test showed two dominant  frequency peaks at 2.95 Hz and 5.85 Hz. The locations of peak amplitudes as well as the intensity of amplitudes are well matched in this configuration. This is supported by the coherence plot in Figure 6.7, which shows that the average coherence coefficient between the frequencies of 0 Hz and 10 Hz is 0.75 for the two signals.  78  Comparison of Results  Power Spectral Density Analysis of Results for 6-Pin Landers 5.85  test results FRAME results  6  4  frequency(Hz)  Figure 6.6 - Power Spectral Density Comparison of Results for Configuration #1.  Coherence for 6-Pin Landers Results  L  o u  avg.=  I  -c o o  1  4  6 frequency(Hz)  Figure 6.7 - Coherence of Displacement Results for Configuration #1.  79  10  12  Comparison of Results  The power spectral density analysis results for configuration #2 (3-dowel with Kobe earthquake record) are shown in Figure 6.8. As initially observed in the comparison of the displacement results in Figure 6.2, the results from F R A M E and the shake table tests are the most dissimilar for this configuration. The PSD comparison shows that the test results have a dominant frequency content occurring at 2.88 Hz while the results from F R A M E showed a very small frequency peak occurring at 3.55 Hz. Though differences in amplitude are quite apparent, they tend to over emphasize the actual differences in the amplitudes of the Fourier spectrum because the PSD squares the amplitudes. In this respect, agreement of the frequencies is more important than the magnitude of the amplitudes. The coherence plot shown in Figure 6.9 shows that the average coherence coefficient is 0.65, which still shows reasonable correlation between the two signals.  Power Spectral Density Analysis of Results for 3-Pin Kobe  3000 i  2.8  test results FRAME results o  o  2  6  4  8  frequency(Hz)  Figure 6.8 - Power Spectral Density Comparison of Results for Configuration #2.  80  10  Comparison of Results  rv  Coherence for 3-Pin Kobe Results  1  <D O  0  0  2  avg. = 0.65  4  6  8  10  12  frequency(Hz)  Figure 6.9 - Coherence of Displacement Results for Configuration #2.  The power spectral density analysis results for configuration #3 (3-dowel with Landers earthquake record) are shown in Figure 6.10.  The dominant frequency in the two  displacement results is quite evident from the large peaks occurring near 3 Hz. F R A M E results had a primary frequency content of 2.95 Hz while the shake table test results had a frequency o f - 2 . 9 Hz. The coherence plot in Figure 6.11 shows that the frequency agreements of the two signals are very good since the average coherence coefficient between 0 Hz and 10 Hz is approximately 0.75. Thus, F R A M E was able to adequately predict the frequency response of this connection configuration to the input acceleration record.  81  Comparison of Results  Power Spectral Density Analysis of Results for 3-Pin Landers 2000  1000  test results FRAME results 6  4  8  10  Figure 6.10 - Power Spectral Density Comparison of Results for Configuration #3.  Coherence for 3-Pin Landers Results  avg. = 0.75  0  4  6  8  frequency (Hz)  Figure 6.11 - Coherence of Displacement Results for Configuration #3.  82  10  12  Comparison of Results  The power spectral density analysis results for configuration #4 (6-dowel with Kobe earthquake record) are shown in Figure 6.12. The corresponding coherence plot is shown in Figure 6.13. The PSD of the displacements from the test results show multiple frequency peaks between 2.5 H z and 7 Hz. The results from F R A M E were able to correspond reasonably well to those peak locations. From Figure 6.12, the fundamental frequencies are assumed to be near 3 Hz for the first and 6 Hz for the second. Figure 6.13 shows that the average coherence coefficient is above 0.8 between the frequencies of 0 to 10 Hz. These results also show that good correlation was achieved between the two signals.  Power Spectral Density Analysis of Results for 6-Pin Kobe 75  -j  frequency(Hz) Figure 6.12 - Power Spectral Density Comparison of Results for Configuration #4.  Coherence for 6-Pin Kobe Results  ave. = 0.8  0 -I 0  ,  2  ,  4  ,  ,  ,  6  8  10  frequency(Hz) Figure 6.13 - Coherence of Displacement Results for Configuration #4.  83  Comparison of Results  6.6 Hysteresis Results for A Connector The hysteresis plots for a single connector are shown in Figures 6.14 - 6.15. F R A M E has an option which allows the user to specify a particular connector in which to calculate its hysteretic response. Calculation of the hysteretic response of the test specimens was done by equating the overturning moment of the connection about the centroid of the connections with the resisting forces of each dowel and then plotting this force against the relative slip of a particular connector (pgs 124-126 in Appendix C). Connector 2 corresponds to the vertical response of the right-most dowel (which is horizontally in-line with the centroid of the connection) in each connection configuration.  Hysteresis of the first connection configuration is shown in Figure 6.14, with the horizontal axis showing the slip (mm) and the vertical axis showing the force (kN). The displacement and force magnitudes are modeled very well. Also, the stiffness of the connection is modeled accurately. This is shown by the similar slope of the hysteresis loops. F R A M E tends to underestimate the formation of gaps as shown by the larger loops around the zero displacement zone.  