2001

CUREE Publication No. W-02

Earthquake Hazard Mitigation of Woodframe ConstructionFunded by the Federal Emergency Management Agency through a Hazard Mitigation Grant Program

award administered by the California Governor’s Office of Emergency ServicesCUREE

the CUREE-Caltech Woodframe Project

Development of a Testing Protocol forWoodframe Structures

Helmut KrawinklerFrancisco Parisi

Luis IbarraAshraf Ayoub

Ricardo Medina

Department of Civil and Environmental EngineeringStanford University

The CUREE-Caltech Woodframe Project is funded by the Federal Emergency Management Agency(FEMA) through a Hazard Mitigation Grant Program award administered by the CaliforniaGovernor’s Office of Emergency Services (OES) and is supported by non-Federal sources fromindustry, academia, and state and local government. California Institute of Technology (Caltech)is the prime contractor to OES. The Consortium of Universities for Research in EarthquakeEngineering (CUREE) organizes and carries out under subcontract to Caltech the tasks involv-ing other universities, practicing engineers, and industry.

the CUREE-Caltech Woodframe Project

CUREE

Disclaimer

The information in this publication is presented as apublic service by California Institute of Technology andthe Consortium of Universities for Research in EarthquakeEngineering. No liability for the accuracy or adequacy ofthis information is assumed by them, nor by the FederalEmergency Management Agency and the CaliforniaGovernor’s Office of Emergency Services, which providefunding for this project.

CUREe

CUREEConsortium of Universities for Research in Earthquake Engineering

1301 S. 46th St.Richmond, CA 94804-4698

tel.: 510-231-9557 fax: 510-231-5664e-mail: curee@curee.org website: www.curee.org

Development of a Testing Protocol forWoodframe Structures

CUREE Publication No. W-02

2001

Helmut KrawinklerFrancisco Parisi

Luis IbarraAshraf Ayoub

Ricardo Medina

Department of Civil and Environmental EngineeringStanford University

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Preface The CUREE-Caltech Woodframe Project originated in the need for a combined research and implementation project to improve the seismic performance of woodframe buildings, a need which was brought to light by the January 17, 1994 Northridge, California Earthquake in the Los Angeles metropolitan region. Damage to woodframe construction predominated in all three basic categories of earthquake loss in that disaster:

Casualties: 24 of the 25 fatalities in the Northridge Earthquake that were caused by building damage occurred in woodframe buildings (1);

Property Loss: Half or more of the $40 billion in property damage was due to damage to woodframe construction (2);

Functionality: 48,000 housing units, almost all of them in woodframe buildings, were rendered uninhabitable by the earthquake (3).

Woodframe construction represents one of society’s largest investments in the built environment, and the common woodframe house is usually an individual’s largest single asset. In California, 99% of all residences are of woodframe construction, and even considering occupancies other than residential, such as commercial and industrial uses, 96% of all buildings in Los Angeles County are built of wood. In other regions of the country, woodframe construction is still extremely prevalent, constituting, for example, 89% of all buildings in Memphis, Tennessee and 87% in Wichita, Kansas, with "the general range of the fraction of wood structures to total structures...between 80% and 90% in all regions of the US….” (4). Funding for the Woodframe Project is provided primarily by the Federal Emergency Management Agency (FEMA) under the Stafford Act (Public Law 93-288). The federal funding comes to the project through a California Governor’s Office of Emergency Services (OES) Hazard Mitigation Grant Program award to the California Institute of Technology (Caltech). The Project Manager is Professor John Hall of Caltech. The Consortium of Universities for Research in Earthquake Engineering (CUREE), as subcontractor to Caltech, with Robert Reitherman as Project Director, manages the subcontracted work to various universities, along with the work of consulting engineers, government agencies, trade groups, and others. CUREE is a non-profit corporation devoted to the advancement of earthquake engineering research, education, and implementation. Cost-sharing contributions to the Project come from a large number of practicing engineers, universities, companies, local and state agencies, and others. The project has five main Elements, which together with a management element are designed to make the engineering of woodframe buildings more scientific and their construction technology more efficient. The project’s Elements and their managers are: 1. Testing and Analysis: Prof. André Filiatrault, University of California,

San Diego, Manager; Prof. Frieder Seible and Prof. Chia-Ming Uang, Assistant Managers 2. Field Investigations: Prof. G. G. Schierle, University of Southern California, Manager 3. Building Codes and Standards: Kelly Cobeen, GFDS Engineers, Manager; and James Russell,

Assistant Managers 4. Economic Aspects: Tom Tobin, Tobin Associates, Manager 5. Education and Outreach: Jill Andrews, Southern California Earthquake Center, Manager

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The Testing and Analysis Element of the CUREE-Caltech Woodframe Project consists of 23 different investigations carried out by 16 different organizations (13 universities, three consulting engineering firms). This tabulation includes an independent but closely coordinated project conducted at the University of British Columbia under separate funding than that which the Federal Emergency Management Agency (FEMA) has provided to the Woodframe Project. Approximately half the total $6.9 million budget of the CUREE-Caltech Woodframe Project is devoted to its Testing and Analysis tasks, which is the primary source of new knowledge developed in the Project.

Woodframe Project Testing and Analysis Investigations

Task # Investigator Topic

Project-Wide Topics and System-level Experiments

1.1.1 André Filiatrault, UC San Diego

Kelly Cobeen, GFDS Engineers

Two-Story House (testing, analysis)

Two-Story House (design)

1.1.2 Khalid Mosalam, Stephen Mahin, UC Berkeley Bret Lizundia, Rutherford & Chekene

Three-Story Apt. Building (testing, analysis) Three-Story Apt. Building (design)

1.1.3 Frank Lam et al., U. of British Columbia Multiple Houses (independent project funded separately in Canada with liaison to CUREE-Caltech Project)

1.2 Bryan Folz, UC San Diego International Benchmark (analysis contest)

1.3.1 Chia-Ming Uang, UC San Diego Rate of Loading and Loading Protocol Effects

1.3.2 Helmut Krawinkler, Stanford University Testing Protocol

1.3.3 James Beck, Caltech Dynamic Characteristics

Component-Level Investigations

1.4.1.1 James Mahaney; Wiss, Janney, Elstner Assoc. Anchorage (in-plane wall loads)

1.4.1.2 Yan Xiao, University of Southern California Anchorage (hillside house diaphragm tie-back)

1.4.2 James Dolan, Virginia Polytechnic Institute Diaphragms

1.4.3 Rob Chai, UC Davis Cripple Walls

1.4.4.4 Gerard Pardoen, UC Irvine Shearwalls

1.4.6 Kurt McMullin, San Jose State University Wall Finish Materials (lab testing)

1.4.6 Gregory Deierlein, Stanford University Wall Finish Materials (analysis)

1.4.7 Michael Symans, Washington State University Energy-Dissipating Fluid Dampers

1.4.8.1 Fernando Fonseca, Brigham Young University Nail and Screw Fastener Connections

1.4.8.2 Kenneth Fridley, Washington State University Inter-Story Shear Transfer Connections

1.4.8.3 Gerard Pardoen, UC Irvine Shearwall-Diaphragm Connections

Analytical Investigations

1.5.1 Bryan Folz, UC San Diego Analysis Software Development

1.5.2 Helmut Krawinkler, Stanford University Demand Aspects

1.5.3 David Rosowsky, Oregon State University Reliability of Shearwalls

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Not shown in the tabulation is the essential task of managing this element of the Project to keep the numerous investigations on track and to integrate the results. The lead management role for the Testing and Analysis Element has been carried out by Professor André Filiatrault, along with Professor Chia-Ming Uang and Professor Frieder Seible, of the Department of Structural Engineering at the University of California at San Diego. The type of construction that is the subject of the investigation reported in this document is typical “two-by-four” frame construction as developed and commonly built in the United States. (Outside the scope of this Project are the many kinds of construction in which there are one or more timber components, but which cannot be described as having a timber structural system, e.g., the roof of a typical concrete tilt-up building). In contrast to steel, masonry, and concrete construction, woodframe construction is much more commonly built under conventional (i.e., non-engineered) building code provisions. Also notable is the fact that even in the case of engineered wood buildings, structural engineering analysis and design procedures, as well as building code requirements, are more based on traditional practice and experience than on precise methods founded on a well-established engineering rationale. Dangerous damage to US woodframe construction has been rare, but there is still considerable room for improvement. To increase the effectiveness of earthquake-resistant design and construction with regard to woodframe construction, two primary aims of the Project are:

1. Make the design and analysis more scientific, i.e., more directly founded on experimentally and theoretically validated engineering methods and more precise in the resulting quantitative results.

2. Make the construction more efficient, i.e., reduce construction or other costs where possible,

increasing seismic performance while respecting the practical aspects associated with this type of construction and its associated decentralized building construction industry.

The initial planning for the Testing and Analysis tasks evolved from a workshop that was primarily devoted to obtaining input from practitioners (engineers, building code officials, architects, builders) concerning questions to which they need answers if they are to implement practical ways of reducing earthquake losses in their work. (Frieder Seible, André Filiatrault, and Chia-Ming Uang, Proceedings of the Invitational Workshop on Seismic Testing, Analysis and Design of Woodframe Construction, CUREE Publication No. W-01, 1999.) As the Testing and Analysis tasks reported in this CUREE report series were undertaken, each was assigned a designated role in providing results that would support the development of improved codes and standards, engineering procedures, or construction practices, thus completing the circle back to practitioners. The other elements of the Project essential to that overall process are briefly described below. To readers unfamiliar with structural engineering research based on laboratory work, the term “testing” may have a too narrow a connotation. Only in limited cases did investigations carried out in this Project “put to the test” a particular code provision or construction feature to see if it “passed the test.” That narrow usage of “testing” is more applicable to the certification of specific models and brands of products to declare their acceptability under a particular product standard. In this Project, more commonly the experimentation produced a range of results that are used to calibrate analytical models, so that relatively expensive laboratory research can be applicable to a wider array of conditions than the single example that was subjected to simulated earthquake loading. To a non-engineering bystander, a “failure” or “unacceptable damage” in a specimen is in fact an instance of successful experimentation if it provides a valid set of data that builds up the basis for quantitatively predicting how wood components and systems of a wide variety will perform under real earthquakes. Experimentation has also been conducted to improve the starting point for this kind of research: To better define what specific kinds of simulation in the laboratory best represent the real conditions of actual buildings subjected to earthquakes, and to develop protocols that ensure data are produced that serve the analytical needs of researchers and design engineers.

vi

Notes (1) EQE International and the Governor’s Office of Emergency Services, The Northridge Earthquake of January

17, 1994: Report of Data Collection and Analysis, Part A, p. 5-18 (Sacramento, CA: Office of Emergency Services, 1995).

(2) Charles Kircher, Robert Reitherman, Robert Whitman, and Christopher Arnold, “Estimation of Earthquake

Losses to Buildings,” Earthquake Spectra, Vol. 13, No. 4, November 1997, p. 714, and Robert Reitherman, “Overview of the Northridge Earthquake,” Proceedings of the NEHRP Conference and Workshop on Research on the Northridge, California Earthquake of January 17, 1994, Vol. I, p. I-1 (Richmond, CA: California Universities for Research in Earthquake Engineering, 1998).

(3) Jeanne B. Perkins, John Boatwright, and Ben Chaqui, “Housing Damage and Resulting Shelter Needs: Model

Testing and Refinement Using Northridge Data,” Proceedings of the NEHRP Conference and Workshop on Research on the Northridge, California Earthquake of January 17, 1994, Vol. IV, p. IV-135 (Richmond, CA: California Universities for Research in Earthquake Engineering, 1998).

(4) Ajay Malik, Estimating Building Stocks for Earthquake Mitigation and Recovery Planning, Cornell Institute for

Social and Economic Research, 1995.

vii

Contents

Summary ix

Objective and Scope 1

Proposed Testing Protocols

1. Testing Protocol for Deformation Controlled Quasi-Static 2 Cyclic Testing

1.1 Loading History for Ordinary Ground Motions (Basic Loading History) 2 1.2 Loading History for Near-Fault Ground Motions (Near-Fault Loading History) 5 1.3 Specimen Fabrication, Testing and Instrumentation Issues 6 1.4 Documentation of Specimens and Test Results 7

2. Testing Protocol for Force Controlled Quasi-Static Cyclic Testing 9 2.1 Loading History for Force Controlled Quasi-Static Cyclic Testing 9 2.2 Specimen Fabrication, Testing, Instrumentation, and Documentation 10

3. Time Histories for Shaking Table Testing 10

Figures 12

Commentary to Proposed Testing Protocols 19

C1. Considerations in Development of Loading Protocols 19 C1.1 Considerations for Deformation-Controlled Quasi-Static Cyclic Testing 19 C1.2 Considerations for Force Controlled Quasi-Static Cyclic Testing 20 C1.3 Considerations for Shaking Table Testing 21

C.2 Selection of Ground Motion Records 21 C.2.1 Set of Ordinary Records for Development of Basic Loading History 23 C.2.2 Small Events Preceding a Performance Assessment Event 24 C.2.3 Set of Near-Fault Records for Development of Near-Fault Loading History 25

C.3 Selection of Structural Systems for Prediction of Response 26 C.3.1 Common System Parameters 27 C.3.2 Hysteresis Models 28

C.4 Maximum Response Values 29 C.4.1 Results for Ordinary Ground Motions 29 C.4.2 Results for Near-Fault Ground Motions 31

C.5 Cumulative Damage Considerations 32 C.5.1 Cumulative Damage Issues 32 C.5.2 Process for Incorporating Cumulative Damage Effects into Loading History 34

C.6 Development of Representative Loading Histories 35 C.6.1 Development of Deformation Controlled Basic Loading History 36 C.6.2 Development of Deformation Controlled Near-Fault Loading History 38 C.6.3 Development of Force Controlled Loading History 39

C.7 Representative Input for Shaking Table Studies 40

Acknowledgements

viii

41

References 42

Tables 44

Figures 50

Appendix A – Representative Results for Response to Ordinary Ground Motion 74

ix

Summary

This report offers recommendations for a protocol for quasi-static experimentation on components of woodframe structures and for shaking table experimentation on wooden houses. The emphasis is on the development of loading histories useful for a performance assessment at various performance levels, for the evaluation of various failure modes, and for the development of design equations and analytical models. Recommendations are made also on specimen fabrication, testing procedures, specimen instrumentation, and documentation of test results. The development of loading histories is based on results of nonlinear dynamic analysis of representative hysteretic systems subjected to sets of ordinary and near-fault ground motions. Cumulative damage concepts are employed to transform time history responses into representative deformation and force controlled loading histories. The results of the work are

• a testing protocol for deformation controlled quasi-static cyclic testing, consisting of proposed loading histories for ordinary and near-fault ground motions, and recommendations for fabrication and instrumentation of test specimens and documentation of test data,

• a testing protocol for force controlled quasi-static cyclic testing, and

• a series of recommendations on input ground motions for shaking table studies.

x

1

Objective and Scope Several testing protocols have been proposed, and are in use, for the monotonic and cyclic testing of woodframe structural components (ASTM 1995a, ASTM 1995b, CEN 1995, CoLA/UCI Committee 1999, Foliente et al. 1998, ISO 1999, SAA 1997, Shepherd 1996). The objective of the work summarized in this report is to establish common testing protocols for all component tests and shake table tests of the CUREE/Caltech Woodframe Project. The loading histories in these testing protocols should represent the seismic demands imposed by Californian earthquakes on woodframe buildings. For shake table testing, single- and multi-axis excitations for different seismic hazard levels should be established. The loading histories should be representative of the seismic demands imposed on components and structures for the following conditions:

• Ordinary ground motions that represent design events envisioned by present codes. • Near-fault ground motions. • Multiple earthquakes occurring in the lifetime of the structure. The development of loading histories for seismic performance testing requires the execution of time history analysis that captures the demand characteristics peculiar to the wood structures of interest in the Woodframe Project. These demands are evaluated through simulation studies in which analytical models of representative woodframe structures are subjected to ground motions of various characteristics. The demands are then represented in loading histories that simulate, in a cumulative manner, the damage experienced by the structural systems. To perform this work, models of structures of different periods are subjected to various sets of ground motions, utilizing a versatile force - deformation model capable of representing the hysteretic characteristics of typical wood components and systems. Specifically, the following tasks are addressed: • Develop a basic loading history for component tests, considering ordinary ground motions

representative of California (in particular Los Angeles) conditions. • Develop a near-fault loading history for component tests. • Develop a loading history for force-controlled elements (e.g., certain types of hold-downs). • Develop a testing protocol that addresses common issues of testing and documentation of

component test results. • Establish sequential ground motions for shaking table tests. • Establish near-fault ground motions for shaking table tests.

2

Proposed Testing Protocols Testing protocols are concerned with the construction and instrumentation of test specimens, the planning and execution of experiments, the loading history to be applied to a test specimen, and the documentation of experimental results. In this report the emphasis is on the development and documentation of loading histories for deformation and force controlled component testing and of time history inputs for shaking table testing. All other aspects of testing protocols are summarized as needed and, whenever feasible, adopted by reference to existing documents. 1. Testing Protocol for Deformation Controlled Quasi-Static Cyclic Testing This protocol is intended to apply for all component tests in which a deformation parameter (displacement, rotation, angle of shear distortion, etc.) can be identified that relates the component response to the response of the structural system of which the component is part. A typical example is the test of a plywood shear wall panel in which the racking distortion can be related to the story drift. An example to which this protocol should not be applied is the test of anchor bolts in which the interest is not in the load-deflection behavior as much as in the sudden brittle failure that would limit the system capacity.

1.1 Loading History for Ordinary Ground Motions (Basic Loading History) The primary objective of this loading history is to evaluate capacity level seismic performance of components subjected to ordinary (not near-fault) ground motions whose probability of exceedance in 50 years is 10 percent. Deformation cycles due to smaller events prior to the capacity level event are included in the history’s deformation history. Applicability of the loading history to limit states other than capacity is not specifically addressed. The commentary provides further discussion of the objectives. The loading history for a basic cyclic load test should follow the pattern given in Fig. 1. The history is defined by variations in deformation amplitudes, using the reference deformation ∆ as the absolute measure of deformation amplitude. The history consists of

• initiation cycles, • primary cycles, and • trailing cycles.