The hysteresis for configuration #3 is  comparable to the results from configuration #1 (Figure 6.16). Slender loops with similar maximum displacement and force magnitudes are shown. This is expected since the 6dowel configurations should show a stiffer response than the 3-dowel configurations. The hystersis plots for configurations #2 and #4 are shown in Figures 6.15 and 6.17. More variability is evident in these comparisons between the analytical and experimental results. Of particular importance is F R A M E ' S overestimation of energy dissipation in the connector. This is evident by the large oval-like loops of the F R A M E results. Maximum displacements as well as maximum forces in the connector are still relatively comparable. Generally, the response of the 3-dowel configurations (Figures 6.15 & 6.16) should be similar i f the demands on the connector are comparable, which they are. Although the Kobe and Landers earthquakes are not the same, the displacement demands on the connector are quite similar, as shown by the similar maximum forces and displacements in the hysteresis plots. This is also true for the 6-dowel configurations (Figures 6.14 & 6.17). Overall, these hystersis plots show that F R A M E is quite accurate in modeling the stiffness of a connector in a connection.  84  Comparison of Results  Hysteresis for Connector 2 6-Dowel Landers  displacem ent(mm)  Figure 6.14 - Comparison of Hystersis for Configuration #1.  Hysteresis for Connector 2 3-Dowel Kobe  displacementfmm)  Figure 6.15 - Comparison of Hysteresis for Configuration #2  85  Comparison of Results  Hysteresis for Connector 2 3-Dowel Landers  displacementfmm) Figure 6.16 - Comparison of Hysteresis for Configuration #3  Hysteresis for Connector 2 6-Dowel with Kobe 20  displacementfmm) Figure 6.17 - Comparison of Hysteresis for Configuration #4.  86  Comparison of Results  6.7 Concluding Remarks By comparing the analytical results from F R A M E to results of real shake table tests, several conclusions can be drawn. Since the hysteresis of a connection is not a property of the material, but history dependent, the results of the displacements show that F R A M E is capable of adapting to different connection configurations as well as different input acceleration records.  This is proven by the generally good correlation, coherence and  power spectral density results.  F R A M E is also capable of accurately modelling the  amount of damage (ie. crushing of wood around dowels and/or the yielding of the dowels) as shown from the good correspondence of the values of the residual displacements in the results. The comparison of the hysteresis plots show that F R A M E is able to accurately model the stiffness of a connection regardless of its history.  The discrepancy in values could have been the result of many factors. These included: loose bolts in the attachment of the mass to the column, friction between the side plates and the column, initial pressure on the dowels from the surrounding wood medium, and friction between the dowels and the wood medium. Increased friction in the experimental tests would increase the stiffness of the connection, thus reducing the amount of deflection the connection would experience. Although friction may have played a role in * increasing the possibility of error, comparison of the results show that in some cases the experimental results were larger than those predicted by F R A M E . This shows that even though friction is not accounted for in this version of F R A M E , the discrepancy in results cannot be mainly attributed to friction. Also, basic material property tests were not performed on the Parallam® used to construct the test specimens. However, steel tensile tests of the dowels were performed prior to the shake table tests. Normally, to reduce the amount of error due to material variability, basic material tests are performed on the batch of materials immediately prior to shake table tests. For these tests, however, time constraints did not allow for basic material tests of the Parallam® used for the shake table test specimens. Instead, compression(embedment) material properties for the Parallam® used in F R A M E were average properties obtained from a previous study at U B C .  87  Conclusions and Future Considerations  Chapter 7  CONCLUSIONS AND FUTURE CONSIDERATIONS The experimental verification of F R A M E , an analysis based on a non-linear finite element hysteresis model for dowel-type connections, was completed. included three specific parts:  This study  the experimental testing of full-scale dowel connections of  different configuration on the U B C Earthquake Shake Table, the analytical prediction of the connection response to earthquakes using F R A M E , and the comparison of the results from the analytical model to the seismic response monitored during the shake table tests.  7.1 Conclusions 7.1.1 Analytical Predictions The dowel connector program called F R A M E , was developed by Foschi (1999). It is based on a non-linear finite element hysteresis model for an elasto-plastic beam supported in a nonlinear medium that acts only in compression.  One of its main  advantages is that the model only needs basic material properties to simulate the dynamic response of dowel type connections, thus removing the need perform full-scale tests to obtain a connection's seismic response.  Analytical results from F R A M E were first used to determine appropriate connection geometries and masses for the experimental shake table tests based on the material properties of the connection and experimental limitations. Initially, a 3-dowel, 4-dowel, or a 6-dowel configuration was used with F R A M E , with two different earthquakes and using different amounts of mass. Based on these results, a 3-dowel and a 6-dowel configuration with 2000 lbs. of weight were chosen for the shake table tests.  Using the acceleration input records obtained directly from the shake table tests, F R A M E was used to predict the displacement response of each connection configuration to the two earthquake records. The results showed that F R A M E was able to adapt to different connection configurations subjected to different earthquake time history records. Also, FRAME  predicted that some damage might occur, particularly in the 3-dowel  88  Conclusions and Future Considerations  connections, as shown by the residual displacements that were evident in the results. The comparisons of the hysteresis plots were able to confirm this. As expected, F R A M E predicted that the 6-dowel connections were suffer than the 3-dowel connections and showed smaller maximum longitudinal displacements. Overall, the general shapes of the connection responses matched those of the input acceleration records, which gave an initial indication that F R A M E was producing good results.  7.1.2 Experimental Results  Shake table tests on two different connection configurations (3-dowel and 6-dowel) using two different earthquake records (Landers (1992) and Kobe (1995)) were performed to validate the analytical results of F R A M E . The connections were made from Parallam®, steel plates and mild steel dowels.  The horizontal displacement of the connection mass was recorded during the shake table tests. Out-of plane accelerations were also recorded to investigate if they were sufficient to distort the results of the horizontal displacements. They proved to be negligible. A l l connection configurations showed good ductility and evidence of damage was apparent in each connection. As expected, the 3-dowel connections suffered the most damage and responded the most violently to the shake table tests. This is supported by the results from the spectral analysis of the impact tests that were performed before and after each connection test. These clearly show that the fundamental frequencies of each connection decreased after each test, confirming the occurrence of damage, in which case the connections lose stiffness and thus have lower fundamental frequencies.  7.1.3 Comparison of Results  Analytical results from F R A M E were compared with the experimental results from the shake table tests. Comparison of the results was completed using different methods. These included direct numerical comparison of the displacement values at each time step, a correlation comparison of the results, a power spectral density analysis of the resulting signals and a coherence analysis.  Direct numerical comparison of the displacement  results showed that with the exception of test configuration #2, F R A M E was able to  89  Conclusions and Future Considerations i  -adequately model the response of each connection to different earthquake records. The response  shapes, maximum displacements, and residual displacement values all  corresponded reasonably well between the analytical and experimental results. Further analysis using correlation, coherence and power spectral density functions also showed that F R A M E was adequately accurate in modeling the response of the dowel connections. The power spectral density and coherence analyses showed that the frequency contents of each connection configuration were well matched while the correlation results showed that the values between the two signals were associated with each other.  Also,  comparison of the residual displacement results showed that all values were within 15 % of each other.  In conclusion, this study has given sufficient support to F R A M E as adequately modeling the response of a dowel-type connector subjected to a time-history, earthquake record. The model is able to adapt to connections of different configurations and respond to different earthquake excitations accurately. Successful verification of this model is the first step towards removing the necessity to perform expensive labor and time-intensive full-scale tests on dowel-type timber connections.  7.2 Future Considerations The response of a timber structure to seismic loads is mainly controlled by the hysteretic behaviour of its connections. The promising results obtained from this study regarding the experimental verification of the hysteretic model F R A M E show that the knowledgebase for the seismic response of semi-rigid timber connections is following the right direction. Further research is still required, to fully understand the behaviour of complete timber structures during earthquakes. Future research topics recommended include:  •  Refinements of the current algorithms in F R A M E (eg. formation of gaps in wood, accounting for friction between the connector-medium interface, and initial confining pressure on the connector by the surrounding medium due to tight-fitting dowels);  90  Conclusions and Future Considerations  •  Instrumentation of individual dowels so that hysteresis loops predicted by F R A M E can be experimentally verified;  •  Development of hysteresis models for other types of connectors used in wood  •  Development of analytical models for the analysis of the response of complete timber structures, including shear walls, and frames in 2 and 3-dimensions;  •  Development of a reliability analysis program that will enable designers to design timber structures to a certain level of seismic reliability  91  References  REFERENCES  Bakhtavar M.S., (1998) Behaviour of Timber Structures in Earthquakes, Paper for  Civil 516 course: Behaviour of Timber Structures, University of British Columbia, Canada. Bimha, R., Bagga, G., Elwood K., Foschi R., Prion, H . G. L., (1996) Final Report: Behaviour of Heavy Timber Connections Under Cyclic Loadings, Technical  Paper, University of British Columbia, Canada. Blass H.J., (1989) Earthquake Bracing of Multi-Storey Timber Skeleton Structures,  Proceedings of a Workshop: Structural Behaviour of Timber Constructions in Seismic Zones, University of Florence, Italy, 175-189. Blass H.J., (1994) Basic Behaviour of Joints, Timber Structures in Seismic Regions: R I L E M State-of-the-art Report, Materials and Structures, 27, 173-175. Blass H.J., (1994) Behaviour of Timber Joints Under Cyclic Loading, Timber Structures  in Seismic Regions: R I L E M State-of-the-art Report, Materials and Structures, 27, 175-177. Booth E.D., (1998) Earthquake Engineering in the 1990's: achievements, concerns and  future directions, Proc. Institution of Civil Engineers, Structures and Buildings, Paper 11449,128, 154-166. Buchanan A . H . (1984) Wood Properties and Seismic Design of Timber Structures,  Proceedings of the Pacific Timber Engineering Conference, Auckland, New Zealand, 462-469. Buchanan A . H . , Dean J.A., (1988) Practical Design of Timber Structures to Resist  Earthquakes, Proceedings of the International Conference on Timber Engineering, Seattle, USA, 813-822. Buchanan A . H . , (1989) Earthquake Resistance of Timber Buildings in New Zealand,  Proceedings of a Workshop: Structural Behaviour of Timber Constructions in Seismic Zones, University of Florence, Italy, 209-223 Canadian Wood Council (CWC), (1995) Wood Design Course Notes, CWC, Ottawa, Canada.  