Initiation cycles are executed at the beginning of the loading history. They serve to check loading equipment, measurement devices, and the force-deformation response at small amplitudes. A primary cycle is a cycle that is larger than all of the preceding cycles and is followed by smaller cycles, which are called trailing cycles. All trailing cycles have an amplitude that is equal to 75% of the amplitude of the preceding primary cycle. All cycles are symmetric, i.e., they have identical positive and negative amplitudes. Deformation control should be used throughout the experiment. The following sequence of cycles is to be executed:

3

• Six cycles with an amplitude of 0.05∆ (initiation cycles) • A primary cycle with an amplitude of 0.075∆ • Six trailing cycles • A primary cycle with an amplitude of 0.1∆ • Six trailing cycles • A primary cycle with an amplitude of 0.2∆ • Three trailing cycles • A primary cycle with an amplitude of 0.3∆ • Three trailing cycles • A primary cycle with an amplitude of 0.4∆ • Two trailing cycles • A primary cycle with an amplitude of 0.7∆ • Two trailing cycles • A primary cycle with an amplitude of 1.0∆ • Two trailing cycles • Increasing steps of the same pattern with an increase in amplitude of 0.5∆, i.e., one

primary cycle of amplitude equal to that of the previous primary cycle plus 0.5∆, followed by two trailing cycles.

Reference deformation ∆: This deformation is the maximum deformation (displacement, drift angle, rotation, etc.) the test specimen is expected to sustain according to a prescribed acceptance criterion, and assuming that the proposed basic loading history has been applied to the test specimen. It is a measure of the deformation capacity of the specimen. Thus it will be necessary to estimate the deformation capacity prior to the test. This estimate can be based on previous experience, the execution of a monotonic (or near-fault) test to assist in this estimate, or a consensus value that may prove to be useful for comparing tests of different details or configurations. The choice of the reference value ∆ may vary from component to component or may be fixed for a specific testing program. The Woodframe Project management will provide input to this decision. The general guidelines are as follows: • Perform a monotonic test, which provides data on the monotonic deformation capacity, ∆m.

This capacity is defined as the deformation at which the applied load drops, for the first time, below 80% of the maximum load that was applied to the specimen, see Fig. 2.

• Use a specific fraction of ∆m, i.e. γ∆m, as the reference deformation for the basic cyclic load test. At this time, a value of ∆ = 0.6∆m is suggested. The factor γ should account for the difference in deformation capacity between a monotonic test and a cyclic test in which cumulative damage will lead to earlier deterioration in strength. This factor is subject to change based on information acquired in the Woodframe project testing programs.

Additional considerations: 1. The test should be continued in the predetermined pattern until the maximum load applied in

a cycle decreases to a small fraction of the maximum load.

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2. If deemed useful, the execution of small amplitude cycles after the execution of each “step” (a primary cycle followed by trailing cycles) should be contemplated.

3. A final definition of acceptable performance is not provided here. Testing and analysis will tell what acceptability criteria should be used. The emphasis for acceptance should be on a threshold for unacceptable deterioration in strength. A basic concept is to define the deformation level associated with acceptable performance as that primary cycle amplitude at which both of the following criteria are fulfilled for the last time:

(a) At both the positive and negative peak deformation of the primary cycle (points A and B in Fig. 1) the load does not drop below a predetermined fraction α of the maximum load that was applied to the specimen in the respective direction. For reporting of CUREE/Caltech Woodframe test results a value of α = 0.8 is recommended.

(b) After the trailing cycles have been completed and the next larger primary cycle is attempted, the maximum load at deformations ≥ the deformation amplitude of the previous primary cycle (point C in Fig. 1) should not be less than a predetermined fraction β of the maximum load that was applied to the specimen in the positive direction. For reporting of CUREE/Caltech Woodframe test results a value of β = 0.4 is recommended.

4. The deformation level associated with acceptable performance does not have to be equal to the selected value of ∆. It may be associated with a smaller or larger deformation amplitude.

5. Acceptable performance for a variety of performance objectives other than collapse will be addressed later in the Woodframe project. Consideration might include strength, damage control, and cost of repair. Test performed in accordance with this protocol should prove useful for these performance objectives.

Potential Variations to Basic Loading History The basic loading history, like the other loading histories described next, has been developed with an emphasis on performance evaluation. Thus, emphasis is placed on a conservative but realistic simulation of cycles that contribute significantly to damage at the 10/50 hazard level, as well as on adequate simulation of potentially damaging cycles at hazard levels associated with higher performance levels (e.g., continuous operation under more frequent events). The former necessitates the distinction between primary and trailing cycles, and the latter necessitates the execution of a large number of relatively small cycles. Both considerations render the basic loading history more complicated. The following two options are presented as potential simplified alternatives to the basic loading history: Abbreviated Basic Loading History. This loading history, which is illustrated in Fig. 3, has fewer smaller cycles. Compared to the basic loading history, the following simplifications are incorporated:

• There are four cycles with an amplitude of 0.05∆ (rather than six cycles) • There are four (rather than six) trailing cycles following the primary cycle of amplitude

0.075∆

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• There are four (rather than six) trailing cycles following the primary cycle of amplitude 0.1∆

• There are two (rather than three) trailing cycles following the primary cycle of amplitude 0.2∆

• There are two (rather than three) trailing cycles following the primary cycle of amplitude 0.3∆

The expectation is that the smaller number of small cycles will not have a large influence on performance. Calibration testing will be needed to tell the extent of the influence. Simplified Basic Loading History. This loading history is illustrated in Fig. 4. In this history, the trailing cycles, which have an amplitude of 75% of the preceding primary cycle, are replaced by cycles of an amplitude equal to that of the preceding primary cycle. Thus, several cycles of equal amplitude are being executed at each step. This simplification facilitates the execution of the test and the test interpretation, and may be more useful for the development of analytical models. But it must be recognized that it will overestimate the extent of damage, particularly for large amplitude cycles. The extent to which the damage is overestimated is not known. Only testing will tell. Permissible Deviation for Acceptance Testing: The basic loading history is a realistic and conservative representation of the cyclic deformation history to which a component of a wood structure likely is subjected in earthquakes. At relatively large deformations (primary cycles exceeding an amplitude of 0.4∆), the amplitude of the primary cycles increases by steps ≥ 0.3∆. These large steps are based on statistics of response deformation demands. In a testing program in which the results of different tests have to be compared and evaluated for analytical modeling, the proposed loading history should be followed rigorously and without deviations. However, if the purpose of the experiment is acceptance testing, then it is permissible to reduce the step size of the primary cycles with amplitude > 0.4∆ at the discretion of the experimentalist. But even with smaller step sizes, every primary cycle must be followed by two trailing cycles of amplitude equal to 0.75 of the preceding primary cycle. Smaller step sizes close to failure (according to an established acceptance criterion) may result in a larger capacity (largest amplitude at which the acceptance criteria are passed), even though they will result in larger cumulative damage. The reason is that the large step sizes of the basic loading history permit evaluation of acceptance only at large amplitude intervals. This is illustrated in the test example presented in Fig. 9, in which the acceptance test is passed at the target amplitude ∆ = 0.6∆m, but is not passed at the amplitude 1.5x0.6∆m = 0.9∆m, because of the very large deterioration before the primary cycle at this amplitude is completed. But it is quite conceivable that the acceptance test would have been passed at an amplitude between 0.6 and 0.9∆m. 1.2 Loading History for Near-Fault Ground Motions (Near-Fault Loading History) The loading history for the near-fault cyclic load test should follow the history given in Fig. 5. The history is defined by variations in deformation amplitudes, using the reference deformation ∆n as the absolute measure of deformation amplitude. The history consists of the following sequence of cycles:

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• Four cycles with an amplitude of 0.025∆n • Four cycles with an amplitude of 0.05∆n • A primary cycle with an amplitude of 0.1∆n • Two trailing cycles of amplitude 0.075∆n • A primary cycle with an amplitude of 0.6∆n • One trailing cycle with an amplitude of 0.2∆n • A primary positive excursion to 1.0∆n • A reversal to zero deformation • A positive excursion to 0.8∆n • Two cycles with amplitude 0.1∆n and mean deformation of 0.7∆n • One positive excursion to the maximum deformation the test specimen can sustain

without causing harm to the test facility. Reference deformation ∆n: This deformation is the maximum deformation (displacement, drift angle, rotation, etc.) the test specimen is expected to sustain according to a prescribed acceptance criterion, and assuming that the proposed near-fault loading history has been applied to the test specimen. It is a measure of the deformation capacity of the specimen. It will be necessary to estimate this deformation capacity prior to the test. The present recommendation is to take this deformation capacity from a monotonic test, i.e., setting it equal to the monotonic deformation capacity ∆m as defined in Section 1.1. It is hypothesized that the deformation capacity under the near-fault loading history will be close to that under monotonic loading. Pilot tests will be performed to test this hypothesis. Should this hypothesis prove to be reasonable, the much simpler monotonic test can be used in lieu of the more complex near-fault test. Thus, at this time the near-fault loading history is of secondary importance except for correlation tests with monotonic loading. Additional considerations: 1. Loading should be continued monotonically after point C until the maximum load applied

decreases to a small fraction of the maximum load.

2. A final definition of acceptable performance is not provided here. Testing and analysis will tell what acceptability criteria should be used. The emphasis for acceptance should be on a threshold for unacceptable deterioration in strength. A basic concept is to tie acceptability to the maximum deformation associated with a residual strength of a predetermined fraction γ of the maximum load that was applied to the specimen in the direction of the pulse (Point A in Fig. 5). For reporting of CUREE/Caltech Woodframe test results a value of γ = 0.8 is recommended.

1.3 Specimen Fabrication, Testing and Instrumentation Issues Recommendations for the fabrication of test specimens, for material testing, planning and execution of experiments, test control, and specimen instrumentation should be taken from

7

existing standards and guidelines for testing of components and materials applicable to the woodframe project. Materials of interest are wood (for framing and panel elements), stucco and gypsum (for panel elements), and steel (light gage metal studs, hold-downs, and nails). Summarized here are a few general considerations, but they should not be considered as comprehensive. Specimen fabrication. Test specimens should replicate in-situ conditions so that material properties, standard construction techniques, and boundary conditions are properly simulated. Specimens should be as close as possible to full size in order to minimize size effects. Material Testing. Salient properties of materials used as part of the test specimens should be measured and documented, so that the sources of damage and failure modes can be evaluated and quantified. For wood the species and grade should be recorded and the moisture content should be determined (ASTM (1995a) Section 15.5). Test Set-up and Test Procedure. The test set-up should be configured so that the specimen boundary conditions and load application simulate in-situ conditions as closely as possible. Applicable guidelines are given in CoLA/UCI Committee (1999) Section 3, ASTM (1995a) Section 5, and ASTM (1995b) Section 14. The test procedure should follow the loading history recommendations given in this report. No specific recommendations on loading rate are given here, but reference is made to ISO (1999) which recommends a displacement rate between 0.1 and 10 mm/sec. Instrumentation. Measurement should be made of all force and deformation parameters that significantly affect specimen behavior and are needed to evaluate and quantify important failure modes. Pertinent guidelines are given in ATC (1992) Section 3.4, CoLA/UCI Committee (1999) Section 3, and ASTM (1995a) Section 6. 1.4 Documentation of Specimens and Test Results Experiments should be documented to the extent that an individual not involved in the testing program can carry out an objective interpretation of test results, considering all variables that significantly affect specimen behavior. Specific guidelines for documentation are given in ATC (1992) Section 5, and CoLA/UCI Committee (1999) Section 4. The following summary recommendations are taken primarily from ATC-24 (ATC 1992). For each experiment the following information should be documented: 1. Geometric data and important details of the test specimen, including fabrication/construction

details (including joining and hold-downs), boundary conditions, constraints, and applied loads (e.g., location and magnitude of gravity loading).

2. Locations of instruments for the measurement of primary response parameters (parameters

needed to evaluate the performance of the test specimen). 3. All material test data needed for performance evaluation.

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4. The following data for the force and deformation control parameters. [The terminology used

here is that employed in ATC-24.]

• A schematics of the deformation control history with sequential cycle numbers indicated at the positive peaks.

• A trace of the force-deformation history that shows all important aspects of the response, including points of maximum response and other important points that are needed to define salient hysteretic characteristics of the specimen.

• The deformation (force) value(s) that have been used for deformation (force) control.

• Numerical values of the following measurements for the positive and negative excursions of individual cycles, with appropriate sign. An excursion is part of a cycle and extends from zero force to maximum deformation to zero force (see Figure 1 of ATC-24 for definitions of force and deformation parameters). Only those data points that show an appreciable change compared to previously registered values need to be documented.

• Peak deformation, δi+

and δi-

• Deformation at start of excursion, δ0,i+

and δ0,i-

• Measured plastic deformation range, (∆δpm)i+

and (∆δpm)i-

• Force at peak deformation, Qi+

and Qi-

• Maximum force in excursion, Qmax,i+

and Qmax,i-

• Force in primary excursion at the peak deformation of the previous primary excursion

• Slope of Q-δ diagram at start of loading, K0,i+

and K0,i-

• Slope of Q-δ diagram at start of unloading Ki+

and Ki-

• Area enclosed by Q-δ diagram of excursion, Ai+

and Ai-

5. Observations made during the test and identification of any problems that may affect the

interpretation of the data. 6. Data similar to those listed under 4. should be documented for other response parameters to

the extent needed for a performance evaluation. No documentation is requested at this time for elastic stiffness and yield force, because for timber structures these quantities are not yet well defined and numerical values depend on the definition used.

9

2. Testing Protocol for Force Controlled Quasi-Static Cyclic Testing 2.1 Loading History for Force Controlled Quasi-Static Cyclic Testing The following loading history should be applied to components whose behavior is controlled by forces rather than deformations (see Fig. 6):

• Five cycles* with an amplitude of 0.5Q0 • Five cycles with an amplitude of 0.7Q0 • A primary cycle with an amplitude of 0.8Q0 • Two trailing cycles (amplitude = 0.6Q0) • A primary cycle with an amplitude of 0.9Q0 • Two trailing cycles (amplitude = 0.675Q0) • A primary cycle with an amplitude of 0.9Q0 • Two trailing cycles (amplitude = 0.675Q0) • A primary cycle with an amplitude of 1.0Q0 • Two trailing cycles (amplitude = 0.75Q0) • A primary cycle with an amplitude of 1.0Q0 • Two trailing cycles (amplitude = 0.75Q0) • Additional steps of the same pattern with an increase in force amplitude of 0.1Q0, i.e.,

two sequences of one primary cycle of amplitude equal to that of the previous primary cycle plus 0.1Q0, followed by two trailing cycles of amplitude equal to 0.75 times that of the last primary cycle.

* For components that do not undergo load reversals (e.g., anchor bolts), the term “cycle” refers

to a half cycle from zero load to maximum load followed by unloading. Reference force Q0: This force is the maximum force the test specimen is expected to experience in the maximum considered earthquake, and assuming that the proposed basic loading history has been applied to the test specimen. It is a measure of the force capacity of the specimen. It will be necessary to estimate this force capacity prior to the test. This estimate can be based on previous experience or the execution of a monotonic test. Additional considerations: 1. The test should be continued in the predetermined pattern until the maximum load applied in

a primary cycle can no longer be increased by the increment 0.1Q0.

2. A final definition of acceptable performance is not provided here. Testing and analysis will tell what acceptability criteria should be used. A basic concept is to define the force level associated with acceptable performance as that primary cycle amplitude at which both of the following criteria are fulfilled for the last time:

(a) The two sequences of one primary cycle followed by two trailing cycles can be executed without brittle failure.

10

(b) After the last trailing cycle the specimen can still be loaded to a predetermined fraction η of the maximum force amplitude (point C in Fig. 6). For reporting of CUREE/Caltech Woodframe test results a value of η = 0.8 is recommended.

3. The force level associated with acceptable performance does not have to be equal to the selected value of Q0. It may be associated with a smaller or larger deformation amplitude.

2.2 Specimen Fabrication, Testing, Instrumentation, and Documentation A near-fault loading history is not provided for force-controlled testing. The reason is that the force demands under near-fault ground motions will not differ much from those under ordinary ground motions, and the cumulative damage effects will be smaller. In all respects, except loading history, a force controlled cyclic test should follow the guidelines provided in Sections 1.3 and 1.4 for deformation controlled cyclic testing. 3. Time Histories for Shaking Table Testing Time histories should be applied in a manner that permits evaluation of the performance of wooden houses at various performance levels. The performance levels of interest may be taken from guidelines such as FEMA 273 or SEAOC Vision 2000. They are associated with ground motions of specific return periods. Widely used return periods are 2475, 475, 72, and 43 years. The following time history records are recommended to simulate seismic conditions at various return periods: At return periods of 475 years and smaller: A typical ordinary (not near-fault) record that represents, in shape, the NEHRP design spectrum for soil type D in the period range of interest (from about 0.1 to 1.0 sec.). The preferred choice is the Northridge 94 Canoga Park record. The acceleration response spectra of the two horizontal components of this record are shown in Fig. 7. An alternative choice is the Loma Prieta 89 Hollister Differential Array record.

• Whenever feasible the two horizontal components of the record should be applied simultaneously (preferable together with the vertical components).

• If only one component can be applied, then the larger of the two components should be used.

• Scale factors for the ground motions should be determined such that the acceleration response spectrum of the larger of the two horizontal components provides, in the vicinity of the fundamental period of the structure to be tested, a reasonable match with the NEHRP soil type D spectrum for the appropriate return period.

• The NEHRP soil type D spectra for the selected return periods need to be determined first. The basis for these spectra are USGS hazard curves for spectral acceleration at periods of 0.2 and 1.0 seconds, which are used to construct the constant acceleration and constant velocity branches of the NEHRP spectra. The spectral accelerations should be taken directly from USGS maps for the appropriate location and return period. If maps for selected return

11

periods are not available, then the 475 year return period map should be taken as the basis, and scale factors for spectral accelerations, Sf, should be determined in accordance with the following equation proposed in FEMA 273:

Sf = (return period / 475)0.44 (1)

• Since the so determined scaled spectra are for USGS site category SB/SC, the spectral values for appropriate return periods should be modified to soil type D conditions by using the NEHRP’97 Fa and Fv site coefficients. This provides the spectral amplitudes and shapes that should be used to scale the ground motion record.

At very long return periods (in the order of 2500 years): At very long return periods it is appropriate to assume that the seismic hazard in an area like Los Angeles is controlled by fault ruptures close to the site. Thus, a near-fault record should be used to simulate seismic conditions associated with this hazard. The preferred choice is the Northridge 94 Rinaldi Receiving Station record. The acceleration response spectra of the two horizontal components of this record are shown in Fig. 8. An alternative choice is the Kobe 95 Takatori Station record.

• Whenever feasible, the two horizontal components of the record should be applied simultaneously (preferable together with the vertical components).

• If only one component can be applied, then the fault-normal component should be used.

• The near-fault record should not be scaled because insufficient knowledge exists at this time to scale near-fault records to return period specific hazard levels.