Canadian Wood Council, (1995) Wood Design Manual, CWC, Ottawa, Canada.  92  References  Canadian Wood Council (1998) PSL ProductInfo, CWC, Ottawa, Canada, http://wwwxwc.memcsxom/psl.html. Carr, A.J., (1998) Ruaumoko: Inelastic Dynamic Analysis Program, Department of Civil  Engineering, University of Canterbury, Christchurch, New Zealand, 226 pp. Ceccotti A., Vignoli A . , (1988) The Effects of Seismic Events on the Behaviour of SemiRigid-Joint Timber Structures: A Simulation of the Influence of Structural  Scheme and ofJoint Characteristic, Proceedings of the International Conference on Timber Engineering, Seattle, USA, 823-837. Ceccotti A., Vignoli A . , (1989) A Hysteretic Behavioural Model For Semi-RigidJoints,  European Earthquake Engineering, Vol. 3,3-9. Ceccotti A., (1994) Modelling Timber Joints, Timber Structures in Seismic Regions: R I L E M State-of-the-art Report, Materials and Structures, 27, 177-178. Chopra A . K . , (1995) Dynamics of Structures, Prentice-Hall Inc., New Jersey, U S A Chui Y . H . , N i C , Jiang L., (1998) Finite-Element Modelfor Nailed Wood Joints Under  Reversed Cyclic Load, Journal of Structural Engineering, ASCE, Vol. 124, No. 1, 96-103. Cooney R.C., (1979) The Structural Performance of Houses in Earthquakes, National  Society of Earthquake Engineering, Bulletin, New Zealand, 12(3): 223-237. Dean J.A., Deam B.L., Buchanan A . H . , (1989) Earthquake resistance of timber structures, Technial Article: Reprinted from Proceedings from IPENZ Annual Conference. Foliente G . C , (1995) Hysteresis Modelling of Wood Joints and Structural Systems,  Journal of Structural Engineering, ASCE, Vol. 121, No. 6, 1013-1022. Foschi R.O., (1999) FRAME, Analytical Hysteresis Model for Dowel-Type Timber Connections, Computer Program, Department of Civil Engineering, University of British Columbia, Canada. Foschi R.O., (1998) Modelling Hysteretic Response for Reliability-Based Design in  Earthquake Engineering, Technical Paper, Department of Civil Engineering, University of British Columbia, Canada. Frenette C D . , (1996) Dynamic Behaviour of Timber Frame with Dowel Type  Connections, Proceedings of the International Wood Engineering Conference, New Orleans, USA, Vol. 4, 89-96.  93  References  Frenette C D . , (1997) The Seismic Response of a Timber Frame With Dowel Type Connections, Master's Degree Thesis, University of British Columbia, Canada. Jorissen A., (1996) Multiple Fastner Timber Connections with Dowel Type Fasteners, Proceedings of the International Wood Engineering Conference, New Orleans, USA, Vol. 4, 189-196.  Khan A., (1995J A new, nonlinear analysis of layered soil-pile-superstructure seismic Response, Master's Degree Thesis, University of British Columbia, Canada. Ohlsson S., (1994) Damping in Timber Structures Under Seismic Actions, Timber Structures in Seismic Regions: R I L E M State-of-the-art Report, Materials and Structures, 27, 164-166. Prakash V., Powl G.H., Campbell S., (1993) DRAIN-2DXBase Program Description and User Guide, Version 1.10, University of California, USA. Prion H.G.L., Foschi R.O., (1994) Cyclic Behaviour of Dowel Type Connections, Proceedings of the Pacific Timber Engineering Conference, Gold Coast, Australia, Vol. 2, 19-25. Riahl S.S., (1983) Behaviour of Timber Building Structures during the Coalinga, California Earthquake of May 2, 1983, Paper 222, California Polytechnic State University, Californa, USA. Stewart, W.G., (1987) The Seismic Design of Plywood Sheathed Shear Walls, Ph.D. Thesis, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand, March, 395 pp. Wood J.H., Cooney R.C., Potter S.M., (1976) Cyclic Testing of Connections for Light Timber Construction, Central Laboratories Report 5-76/12, Ministry of Works and Development, New Zealand.  94  APPENDIX A  95  Degrees of freedom of a beam element: In the direction perpendicular to the beam element: w = A / Q (%>) - Displacement w'=Mf ( \ ) a - Slope a  w"= Mj (I,)-a - Curvature In the direction parallel to the beam element: u = Nl (t,)-a -Displacement u'=Nf ($)-a -Slope Shape functions:  M (l,y=(8-15J i+104 - 3 4 )/i6 3  5  0  M,  -64 H-104 +4 -34 )(A/32)  M  24  3  2  0  4  5  +24 +4 - 4 )(A /64)  2  3  4  s  2  M  0  (4,4) = 0 (5,4) = 0 (6,4) = (8+ 15 4-io4 +34 )/i6  M  Q  (7,4) = (-5-74+64 +104 - 4 - 3 4 )(A/32)  M» M  o.  ;  3  5  2  (8,4) = (i + 4- 24 M (9,4) = o M (10,4) = o M  3  4  s  - 2 4 +4 +4 )(A /64)  2  3  0  4  5  2  Q  0  M , (1,4) = (-15+ 304  2  -154 )(2/16A ) 4  :  ;  /r,  M (2,4) = (-7-124 + 304 +44 -154 )/16 2  3  4  1  Mi (3,4) = (-l-44+64 +44 - 5 4 )(A/32) 2  3  4  3/, (4,4) = o A/, (5,4) = o M  i  (6,4) = (l5-304 +154 )(2/16A) .. 2  4  M , (7,4) = (-7+ 124+304 -44 -154 )/16 2  3  4  M (8,4) = ( l - 4 4 - 6 4 +44 +54 )(A/32) 2  x  M (9,4) = 0 " x  3f,(l0,4) = 0  3  4  '"..Ji'"" 7- . V  ;  /  ;:  ;;  96  = ( ^«H 6 0  z  M  M  3  )(4/16A ) 2  (2,$) = (-12 + 60$+12$ - 6 0 £ 2  2  AT, ( 3 , £ ) = (-4 +12$ + lit, -20$  )/16  3  2  )(2/16A)  3  3/ (4,$) = 0 3/ (5,$) = 0 2  2  M (6,$) = (-60$ + 60$ )(4 /16 A ) 3  2  2  M (7,$) = (l2 + 60$-12$ - 6 0 $ )(2/16A) 2  3  2  M (8,$) = (-4-12$ + 12$ + 20$ )/16 2  3  2  M (9,$) = 0 M (l0,$)-0 2  ,  2  .  :  ,  M  JV (2,$) = 0 iV (3,$) = 0 0  0  JV (4,$) = (2-3$ + $ ) / 4  , •  3  0  /V (5^) = ( l - ^ - ^ ^ ) ( A / 8 )  .  3  0  +  N  i  (6,^)=o  0  ^o(7,^) = 0  '  * ( >0 = ° 8  0  iV (9,0 = (2 + 3 ^ - ^  3  0  JV (lO,$) = ( - l - $ + $ 0  2  )/4  )  + $ )(A/8) /  :  3  ^,(U) = 0 A i(2,^) = 0 r  ^i(3,^) = 0  AT (4,$) = (-3 + 3 $ ) / ( 2 A )  '  2  1  JV (5,$) = ( - l - 2 $ + 3 $ ) / 4 2  1  (6,5) = 0  AT,  ^,(8,0 = 0  ,  ..  * i (9,$) = ( 3 - 3 $ ) / ( 2 A ) 2  -  ;  /V (l0,$) = ( - l + 2 $ + 3 $ ) / 4 2  1  97  MATERIAL PROPERTIES INPUT FILE 5(#  o f gauss p o i n t s i n x)  16(# o f gauss p o i n t s i n y)  1(#  of different  types)  fastener  E)  0.2000E+03(modulus o f e l a s t i c i t y ,  0.3550E+00(yield s t r e n g t h ,  a) y  0.8900E+02(length o f connector) 1(cross  s e c t i o n type:  l=circular)  0.1270E+02(diameter o f connector)  1(number o f l a y e r types) (compression p r o p e r t i e s ' p a r a l l e l t o g r a i n ) 0. 6900E+00 (Q ) 0 . 9380E-02 (Qi) 0 . 5000E+00 (Q ) 0. 6900E+00 (K) . lSOOE+O^.d?^) 0  0.3953E+01 (Q )  2  3  (compression p r o p e r t i e s ..perpendicular..to ..grain) 0.4700E+00 2(#  0.4700E-01  of different  1(fastener  type)  1 ( l a y e r type)  0.5000E+00  0.1942E+01  0.3100E+00  0.1125E+02  types o f c o n n e c t i o n s ) 1(# o f l a y e r s )  2(perp. l a y e r p r o p e r t i e s used)  6 ( l a s t element i n l a y e r ) 2(nodes with boundary  s t  element i n l a y e r )  .0.8900E+02(thickness o f l a y e r )  conditions)  1 (node « with) boundary., conditions/). 7(node # with b.c.'s)  l(l  2 (#• of b.c.'s)  3 (#)l(u=0)  2(w=0)  _  1 ( l a y e r displacement - s p e c i f i e d ; I ( l a y e r # f o r displacments). •• •  3(w'=0)  o , 9.;'"?- , i  '  2 (w=C>;..: 3 (w' =0)  ;  2l~--'.7.-<'-':  "  .."•>•••'  •• •  0.2000E+01(load m u l t i p l i c a t i o n f a c t o r ) {same as above but f o r p a r . l a y e r p r o p e r t i e s ) I  V;-...;.  I  1 1 2 1 2 7 3  1  6  2 3 1 2  0.2000E+01  0.';v00;>02  .8900E+02  3  > "  "  ;  i  0.1000E-01 ( t o l x - r e l a t i v e rcqnyer.gence t o l e r a n c e ••,for -vector; ,X ) 0  0.1000E-03(tolf - a b s o l u t e  convergence t o l e r a n c e  f o r force)  :  -.'?  , • GEOMETRY INPUT FILE 0. 9 0 7 2 4 E - 0 3 (mass) 9810 ( a c c e l e r a t i o n  0.84900E+03(eccentricity 1500  of- g r a v i t y )  o f mass)  (length o f mass-for i n e r t i a  calcs.)  12(# o f c o n n e c t o r s - b o t h d i r e c t i o n s ) l ( c o n n e c t o r t y p e no.)  1 ( f o r c e o r i e n t . . code: l = p a r a l l e l t o x, 2 = p a r a l l e l t o y)  0.10500E+03(x-coordinate)  0.00000E+00(y coordinate)  2  2 0.10500E+03 1 1 0.52500E+02 2 2 0.52500E+02 1 1 -0.52500E+02 2 2 -0.52500E+02 1 1 -0.10500E+03 2 2 -0.10500E+03 1 1 -0.52500E+02 2 2 •0.52500E+02 1 1 0.52500E+02 2 2 0.52500E+02  0 .00000E+00 0 .90930E+02 0 .90930E+02 0 90930E+02 0 90930E+02 0 00000E+00 0. 00000E+00 - 0 . 90930E+02 - 0 . 9.0930E.H-02; - 0 . 90930E+02 - 0 . 90930E+02  0 . 0 0 5 ( g l o b a l convergence t o l e r a n c e ) 1 ( h o r i z o n t a l input a c c e l e r a t i o n present) joshua6.txt(name o f h o r i z o n t a l " a c c e l e r a t i o n r e c o r d ) 0 ( v e r t i c a l i n p u t a c c e l e r a t i o n (hot p r e s e n t ) 0.20000E+03(maximum  a c c e l e r a t i o n step)  4 . 40 (damping c o n s t a n t ,:s-d)iiJ Jlr*u:o :  h y s t e r ( n a m e o f .output f i l e :for. c o n n e c t o r 2 ( c o n n e c t o r number f o r h y s t e r e s i s  output)  *input values are i n b o l d * e x p l a n a t i o n s a r e i n p a r e n t h e s i s ()  99  hysteresis)  Sample F R A M E output for prediction of hysteresis of one connector  time(s) 2.3E-05 0.000117 •0.000245 0.000318 0.000472 0.000896 0.001385 0.001726 0.002201 0.002938 0.003559 0.004006 0.004529 0.004918 0.005054 0.005342 0.005688 0.00566 0.005656 0.005989 0.006011 0.005667 0.005618 0.005448 0.004626 0.004079 0.004165 0.00379 0.003012 0.002836 0.002751 0.00214 0.001839 0.00185 0.001242 0.000669 0.000925 0.001056 0.00084 0.001276 0.001841 0.001698 0.001827 0.002501 0.002542  hyst 0.000674 0.003429 0.007168 0.009313 0.013825 0.026255 0.04058 0.050571 0.064483 0.08608 0.10426 0.11736 0.13266 0.14403 0.14803 0.15645 0.16656 0.16576 0.16564 0.17536 0.17601 0.16594 0.16451 0.15951 0.13543 0.11939 0.12192 0.11092 0.088103 0.082949 0.08046 0.062535 0.053726 0.054035 0.03623 0.01943 0.026918 0.030765 0.024443 0.037218 0.053793 0.049582 0.053362 0.073113 0.074336  100  Sample F R A M E output for prediction of relative rotation of connection time(s)  rot. (rads)  5.00E-03 1.00E-02 1.50E-02 • 2.00E-02 2.50E-02 3.00E-02 3.50E-02 4.00E-02 4.50E-02 5.00E-02 5.50E-02 6.00E-02 6.50E-02 7.00E-02 7.50E-02 8.00E-02 8.50E-02 9.00E-02 9.50E-02 1.00E-01 1.05E-01 1.10E-01 1.15E-01 1.20E-01 1.25E-01 1.30E-01 1.35E-01 1.40E-01 1.45E-01 1.50E-01 1.55E-01 1.60E-01 1.65E-01 1.70E-01 1.75E-01 1.80E-01 1.85E-01 1.90E-01 1.95E-01 2.00E-01 2.05E-01 2.10E-01 2.15E-01 2.20E-01 2.25E-01 2.30E-01 2.35E-01 2.40E-01  2.19E-07 1.11E-06 2.33E-06 3.03E-06 4.49E-06 8.53E-06 1.32E-05 1.64E-05 2.10E-05 2.80E-05 3.39E-05 3.82E-05 4.31 E-05 4.68E-05 4.81 E-05 5.09E-05 5.42E-05 5.39E-05 5.39E-05 5.70E-05 5.72E-05 5.40E-05 5.35E-05 5.19E-05 4.41 E-05 3.88E-05 3.97E-05 3.61 E-05 2.87E-05 2.70E-05 2.62E-05 2.04E-05 1.75E-05 1.76E-05 1.18E-05 6.35E-06 8.79E-06 1.01 E-05 8.01E-06 1.21 E-05 1.75E-05' 1.62E-05 1.74E-05 2.38E-05 2.42E-05 2.26E-05 2.88E-.05 3.34E-05  v;:,;lj';nLt'^:.!irad.K  :  ' ' ^  !  101  Sample F R A M E output for prediction of relative vertical displacement of connection  time(s)  vert, (mm)  5.00E-03 0.00E+00 1.00E-02 9.06E-11 1.50E-02 9.09E-10 • 2.00E-02 3.02E-09 2.50E-02 5.80E-09 3.00E-02 1.32E-08 3.50E-02 4.04E-08 4.00E-02 9.23E-08 4.50E-02 1.46E-07 5.00E-02 2.10E-07 5.50E-02 3.38E-07 6.00E-02 5.06E-07 6.50E-02 6.22E-07 7.00E-02 6.83E-07 7.50E-02 7.46E-07 8.00E-02 8.19E-07 8.50E-02 9.12E-07 9.00E-02 1.00E-06 9.50E-02 9.96E-07 1.00E-01 9.45E-07 1.05E-01 1.01E-06 1.10E-01 1.09E-06 1.15E-01 1.03E-06 1.20E-01 9.57E-07 1.25E-01 1.03E-06 1.30E-01 1.58E-06 1.35E-01 2.22E-06 1.40E-01 1.55E-06 1.45E-01 1.43E-07 1.50E-01 2.59E-07 1.55E-01 1.85E-06 1.60E-01 2.39E-06, 1.65E-01 9.35E-07 1.70E-01 -3.23E-07 1.75E-01 4.91 E-07 1.80E-01 2.15E-06 1.85E-01 2.16E-06 1.90E-01  4.79E-Q7  1.95E-01 2.00E-01 2.05E-01 2.10E-01 2.15E-01 2.20E-01 2.25E-01 2.30E-01 2.35E-01 2.40E-01  -3.49E-07. 