12

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20

Commentary to Proposed Testing Protocols

C1. Considerations in Development of Loading Protocols

C1.1 Considerations for Deformation-Controlled Quasi-Static Cyclic Testing It is recognized that quasi-static cyclic load testing may be performed for a variety of objectives, and that the objective should drive the loading history to be applied to a test specimen. Basic objectives of experimentation may be as follows:

• Provide knowledge on cyclic behavior characteristics useful for • analytical modeling • improvement of detailing • development of design equations

• Permit acquisition and documentation of consistent data that can be used for performance assessment at various performance levels and for comparison of performance of different components tested in different laboratories.

Each objective may provide strong arguments for the adoption of different loading protocols. It can be argued that the applied loading history should follow behavior and should not be predetermined at all, because behavior is the unknown quantity that necessitates testing. It can be argued further that a single loading protocol cannot cover all components, behavior modes, and failure modes that may be experienced by woodframe structures. These are strong arguments that have prevailed for many years. However, recently it has been recognized that the need exists to share experimental results on a worldwide basis to facilitate advancement of knowledge. Moreover, international trade agreements make it necessary to establish protocols for testing that can be employed universally for performance assessment that is independent of subjective opinions and national codes or guidelines. Modern information technology, which so much facilitates sharing of data, has given another large boost to the need for consistency in data in order to make experiments useful to more than the one who has performed them. The one aspect that makes consistent interpretation of data of past experiments a difficult task is the fact that the performance of components under cyclic loading is history dependent, because of cumulative damage considerations. If this were not the case, any test to very large deformations would suffice, even a monotonic one. Because of the dependence on “loading” (more realistically, “deformation”) history, a pattern needs to be established that permits consistent interpretation of data, preferably for all the aforementioned objectives. With this preamble in mind, as well as with the recognition that no one history can serve all purposes, the following criteria are used as the background for the development of the loading protocols presented here:

• Cumulative damage concepts should be considered to the extent necessary to make testing representative for seismic response behavior.

21

• Testing should be performed under loading histories that represent, in a conservative manner (this implies the use of 84th percentile values in most cases), the demands imposed by ground motions on structures of which the tested component (or assembly) is part.

• With the recent emphasis on performance-based design, the loading histories should be structured such that performance assessment at various performance levels is feasible. Since cumulative damage is expected to have a larger effect at low performance levels (e.g., collapse prevention), the weight in the loading history development is on the response to severe ground motions.

• The loading histories should account for earthquakes of different characteristics and return periods. The following criteria are set:

• Emphasis should be placed on performance assessment of components at ground motion levels associated with long return periods, i.e., • 10/50 events [475 years return period] • 2/50 events [2475 years return period] The records associated with these events are called “performance assessment records”.

• It was decided to describe the 10/50 hazard by ordinary (non-near-fault) ground motions that are scaled to appropriate spectral values.

• It was decided to describe the 2/50 hazard by near-fault ground motions. • Ground motions should represent conditions typical for Los Angeles conditions. [It is

assumed that these ground motions are also representative for other regions of high seismicity in California.]

• Multiple events should be considered, i.e., short return period records preceding the performance assessment records should be included.

C1.2 Considerations for Force Controlled Quasi-Static Cyclic Testing Force controlled testing should be performed only if a suitable deformation parameter cannot be found. In general this means components whose behavior is controlled by brittle failure modes. An example is pull-out of an anchor bolt from the foundation – provided that pull-out is indeed the failure mode. Deformation controlled testing should be performed if bolt yielding is the failure mode. Force-controlled testing has the purpose of quantifying the strength of a component. Since cumulative damage considerations may again enter, the test should be performed under a loading history that simulates that experienced by the component in an earthquake. Like in deformation controlled testing, the reference value for force application should be the maximum force the component will experience in an earthquake. All other force amplitudes should be a function of the maximum one. The function of force controlled components is usually that of a joining or connection medium. As such, there is no need to make a distinction between the force capacity under ordinary and near-fault ground motions. Since the cumulative damage effects are larger under ordinary ground motions, it was decided to develop only one force controlled loading history and to

22

structure it according to the response to ordinary ground motions of the type associated with the 10/50 events identified in the previous section. Little attention needs to be paid to performance at high performance levels (e.g., continuous operation) because it can be assumed that design and behavior of a brittle component will be governed by severe ground motions. In order to establish criteria for a force controlled loading history, response statistics must be obtained on the force amplitudes a component may experience under sets of representative ground motions. The results from the nonlinear dynamic analyses carried out for the development of the deformation controlled loading history, together with the relative deformation amplitudes of the basic deformation controlled loading history, are utilized to develop the force controlled loading history (see Section C.6.3).

C1.3 Considerations for Shaking Table Testing Shaking table testing is an expensive proposition, even for relatively simple structural systems. In the context of the woodframe project, it is assumed that only one (or maybe two) house(s) can be tested, and that the objective is a comprehensive performance evaluation under various levels of seismic input, spanning from ground motions associated with a frequent events (say, 50/30 or 50/50), to design ground motions associated with a 10/50 event, to motions associated with a 2/50 event. However, it must be considered that damage under an earlier event may affect the dynamic response under later events, and that it is very unlikely that a wooden building will be subjected to a 10/50 event as well as a 2/50 event in its lifetime. To maximize the benefit of a shaking table testing program, it is prudent to perform tests under various levels of ground motion intensities but preserve the condition the structure is likely to be in before a severe earthquake (either a 10/50 or a 2/50 event) occurs. The latter implies the execution of shaking table test series that include the history of earthquakes the structure is likely to experience before the severe event occurs. It also implies the need for repair if a test series brings the structure to a state of damage that is not representative of realistic initial conditions. In the selection of ground motion records the consideration of cumulative damage effects invites the selection of a sequence of records in accordance with the criteria outlined in Section C.1.1 and elaborated in Section C.2. Separate test series should be executed for the 10/50 and 2/50 performance assessments, with intermittent repair performed as needed to preserve realistic initial conditions. It is not advisable to perform an arbitrary number of small amplitude tests unless it is clear that such tests will not cause cumulative damage that inappropriately affects the failure mode in severe tests (e.g., nail fatigue versus pullout). C.2 Selection of Ground Motion Records

Representative ground motion records are needed for analytical simulation studies on which to base the loading histories for quasi-static cyclic testing and the selection of records for shaking table tests. The following sets were developed for the purposes summarized in Section C.1:

• A set of 20 “ordinary” records (performance assessment records) representative of the 10/50 hazard level for Los Angeles conditions, on which to base the development of the

23

basic loading history. [“Ordinary” implies that these ground motions are recorded far enough from the fault rupture to be free of typical near-fault pulse characteristics.]

• A series of low amplitude “ordinary” records that represent small ground motions preceding the performance assessment records.

• A set of near-fault records representing rare events with a return period of 2475 years (2/50 hazard) for Los Angeles conditions. It is assumed that the 2/50 hazard is controlled by near-fault ground motions.

For all cases it is assumed that the structure is located at a site with NEHRP soil type D. Soft soil ground motions are not considered. As a starting point the USGS spectral values at various hazard levels, which were used in the SAC steel project (Somerville et al., 1997), are employed to develop target spectra for the selection and, when necessary, scaling of records. Target spectra are defined as spectra that represent the site hazard for soil type D at specified hazard levels (return periods), and within the period range of interest, which for woodframe structures is assumed to be from 0.2 to 1.0 seconds. In the SAC steel project, USGS hazard mapping information has been used to establish anchor points for uniform hazard spectra for a typical LA site at various hazard levels (Somerville et al., 1997). The USGS spectral acceleration values at periods of 0.1, 0.2, 0.3, 0.5, and 1.0 sec. and for the 10/50 and 2/50 hazard levels are used here as starting points. These values, which come from USGS hazard maps, are for USGS site category SB/SC. They are modified to soil type D conditions by using the NEHRP’97 Fa and Fv site coefficients. The so obtained target spectra for the 10/50 and 2/50 hazard levels are shown in dashed lines in Fig. C.1. These spectra are obtained by fitting a curve to the five data points, which results in irregular spectral shapes. In the NEHRP’97 (IBC’2000) design approach, spectral acceleration values at 0.2 and 1.0 sec. are used as anchor points for the maximum considered earthquake (MCE) and for the design spectrum. According to (Somerville et al., 1997) the 2/50 and MCE spectral values are very close for the selected sites. Thus, a horizontal line through the 0.2 sec. value and a 1/T line through the 1.0 sec. value can be used to construct the MCE spectrum, and multiplying this spectrum by 2/3 provides the NEHRP’97 design spectrum (FEMA 302, 1997). As Fig. C.1 shows, the MCE spectrum is a good representation of the 2/50 uniform hazard spectrum, and the NEHRP design spectrum is a good representation of the 10/50 uniform hazard spectrum in the period range of primary interest. Thus, the NEHRP design spectrum (i.e., 2/3 times the MCE spectrum) is used here as the reference spectrum for scaling of ordinary ground motions and deciding on the strength of the structural systems to be analyzed. In the subsequent analyses, all ordinary records are scaled to the spectral acceleration of the NEHRP design spectrum at the first mode period of the structure. For the later discussed near-fault ground motions this scaling process was not applied, because it is questionable whether the MCE spectrum is a reasonable representation of the 2/50 seismic hazard near a fault. For these ground motions the MCE spectrum is used only for comparison and not for scaling.

24

C.2.1 Set of Ordinary Records for Development of Basic Loading History This set is assumed to be representative of the 10/50 hazard level in Los Angeles. The following criteria are employed to arrive at a representative set:

• These records are to represent “ordinary” ground motions at the design level. Ordinary implies that they are not near-fault ground motions.

• A minimum of 20 records is deemed to be necessary in order to obtain stable statistical estimates (median, 84th percentile).

• To arrive at a well defined set of records the following constraints were placed on the selection process:

• Only California earthquakes are considered with a moment magnitude range of 6.7 ≤ Mw ≤7.3.

• The closest distance to the fault is bracketed between 13 km ≤ R ≤ 25 km.

• As much as feasible, only soil type D records are to be selected.

• Records from several earthquakes are to be selected, without regard to the faulting mechanism.

Records were chosen from the Pacific Engineering and Analysis Strong-Motion Catalog of 10/06/98 (Silva’s database). The records come from the following five events: Superstition Hills(3), Northridge(7), Loma Prieta(6), Cape Mendocino(2) and Landers(2). From Silva’s database 23 suitable records could be found. Only 13 of these are of soil type D the other 10 are of soil type C. Three of the soil type C records were discarded based on achieving balanced contributions from different earthquakes. The elimination of these 3 records did not have a significant effect on the median and 84% spectra of the set. Random horizontal components were chosen in order to eliminate any bias in the selection process. The final set of records and their properties are listed in Table C.1. Acceleration response (elastic strength demand) spectra for the selected unscaled 20 records are shown in Fig. C.2 together with the median and 84th percentile spectra. [Median is defined as the geometric mean (exponential of the average of the natural log values) of the data points, and the 84th percentile is defined as the median multiplied by the exponent of the standard deviation of the natural log values of data points.] The statistical spectra follow expected patterns, but show relatively small values compared to the spectrum representing the LA 10/50 hazard. Thus, scaling of records is an important issue. As discussed previously, all spectra are scaled to the 10/50 spectral acceleration value of the period of the structural model. Scaled spectra for the case of T = 0.5 sec. are shown in Fig. C.3, together with the USGS 10/50 spectral target values at T = 0.1, 0.2, 0.3, 0.5, 1.0 and 2.0 sec. The median spectrum matches the USGS values rather well in the period range from 0.1 to 1.0 sec. It is noted that the dispersion, which is zero at T = 0.5, grows rapidly for periods longer than 0.5 sec. (the period elongation range) and shorter than 0.5 sec (the higher mode range, which is not relevant in this context). Because of this significant dispersion, large differences have to be expected between median and 84th percentile response values for inelastic systems.

25

C.2.2 Small Events Preceding a Performance Assessment Event This issue may be important (or at least its importance needs to be evaluated) because earthquakes that will occur before the performance assessment event may cause damage whose cumulative effect may significantly alter the initial conditions (initial stiffness and state of damage) at the time of the performance assessment event. Realistic simulation of previous earthquake(s) is needed because

• disregard may underestimate the demands imposed by the performance assessment event, and

• overestimation may create an unrealistic state of damage that may misrepresent the likely mode of failure.

The following reasoning can be employed to establish a “probable” sequence (train) of ground motions (from C.A. Cornell, private communication). The question is what are reasonable representative record amplitudes for smaller events that might precede the performance assessment event at a site? Analogous to reasoning used in, e.g., API guidelines for coincidental (secondary) load effects and in FORM-SORM “design point” definition, we ask for the conditional mean (or conditional expected) values of secondary effects given the value of the primary effect. We ask therefore, given that the structure experiences its “performance assessment event” at some point in its life, what do we “expect” has preceded it? . To a first approximation, we assume it is a sequence of events of various first mode spectral acceleration levels (same spectral shape). [This reduces the problem to a scalar representation of the ground motion.] Now we start using “expected” or “associated” arguments given the performance assessment event occurrence. We “expect” it to occur in the middle of the 50 year life, i.e., at year 25. Then we ask what is the spectral acceleration of the event we “expect” to occur once prior to the performance assessment event. The expected number of events per year is λx, where λx is the mean rate of occurrence of events with Sa ≥ x. This rate is equal to the “annual probability” of the event. In 25 years we “expect” λx*25 such events. Setting this equal to unity, we get λx = 1/25 = 0.04, i.e., the associated event to occur once prior to the performance assessment event is one with a mean return period of (1/ λx) equal to 25 years [in 50 year basis: p50 = 1 – e-λ50 = 1 – e-50/25 = 1 – e-2 = 0.86, i.e., 86% in 50 years.] Analogously, for the size of event we expect to occur twice before the performance assessment event, we set λx*25 = 2 and get λx = 2/25 = 0.08, i.e., a 12.5 year mean return period [in 50 year basis: p50 = 1 – e-λ50 = 1 – e-50/12.5 = 1 – e-4 = 0.98, i.e., 98% in 50 years.] Thus, spectral accelerations are needed at the periods of interest corresponding to return periods of 25 and 12.5 years. USGS uniform hazard curves at these return periods are not available, however, the USGS hazard values given in (Somerville et a., 1997) for the 2/50, 10/50, 50/50, and 30/50 hazards can be used to extrapolate to the referenced return periods. The extrapolated values at T = 0.2 and 1.0 sec. are 0.23g and 0.09g for the 25 year return period, and 0.14g and 0.055g for the 12.5 year return period. These values are modified for soil type D conditions with

26

NEHRP’97 Fa and Fv site coefficients, resulting in the following target spectral acceleration values: For T = 0.2 sec: 25 year Sa = 0.37g 12.5 year Sa = 0.23g (same for 0.5 sec.) For T = 1.0 sec: 25 year Sa = 0.21g 12.5 year Sa = 0.13g It can be seen that these spectral accelerations are not negligible compared to the acceleration values for the design ground motions. However, their values are relatively small and it is reasonable to assume that their effect on initial conditions and cumulative damage will be small. Nevertheless, in all analysis cases with ordinary ground motions (10/50 hazard level) a sequence of three smaller ground motions (12.5, 25, and 12.5 years return period, respectively) precedes the ordinary ground motion that represents the 10/50 hazard level. The WN87hol (Whittier Narrows, Hollywood Storage) record is utilized to represent the 12.5 year return period, and the LV80kod (Livermore, San Ramon-Eastman Kodak) record is used to represent the 25 year return period. The records are scaled so that the spectral acceleration at the structural period is equal to the value given above. A typical acceleration time history train (the three smaller records followed by an performance assessment record) is shown in Fig. C.4.

C.2.3 Set of Near-Fault Records for Development of Near-Fault Loading History Recordings from recent earthquakes have provided much evidence that ground shaking near a fault rupture is characterized by pulses with very high energy input. This holds true particularly in the “forward” direction, where the propagation of the fault rupture towards a site at a velocity close to the shear wave velocity causes most of the seismic energy from the rupture to arrive in a single large pulse. Large pulses amplify the maximum interstory drift for elastic structures and more so for inelastic structures. Many studies are in progress on near-fault ground motion characterizations, and several studies are concerned with an evaluation of the effect of these motions on structures (Somerville et al. 1999, Krawinkler and Alavi 1998, Alavi and Krawinkler 1999). The following observations summarize salient characteristics of near-fault ground motions in the context of this project:

• In many cases it is feasible to describe near-fault records by an equivalent square pulse of period Tp and an effective velocity which is close to the PGV.

• The pulse period increases with earthquake magnitude, and the effective velocity increases with magnitude and closeness to fault.

• The response of structures is sensitive to T/Tp, with T being the fundamental structure period.

• In particular, the response of structures with T/Tp < 1.0 is very different from those with T/Tp > 1.0. This may not be evident for elastic SDOF systems (see Fig. C.5(a)), but is clearly evident for inelastic SDOF (and MDOF) systems (see Fig. C.5(b), which shows

27

displacement response histories for inelastic systems (µ = 6) subjected to a near-fault record).

• Wood structures usually have short fundamental periods. Thus, the primary range of interest is T/Tp < 1.0.

Based on these observations, and focusing on records whose pulse period is relatively short (close to the period range of interest for woodframe structures), the six records listed in Table C.2 are selected for this study. Figure C.6 presents the acceleration, velocity, and displacement spectra of the individual records together with the median spectrum. Only the fault-normal component of the ground motion is shown and is used in the analysis. A pilot study (Alavi and Krawinkler 1999) has indicated that rotation by 45 degrees does not make the larger of the two rotated components much smaller than the fault-normal component. Thus, the fault-normal component is a reasonable representation of the larger of two orthogonal components in a random direction. The records are not scaled to a common spectral acceleration; they are used as recorded. Figure C.6 indicates that the records are very severe. The median values are somewhat smaller than the Los Angeles MCE spectral values for periods shorter than about 0.6 sec., but they exceed the MCE values considerably for longer periods. Considering that woodframe structures have a relatively short period but are expected to undergo large inelastic deformations under these ground motions, the period elongation is expected to drive the structures into the range of very large demands. C.3 Selection of Structural Systems for Prediction of Response The development of representative loading histories requires information on response behavior of representative structural systems subjected to various types of ground motions. Woodframe buildings are mostly from one to three stories high, and more often than not of one story only. The emphasis is placed here on single story buildings. Two and three story buildings will show relatively small higher mode effects and their response is not expected to differ much from that of single story buildings. The case of more than one story with a soft first story deserves special consideration. But because of its undesirable characteristics it is not considered a structure type that should control the development of representative loading histories. It is a matter of a separate demand evaluation study to assess the amplified demands for a soft story building. Basic information on demands for soft (weak) first story buildings is available in (Seneviratna and Krawinkler, 1997). For the reasons just quoted, it was decided to focus on the prediction of response for SDOF systems and place emphasis on the type of hysteretic force-deformation behavior of the system. Baseline studies are performed with bilinear SDOF systems, but the emphasis is on hysteretic systems that are representative of woodframe buildings whose primary lateral load resistance comes from plywood shear wall panels. A literature survey shows clearly that this comprises a wide range of systems, ranging from relatively stable multi-linear systems to complex and deteriorating curvilinear systems. The database established on the University of California at Irvine CoLA test series proved to be most helpful in assessing structural behavior and deciding

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on basic parameters. In the context of loading history development it serves little purpose to evaluate a large range of systems which will result in a huge variation of response parameters. There is little to be gained because for a given type of ground motion (ordinary or near-fault) a single history needs to be chosen at the end. Nevertheless, it is prudent to conduct parameter studies in order to evaluate the range of results that can be expected.