9.28E-07 2.35E-06 1.78E-06 9.64E-08 -1.55E-07 1.38E-06 2.54E-06 i.73E-0d 3.14E-08  102  Sample F R A M E output for prediction of relative horizontal displacement of connection time(s)  hor. (mm)  5.00E-03 -3.12E-04 1.00E-02 -1.06E-03 1.50E-02 -1.68E-03 2.00E-02 -2.47E-03 2.50E-02 -4.38E-03 3.00E-02 -7.15E-03 3.50E-02 -1.02E-02 4.00E-02 -1.37E-02 4.50E-02 -1.80E-02 5.00E-02 -2.24E-02 5.50E-02 -2.72E-02 6.00E-02 -3.18E-02 6.50E-02 -3.51 E-02 7.00E-02 -3.74E-02 7.50E-02 -4.00E-02 8.00E-02 -4.21 E-02 8.50E-02 -4.30E-02 9.00E-02 -4.41 E-02 9.50E-02 -4.55E-02 1.00E-01 -4.57E-02 1.05E-01 -4.55E-02 1.10E-01 -4.56E-02 1.15E-01 -4.38E-02 1.20E-01 -4.01 E-02 1.25E-01 -3.68E-02 1.30E-01 -3.40E-02 1.35E-01 -3.07E-02 1.40E-01 -2.80E-02 1.45E-01 -2.58E-02 1.50E-01 -2.23E-02 1.55E-01 -1.91 E-02. 1.60E-01 -1.79E-02 1.65E-01 -1.58E-02 1.70E-01 -1.20E-02 1.75E-01 -9.50E-03 1.80E-01 -8.11E-03 1.85E-01 -6.27E-03 1.90E-01 -6.44E-03 1.95E-01 -9.01 E-03 2.00E^01 -1.05&02, 2.05E-01 -1.15E-02 2.10E-01 -1.43E-02 2.15E-01 -1.65E-02 2.20E-01 -1.70E-02 2.25E-01 -1.89E-02 2.30E-01 -2.17E-02 2.35E-01 -2.29E-02 2.40E-01 -2.47E-02  103  APPENDIX B  104  CALCULATION OF DAMPING COEFFICIENT, a  Obtain the natural period o f the signal from the free vibration portion o f the response. /=  1/T  co = 2*71*/  cc = 2*4*co where T , is the period (s) co, is the natural frequency (rads/sec) f, is the natural frequency (Hz) 4, is the damping ratio (Calculated from degradation o f signal from hammer tests, M  ' s h o w n below." ImtiaTassumption o f 5% (Chopra, 1995.))  The assumption o f 5% is fairly accurate as the hammer test results show 3 degrading signals each with approximately 20 amplitude peaks from the maximum amplitude to the steady-state amplitude.  Impact Test Results for 3-pin Configuration 0.1  n  0J08 0.06 &  0.04  "35  C  0.02  -0.06 ->  time(s)  105  Bulletins from The Northwest Division TRUS JOIST MACMILLAN PRODUCT APPLICATION ASSURANCE March 30, 1993  PAA BULLETIN No. 4.1.0 Page 4 of 6  LIMIT STA' MICRO=LAM® LVL AND P. ORED STRENGTHS AND STIFFNESS (A Resistance Factor of 0.90!is Included) 1.8E WS MICRO • LAM* 2.0E WS PARALLAM* LVL PSL Shear Modulus of Elasticity  G  177,800  197.600  Modulus of Elasticity  E  1.8x10°  2.0x10'xj.lv.  Flexural Stress  F>  4110  4585% 1,35"^  Fc  1185  Compression Perpendicular . to Grain, Parallel to Glue Line Compression Parallel to Grain Horizontal Shear Perpendicular to Glue Line  4  Fc  n  Fv  Ill For a 12 inch depth, d, for others ers *&i!piy by  m  w  1030  3890  4685  450  460  (PARALLAM PSL Only) 0 1  •  •  •  F C i shall not be increased for duration of load  •  W  V,  2(  ?£c^O  For a 12 inch depth, d, for others multiply by J  l K  01  (MICRO =LAM* LVL Only] 01  /'  1,0,  "Jr.  APPENDIX C  107  Orientation #2 for bearing parameters of steel dowels in Parallam® (from Table 5.1) .  :  i  ;  .  O d  (Nftfe>JOf  The position measurement workhorse.  Specifications:  The PT101 is our most widely used transducer. Such diverse . applications as positioning shutter dampers for draft control, testing automotive suspension systems and monitoring lateral drift in space shuttle launching operations are common. Designed for linear measurements up'to'75CJ inch put by means of a precision potentiometer. A more rugged ver- :' sion is available for use in extreme environments. This instrument will endure shock of 2.000 G's for six milliseconds and vibration • of 10 G's from 10 to 2000 Hz without damage or changes in ,..„• calibration. • • :' , . "-f. .'.  GENERAL .Range'  0-2 to 0-750 i n c h e s  Weight Case Material..  20 o u n c e s (to;50 inch  range) Aluminum  Sensing System......  ; Electrical Connector .'  :  FEATURES: • • • • •  0.1% accuracy standard (most ranges) Resolution of 0.002 inch range dependent,.,. Ranges from 0-2 to 0-750 inches full scale . "A" circuit full scale output approximates input' "B" circuit simulates wheatstone bridge with zero at midpoint, for use with strain gauge signal conditioners • Operates to velocities up to 300 in./sec.;  .  . ............. Precision Potentiometer  : MS3102E-MS-6P  (Other c o n n e c t o r s optional)  ELECTRICAL -' Input Resistance: For 2" & 5" units contact factory . "A" .Circuit . . 500 ohms std. Other options available "B". Circuit -.: .1100 ohms std. Other options available v ' Output Resistance: For 2" & 5" units contact factory' "A" Circuit 138 ohms max. std. Other options available "B" Circuit 240 ohms std. Other options available Excitation Voltage . . : . . . 25 volts max.. AC or DC Insulation Resistance 100 meg ohms min. at 100 VDC PERFORMANCE Accuracy:' • • 2 and 5 Inch ranges .±0.25% F.S. typical 10 and 15 Inch ranges ±0.15% F.S. typical 20 inch and greater ±0.10% F.S. typical Resolution: ; 2 and 5 Inch ranges 0.08% F.S. max. . 10 inch and greater . . . . 0.008% F.S. max. Thermal Coeffeclent of Sensing Element : Zero . . . 88 P.P.MV*F Span . 88P.P.M7T Sensitivity " . v ! ; . • : ( . ; . . . . . See Specification'Table ENVIRONMENTAL 0° F to +200° F (-65° F to +250° F available) Temperature Range up to 90% R H Humidity . Vibration. up to 10 G s to 2000 Hz. :  1  Measurement Range  Output •A" Circuit  Output "B" Circuit  Approx. Cable Tension'  Max. Cable* Acceleration  Inches  MWV/lnch  MV/V/lnch  Ounces  Ext. Gs Ret.  2 5 10 15 20 25 30 40 50 60 75 80 100 150 200 250 300 350 500 750  468.5 188.5 94 75 62.55 47.38 38.18 31 32 2464 19 09 16.34 13.19 12.45 9.887 6176 4964 3.882 3.311 2.816 1.978 1310  1.0536 0.4240 0.2131 0.1407 0.1065 08587 . .07043 . 05542 04294 03676 02967, .02799 02224 .01389 .01116 008731 N.A. N.A. N.A, N.A.  :  ,. 37 . ; 7 5 ' 37 "'• 25 ' 1 5 - ; • 10 •. ; 25 "• ' l~ . .; 37 :•" 7 5 •'• 11 „••' • ' N 10 15 1 :i.8V-r:.12 8 ., 14 factory. . 7 5 • 11 2 6 . ... . 4 . • •• ' .9 • -• 3 3 5- -I '4 , 3 .;• 5 14 . . , . 7 5 • ' 25 6 16 '. ,'A ' 3 5 12 5 . . 3 •103 • 5 3 • -> 5 • •10 5 3 15 5 3 ;' 9 \ . .  27 11 27 18  •' liv  :  ' '.Pit',- "'•  1  20OIA I Y P . . (5081  i  D I M  < ' 66 ' •  l'6?6) ' :  44 (11 101  17 . (4 32) •  R A N G E  2'10" 20  Consult factory (or dimensional information on ranges . over 50" "  15" 30" .5" 25" 40" 50"  • ei Finical, CIRCUIT  A  ELECTRICAL CIRCUIT B -  JI COMMON  •our  no . OROFR INronMATIDM MOnn. M O  '  RANGFIINl  CIRCUIT  Model 3021 Performance  Specifications  C  Supply Current = 1.5 mA & Ambient Temperature = 25°C (Unless otherwise specified)  PARAMETER  ±5G  ±10G  RANGE ±20G  0-350 Hz 0-500 Hz 0-700 Hz 600 Hz 1200 Hz 850 Hz  Frequency Response (-5%) Mounted Resonant Frequency (±15%)  A LL RANGES TYP MAX UNITS NOTES  MIN  PARAMETER  ±50G ±100G 0-1050 Hz 0-1600 Hz 2750 Hz 1800 Hz  30 1 50 mV Full Scale OutDUt SDan ±mV 2 . 1 Zero Acceleration OutDut 3 .707 DamDina Factor • .- •. 4 .±%Span: ":' ' 'IX ?' Non-Linearitv and H y s t e r e s i s . » ±%Span 5 3 Transverse Sensitivity a 6000 4500 InDut & OutDut Resistance 2,5 ±%Span 2.0 Temoerature Coefficient - SDan ±%Span 2.5 1.0 TerrtDerature Coefficient - Zero -"..5""" ±%7°C " "0.22 Temperature Coefficient - Resistance — -1.5 -•—mA ~ - • - 2 Supply -Current" - - — . _ — —~ —• .;2.o '::;.'..:..'-.:hl.„.;.,:.:„ - --5.0 Supply Voltage .-.,„_„„... ...... . ... ..,«. 12,0VDC- :-. •-- 2 — 1 . r. OutDUt Noise" • /•"V'r::.:J;:'...;.;7,^:.;.':,,. ... „ :.1.0 . M-V. D-D 6 MQ 2 Output Load Resistance „20X Rated Acceleration Limits (Any Direction),,,..,.„. Operating Temperature -40°C to +125 °C Storage Temperature -55°G to +150 °C Weight (Excluding C a b l e ) " " " ~ "" T.2 G r a m s " • " " " " " " " " ' ~ " -  :  :  :  : rt  ;  T O  Notes  .  C  . z.;:.rz:r.;:::'t^.".rr"7::::T\T*r::r: ':":7:':''^r^:;:.;'";:": ,  1. From zero to positive acceleration value. ' 2. With external resistors added to reduce zero and spa^ to reduce zero acceleration output,"'ffie"v^ues"'f6r these resistors are supplied with each unit. Compensate constant current and. constant'voltage excitation. Consult factory."'"" 3. Damping factor is. coritrolled-to. within..±10.% over entire temperature range Alternate damping ratios are available on a special order basis. • 4. Best Fit Straight Line linearity. ?'." '". 5. Temperature range: 0-50 C in reference"to 25 °C. 6. Prevents increase of TC-Span due to output loading. 7. Various electrical connections are avairaBlerR""= nbbon cable, P = pins, N = none. 6  Ordering  Information  SRJPcontrol systems ltd.  3021 - 010 - R  I  Represented  Electrical Connection (R,P,N - see Note 7) Acceleration Range Model  62 GUIDED COURT R E X D A L E . ONT. M9V 4K6 (416) 746-0117 TELEX: 065-27418 FAX: (416) 746-0570  211 G O R D O N C R E S C E N T PRINCE G E O R G E . B.C. V2M 4 R : (604) 561-0622TELEX: 04-77133 FAX:(604) 563-4752  IC Sensors products are warranted against defects in materia! and workmanship for 12 months from data of shipment.,Products not I subjected to misuse will be repaired or replaced. THE FOREGOING IS IN LIEU OF ALL OTHER EXPRESSED OR IMPLIED.WARRANTIES. IC ( Sensors reserves the right to make changes/to any/..product.herein and assumes no liability arising out of the application of any produc. i or circuit described or referenced herein.  ICSENSORS  1701 McCarthy Blvd. Milpitas, California 95035 Fax (408) 434-6687 Telex 350066 Phone (408) 432-1800 — —. • ' 7. r—. : : — M3021RO-8709 Printed in USA  PRELIMINARY  OEM Accelerometer Miniature Size Low Cost Features DC Response - Wide Bandwidth • High Sensitivity Built-in Damping • Low Mass Built-in Overrange Stops Solid State Reliaibility? • Piezoresistive Ease of Mounting  i a l J  Description  t.\ilZL~i  The. Model,3021 is the f i r s t In a f a m l I y of general purpose, solid-state, piezoresistive accelerometers and is packaged on a ceramic substrate and is intended for use where small size, exceNeht required. The accelerometer cdnsistS4"of a"rriicromachined silicon mass suspended:^ to an outside frame. Piezoresistors 3 located ' i h ' t h e r e s i s t a n c e as the.motion of the suspended mass changes the .strain in the beams. Silicon caps on the top and the bottom of the device are added to provide overrange stops and increased durability. As a result of this unique three-layer silicon structure, accelerometers with a very low profile and low mass can be batch fabricated at a very low cost. An added feature is the built-in damping, which allows a wide useable bandwidth to be achieved. The damping factor is contr6iied<fo within ± 10% over the entire operating temperature range. ..."„...,."  • pica! Applications Automotive Suspension Control Automotive Braking Control. Machine Tool Monitoring Industrial Vibration Monitoring Computer Peripherals o-rv.-u ;csran:;ic: -SiliiiHt'ais • -;:nd :s !.'.;'...sn'.:;."d; ---jr.?-.--. rr.~ Modal Analysis - . ; Theidevice is' available .^acceleration: ranges from ± 5 G to ± Security Systems Motion Detection 100 G. Device performance characteristics and packaging can be easily tailored to meet the requirements of specific applications. Aerospace Flight Navigation Connections/Dimensions Robotic Motion Control ., ....... . Medical Patient Activity Monitoring Appliance Control Military Arming and Fuzing ;  60 . 55 •  .bu •  i  i.  C*«L£' 5 Conductors . 050 Centeri '  R I B B O N  5G 10G 20 G 50 G 100 G  nTTTT  3021-R  3021-P * Accetf  aww  •Acceleration  r  I  -•• •  c — t  O  U  S  U  O  Standard Ranges ± ± ± ± • -  f—o  2 . T Y P  ALL DIMENSIONS IN INCHES  112  U  T  T  P  P  P  U P  U  EXPERIMENT F O R M :  DATE:  3L  Fe(b £ Y  PROJECT: PROJECT NUMBER:  ACCOUNT NUMBER:  PROJECT SUPERVISOR STUDENTS NAMES: DATA ACQUISITION SYSTEM LAB VIEW FILE DIRECTORY NAME : p , ' \ Pot^er«J?3\3 \i^s. SAMPLING FREQUENCY :. - • Z o z > s / ? ..  GLOBAL LAB "D:\poco^f?  \ 3 W^i\  z^aVr  :  <S{- - o • 00 f  CHANNEL LAB VIEW  0  ALLOCATION :  ;  16  Cfo  2 CrM  11  17 18  4 o/y  /?  20  zft  11diV  36  .3..c#../.£ 4  CA«/<?  7 c/s*<~  24  25. 26 27 28 29 30 31  10 eft zt  T2cff 3/  13_ 14_ 15  NOTES:  2 cA*VS-  21 22 23  5 cj£_3±_ ctf z r 1 c±_a±  9 Iff  :1 clL*.  19  6  GLOBAL  ; r ^ . ; ^ T  f(LC  PRC  10 12 44  15 0  it*/)AG-T  ^_  C^>  6U>bA<^  LAP)-  £  >  113  M  LAB  ^^A55.  DATE: / g g £3 /??  Douet CTo^-oe*:77^  PROJECT:  -  PROJECT NUMBER:  ACCOUNT NUMBER:  PROJECT SUPERVISOR  ' ^ r ^ - ^ r : : - - - ^ . - J - 4 ^ r - , : - - - - . - -  STUDENTS NAMES:  0 CrVo  ~ + LjeTT/ftnt'16-  -fane brs'f  ' CH Ka irvsss  £p&rueorini*'mm  2 " ( 7 ' mio••*-a-Asf  act fwr  efoH-ti £  0 CT/^ 2 £ FHASS .40£ e « r 7 <7-rV/  A»c e*Sr  17  ..  