C.3.1 Common System Parameters The period range of interest is assumed to be from 0.2 to 1.0 seconds. Specific values used in the demand prediction are 0.2, 0.35, 0.5, and 1.0 seconds. Shorter periods are conceivable (see Table C.3, taken from Foliente and Zacher, 1994), but likely will exist only for small amplitude vibrations. It was decided to focus on relatively simple hysteretic systems that have a bilinear skeleton curve in common. For a given mass the elastic stiffness, Ke, defines the period T of the system. The second slope of the bilinear skeleton is defined by a material strain hardening ratio α = Ks/Ke, which was selected as 0.08. This value is approximately a mean minus sigma value of judgmental slopes placed on the UCI/CoLA test results. It is assumed that P-delta causes an effective softening of 3%, i.e., the value of (Pδ/h)/V is taken as 0.03. This value is estimated from the following approximate scenario:

• Single story building with T = .5 sec., V = 0.23W (corresponds to R = 5). • For Vy = 0.23W and µ = 5, δy = 7.0/5 = 1.4 cm • Panel height = 8 ft = 240 cm • V’/V = (Pδ/h)/V = (Wx1.4/240)/0.23W = approximately 0.03

The yield strength Vy of the system is determined from the NEHRP’97 design spectrum (see Fig. C.1), by taking the elastic strength demand value, Ve = (Sa/g)W, and dividing this value by a reduction factor R. Thus, the strength is defined by the strength parameter η given as

η = Vy/W = Ve/(RW) = (Sa/g)/R (C.1)

Global response parameters are determined for a range of R values from 1 to 6, and the response parameter of importance for the loading history development are determined for R factors of 5.0 (close to minimum required strength) and 2.0 (significant overstrength, which is present in most cases). It is important to note that the yield strength is computed with R factors that refer to the NEHRP’97 design spectrum, which is associated with the 10/50 set of ordinary records. Since all these records are scaled to a common Sa (at the first mode period), the strength value η associated with a given R-factor will be identical for each record. When the near-fault records are used in the analysis, the same strength values η are used, i.e., the relationship between η and Sa (i.e, the R-factor) varies between near-fault records. A choice had to be made also on the value of equivalent damping. A viscous damping coefficient of ξ = 5% is assumed in all cases, recognizing that this value is expected to fluctuate significantly dependent on the structural configuration.

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In summary, the following system parameters are employed:

1. The structure period, T, is assumed to be defined by an elastic stiffness Ke and the mass of the system. Values of T = 0.2, 0.35, 0.5, and 1.0 are used.

2. All systems have a bilinear skeleton defined by Ke and Ks.

3. A material strain hardening ratio of α = 0.08 is used in all cases.

4. A P-delta softening ratio of 0.03 is used in all cases.

5. The system yield strength is defined by the base shear coefficient η = Vy/W = (Sa/g)/R. The 10/50 values of Sa are those obtained from the NEHRP’97 design spectrum.

6. Viscous damping of ξ = 5% is used in all cases.

C.3.2 Hysteresis Models Even with the constraint of a bilinear skeleton, great variations of hysteretic models can be employed, which may include various options of stiffness degradation or strength deterioration. In this study the following options are considered:

1. The basic (non-degrading) bilinear model (even though rather unrealistic for wood structures) serves as a reference structural model.

2. The basic peak-oriented (Clough) model serves as an intermittent model between the bilinear model and the pinching model.

3. The basic pinching model of the type illustrated in Fig. C.7 serves as the primary model for demand evaluation. With appropriate parameters it is capable of closely simulating typical load-deformation behavior of plywood shear wall panels. Compared to the basic bilinear model the basic pinching model needs only one additional parameter, called κ, to define the point targeted after unloading. There are several variations to this model; the one implemented here is illustrated in Fig. C.7. It is noted that the value of κ defines the pinching strength as a function of the maximum (not yield) strength in the direction of loading, and defines the pinching stiffness as a function of the maximum residual deformation in the direction of loading. In this study κ values of 0.25 and 0.5 are selected. Based on the UCI/CoLA tests, a value of κ = 0.5 appears to be representative, in average, of plywood shear wall behavior.

4. Stiffness degradation and strength deterioration are phenomena that can be applied to the three basic models. Stiffness degradation may apply to the unloading stiffness or the reloading stiffness. Strength deterioration implies that the strain hardening branch of the skeleton curve moves inwards (translates towards the zero resistance axis). It also implies that this branch will degrade (the slope will become negative or smaller positive) at large deformations (or after many cycles). Any combinations of these phenomena can be considered simultaneously, and any of these phenomena can be described by a single deterioration parameter of the type (Rahnama and Krawinkler, 1993)

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c

i

jjt

ii

EE

E

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−=

∑=1

β (C.2)

in which βi = parameter defining the deterioration in excursion i

Ei = hysteretic energy dissipated in excursion i

Et = hysteretic energy dissipation capacity = γFyδy

Ej∑ = hysteretic energy dissipated in all previous excursions

c = exponent defining the rate of deterioration This parameter can be applied to strength deterioration or stiffness degradation, i.e.,

( ) 111 −− =−= isiii FFF ββ

or 111 −− =−= i,uui,uii,u KK)(K ββ

This model was investigated for different loading histories, and parameters were adjusted based on load-deformation data obtained from experiments (Fig. C.8). The upshot of analysis studies is that these deterioration phenomena have either benign or domineering effects, depending on the selected deterioration parameters and the intensity of the ground motion. From the perspective of loading history development, this adds another dimension that makes the choice of a representative loading history more ambiguous. The simple, and only feasible, way out is to assume that within the range of acceptable performance the specimen should sustain the imposed deformation cycles without much strength deterioration. If this is the case, then the deterioration phenomena can be ignored in the demand predictions that serve as a basis for loading history development.

C.4 Maximum Response Values

C.4.1 Results for Ordinary Ground Motions The response of SDOF systems to the set of 20 ordinary design level ground motions is evaluated at periods of 0.2, 0.35, 0.5, and 1.0 seconds for various hysteresis systems. For this purpose the ordinary (10/50) ground motions are scaled to the target spectral acceleration values of the LA NEHRP’97 design spectrum (see Fig. C.1) at the respective periods. The scaled individual spectra and the median and 84th percentile spectra at 0.2, 0.5, and 1.0 sec. are shown in Fig. C.9. It can be seen that the median spectra of the scaled records for T = 0.2 and 0.5 sec. are almost identical, but that the median spectrum of the scaled records for T = 1.0 sec. is substantially higher in the short period range. The reason is that the selected records have, in average, a spectral shape that has a relatively low spectral acceleration at 1.0 sec. (compared to the design spectrum). In all three cases the dispersion (which is zero at the selected periods) grows rather quickly at periods larger than the selected one. This implies that significant scatter has to be expected in the response parameters for inelastic systems.

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Results for median values and 84th percentile values of selected response parameters are presented in Figs. C.10 to C.12. Most of the graphs are for the pinched hysteresis model with κ = 0.5, which is used as the primary model for demand evaluation. The graphs in Fig. C.12 show the sensitivity of the results to the type of hysteresis model. Strength – Ductility (η – µ and R – µ) Graphs. These graphs provide a picture of the strength-ductility relationships of the analyzed systems. A comprehensive graph for pinching systems (κ = 0.5) of various periods is presented in Fig. C.10. The figure shows ductility demand µ on the horizontal axis, and the strength parameter η = (Sa/g)/R (left scale) as well as the strength reduction factor R = (Sa/g)/η (right scale) on the vertical axis. For µ = 1.0 the elastic strength demand is η = (Sa/g) and the R-factor is 1.0. The η – µ curves show the increase in ductility demand with a decrease in provided strength, and the R – µ curves show the increase in ductility demand with an increase in the R-factor. The R-µ curves show a close to linear relationship between R and µ for T = 0.5 and 1.0 seconds. In fact the relationship is close to R = µ in the median. The slope of the R-µ curve for T = 0.2 sec. is much flatter, indicating a rapid increase in ductility demand with a decrease in strength. As expected, the dispersion in the results is significant, as indicated by the differences in the median and 84th percentile curves. Strength – Displacement (η – δ) Graphs. These graphs show the variation in displacement demands with a decrease in provided strength. A comprehensive graph for pinching systems (κ = 0.5) of various periods is presented in Fig. C.11. The marked data points correspond to R-factors of 1.0, 1.5, 2.0, 3.0, 4.0, 5.0, and 6.0. If the displacement curve bends to the right, the inelastic displacement is larger than the elastic one, and vice versa. For T = 0.2 sec. the inelastic displacements are much larger than the elastic ones (by a factor larger than 3 for R = 6, in the median), whereas for T = 0.5 and 1.0 sec. the inelastic displacements are about equal to the elastic ones (in the median). These results are expected for ordinary ground motions. Figure C.12 illustrates the variation of displacement demands for different hysteresis models at T = 0.2 seconds. The pattern seen here is observed, but less pronounced, also for longer periods. The displacements increase successively as the model is changed from bilinear to Clough to piching with κ = 0.50 to pinching with κ = 0.25. This illustrates the sensitivity to details of the hysteresis model. This sensitivity is by far the largest for T = 0.2, and is becoming much smaller for longer periods. For T = 0.2 sec. the displacement demands are smallest, but the ductility demands are largest. If ductility is the best measure for performance of woodframe structures (this is not a foregone conclusion because ductility is not well defined for wooden structures), then the T = 0.2 second case becomes critical. On the other hand, if displacement is the best measure, then the T = 0.2 sec. structure is well protected. The η-δ curves facilitate visualization of the load-displacement response of the SDOF system. Connecting the origin with the top point (elastic strength demand) for each curve provides the elastic load-displacement response of the system. Horizontal lines at the various η values (which correspond to R = 1.5, 2, 3, 4, 5, and 6) provide the inelastic load-displacement response for

32

systems whose elastic strength demand is reduced by specific R-factors. The ductility demands can also be deduced by dividing the total displacement demands by the corresponding elastic displacement. These graphs provide important input to the decision process for selecting representative systems for loading history development. They show that the displacement demands are strongly dependent on the period of the system. This makes the displacement capacity (rather than a relative measure such as story drift capacity) a critical parameter for the loading history, and it makes it necessary to pattern the amplitude of all other cycles in the loading history around the amplitude of the largest cycle. This is different from most loading protocols used in the past, which use a “yield” displacement (which is very difficult to define for wooden test specimens) to decide on the amplitude of cycles that ramp up to the largest cycle. A disconcerting observation is that the analytically predicted displacement demands under the 10/50 ground motions are very large. Using the 84th percentile value for R = 5 (second lowest η value) at 0.5 sec. as a representative value, it is seen from Fig. C.11 that the displacement demand for a pinching system with κ = 0.5 is about 13 cm. Using a plywood panel of 8 ft = 240 cm, this corresponds to a drift angle of 13/240 = 0.054. This value is large already and, as will be seen later, is much exceeded by the predicted value for the 2/50 ground motions. Variation of Peak Parameters with Period. For the loading history development the yield strengths corresponding to R = 5 and R = 2 are selected. For these specific R-factors, and for the pinched hysteresis model, the variation with period of maximum displacement, δmax, maximum ductility, δmax/ δy, and normalized hysteretic energy dissipation (NHE = HE/(Fyδy)) is illustrated in Figs. C.13 to C.15. These figures serve as documentation of statistical values of parameters that are much needed in the decision process for the loading protocol. Values are also shown for the near-fault responses (NF). For these cases the R designation is replaced with η values since the R-factors refer to the NEHRP’97 design spectrum and not to the near-fault records. The NHE curves are shown merely to assist in judgment. For bilinear systems the normalized hysteretic energy NHE, defined as NHE = HE/(Fyδy), is equal to the sum of all plastic deformation ranges (normalized to yield displacement) the structure experiences in the earthquake. For these cases, NHE is a direct measure of cumulative damage effects if it assumed that only inelastic excursions contribute to damage. This assumption is not made in this study, but the NHE values are useful, nevertheless, for preliminary assessment of cumulative damage effects. For instance, for R = 5 the median NHE value is much larger for T = 0.2 sec. than for T = 0.5 and 1.0 seconds. This indicates comparable cumulative damage effects for systems with T = 0.5 and 1.0, but much larger cumulative damage effects for T = 0.2 seconds.

C.4.2 Results for Near-Fault Ground Motions Since these ground motions are not scaled to a common spectral acceleration, the R-factor has no meaning in this analysis. Thus, strength is referred to here by the base shear coefficient η = Vy/W only. A series of η values are chosen, somewhat arbitrarily, to establish η−δ and η−µ relationships for individual records. Graphs of medians and 84th percentile values are shown in Figs. C.16 and C.17. Because of the small sample set, the statistical values are obtained as

33

“counted” rather than “computed” values. As can be seen from the graphs, the inelastic displacement becomes much larger than the elastic one as the strength of the structure decreases, particularly for T = 0.2 seconds. Moreover, the displacements are very large, even for T = 0.2 seconds, and the ductility ratios take on “irrational” values for small η values. But, given a low strength, the large values are believed to be realistic. The reason for the extremely large displacements and ductility ratios lies in the shape of the acceleration response spectra (see Fig. C.6), which shows that the period elongation drives the short period structures into the region of high demands around 0.6 to 1.2 seconds. The extremely large displacement demands associated with relatively low strength are disconcerting. They are also evident from Figs. C.13 and C.14, in which the maximum displacements are shown together with those for the 10/50 ordinary set of records, for the strength values associated with R = 5 and R = 2 for the NEHRP’97 design spectrum. Since these R-factors are meaningless for the near-fault records, they are replaced by the corresponding base shear strength coefficient η. For T = 0.2 and T = 0.5 sec., the η values corresponding to R = 5 and 2 are 0.225 and 0.563, respectively. It can only be hoped that tests will show that the actual strength of woodframe structures is much larger than the minimum code design strength (which is in the order of η = 0.2). There are good reasons to believe that this is the case because of the presence of stucco and gypsum board walls and other elements that contribute to strength and stiffness. Thus, the results for η = 0.225 are believed to be of academic value, and more weight in the development of the near-fault loading history is placed on the results for η = 0.563. C.5 Cumulative Damage Considerations

The results discussed so far are peak values of displacement amplitudes and corresponding ductility ratios. They provide information on the demands imposed by different types of ground motions, and may serve as an anchor for the period dependent displacement capacity the test specimens should sustain in order to be acceptable according to specified performance criteria. However, they do not give insight into the loading history that should be applied in a test.

C.5.1 Cumulative Damage Issues The response of structures to earthquake ground shaking depends on the force-deformation characteristics of its constituent components. These characteristics are defined by loading history dependent stiffness and strength properties. It is expected that for wood components (e.g., plywood wall panels) significant stiffness degradation occurs relatively early, whereas significant strength deterioration (defined here as a decrease in strength under cyclic loading compared to the strength attained under monotonic loading) occurs relatively late in the loading history. Both stiffness degradation and strength deterioration under cyclic loading are caused by damage mechanisms, which makes cumulative damage concepts the basis for the development of a representative loading history. The following general concepts/observations are employed in this development:

• Damage is cumulative and is defined, amongst others, by the full deformation range (peak to peak) of damaging excursions.

34

• All excursions above a certain threshold level contribute to damage and deserve consideration in the loading history development. In this work the threshold level on the excursion range is set as the smaller of δy or ∆δmax/20, where ∆δmax is the maximum range the component is predicted to experience without significant deterioration in strength.

• The response to ground motions is random (not in a sequence of cycles of increasing or decreasing amplitude). Thus, individual excursions need to be extracted from response histories and ordered by means of a “cycle counting method”. The rainflow cycle counting method is employed (other methods will give similar results).

• Large excursions cause much larger damage than small excursions, thus, their representation in the loading history is emphasized.

• In general, the relative amount of damage caused by an excursion depends on the deformation range of the excursion, ∆δ, the mean deformation of the excursion (a measure of symmetry with respect to the undeformed configuration), and the sequence in which large and small excursions are applied to the component (sequence effects). In the response to ordinary ground motions the mean effects are usually small and are ignored.

• The number of damaging excursions, N, the deformation ranges of the excursions, ∆δi, and the sum of the deformation ranges, Σ∆δi are important parameters that should be simulated in the loading history.

• These parameters, and others discussed below, are obtained from statistical evaluation of the response of representative systems to sets of representative ground motions.

• Basic information is obtained by evaluating medians and 84th percentile values of the largest, second largest, third largest, etc., response excursion obtained from time history analysis and rainflow cycle counting.

For components of woodframe structures, such as plywood shear wall panels, the following additional concepts/observations are considered in the development:

• The hysteretic response of wood elements is usually of a pinched nature, as shown in the test results reproduced in Fig. 2, and is defined by the following characteristics that are illustrated in Figs. C.18 (arbitrary deformation history) and C.19 (illustration of corresponding load-deformation response):

• Cumulative damage (measured by the energy dissipated in an excursion) is caused primarily by excursions that have a branch on the skeleton and “widen” the envelope of the load-deformation response (e.g., from origin to point 3, and from point 3 to point 11).

• Smaller excursions, such at the one from the origin to point 1, can be considered interruptions of a subsequent larger excursion (e.g., origin to point 3).

• A smaller excursion following a large excursion does little cumulative damage even if its range is relatively large (e.g., all excursions after point 14). Such excursions are called here “trailing” excursions.

35

• Trailing excursions will not reach the previous peak load, because the previous peak load will only be attained (or exceeded) in an excursion that attains (or exceeds) the previous maximum amplitude.

• Thus, the response history should be separated into primary excursions (those that widen the envelope [without being considered an interruption] and cause most of the damage) and trailing excursions.

• All excursions occurring after the latter of the maximum positive and negative deformation peaks are trailing excursions. These excursions will do little cumulative damage. Thus, the pre-peak response history (all excursions before the latter of the maximum positive and negative peaks is reached) should be considered separately from the post-peak history.

Based on these observations it is considered important to base a representative loading history on the evaluation of primary excursions, and to have these excursions followed by smaller trailing excursions. All post-peak excursions should be given secondary consideration; they may be appended in some fashion to the loading history after the largest excursion has been executed. Thus, much attention is paid in the development of the loading history to a careful evaluation of primary excursions and to separate evaluations of all excursions for (a) the full response history, and (b) the pre-peak response history. Definition of Primary Excursions. Primary excursions are excursions that widen the envelope of response in the positive or negative direction and are primary contributors to cumulative damage. They need to be represented in the loading history and are to be followed by an appropriate number of smaller trailing excursions. A positive excursion with an amplitude larger than all the preceding positive excursions is a candidate for a primary excursion. It becomes a primary excursion if it is followed by a negative excursion with an amplitude that is larger than the largest previous negative amplitude before it is followed by a positive excursion that exceeds the amplitude of the candidate excursion. If the latter occurs first, the candidate excursion is considered an interruption of the excursion with larger amplitude. In the example illustrated in Fig. C.18, the ranges 0-3, 3-10, 10-11, and 11-14 constitute the four primary excursions of the response history. These excursions are shown in bold lines in Fig. C.19. All other excursions are trailing excursions.

C.5.2 Process for Incorporating Cumulative Damage Effects into Loading History The issues outlined in the previous section drive the development of representative loading histories. The following general process is implemented to derive statistical information that can be employed to decide on number and sequence of excursions that will make up the loading history.

36

1. Using structural systems of various periods and hysteretic characteristics, nonlinear time history analysis is performed with the ground motion sets previously identified.

2. Three evaluations are performed. One for the entire displacement response history, one for the pre-peak response history, and one for the primary excursions alone.

3. For each time history response rainflow cycle counting is performed, which provides a series of excursions with well defined properties (deformation range and mean deformation). These excursions can be evaluated for mean and sequence effects, which is critical for near-fault responses. For the responses to ordinary ground motions the mean effects are expected to be small and are ignored. The individual excursion ranges are ordered in decreasing magnitude. Only excursions larger than the previously defined threshold value will be considered from here on.

4. For each structural system and set of ground motions, statistical values (median and 84th percentile, as defined in Section C.2.1) of the following response parameters are obtained:

• Deformation range (∆δi) and mean deformation (∆δmean,i) of each of the ordered excursions (largest, 2nd largest, 3rd largest, etc.).

• Cumulative deformation ranges for partial sums of ranges (∆δmax + ∆δ2, ∆δmax + ∆δ2 + ∆δ3, etc., up to the sum of all damaging excursions).

5. This statistical information is presented in graphical and table forms (see Section C.6). It is employed, together with judgment that will result in conservative decisions, to develop representative loading histories.

6. A backward process is utilized to structure the loading history. This implies that the largest deformation amplitude (generically called ∆) becomes the parameter on which all smaller excursions are based, i.e., all smaller excursions preceding the largest excursion are expressed as fractions of ∆.

The implementation of this process is demonstrated in the next section for the basic and near-fault loading histories. C.6 Development of Representative Loading Histories Loading histories for quasi-static cyclic testing are developed based on the considerations summarized in Section C.5 and on data derived from nonlinear dynamic analyses of various SDOF systems covering a suitable range of hysteretic systems, periods, and strength values. Bilinear, peak oriented (Clough), and pinching models with κ = 0.25 and 0.5 are used to represent structural cyclic characteristics. Systems with periods of 0.2, 0.35, 0.5, and 1.0 sec. are analyzed. Strength values that correspond to R-factors of 5 (close to code minimum strength) and 2 (considerable overstrength) are employed. Using the NEHRP “design” spectrum shown in Fig. C.1 (2/3 of NEHRP MCE spectrum), R = 5 results in η = Vy/W = 0.225 for T < 0.6 sec. and η = 0.133 for T = 1.0 sec., and R = 2 results in η = 0.583 and 0.333, respectively, for these periods.

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For each case (i.e., combination of period, structural system, and strength), time history analysis is performed using the appropriate set of ground motions (ordinary records for basic history and near-fault records for near-fault history). Rainflow cycle counting is applied to all excursions and the pre-peak excursions, and the primary excursions are extracted from the response history. Pertinent results from the analysis are organized into spreadsheets, evaluated statistically (medians and 84th percentile), and plotted as needed to assist in the development of the loading histories. A representative set of data for one case (T = 0.5 sec., pinching model with κ = 0.25, and η = 0.225) is documented in Appendix A. The process of extracting information for the loading histories is discussed in the next section. One decision had to be made up-front, based on the fact that the displacements (and ductility ratios) are strongly dependent on the selected period (and in some cases the selected strength values). This is the need to use the maximum response displacement as the anchor point for the amplitudes of individual cycles (or excursions) rather than the yield displacement. This decision is reinforced by the observation that the response of components of woodframe buildings rarely exhibits characteristics that can be associated with a yield point or, for that matter, any other break-point in the force-deformation response. It is important to point this out now because many of the decisions made later depend on this up-front decision.

C.6.1 Development of Deformation Controlled Basic Loading History The set of 20 ordinary ground motion records (10/50 record set) is employed to derive information on which to structure the individual cycles of the basic loading history. Statistical data on the period dependence of several parameters is presented in Figs. C.20 to C.25 (for the pinching models with κ = 0.5). Figure C.20 shows the rapid increase with period of the maximum displacement range (∆δmax), whereas Fig. C.21 shows the decrease with period of the cyclic ductility ratio (∆δmax/δy). The latter figures indicate that it will not be productive to use yield displacement (or ductility ratio) as an anchor point for decisions on individual cycles of the loading history. Figure C.22 shows the variation with period and yield strength of the number of pre-peak excursions whose displacement range is larger than δy. The significant dependence of this parameter on period and yield strength again points out that the need to make the loading history independent of yield strength. For this reason, the number of damaging excursions is also counted as the number of excursions whose range exceeds ∆δmax /20. This limit controlled the number of damaging excursions for all systems with T = 0.5 and 1.0 sec., whereas the first limit (excursion range > δy) controlled for all systems with T = 0.2 and 0.35 seconds. The number of primary excursions, which is shown in Fig. C.23, is small in all cases. Even for T = 0.2 sec. and R = 5, the 84th percentile value is less than 10. The conclusion is that only few of the excursions in a typical response history widen the envelope to an extent that causes significant damage, and that in all cases the number of trailing excursions is much larger than the number of primary excursions. It is judged that only about 1/6th of all excursions are primary excursions.

38

The dependence of cumulative pre-peak deformation ranges (Σ∆δi/δy) on period and yield strength is also very large, see Fig. C.24, as is the dependence of the cumulative primary deformation ranges (Σ∆δpi/δy), see Fig. C.25. But from these two figures it is judged that the cumulative primary excursion ranges are about 1/3rd of the total cumulative deformation ranges, rather than the 1/6th factor judged for the number of excursions. This implies that for large excursions the difference between the number of pre-peak and primary excursions must be smaller than for small excursions. The 1/3rd factor is confirmed from other observations and is used to structure the relative number of pre-peak and primary excursions in the basic loading history. Basic information on the relative amplitude of excursions is derived from plots of the type presented in Figs. C.26 and C.27. They show the 84th percentile of the magnitudes of ordered deformation ranges, obtained after rainflow cycle counting and computing statistical values of the ordered ranges of individual analysis cases. For instance, the fourth line is the 84th percentile of the fourth largest excursion range of each analysis case of the specific system subjected to the 20 records of the 10/50 record set. Statistical values are obtained of the excursion range as well as the mean (midpoint) displacement of each excursion. The mean displacement is small for the response to ordinary records and is ignored in the development of the basic loading history; i.e., the excursion ranges are centered with respect to the zero displacement line. Figures C.26 and C.27 show 84th percentiles of the ranges for all excursions (pre- and post-peak) and pre-peak excursions, respectively. By superimposing the two graphs we can evaluate the relative importance of pre- and post-peak excursions. In the development of the basic loading history only the pre-peak excursions are considered explicitly. The post-peak excursions, which are believed to contribute little to cumulative damage (presuming that little to no deterioration has occurred at the end of the pre-peak history) are not considered individually in the loading history; they are lumped into the five large trailing excursions (2.5 cycles) that have to be executed after the largest primary cycle. Information of the type presented in Fig. C.27, together with information on primary excursions and on the number of excursions is used to construct the basic loading history. The 84th percentile values rather than median values are used in order to arrive at conservative (high) estimates of the number of excursions and their relative ranges. The relevant data (84th percentile values) for all the cases evaluated are summarized in Tables C.4 to C.6. Table C.4 summarizes the 84th percentiles of the ordered pre-peak excursion ranges, normalized to the yield displacement δy. Looking at the first row, it is evident that the normalized ranges are very much dependent on period and yield strength, with the maximum range varying from 53.9 to 4.3. The values tabulated in each column include all ranges whose magnitude is larger than the smaller of δy and ∆δmax/20. For strong systems (R = 2) the number of excursions > δy is small (values larger than 1.0), but the number of excursions > ∆δmax/20 is large. The reverse is true for weak short period systems. A more relevant presentation of the same data is given in Table C.5. This table shows the same information, but in this case the excursion ranges are normalized to half of the 84th percentile of the maximum range ∆. The value of ∆ for each case is listed in the column heading. For each

39

case, the individual excursion ranges as well as the cumulative excursion ranges are tabulated. When individual rows are compared across all cases, it can be seen that the large differences between cases have disappeared, particularly for the large excursion ranges (top rows), and rather uniform values of individual and cumulative ranges are obtained. These normalized ranges, together with the equivalent information for the primary ranges given in Table C.6, are used to construct the basic loading history. The individual and cumulative ranges of the basic loading history are tabulated in the second and third column of Table C.5. An inspection of each row, all the way across from the basic loading history column to the last of the cases, provides an indication of the “goodness of fit” between the excursions of the basic loading history and the various cases investigated through time history analysis. The fit is very good at large excursions (those that contribute most to cumulative damage) and becomes less accurate as the number of excursion increases. But the concept of providing a conservative estimate of cumulative excursion ranges is maintained for most cases and most excursions. The separation into primary and trailing excursions is accomplished by taking advantage of the information presented in Table C.6. Again, the second and third columns list the primary ranges of the proposed basic loading history, and the columns to the left list the 84th percentile values obtained from time history analysis. Only the most demanding case (T = 0.2 sec., η = 0.225, pinching with κ = 0.25) comes close to the demands on primary excursions imposed by the proposed loading history. Thus, conservatism is maintained. The large number of relatively small cycles is incorporated to permit a performance evaluation at higher performance levels (e.g., immediate occupancy, continuous operation). Several of these cycles come from smaller events preceding the 2/50 events). If these smaller events can be ignored, then several of the smaller cycles can be omitted (see Section 1.1). One can argue that the proposed loading history does not provide a close fit to any of the cases on which its derivation is based. This cannot be disputed, but lies in the nature of the problem. No single predetermined history can represent any or all realistic cases, but in a global sense and in the context of cumulative damage, the match between the proposed basic loading history and the individual cases represented by the selected ground motion records and structural systems is deemed to be satisfactory in general and in many cases actually very good.

C.6.2 Development of Deformation Controlled Near-Fault Loading History The development of the near-fault loading history is based mostly on the same concepts as those discussed in the previous section. The differences are in the set of selected ground motions and in the need to consider mean effects (the fact that the mean [midpoint] displacement of individual excursions is not close to zero). The pulse type near-fault ground motions will result in a response characterized by a small number of large excursions with significant mean displacement and by a large residual displacement at the end of the record. This type of response is illustrated in Fig. C.28, which shows the time history response of a strong (η = 0.563) system subjected to three of the six selected near-fault records. A typical

40

force-displacement response is shown in Fig. C.29. In both figures the pulse-type nature of the response can be seen, but it is also evident that the response usually consists of more than just the single large excursion that is so often observed in the near-fault response of long period structures (Alavi and Krawinkler 1999). The reason is that for wood structures the fundamental period T is usually smaller than the pulse period Tp. This observation complicates the development of a near-fault loading history because of the need to simulate more than a single large excursion. Figures 30 and 31 present typical results on which the development of the near-fault loading history is based. They show medians of ordered (in magnitude) excursion ranges for all damaging excursions and pre-peak excursions. In this case median values rather than 84th percentile values are used because the employed ground motions represent already very rare events (2/50) and the use of 84th percentile response values is deemed to be too conservative. Because mean effects cannot be neglected in this case, the ordered ranges are shown displaced by the median of the mean (midpoint) displacement. Information equivalent to that shown in Table C.5 for ordinary ground motions is shown in Table C.7 for near-fault ground motions. Three columns are shown for the proposed loading history and the individual cases; the individual ranges (normalized to the median of the maximum amplitude [not range]), the means (midpoints) of the ranges, and the cumulative ranges. The median of the maximum amplitude is the anchor point for the amplitudes of individual cycles A graphical presentation of the proposed near-fault loading history is shown in Fig. 5. Decisions on the ranges and mean values for the loading history are based on the data presented in Table C.7 and additional information on primary excursions. A relatively small number of small cycles is imposed at the beginning of the loading history because this loading history is intended only for performance evaluation at the collapse prevention level and is not intended to be applied for performance evaluation at higher performance levels. The type of hysteretic response that will be obtained by implementing the basic and near-fault loading histories is illustrated in Figs. C.32 and C.33. These response curves are obtained by subjecting a non-deteriorating pinched hysteretic system (with κ = 0.5 and 5% strain hardening) that experiences a maximum ductility ratio of 10 to the basic loading history (Fig. C.32) and the near-fault loading history (Fig. C.33).

C.6.3 Development of Force Controlled Loading History As mentioned in Section C.1.2, force controlled testing should be performed only if the component is expected to behave brittle and a suitable deformation parameter cannot be found. The reference value on which to base the amplitudes of individual cycles is the maximum force to which the component may be subjected in a severe earthquake. There is no compelling reason to distinguish between ordinary and near-fault ground motions, and the 10/50 set of ordinary ground motions is used as the basis for decisions on relative force amplitudes of individual cycles.

41

A typical force response of a pinching system with κ = 0.25 is presented in Fig. C.34. The upper graph shows the complete response history and the lower graph shows the pre-peak response history. Statistics on force level crossings obtained from time history analyses of the cases evaluated for development of the basic loading history are used to provide guidance for the selection of relative force levels. Table C.8 lists medians and 84th percentile values of level crossings at force levels of yield force Fy, at 0.75Fy, and at 0.5Fy. Values for all excursions and pre-peak excursions are tabulated. The tabulated values include all crossings, i.e., the sum of crossings in the positive and negative directions. The number of crossings was found to be rather symmetric, i.e., the number of crossings in the positive or negative direction is close to half of the tabulated values. The tabulated level crossings are of some, but limited, value in the decision process on relative force levels for the loading history. The reason is that the number of level crossings depends strongly on the ductility demand of the system investigated. If the ductility demand is large (e.g., 21), then the maximum force for a system with 5% effective hardening is large (e.g., 2Fy), whereas it is only 1.1Fy for a system with a maximum ductility of 3. For this reason the level crossings are used only as guidelines but not as a tool for final decisions. The relative peak force values of individual cycles of the force controlled loading history shown in Fig. 6 are obtained from the abbreviated deformation controlled loading history presented in Fig. 3 and the graph shown in Fig. C.35. In this graph three bilinear systems with ductility ratios of 4, 10, and 20 and a strain hardening ratio of 0.05 are normalized to the maximum force and displacement value. Recognizing that components of wood structures usually are not of bilinear nature, curved skeleton curves are fit by judgment to the three bilinear diagrams. The envelope of these curves, together with the displacement amplitudes of the primary cycles of the basic loading history (e.g., displacement values of 0.7, 0.4, 0.3, 0.2, 0.1, 0.075, and 0.05) are used to obtain corresponding force values of primary force excursions (dashed horizontal lines). The so obtained force values are rounded up to obtain force amplitudes of the primary cycles of the force controlled loading history shown in Fig. 6. The trailing cycles are assumed to be of an amplitude of 0.75 times the preceding primary cycle. Since small cycles are believed to have negligible effects on the force capacity, all cycles with a force amplitude smaller than 0.7 (which corresponds to a displacement amplitude of 0.1) could be omitted in the loading history. Nevertheless, five cycles with force amplitude equal to 0.5Q0 are recommended to be executed. C.7 Representative Input for Shaking Table Studies Several ground motion records were evaluated for the purposes summarized in Section 3. The objective of record selection is to identify ground motions that are representative, in average, of the shaking a woodframe structure would experience at the hazard levels (or return periods) identified in Section 3. “In average” implies also that the records should be representative of the cumulative damage contained in the proposed loading histories. The following time history records are recommended to simulate seismic conditions at various return periods:

42

At return periods of 475 years and smaller: Select a typical ordinary (not near-fault) record that represents, in shape, the NEHRP design spectrum for soil type D in the period range of interest (from about 0.1 to 1.0 sec.). The preferred choice is the Northridge 94 Canoga Park record. The acceleration response spectra of the two horizontal components of this record are shown in Fig. C.36(a), together with the LA NEHRP design spectrum on which the development of the basic loading history is based. It can be judged that the larger of the two components may be sufficiently severe to represent the design spectrum at a period of about 0.6 seconds. For shorter periods some scaling will be necessary. If a scale factor of 1.3 is applied to the two components, the spectra shown in Fig. C.36(b) are obtained. Now the match in the short period range (less than about 0.35 sec.) is adequate, but the spectrum of the “larger” component is somewhat high from about 0.35 to 0.7 seconds. Because of the frequency characteristics of recorded ground motions it will never be possible to obtain uniform matching for the full period range of interest and compromises will have to be made. In judging these compromises it is necessary to consider more than just the elastic shape of the spectra because of inelastic response characteristics. It would be desirable to consider the shape of inelastic spectra together with that of the elastic spectrum. For instance, the hump around the period of 0.6 sec. in the elastic spectrum of the larger component of this record will affect the inelastic response of a structure with a shorter period. In the writer’s judgment it is preferable to use recorded ground motions together with sound judgment for scaling rather than simulated ground motions whose spectrum could be matched well over a wide range of periods but whose frequency content may not be representative of realistic ground motions. An alternative choice to the Northridge 94 Canoga Park record is the Loma Prieta 89 Hollister Differential Array record. The spectra of the two components of this record, unscaled and scaled by a factor of 1.3, are presented in Fig. C.37. Again, it is evident that good judgment will be needed to scale the record to provide the most appropriate match with the target spectrum. At very long return periods (in the order of 2500 years): At very long return periods it is appropriate to assume that the seismic hazard in an area like Los Angeles is controlled by fault ruptures close to the site. Thus, a near-fault record should be used to simulate seismic conditions associated with this hazard. The preferred choice is the Northridge 94 Rinaldi Receiving Station record whose acceleration, velocity, and displacement spectra are shown in Fig. C.38. The large difference in spectral shape and ordinates between the fault normal and fault parallel component is characteristic of near-fault records. An alternative to this record, although not a very desirable one because of the presence of multiple humps in the spectra, is the Kobe 95 Takatori Station record whose spectra are presented in Fig. C.39.