v  ;  .-. .. • •  ;  (-0- ^  19 20 21  -  1  12 13 14 15 NOTES:  , U H f s  ^  ; -  R  ,:..^^!:--cArv:- I O / o v/,,,,;: . 2 cff rt'' "  . • ,/:,;,,  .-- :, .  _ _ e 5  C//«?.r..  ... •  <?//  .6^  ,22  -fiC.: • 3/  7  23 24 ; , ^ _ _ ^ „ _ _ _ _ ,25 •26 . ::.- . ••cn f 27 . • 12cA ^ 28 ^ ~ — — 2 9 • -  Uf?W 0  e^r  30  1  * — — 7 :  3 1 — - r — ~  ">5_<_r7.u-o^  114 _  ____  ^ g p - 5/5 -  -  '  -  - £/*•• 30 -  30  m--.  0  "  " -~~  ;  1 Q / 3 / mete Me  fc  -"fry p o u ^ H l ^ L  £ \ . - .  r  ' —jrjfiZ  s  5 off  :  ^ :  GLOBAL LAB  CHANNEL ALLOCATION  .4^Y/p_., -  / 3  .-  r:rr::•^••::,-^-y:-=:-,  DATA ACQUISITION SYSTEM LABVIEW . FILE DIRECTORY NAME : -.•.•OAfo.- e t p ? \ 4 £ SAMPLING FREQUENCY : e p o ^  ^  -'  •" - ••  Channel Locations for Shake Table Tests  MathCAD worksheet for spectral analysis of results of configuration #1  fileinpe := "expl.inp" exp := READPRN(fileinpe) N := 7999  At := 0.005  fileinpa := "predl.inp"  fileout  := "pair 1.out"  n:=i.N  time .-(n-l)At n  ana := READPRN(ffleinpa) last(ana)= 7.999-10  last(exp) = 7.999-10  J  40  exp  MI  1  1  J  -  J  1  1  1  20  u l l l i l l 1 l"MliftiiiiUirr  —HW|^^  ifif  "20  10  ji  15  j i\ieoui.;-=  .20  "pairi.oi  l  " 1 ^  25  1 35  30  time"'  cor_coef := corr(exp, ana)  cor^cc43f= 0.766 l  ni:=2  !  mavg := 6  1  Af := !  N  At— ni  exp := detrend(exp)  k:= 1- — + • 1 2-ni  f. :=(k- l)Af-  over:=0  ^ ../max(f) ~finax:= ceil  ana := detrend(ana)  psdEX := pspectTum(exp, ni,' over) '  psdEX :=movavg(psdEX, mavg) >  psdAN := pspecunim(ana, ni, over)  psdAN := movavg(psdAN, mavg)  ,;  :  2000  psdEX psdAN1000 1  h  116  40  Configuration # 1 cont'd  Coherence function  coh := coherence(exp, ana, ni, over)  cob. := movavg(coh, mavg)  sigbur^^^f^n-sigbu^j := psdEX^; sigout^ := psdAN^ sigout,^ := cor^ 3  WRITEPRN(fileout) := sigout  \  'f \ j \ / " V  117  V  1  > -  . /W •  vw:  < ' *•  MathCAD worksheet for spectral analysis of results of configuration #2  fileinpe= "exp2.inp"  exp := READPRN(  N:= 7999  fi!einp$  it := 0.005  fileinpa= "pred2.inp"  fileout=  "pair2.out"  n := 1.. N  tim^ := ( n - 1) it  ana := READPRN( fileinp^ las<ana) = 8.001 10  T  .  1  r  1  las<e.\p) = 8.001 10"'  3  T  r  -  j i . L -1 T o io~_ srs;.:,. ;r:'!i 10 .•;••• - - i s .•' ':.2o..•;: . 2 5 . • •. 3q '.:>;:.;;35time ;  cdr_coef:= corr(exp, ana)  Af:=  expi= detrend(exp)  cor_coef= 0.522  hi := 2  k:=l..—-hi •. ...2ni  mavg:= 6  t:=(k-l)-4f *  ana:= detrend(ana)  :  over:=0  frnax" c e i / " " " ^ \ 2 /  "  Power Spectral Densities of Model Base Accelerations relative to the Shake Table: psdEX := pspectrurr(exp, ni, oyer) psdAN := pspectruir(ana, ni, over)  Coherence function '  •j>sidEX,:=iriqvavg[psdEX,may$ :  i  K  psdAN := movavg; psdAN, mav$  cbh':= coherence; e'xpi aha,'ni,'over) '''''coh^'mbvav^coh.rnavg  MathCAD worksheet for spectral analysis of results of configuration #3  fileinpe := "exp3.inp"  exp :=READPRN(fileinpe) -  N:=8000  At:= 0.005  fileinpa := "pred3.inp  n  fileout:= "pai* - *" 3  n:=l-N  011  time := ( n - l ) A t  ana := READPRN(fileinpa) last(ana) = 8.001 10  J  last(exp) = 8.001  20  -10  corcoef := corr(exp, ana)  cbVcoef = 0.543  ni:=2  mavg := 6  over := 0  1  Af :=  . N At-— ni exp :=-detrend(exp)  -k:=:1. — - I ' 2-ni  1  f. :=(k- l ) A f • ' k  ana := detrend(ana).  p s d E X p s p e c t r u m (exp, ni, over)  psdEX :^movavg(psdEX, mavg)  psdAN := pspectrum (ana, ni, over)  psdAN := mdvavg(psdAN, mavg)  psdEX ' psdAN : |  •JmaxCQ  finax •:= ceil  \  2  W  Configuration #3 cont'd  Coherence function  cob. := coherence (exp, ana, ni, over)  cob. := movavg(coh, mavg)  f  sigout j := k)  sigout^ :- psdEX^ sigoul^. : := psdAN . sigout^ .:= cob^ 2  3  k  WlUTEPRN(fileout) := sigout  120  4  MathCAD worksheet for spectral analysis of results of configuration #4  fileinpe := "exp4.inp" exp := READPRN(fileinpe) N:=8000 fileinpa := "pred4.inp"  At:=0.005  fileout  n:=l-N  := "pair4.out" time  := (n-l)At  n  ana := READPRN(fileinpa) last(ana) = 8-10  last(exp) = 8-10  3  101  0  1  1  1  5  10  15  1  1  20  23  timo  cor_coef := corr(exp, ana)  cor_coef = 0.564  k:= l.iL  exp.:= detrend(exp) ...,.....„,.,.  f :=(k-l)Af k  : — i  1  30  mavg := 6  -  over := 0  finax^ceii^^)  ana•:= detrend(ana)  Power Spectral Densities of Model Base Accelerations relative to the Shake Table: psdEX := pspectrum (exp, ni, over)  psdEX : = movavg( psdEX, mavg)  psdAN := pspectrum(ana, ni, over)  • psdAN := movavg(psdAN, mavg)  psdEX psdAN  121  35  n  ni:=2  +l  3  40  Configuration #4 cont'd  Coherence function  coh := coherence(exp, ana, ni, over)  cob. := movavg(coh,mavg)  f  sigout^ j :=  sigout,^ := psdEX^ sigout^ := psdAN sigout,^ := cob^. (3  k  WRTIEPKN(fileout) := sigout  122  I I  N.  UJ  8 j  70 • 3 5 0  •  532  535  535  •  53  530  io  !  i  j.  |  f!0A co j  535  •  530  530  524  CO  i65 • 3 4 5  •  *  528  8.  2  .  65  542  ! 3  s  8 -  i• '•  11 .j. i  i  123  •  v  425  83  <*3  439  8  •  £  CO  529  IS  a  2  CO  T  CO  530  542  CO  co  8  445  5  IO  2 530  •  IO  S3  S3  528  2  co  440  5  S3  540  65 S3  530  .8  • 335 •  ;  o  5?  3  a-'  r  444  533  5?  o  75  533 -T0 .  535  533  •347  1  co  CM  85  5  S3~  s.  417  530  535  535  • 440  397  5  3  sio  cvi —J io  3  65  535  53  57 • 347 •  •  533  65  437 N.  N.  •9  831 • 333  533  8  • 67  8  Calculation of Hysteresis Plots from Experimental Results  (LoTfVT'O VAX- v \ 0 ' " ' i - ' OF- K V & C T T ' A Or ^ ~ ^ \ T  —>  a,(*)  V  pot  £ c r - > i ° 6 e--ro£  $ wit-c. Be  2,  F o e <• B-  C A t - C - V F ^ O / V l  r  A M U ^ O M M 6CTPC  ( F*ecc- oe-Fi^tS  e 62  w  <?tv_c- f~*  Hot'tot-n^A  AT e  E/V-t-f-|  pie  rue  c o / v / M e O b C >g<?^T roMMe  dT^oO  r n ?  T  € - » ~ - » T £<=>>_>  £ o *-J-F7 6 <-> ^ T R S K J E  0.  Ion itetui-rvwT  M  e  124  r  O  P  Cal. of Hysteresis cont'd  U.C+")  Or  SPcxi-vcV  AT  £ 1<Sverri o r '  3  =e  A F. If  roue  -1 f  125  J  , , , < 0  r  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0050148/manifest

Comment

Related Items