43

Acknowledgements This work was carried out as part of the CUREE/Caltech woodframe research program. The feedback provided by many of the researchers and advisors of this program is gratefully acknowledged. In particular, much helpful feedback was provided by Professors D. Dolan, A. Filiatrault, J. Hall, R. Hanson, G. Pardoen, and C.M. Uang, and by Dr. G. Foliente and Ms. K. Cobeen. This feedback is gratefully acknowledged.

44

References Alavi, B., and H. Krawinkler (1999) Structural Design Implications of Near-Fault Ground Motion, Kajima-CUREE Research Report 1999.12. ASTM (1995a). “Standard Practice for Static Load Test for Shear Resistance of Framed Walls for Buildings,” ASTM E564-95. American Society for Testing and Materials. ASTM (1995b). “Standard Test Methods of Conducting Strength Tests of Panels for Building Construction,” ASTM E72-95. American Society for Testing and Materials. ATC (1992). Guidelines for Cyclic Seismic Testing of Components of Steel Structures,” ATC-24, Applied Technology Council. CEN (1995). “Timber structures – test methods – cyclic testing of joints made with mechanical fasteners,” pr EN 12512, European Committee for Standardization, Brussels, Belgium. CoLA/UCI Committee (1999). “Cyclic Racking Shear Tests for Light Framed Shear Walls – August 1999.” FEMA 302 (1997) “1997 Edition: NEHRP Recommended Provisions for Seismic Regulations for New Buildings”, Federal Emergency Management Agency. Foliente, G.C. and Zacher, E.G. (1994). “Performance tests of timber structural systems under seismic loads,” Proceedings of a Research Needs Workshop on Analysis, Design and Testing of Timber Structures Under Seismic Loads, Foliente, G.C. (editor). University of California, Forest Products Laboratory, Richmond, California., pp 21-86. Foliente, G.C., Karacabeyli, E., and Yasumura, M. (1998). “International Test Standards for Joints of Timber Structures Under Earthquake and Wind Loads,” Elsevier Scienc Ltd, Structural Engineering Worldwide 1998, Proceedings of the First International Congress of Structural Engineers, San Francisco. ISO (1999). “Timber Structures – Joints made with mechanical fasteners – Quasi-static reversed-cyclic test method,” draft document ISO TC 165/SC N. Krawinkler, H., and Alavi, B. (1998). “Development of Improved Design Procedures for Near-Fault Ground Motions,” SMIP98 Seminar on Utilization of Strong-Motion Data, Oakland, pp. 21-41. Rahnama, M., and Krawinkler, H. (1993). “Effects of Soft Soils and Hysteresis Model on Seismic Demands,” John A. Blume Earthquake Engineering Center Report No. 108, Department of Civil Engineering, Stanford University.

45

SAA (1997). “Timber – Methods for evaluation of mechanical joint systems. Part 1: Static loading; Part 2: Cyclonic wind loading; Part 3: Earthquake loading,” Draft AS/NZS BBBB. Standards Association of Australia. Seneviratna, G. D. P. K., and Krawinkler, H. (1997). “Evaluation of Inelastic MDOF Effects for Seismic Design,” John A. Blume Earthquake Engineering Center Report No. 120, Department of Civil Engineering, Stanford University. Shepherd, R. (1996). “Standardized Experimental Testing Procedures for Low-Rise Structures,” Earthquake Spectra, EERI, Vol. 12, No. 1. Somerville, P., et al. (1997). “Development of ground motion time histories for phase 2 of the FEMA/SAC steel project,” SAC Background Document, Report No. SAC/BD-97/04. , EERC – University of California at Berkeley. Somerville, P.G., H. Krawinkler, and B. Alavi. (1999). Development of Improved Ground Motion Representation and Design Procedures for Near-Fault Ground Motions, CSMIP Data Utilization Program, Report CSMIP/99-xx.

46

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47

Table C.2. Identification and Properties of Selected Near-Fault Records

Designation Earthquake Station Magnitude Distance Tp (sec) PGV (cm/sec)

LP89lex Loma Prieta, 1989 Lexington 7.0 6.3 1.0 179NR94rrs Nothridge, 1994 Rinaldi 6.7 7.5 1.0 174NR94newh Nothridge, 1994 Newhall 6.7 7.1 1.3 119KB95kobj Kobe, 1995 JMA 6.9 0.6 0.9 160KB95tato Kobe, 1995 Takatori 6.9 1.5 2.0 174MH84cyld Morgan Hill, 1984 Coyote L D 6.2 0.1 0.8 65

Table C.3. Summary of natural periods and frequencies of wood and wood-based buildings from experiments and calculation estimates [Foliente, G.C. and E.G. Zacher (1994)]

Building Type Natural Period

Tn (sec)

Natural

Frequency (1/Tn)

(Hz)

Reference(s)

One- and two-story New Zealand residential 0.1 to 0.6 1.7 to 10.0 (24)

One-story truss-frame residential 0.14 to 0.26 3.8 to 7.2 (33)

Two- and three-story N. American residential 0.14 to 0.32 3.0 to 7.0 (62)

Two-story residential (Greece) 0.18 to 0.22 4.5 to 5.6 (67)

Two-story base-isolated residential 0.48 to 1.25 0.8 to 2.1 (55)

One-, one and a half-, and two-story N. American

residential and school buildings

0.06 to 0.25

4.0 to 18.0

(61)

One-, two- and three-story Japanese residential 0.11 to 0.33 3.0 to 9.0 (3)

Three-story Japanese residential 0.16 to 0.20 4.7 to 6.2 (51, 70)

One-, and two-story commercial/industrial (plywood

roof diaphragm and concrete/masonry walls)

0.20 to 0.80

1.2 to 5.1

(11)

Range of Values* 0.06 to 0.80 1.2 to 18.0

* Excluding the two-story base-isolated building.

48

Table C.4 84th Percentile Values of Ordered Pre-Peak Excursions (Normalized by δy), Ordinary Ground Motions

Ordinary Ground MotionsOrdered Pre-Peak Excursions - 84th Percentile Values

Normalized to yield displacement, δyT = 0.2 sec T = 0.2 sec T = 0.2 sec T = 0.2 sec T=0.35 sec T=0.5 sec T = 0.5 sec T= 1 sec. T=0.2 sec T =0.5 sec.R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.133 R = 2 η = 0.563 R = 2 η = 0.563Pinching, κ = 0.25 Pinching, κ = 0.5 Bilinear Clough Pinching, κ = 0.5 Pinching, κ = 0.25 Pinching, κ = 0.5 Pinching, κ = 0.5 Pinching, κ = 0.5 Pinching, κ = 0.5δy = 0.224 cm δy = 0.224 cm δy = 0.224 cm δy = 0.224 cm δy = 0.685 cm δy =1.398 cm δy =1.398 cm δy =3.315 cm δy =0.559 cm δy =3.495 cm

53.8935 48.3743 36.8249 41.6980 28.2514 16.3936 15.0030 12.6617 8.4202 4.315140.7126 33.0888 24.0517 24.9365 20.9693 14.1848 11.7575 10.3608 7.0723 3.611633.4014 25.6747 17.5935 18.8929 13.9723 11.9693 9.7949 7.5406 5.8227 3.187328.5754 22.9929 14.1685 17.4589 12.6747 10.6240 8.5958 6.4527 5.5045 2.984425.4161 18.9273 8.8989 12.4336 9.4112 8.6961 6.6774 5.5977 4.6083 2.629223.2201 16.2160 6.6030 11.9354 8.6534 7.9871 6.2482 4.5906 4.1723 2.508019.9058 14.3563 6.2852 9.2020 7.4122 7.0305 5.5934 3.6538 3.4178 2.157017.6175 12.4266 5.5133 8.0942 6.7555 6.3476 4.9177 3.3781 3.0390 1.986115.1289 10.5897 4.8212 7.6748 6.4050 5.7980 4.7039 3.0496 3.0390 1.878313.4765 9.5678 4.6429 7.3693 6.0180 4.8779 4.2699 2.8576 2.8025 1.716711.7281 8.7070 4.3918 6.6612 5.7103 4.6460 4.0257 2.6107 2.7618 1.629210.7125 7.9810 4.1280 6.3049 5.4525 4.1117 3.7365 2.4672 2.7283 1.5221

9.3185 6.9785 3.9298 5.4788 4.7591 3.6366 3.6205 2.3125 2.6446 1.38478.3547 6.3632 3.8194 4.9351 4.5817 3.4674 3.2518 2.1990 2.4679 1.31757.9261 5.9806 3.3878 4.7485 4.0365 3.3114 3.1713 2.0583 2.4125 1.28347.0927 5.6827 3.3520 4.5655 3.5992 3.1826 2.9153 1.9474 2.3958 1.22546.7430 5.3044 3.0509 4.4083 3.3369 3.0404 2.7686 1.8932 2.2695 1.19066.5697 5.0251 3.0277 4.1789 3.0500 2.8440 2.5091 1.8366 2.2695 1.12636.2991 4.8732 2.8311 4.1001 2.9682 2.7062 2.3672 1.7435 2.1417 1.10685.9625 4.6941 2.8116 3.9195 2.5843 2.4937 2.2976 1.6632 2.1417 1.04885.5784 4.3697 2.6875 3.5371 2.5512 2.4337 2.1913 1.6432 2.0448 1.02625.0907 4.0185 2.6634 3.0847 2.5512 2.3123 2.1536 1.5456 2.0340 0.97584.6278 3.6611 2.5832 3.0119 2.5005 2.1467 2.0573 1.4970 2.0340 0.92814.4284 3.5782 2.5156 2.7192 2.5005 2.0923 2.0171 1.4563 2.0210 0.91674.2734 3.4241 2.4622 2.6733 2.4839 1.9979 1.9385 1.3878 1.8469 0.86944.0795 3.3010 2.4014 2.6369 2.4839 1.9515 1.8999 1.3536 1.8469 0.80663.9297 3.1002 2.3523 2.6369 2.2667 1.8677 1.7997 1.3286 1.6924 0.79473.6629 2.9566 2.2911 2.4720 2.2667 1.7149 1.7563 1.3072 1.6924 0.77893.5776 2.9063 2.1203 2.4720 1.9343 1.6698 1.6551 1.2603 1.4679 0.75983.4946 2.4801 2.1203 2.0897 1.9343 1.6082 1.5830 1.2171 1.4679 0.73133.4115 2.4801 2.0432 2.0897 1.8385 1.5782 1.5430 1.2014 1.3898 0.71623.2749 2.1263 2.0432 2.0819 1.8385 1.5165 1.4964 1.1381 1.2215 0.69263.2480 2.1263 1.9011 2.0819 1.7996 1.4764 1.4524 1.1136 1.2215 0.67603.1176 1.9465 1.8913 1.8468 1.5928 1.4386 1.3951 1.0789 1.1289 0.66043.0221 1.9465 1.8599 1.8468 1.5928 1.3718 1.3766 1.0326 1.1289 0.65222.8113 1.9078 1.8417 1.7443 1.5362 1.3138 1.3240 0.9751 1.0864 0.62802.7476 1.8949 1.7857 1.7443 1.2695 1.3001 1.3205 0.9615 1.0864 0.61062.6724 1.8949 1.7857 1.6785 1.2695 1.2371 1.2104 0.9092 1.0092 0.60492.6003 1.8785 1.7592 1.6785 1.2015 1.2118 1.1732 0.8734 1.0092 0.59192.4757 1.8785 1.7592 1.6638 1.2015 1.1739 1.1361 0.8462 1.0001 0.58472.3944 1.7348 1.7439 1.6638 1.1599 1.1367 1.1131 0.8071 0.9373 0.55822.3605 1.7348 1.7439 1.5121 1.1599 1.1079 1.0424 0.7874 0.9373 0.54862.3062 1.5651 1.7232 1.4089 1.0984 1.0745 1.0225 0.7400 0.9051 0.53442.2555 1.5651 1.7232 1.3648 1.0891 1.0463 1.0001 0.7257 0.9051 0.53182.2046 1.5546 1.7232 1.3648 1.0891 1.0321 0.9473 0.6912 0.7798 0.51932.1875 1.4904 1.7232 1.3105 1.0777 1.0177 0.9204 0.6843 0.7798 0.51091.9672 1.4904 1.5958 1.2584 1.0777 1.0010 0.8865 0.6646 0.7652 0.49261.9672 1.4351 1.5958 1.2533 1.0422 0.9805 0.8723 0.6557 0.7652 0.48381.7675 1.4351 1.5458 1.2059 1.0422 0.8999 0.8423 0.7388 0.47071.7348 1.3688 1.4917 1.1916 1.0209 0.8836 0.8277 0.7388 0.45701.7348 1.3688 1.4917 1.1329 1.0209 0.8437 0.7976 0.7045 0.44831.6859 1.3181 1.4811 1.0643 0.8330 0.7807 0.7045 0.44101.6859 1.3181 1.4811 1.0550 0.7602 0.6884 0.42921.6358 1.2941 1.3309 1.0228 0.7515 0.6884 0.42441.5602 1.2533 1.3309 1.0228 0.6884 0.41681.5602 1.2533 1.2867 0.6884 0.40361.5397 1.1541 1.2762 0.6733 0.40111.5150 1.1541 1.2762 0.6733 0.39861.5150 1.1329 1.2439 0.6649 0.38831.5148 1.0676 1.2439 0.6649 0.38701.5017 1.0676 1.2285 0.6175 0.37711.4811 1.0372 1.2285 0.5753 0.36581.4811 1.0372 1.1270 0.5753 0.35901.4811 1.0297 1.1270 0.5682 0.35081.4811 1.0297 1.1269 0.5682 0.34181.4383 1.0090 1.1100 0.5572 0.33661.4219 1.0090 1.1100 0.5572 0.33031.3782 1.0987 0.5169 0.32351.3782 1.0987 0.5154 0.32041.3534 1.0024 0.5140 0.31241.3534 1.0024 0.5140 0.30981.3495 0.5140 0.30481.3451 0.5006 0.30191.3232 0.5006 0.29871.2941 0.4972 0.29181.2861 0.4972 0.28971.2541 0.4526 0.28621.2533 0.4466 0.28041.2439 0.4466 0.27711.2303 0.4451 0.26811.2066 0.4451 0.26511.2066 0.4436 0.26321.1568 0.4436 0.25471.1329 0.4390 0.24941.1014 0.4390 0.24771.1014 0.4373 0.24361.1011 0.4373 0.24021.0264 0.4364 0.23421.0108 0.4364 0.2302

0.22690.22620.2216

49

Table C.5 84th Percentile Values of Ordered Pre-Peak Excursions (Normalized by ∆ = Max. Range/2), Ordinary Ground Motions

Ordinary Ground Motions

Ordered Pre-Peak Excursions - 84th Percentile Values Normalized to half of computed 84th percentile maximum range,∆

Excursion Proposed T = 0.2 sec T = 0.2 sec T = 0.2 sec T = 0.2 sec T = 0.35 sec T=0.5 sec T = 0.5 sec T= 1 sec. T=0.2 sec T =0.5 sec.Number Loading History R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.133 R = 2 η = 0.563 R = 2 η = 0.563

for Pinching, κ = 0.25 Pinching, κ = 0.5 Bilinear Clough Pinching, κ = 0.5 Pinching, κ = 0.25 Pinching, κ = 0.5 Pinching, κ = 0.5 Pinching, κ = 0.5 Pinching, κ = 0.5

Ordinary δy = 0.224 cm δy = 0.224 cm δy = 0.224 cm δy = 0.224 cm δy = 0.685 cm δy =1.398 cm δy =1.398 cm δy =3.315 cm δy =0.559 cm δy =3.495 cm

Ground Motions ∆ (cm)= 12.0560 ∆ (cm)= 10.8213 ∆ (cm)= 6.3198 ∆ (cm)= 9.3278 ∆ (cm)= 19.3522 ∆ (cm)= 22.9216 ∆ (cm)= 20.9772 ∆ (cm)= 41.9734 ∆ (cm)= 4.7069 ∆ (cm)= 15.0812Indiv. Cum. Indiv. Cum. Ind Cum Ind Cum Ind Cum Ind Cum Ind Cum Ind Cum Ind Cum Ind Cum Ind Cum

1 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000

2 1.7000 3.7000 1.5109 3.5109 1.3680 3.3680 1.3063 3.3063 1.1961 3.1961 1.4845 3.4845 1.7305 3.7305 1.5673 3.5673 1.6366 3.6366 1.6798 3.6798 1.6739 3.67393 1.4000 5.1000 1.2395 4.7504 1.0615 4.4295 0.9555 4.2618 0.9062 4.1022 0.9891 4.4736 1.4602 5.1908 1.3057 4.8731 1.1911 4.8277 1.3830 5.0629 1.4773 5.1513

4 1.1000 6.2000 1.0604 5.8108 0.9506 5.3802 0.7695 5.0313 0.8374 4.9396 0.8973 5.3709 1.2961 6.4869 1.1459 6.0189 1.0193 5.8469 1.3075 6.3704 1.3832 6.53455 1.0500 7.2500 0.9432 6.7540 0.7825 6.1627 0.4833 5.5146 0.5964 5.5360 0.6662 6.0371 1.0609 7.5478 0.8901 6.9091 0.8842 6.7311 1.0946 7.4649 1.2186 7.7531

6 1.0500 8.3000 0.8617 7.6157 0.6704 6.8331 0.3586 5.8732 0.5725 6.1085 0.6126 6.6497 0.9744 8.5222 0.8329 7.7420 0.7251 7.4562 0.9910 8.4559 1.1625 8.91557 1.0500 9.3500 0.7387 8.3544 0.5936 7.4267 0.3414 6.2146 0.4414 6.5498 0.5247 7.1745 0.8577 9.3799 0.7456 8.4876 0.5771 8.0334 0.8118 9.2677 0.9997 9.9153

8 1.0500 10.4000 0.6538 9.0082 0.5138 7.9405 0.2994 6.5140 0.3882 6.9381 0.4782 7.6527 0.7744 10.1543 0.6556 9.1432 0.5336 8.5670 0.7218 9.9896 0.9205 10.83589 0.8000 11.2000 0.5614 9.5697 0.4378 8.3783 0.2618 6.7759 0.3681 7.3062 0.4534 8.1061 0.7074 10.8617 0.6271 9.7703 0.4817 9.0487 0.7218 10.7114 0.8706 11.706410 0.7000 11.9000 0.5001 10.0698 0.3956 8.7738 0.2522 7.0280 0.3535 7.6596 0.4260 8.5322 0.5951 11.4568 0.5692 10.3395 0.4514 9.5001 0.6657 11.3771 0.7957 12.5021

11 0.6000 12.5000 0.4352 10.5050 0.3600 9.1338 0.2385 7.2666 0.3195 7.9791 0.4043 8.9364 0.5668 12.0236 0.5366 10.8761 0.4124 9.9124 0.6560 12.0331 0.7551 13.257212 0.6000 13.1000 0.3975 10.9026 0.3300 9.4638 0.2242 7.4908 0.3024 8.2815 0.3860 9.3224 0.5016 12.5252 0.4981 11.3742 0.3897 10.3022 0.6480 12.6811 0.7055 13.9627

13 0.6000 13.7000 0.3458 11.2484 0.2885 9.7523 0.2134 7.7042 0.2628 8.5443 0.3369 9.6593 0.4437 12.9689 0.4826 11.8569 0.3653 10.6674 0.6282 13.3093 0.6418 14.604514 0.6000 14.3000 0.3100 11.5584 0.2631 10.0154 0.2074 7.9116 0.2367 8.7810 0.3244 9.9837 0.4230 13.3919 0.4335 12.2903 0.3473 11.0148 0.5862 13.8955 0.6107 15.215115 0.6000 14.9000 0.2941 11.8525 0.2473 10.2627 0.1840 8.0956 0.2278 9.0088 0.2858 10.2694 0.4040 13.7959 0.4228 12.7131 0.3251 11.3399 0.5730 14.4685 0.5948 15.810016 0.5000 15.4000 0.2632 12.1158 0.2349 10.4976 0.1821 8.2777 0.2190 9.2278 0.2548 10.5242 0.3883 14.1841 0.3886 13.1017 0.3076 11.6475 0.5691 15.0376 0.5680 16.378017 0.4500 15.8500 0.2502 12.3660 0.2193 10.7169 0.1657 8.4434 0.2114 9.4392 0.2362 10.7605 0.3709 14.5551 0.3691 13.4708 0.2990 11.9465 0.5391 15.5766 0.5519 16.9298

18 0.4500 16.3000 0.2438 12.6098 0.2078 10.9247 0.1644 8.6078 0.2004 9.6396 0.2159 10.9764 0.3470 14.9020 0.3345 13.8053 0.2901 12.2366 0.5391 16.1157 0.5220 17.451819 0.4500 16.7500 0.2338 12.8436 0.2015 11.1262 0.1538 8.7616 0.1967 9.8363 0.2101 11.1865 0.3301 15.2322 0.3156 14.1209 0.2754 12.5120 0.5087 16.6244 0.5130 17.9648

20 0.4500 17.2000 0.2213 13.0648 0.1941 11.3202 0.1527 8.9143 0.1880 10.0243 0.1830 11.3695 0.3042 15.5364 0.3063 14.4271 0.2627 12.7748 0.5087 17.1331 0.4861 18.450921 0.4500 17.6500 0.2070 13.2718 0.1807 11.5009 0.1460 9.0602 0.1697 10.1939 0.1806 11.5501 0.2969 15.8333 0.2921 14.7192 0.2596 13.0343 0.4857 17.6188 0.4756 18.9265

22 0.4500 18.1000 0.1889 13.4608 0.1661 11.6670 0.1447 9.2049 0.1480 10.3419 0.1806 11.7307 0.2821 16.1154 0.2871 15.0063 0.2441 13.2784 0.4831 18.1019 0.4523 19.378823 0.4000 18.5000 0.1717 13.6325 0.1514 11.8184 0.1403 9.3452 0.1445 10.4864 0.1770 11.9077 0.2619 16.3773 0.2742 15.2806 0.2365 13.5149 0.4831 18.5851 0.4302 19.809024 0.3000 18.8000 0.1643 13.7968 0.1479 11.9663 0.1366 9.4818 0.1304 10.6168 0.1770 12.0847 0.2553 16.6325 0.2689 15.5495 0.2300 13.7449 0.4800 19.0651 0.4249 20.2339

25 0.3000 19.1000 0.1586 13.9554 0.1416 12.1079 0.1337 9.6155 0.1282 10.7450 0.1758 12.2606 0.2437 16.8763 0.2584 15.8079 0.2192 13.9642 0.4387 19.5038 0.4029 20.636826 0.3000 19.4000 0.1514 14.1068 0.1365 12.2444 0.1304 9.7459 0.1265 10.8715 0.1758 12.4364 0.2381 17.1144 0.2533 16.0612 0.2138 14.1780 0.4387 19.9425 0.3738 21.010627 0.3000 19.7000 0.1458 14.2526 0.1282 12.3726 0.1278 9.8737 0.1265 10.9980 0.1605 12.5969 0.2279 17.3422 0.2399 16.3011 0.2099 14.3878 0.4020 20.3445 0.3683 21.379028 0.3000 20.0000 0.1359 14.3886 0.1222 12.4948 0.1244 9.9981 0.1186 11.1165 0.1605 12.7573 0.2092 17.5514 0.2341 16.5352 0.2065 14.5943 0.4020 20.7465 0.3610 21.7400

29 0.3000 20.3000 0.1328 14.5213 0.1202 12.6150 0.1152 10.1133 0.1186 11.2351 0.1369 12.8943 0.2037 17.7552 0.2206 16.7558 0.1991 14.7934 0.3487 21.0951 0.3521 22.092130 0.3000 20.6000 0.1297 14.6510 0.1025 12.7175 0.1152 10.2284 0.1002 11.3353 0.1369 13.0312 0.1962 17.9514 0.2110 16.9669 0.1922 14.9856 0.3487 21.4438 0.3390 22.431131 0.2000 20.8000 0.1266 14.7776 0.1025 12.8200 0.1110 10.3394 0.1002 11.4356 0.1302 13.1613 0.1925 18.1439 0.2057 17.1725 0.1898 15.1754 0.3301 21.7739 0.3319 22.7630

32 0.1750 20.9750 0.1215 14.8992 0.0879 12.9079 0.1110 10.4504 0.0999 11.5354 0.1302 13.2915 0.1850 18.3289 0.1995 17.3720 0.1798 15.3551 0.2901 22.0640 0.3210 23.084033 0.1500 21.1250 0.1205 15.0197 0.0879 12.9959 0.1033 10.5536 0.0999 11.6353 0.1274 13.4189 0.1801 18.5090 0.1936 17.5656 0.1759 15.5310 0.2901 22.3542 0.3133 23.3974

34 0.1500 21.2750 0.1157 15.1354 0.0805 13.0763 0.1027 10.6563 0.0886 11.7238 0.1128 13.5316 0.1755 18.6845 0.1860 17.7516 0.1704 15.7015 0.2682 22.6223 0.3061 23.703435 0.1500 21.4250 0.1122 15.2475 0.0805 13.1568 0.1010 10.7574 0.0886 11.8124 0.1128 13.6444 0.1674 18.8519 0.1835 17.9351 0.1631 15.8646 0.2682 22.8905 0.3023 24.0057

36 0.1500 21.5750 0.1043 15.3519 0.0789 13.2357 0.1000 10.8574 0.0837 11.8961 0.1087 13.7532 0.1603 19.0122 0.1765 18.1116 0.1540 16.0186 0.2580 23.1485 0.2911 24.296837 0.1500 21.7250 0.1020 15.4538 0.0783 13.3140 0.0970 10.9544 0.0837 11.9797 0.0899 13.8430 0.1586 19.1708 0.1760 18.2877 0.1519 16.1705 0.2580 23.4066 0.2830 24.579838 0.1500 21.8750 0.0992 15.5530 0.0783 13.3924 0.0970 11.0513 0.0805 12.0602 0.0899 13.9329 0.1509 19.3217 0.1614 18.4490 0.1436 16.3141 0.2397 23.6463 0.2804 24.8602

39 0.1500 22.0250 0.0965 15.6495 0.0777 13.4700 0.0955 11.1469 0.0805 12.1408 0.0851 14.0180 0.1478 19.4695 0.1564 18.6054 0.1380 16.4520 0.2397 23.8860 0.2744 25.134540 0.1500 22.1750 0.0919 15.7414 0.0777 13.5477 0.0955 11.2424 0.0798 12.2206 0.0851 14.1030 0.1432 19.6128 0.1514 18.7569 0.1337 16.5857 0.2375 24.1235 0.2710 25.4055

41 0.1500 22.3250 0.0889 15.8302 0.0717 13.6194 0.0947 11.3371 0.0798 12.3004 0.0821 14.1851 0.1387 19.7514 0.1484 18.9052 0.1275 16.7132 0.2226 24.3462 0.2587 25.664242 0.1500 22.4750 0.0876 15.9178 0.0717 13.6911 0.0947 11.4319 0.0725 12.3729 0.0821 14.2672 0.1352 19.8866 0.1390 19.0442 0.1244 16.8376 0.2226 24.5688 0.2543 25.9185

43 0.1500 22.6250 0.0856 16.0034 0.0647 13.7559 0.0936 11.5255 0.0676 12.4405 0.0778 14.3450 0.1311 20.0177 0.1363 19.1805 0.1169 16.9545 0.2150 24.7838 0.2477 26.166244 0.1500 22.7750 0.0837 16.0871 0.0647 13.8206 0.0936 11.6190 0.0655 12.5059 0.0771 14.4221 0.1277 20.1453 0.1333 19.3138 0.1146 17.0691 0.2150 24.9988 0.2465 26.412745 0.1250 22.9000 0.0818 16.1689 0.0643 13.8848 0.0936 11.7126 0.0655 12.5714 0.0771 14.4992 0.1259 20.2712 0.1263 19.4401 0.1092 17.1783 0.1852 25.1840 0.2407 26.6534

46 0.1125 23.0125 0.0812 16.2501 0.0616 13.9465 0.0936 11.8062 0.0629 12.6342 0.0763 14.5755 0.1242 20.3954 0.1227 19.5628 0.1081 17.2864 0.1852 25.3692 0.2368 26.890247 0.1125 23.1250 0.0730 16.3231 0.0616 14.0081 0.0867 11.8929 0.0604 12.6946 0.0763 14.6518 0.1221 20.5175 0.1182 19.6810 0.1050 17.3914 0.1818 25.5510 0.2283 27.1185

48 0.1125 23.2375 0.0730 16.3961 0.0593 14.0674 0.0867 11.9796 0.0601 12.7547 0.0738 14.7256 0.1196 20.6371 0.1163 19.7973 0.1036 17.4949 0.1818 25.7327 0.2242 27.342749 0.1125 23.3500 0.0656 16.4617 0.0593 14.1267 0.0840 12.0635 0.0578 12.8125 0.0738 14.7993 0.1098 20.7469 0.1123 19.9095 17.4949 0.1755 25.9082 0.2182 27.5609

50 0.1125 23.4625 0.0644 16.5261 0.0566 14.1833 0.0810 12.1445 0.0572 12.8697 0.0723 14.8716 0.1078 20.8547 0.1103 20.0199 17.4949 0.1755 26.0837 0.2118 27.772751 0.1125 23.5750 0.0644 16.5905 0.0566 14.2399 0.0810 12.2256 0.0543 12.9240 0.0723 14.9439 0.1029 20.9577 0.1063 20.1262 17.4949 0.1673 26.2510 0.2078 27.980552 0.1125 23.6875 0.0626 16.6530 0.0545 14.2944 0.0804 12.3060 0.0510 12.9751 14.9439 0.1016 21.0593 0.1041 20.2303 17.4949 0.1673 26.4184 0.2044 28.1849

53 0.1125 23.8000 0.0626 16.7156 0.0545 14.3489 0.0804 12.3864 0.0506 13.0257 14.9439 21.0593 0.1013 20.3316 17.4949 0.1635 26.5819 0.1989 28.383854 0.1125 23.9125 0.0607 16.7763 0.0535 14.4024 0.0723 12.4587 0.0491 13.0747 14.9439 21.0593 0.1002 20.4318 17.4949 0.1635 26.7454 0.1967 28.5805

55 0.1125 24.0250 0.0579 16.8342 0.0518 14.4542 0.0723 12.5310 0.0491 13.1238 14.9439 21.0593 20.4318 17.4949 0.1635 26.9089 0.1932 28.773756 0.1125 24.1375 0.0579 16.8921 0.0518 14.5061 0.0699 12.6009 13.1238 14.9439 21.0593 20.4318 17.4949 0.1635 27.0724 0.1871 28.9608

57 0.1125 24.2500 0.0571 16.9492 0.0477 14.5538 0.0693 12.6702 13.1238 14.9439 21.0593 20.4318 17.4949 0.1599 27.2323 0.1859 29.146758 0.1000 24.3500 0.0562 17.0055 0.0477 14.6015 0.0693 12.7395 13.1238 14.9439 21.0593 20.4318 17.4949 0.1599 27.3922 0.1847 29.331459 0.1000 24.4500 0.0562 17.0617 0.0468 14.6483 0.0676 12.8071 13.1238 14.9439 21.0593 20.4318 17.4949 0.1579 27.5502 0.1800 29.5114

60 0.1000 24.5500 0.0562 17.1179 0.0441 14.6925 0.0676 12.8746 13.1238 14.9439 21.0593 20.4318 17.4949 0.1579 27.7081 0.1794 29.690761 0.1000 24.6500 0.0557 17.1736 0.0441 14.7366 0.0667 12.9413 13.1238 14.9439 21.0593 20.4318 17.4949 0.1467 27.8548 0.1748 29.8655

62 0.1000 24.7500 0.0550 17.2286 0.0429 14.7795 0.0667 13.0081 13.1238 14.9439 21.0593 20.4318 17.4949 0.1367 27.9914 0.1695 30.035063 0.1000 24.8500 0.0550 17.2836 0.0429 14.8224 0.0612 13.0693 13.1238 14.9439 21.0593 20.4318 17.4949 0.1367 28.1281 0.1664 30.2014

64 0.1000 24.9500 0.0550 17.3385 0.0426 14.8649 0.0612 13.1305 13.1238 14.9439 21.0593 20.4318 17.4949 0.1350 28.2630 0.1626 30.364065 0.1000 25.0500 0.0550 17.3935 0.0426 14.9075 0.0612 13.1917 13.1238 14.9439 21.0593 20.4318 17.4949 0.1350 28.3980 0.1584 30.522466 0.1000 25.1500 0.0534 17.4469 0.0417 14.9492 0.0603 13.2520 13.1238 14.9439 21.0593 20.4318 17.4949 0.1324 28.5303 0.1560 30.6784

67 0.1000 25.2500 0.0528 17.4996 0.0417 14.9910 0.0603 13.3122 13.1238 14.9439 21.0593 20.4318 17.4949 0.1324 28.6627 0.1531 30.831568 0.1000 25.3500 0.0511 17.5508 14.9910 0.0597 13.3719 13.1238 14.9439 21.0593 20.4318 17.4949 0.1228 28.7855 0.1499 30.9815

69 0.1000 25.4500 0.0511 17.6019 14.9910 0.0597 13.4316 13.1238 14.9439 21.0593 20.4318 17.4949 0.1224 28.9079 0.1485 31.130070 0.0750 25.5250 0.0502 17.6521 14.9910 0.0544 13.4860 13.1238 14.9439 21.0593 20.4318 17.4949 0.1221 29.0300 0.1448 31.2748

71 0.0502 17.7024 14.9910 0.0544 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1221 29.1521 0.1436 31.418372 0.0501 17.7524 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1221 29.2741 0.1413 31.559673 0.0499 17.8024 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1189 29.3931 0.1399 31.6995

74 0.0491 17.8515 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1189 29.5120 0.1384 31.838075 0.0480 17.8995 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1181 29.6301 0.1353 31.9732

76 0.0477 17.9472 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1181 29.7482 0.1343 32.107577 0.0465 17.9938 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1075 29.8557 0.1327 32.2402

78 0.0465 18.0403 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1061 29.9618 0.1300 32.370179 0.0462 18.0864 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1061 30.0679 0.1284 32.498680 0.0457 18.1321 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1057 30.1736 0.1243 32.6229

81 0.0448 18.1769 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1057 30.2793 0.1229 32.745782 0.0448 18.2216 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1054 30.3847 0.1220 32.8677

83 0.0429 18.2646 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1054 30.4900 0.1181 32.985884 0.0420 18.3066 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1043 30.5943 0.1156 33.101385 0.0409 18.3475 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1043 30.6986 0.1148 33.216286 0.0409 18.3884 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1039 30.8024 0.1129 33.329187 0.0409 18.4292 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1039 30.9063 0.1113 33.4404

88 0.0381 18.4673 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1036 31.0099 0.1086 33.548989 0.0375 18.5048 14.9910 13.5405 13.1238 14.9439 21.0593 20.4318 17.4949 0.1036 31.1136 0.1067 33.6556

90 18.5048 14.9439 21.0593 20.4318 17.4949 31.1136 0.1052 33.760891 18.5048 14.9439 21.0593 20.4318 17.4949 31.1136 0.1048 33.865692 18.5048 14.9439 21.0593 20.4318 17.4949 31.1136 0.1027 33.9683

50

Table C.6 84th Percentile Values of Ordered Primary Excursions, Ordinary Ground Motions

Ordinary Ground MotionsOrdered Primary Excursions - 84th Percentile Values

Normalized to half of computed 84th percentile maximum range, ∆

# Exc Proposed T=0.2s T=0.2s T=0.35s T=0.5s T=0.5s T=1s T=0.2s T=0.2s T=0.2s Loading History R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.133 R = 2, η = 0.563 R = 5, η = 0.225 R = 5, η = 0.225 for Pinching, κ=0.25 Pinching, κ=0.5 Pinching, κ=0.5 Pinching, κ=0.25 Pinching, κ=0.5 Pinching, κ=0.5 Pinching, κ=0.5 Clough Model Bilinear Model Ordinary δy = 0.224 cm δy = 0.224 cm δy =0.685 cm δy =1.398 cm δy =1.398 cm δy =3.315 cm δy =3.495 cm δy = 0.22 cm δy = 0.22 cm Ground Motions α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3%

Ind Cum Ind Cum Ind Cum Ind Cum Ind Cum Ind Cum Ind Cum Ind Cum Ind Cum Ind Cum1 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.00002 1.7000 3.7000 1.4649 3.4649 1.1880 3.1880 1.4664 3.4664 1.6374 3.6374 1.4918 3.4918 1.7953 3.7953 1.6517 3.6517 1.1650 3.1650 1.0631 3.06313 1.4000 5.1000 1.1238 4.5887 1.0125 4.2005 0.9224 4.3888 1.4099 5.0473 0.9805 4.4722 1.1020 4.8973 1.0359 4.6876 0.4980 3.6630 0.6521 3.71524 1.1000 6.2000 0.7480 5.3368 0.7087 4.9092 0.8246 5.2133 1.1835 6.2308 0.8379 5.3101 0.8296 5.7268 0.7586 5.4462 0.4617 4.1246 0.3933 4.10845 0.8000 7.0000 0.4597 5.7965 0.4613 5.3704 0.4207 5.6341 0.9378 7.1686 0.7443 6.0544 0.5574 6.2843 0.6848 6.1309 0.3918 4.51656 0.6000 7.6000 0.4372 6.2337 0.4078 5.7782 0.2949 5.9289 0.7989 7.9675 0.3774 6.4317 0.2533 4.76987 0.4000 8.0000 0.3491 6.5827 0.3304 6.1086 0.2716 6.2005 0.6957 8.6631 0.3571 6.7889 0.2000 4.96998 0.3000 8.3000 0.2139 6.7966 0.2351 6.3437 0.2091 6.40969 0.2000 8.5000 0.1618 6.9584 0.1726 6.5163

10 0.1750 8.6750 0.1195 7.077911 0.1500 8.8250

Ordinary Ground Motions

Ordered Primary Excursions - 84th Percentile Values

Normalized to yield displacement, δy

# Exc T=0.2s T=0.2s T=0.35s T=0.5s T=0.5s T=1s T=0.2s T=0.2s T=0.2sR = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.133 R = 2, η = 0.563 R = 5, η = 0.225 R = 5, η = 0.225Pinching, κ=0.25 Pinching, κ=0.5 Pinching, κ=0.5 Pinching, κ=0.25 Pinching, κ=0.5 Pinching, κ=0.5 Pinching, κ=0.5 Clough Model Bilinear Modelδy = 0.224 cm δy = 0.224 cm δy =0.685 cm δy =1.398 cm δy =1.398 cm δy =3.315 cm δy =3.495 cm δy = 0.22 cm δy = 0.22 cmα = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3% α = 8 %, P-∆ = 3%Counted Computed Counted Computed Counted Computed Counted Computed Counted Computed Counted Computed Counted Computed Counted Computed Counted Computed

1 46.8029 53.8935 42.3438 48.3743 25.6939 28.2514 15.2512 16.3960 13.5896 15.0052 13.5405 12.6616 7.8121 8.4202 39.5852 41.6986 36.5373 36.82482 37.6719 39.4750 28.7336 20.7141 13.4236 10.5358 11.1921 11.3657 6.9538 22.6048 24.2890 19.57353 29.0755 30.2832 24.4904 13.0290 11.5585 7.3560 6.9763 4.3611 10.3827 12.00674 20.1569 17.1408 11.6476 9.7021 6.2861 5.2519 3.1939 9.6254 7.24095 12.3880 11.1565 5.9429 7.6877 5.5841 3.5290 2.8829 8.16946 11.7805 9.8632 4.1654 6.5495 2.8311 5.28217 9.4063 7.9911 3.8365 5.7030 2.6795 4.17088 5.7631 5.6862 2.95379 4.3599 4.1752

10 3.2208

Table C.7 Median Values of Ordered Pre-Peak Excursions

(Normalized by ∆ = Max. Amplitude), Near-Fault Ground Motions

Near-Fault Ground Motions Ordered Pre-Peak Excursions - Median Values

Normalized to median of the maximum amplitude, ∆

Excursion Proposed T=0.2 sec T=0.2 sec T=0.5 sec T=0.5 secNumber Loading History η = 0.225 η = 0.563 η = 0.225 η = 0.563

for Pinching, κ = 0.5 Pinching, κ = 0.5 Pinching, κ = 0.5 Pinching, κ = 0.5 Near-Fault δy = 0.217 cm δy = 0.54 cm δy = 1.356 cm δy = 3.39 cm Ground Motions ∆(cm) = 13.852 ∆(cm) = 5.373 ∆(cm) = 33.539 ∆(cm) = 22.867 Indiv. Mean Cum. Indiv. Mean Cum. Indiv. Mean Cum. Indiv. Mean Cum. Indiv. Mean Cum.

1 1.5000 0.2500 1.5000 1.5025 0.2487 1.5025 1.4301 0.2850 1.4301 1.5506 0.2247 1.5506 1.5826 0.2087 1.58262 1.2000 0.0000 2.7000 0.8148 0.1750 2.3173 1.2421 0.1460 2.6722 0.8703 0.2746 2.4209 1.5253 0.2133 3.10793 0.8000 0.2000 3.5000 0.5118 0.2586 2.8291 0.8538 0.0805 3.5260 0.8209 0.2786 3.2418 1.2230 0.3272 4.33094 0.7000 0.2500 4.2000 0.3463 0.4322 3.1754 0.5800 0.2601 4.1060 0.4404 0.1631 3.6823 0.5249 0.0761 4.85585 0.4000 0.0000 4.6000 0.2046 0.1202 3.3800 0.2031 0.0328 4.3090 0.1692 0.0777 3.8514 0.2036 0.2507 5.05946 0.2000 0.0000 4.8000 0.0536 0.0580 3.4335 0.1459 0.0114 4.4549 0.0637 0.0140 3.91517 0.1500 0.0000 4.9500 0.0479 0.0677 3.4814 0.1325 0.0084 4.58758 0.1500 0.0000 5.1000 0.0422 0.0035 3.5236 0.1314 0.0100 4.71899 0.1500 0.0000 5.2500 0.0398 0.0025 3.5634 0.1298 0.0076 4.8486

10 0.1500 0.0000 5.4000 0.0354 0.0064 3.5987 0.1248 0.0062 4.973511 0.1500 0.025 5.5500 0.0329 0.0043 3.6317 0.1213 0.0030 5.094712 0.0283 0.0052 3.6600 0.1133 0.0224 5.208113 0.1058 0.0161 5.3139

51

Table C.8 Force Level Crossings, Ordinary Ground Motions

Ordinary Ground Motions

Force Level Crossings

T=0.2 sec T=0.2 sec T = 0.2 sec T = 0.2 sec T=0.35 sec T=0.5 sec T = 0.5 sec T= 1 sec. T=0.2 sec T =0.5 sec.R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.225 R = 5, η = 0.133 R = 2 η = 0.563 R = 2 η = 0.563Pinching, κ = 0.25 Pinching, κ = 0.5 Bilinear Clough Pinching, κ = 0.5 Pinching, κ = 0.25 Pinching, κ = 0.5 Pinching, κ = 0.5 Pinching, κ = 0.5 Pinching, κ = 0.5δy = 0.224 cm δy = 0.224 cm δ y = 0.224 cm δy = 0.224 cm δy = 0.685 cm δy =1.398 cm δy =1.398 cm δy =3.315 cm δy =0.559 cm δ y =3.495 cm

Parameters Median Median Median Median Median Median Median Median Median Median84th %

Force Crossing Level

Fy, All Excursions 17.6651 28.2455 14.5220 24.2720 24.3874 37.8199 15.6370 26.3267 8.5156 13.7688 7.0307 11.1204 6.1029 9.9820 5.5612 8.3059 3.2028 7.4203 3.4917 5.9682

0.75 Fy, All Excursions 36.0195 57.1176 29.8533 49.0564 77.7687 109.9158 34.6122 51.5338 17.1184 25.7797 14.9642 20.2414 14.4447 19.5294 14.2566 17.6337 12.0585 20.1018 7.7999 13.5894

0.5 Fy, All Excursions 65.3928 107.1982 61.9408 99.3345 124.5473168.8001 69.9239 103.6946 30.3530 46.4274 29.1282 35.6277 27.9592 35.8609 25.0723 29.9839 27.9644 43.9822 16.5766 26.6768

Fy, Pre-Peak 12.6662 20.0316 10.3307 17.4075 13.5003 25.8229 10.3874 19.1581 7.1373 11.2449 6.4877 10.3007 5.5805 9.5525 4.7981 7.5222 2.9570 6.6538 3.3659 5.6841

0.75 Fy, Pre-Peak 25.0585 35.2123 19.0224 29.1881 39.2511 68.5141 20.7458 35.2024 13.6775 20.4987 13.5160 18.6340 12.5691 17.1704 12.6887 15.8898 10.0752 18.3978 6.1297 12.0577

0.5 Fy, Pre-Peak 39.5661 54.0881 36.3288 50.3499 61.3282 97.4049 39.1124 59.8572 22.1120 32.4648 25.1333 31.6636 23.9980 31.4879 21.9450 26.6746 20.9093 37.4133 12.1066 21.5700

84th % 84th % 84th % 84th % 84th % 84th % 84th % 84th % 84th %

52

USGS/NEHRP Response Spectra Soil Type D

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1

T (sec)

Ag

LA 10/50

LA 2/50

MCE NEHRP

LA design NEHRP

Figure C.1 Target Spectra for Scaling of Ground Motion Records

ELASTIC STRENGTH DEMAND SPECTRASet of 20 Unscaled Records ξ=5%

0

1

2

3

0 0.5 1 1.5 2

Period (sec)

Fy

/ W

Figure C.2 Spectra of Unscaled Ordinary records

53

ELASTIC STRENGTH DEMAND SPECTRA Set of 20 records ξ=5% Scaled to USGS LA10/50 Sa(0.5)

0

1

2

3

0 0.5 1 1.5 2

Period (sec)

Fy

/ W

USGS values-LA 10/50

Figure C.3 Spectra for Ordinary Records, Scaled to 10/50 Sa at T = 0.5 sec.

Ground Motion Time History -Northridge-Canoga Park (NOR3) scaled to design Sa at T=0.5 s., with small events

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 20 40 60 80 100

Time (sec)

Gro

und

acce

lera

tion

(g)

Figure C.4 Acceleration Time History of Record Train

54

Normalized Elastic SDOF Displacement Time HistoryNR94rrs, α = 3%, ξ = 2%

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

2.1 2.35 2.6 2.85 3.1 3.35 3.6 3.85 4.1 4.35 4.6

t (sec)

u /

um

ax

T = 0.25 sec

T = 0.375 sec

T = 0.50 sec

T = 1.00 sec

T = 2.00 sec

(a) Elastic Response

Inelastic SDOF Ductility Time HistoryNR94rrs, µ = 6, α = 3%, ξ = 2%

-6

-4

-2

0

2

4

6

2.1 2.35 2.6 2.85 3.1 3.35 3.6 3.85 4.1 4.35 4.6

t (sec)

Du

ctili

ty, µ

T = 0.25 sec

T = 0.375 sec

T = 0.50 sec

T = 1.00 sec

T = 2.00 sec

(b) Response of Inelastic Systems with µ = 6

Figure C.5 Near-Fault Response Histories for SDOF Systems; Northridge Rinaldi Receiving Station

55

Elastic SDOF Strength DemandsNear-Fault Ground Motions, ξ = 5%

0

1

2

3

4

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2T (sec)

Sa

(g)

LP89lex

NR94rrs

NR94newh

KB95kobj

KB95tato

MH84cyld

median

NEHRP MCE LA

Elastic SDOF Velocity DemandsNear-Fault Ground Motions, ξ = 5%

0

100

200

300

400

500

600

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2T (sec)

Sv (c

m /

sec)

LP89lexNR94rrsNR94newhKB95kobjKB95tatoMH84cyld median

Elastic SDOF Displacement DemandsNear-Fault Ground Motions, ξ = 5%

0

40

80

120

160

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2T (sec)

Sd

(cm

)

LP89lexNR94rrsNR94newhKB95kobjKB95tatoMH84cyld median

Figure C.6 Acceleration, Velocity, and Displacement Spectra of Fault-Normal Component of Selected Near-Fault Records

56

F

Fm

κFm

δδr(1-κ)δr

Fig. C.7 Pinching Hysteresis Model

-40

-30

-20

-10

0

10

20

30

40

-80 -60 -40 -20 0 20 40 60 80

Displacement (mm)

Lo

ad (

kN)

Plywood Wall – Test Plywood Wall - Simulation

Figure C.8. Experimentally Obtained and Simulated Hysteretic Responses

57

ELASTIC STRENGTH DEMAND SPECTRANew Set of LA Records ξ=5% Normalized at T=0.2 s.(design spect.)

0

1

2

3

4

5

0 0.5 1 1.5 2

Period (sec)

Fy

/ W

ELASTIC STRENGTH DEMAND SPECTRASet of New LA Records ξ=5% Normalized at T=0.5 s. (design spec.)

0

1

2

3

4

5

0 0.5 1 1.5 2

Period (sec)

Fy

/ W

For Period T = 0.2 sec. For Period T = 0.5 sec.

ELASTIC STRENGTH DEMAND SPECTRASet of New LA Records ξ=5% Normalized at T=1.0 s. (design spec.)

0

1

2

3

4

5

0 0.5 1 1.5 2

Period (sec)

Fy / W

For Period T = 1.0 sec.

Figure C.9 Strength Demand (Acceleration) Spectra Scaled to NEHRP’97 Design Values

Strength vs. Ductility,All Exc.; Pinching ModelLA 10/50; ξ=5%;α=8%; P∆=-3%;κ=0.5

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30µ

η

0

1

2

3

4

5

6

7R

T=0.2 s -median

T=0.2 s - 84th

T=0.35 s - median

T=0.35 s -84th

T=0.5 s- median

T=0.5 s - 84th

T=1.0 s. median

T=1.0 s-84th

Figure C.10 Strength – Ductility (η – µ and R – µ) Curves, Pinching with κ = 0.5

58

Strength vs. Displacement,All Exc.; Pinching ModelLA 10/50; ξ=5%;α=8%; P∆=-3%;κ=0.5

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30δ (cm)

ηT=0.2 s -median

T=0.2 s - 84th

T=0.35 s - median

T=0.35 s -84th

T=0.5 s- median

T=0.5 s - 84th

T=1.0 s. median

T=1.0 s-84th

Figure C.11 Strength – Displacement (η – δ) Curves for Different Periods,

Pinching with κ = 0.5

Strength vs. Displacement, All Exc.; T=0.2 s.LA 10/50; ξ=5%;α=8%; P∆=-3%

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10δ (cm)

η

Pinching k=0.25 -medianPinching k=0.25 -84thPinching k=0.5 -medianPinching k=0.5 -84thClough -median

Clough -84th

Bilinear -Median

Bilinear - 84th

R = =1.5

R = 2

R = 3

R = 4R = 5

R = 6

Figure C.12 Strength – Displacement (η – δ) Curves for Different Hysteresis Models

59

Maximum Displacement vs. Period ; Pinching ModelLA 10/50 & N-R; ξ=5%;α=8%; P∆=-3%;κ=0.5

0

5

1015

20

25

30

35

40

45

50

55

60

65

0 0.2 0.4 0.6 0.8 1 1.2T (sec)

δ max

(cm

)

median R = 5 (eta = 0.225)84th - R = 5 (eta = 0.225)median R = 2 (eta = 0.563)84th - R = 2 (eta = 0.563)median - NF, eta=0.22584th - NF, eta=0.225median - NF, eta=0.56384th - NF, eta=0.563

Figure C.13 Variation of Maximum Displacement with Period, R = 5 and 2

Maximum Ductility vs. Period ; Pinching ModelLA 10/50 & NR; ξ=5%;α=8%; P∆=-3%;κ=0.5

0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1 1.2

T (sec)

δ max

/ δy

median R = 5 (eta = 0.225)

84th - R = 5 (eta = 0.225)

median R = 2 (eta = 0.563)

84th - R = 2 (eta = 0.563)

median - NF, eta = 0.225

84th - NF, eta = 0.225

median - NF, eta = 0.563

84th - NF, eta = 0.563

Figure C.14 Variation of Maximum Ductility with Period, R = 5 and 2

Normalized Hysteretic Energy vs. Period ; Pinching ModelLA 10/50; ξ=5%;α=8%; P∆=-3%;κ=0.5

0

50

100

150

200

250

0 0.2 0.4 0.6 0.8 1 1.2T (sec)

ΝΗ

Ε

median R=5

84th - R=5

median R=2

84th - R=2

Figure C.15 Variation of Normalized Hysteretic Energy Dissipation with Period, R = 5 and 2

60

Str. vs Disp.(counted) - All Exc.; Pinching ModelNear-Fault; ζ=5%, α=8%, P∆=-3%, κ=0.5

0

0.5

1

1.5

2

2.5

0 10 20 30 40δ(cm)

η

Median T=0.2s

84th T=0.2s

Median T=0.5s

84th T=0.5s

Figure C.16 Strength – Displacement (η – δ) Curves for Near-Fault Responses

Str. vs Duct.(counted) - All Exc.; Pinching ModelNear-Fault; ζ=5%, α=8%, P∆=-3%, κ=0.5

0

0.5

1

1.5

2

2.5

0 25 50 75 100µ

η

Median T=0.2s

84th T=0.2s

Median T=0.5s

84th T=0.5s

Figure C.17 Strength – Ductility (η – µ) Curves for Near-Fault Responses

61

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

Def

orm

atio

n

δy

0

Figure C.18 Arbitrary Deformation History

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

F

δ

Figure C.19 Hysteresis Response for Deformation History of Fig. C.18

62

Maximum Displacement Range vs. Period ; Pinching ModelLA 10/50; ξ=5%;α=8%; P∆=-3%;κ=0.5

0

5

10

15

20

25

30

35

40

45

0 0.2 0.4 0.6 0.8 1 1.2T (sec)

∆δm

ax (

cm)

median R=5

84th - R=5

median R=2

84th - R=2

Figure C.20 Variation of Max. Displacement Range with Period, Pinching Model, κ = 0.5

Maximum Cyclic Ductility vs. Period ; Pinching ModelLA 10/50; ξ=5%;α=8%; P∆=-3%;κ=0.5

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2T (sec)

∆δm

ax/ δ

y

median R=5

84th - R=5

median R=2

84th - R=2

Figure C.21 Variation of Cyclic Ductility with Period, Pinching Model, κ = 0.5

63

Number of excursions >δy vs. Period, Pre-peak;Pinching ModelLA 10/50; ξ=5%;α=8%; P∆=-3%;κ=0.5

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1 1.2T (sec)

Νmedian R=5

84th - R=5

median R=2

84th - R=2

Figure C.22 Variation of Number of Pre-Peak Excursions > δy with Period, Pinchg Model, κ = 0.5

Number of Primary Excursions vs. Period, Pre-PeakPinching Model;LA 10/50; ξ=5%;α=8%; P∆=-3%;κ=0.5

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2T (sec)

Ν

median R=5

84th - R=5

median R=2

84th - R=2

Figure C.23 Variation of Number of Primary Excursions with Period, Pinching Model, κ = 0.5

64

Normal. Cum. Excursion Range>δy vs. Period, Pre-PeakPinching Model;LA 10/50; ξ=5%;α=8%; P∆=-3%;κ=0.5

0

50

100

150

200

250

300

350

400

450

0 0.2 0.4 0.6 0.8 1 1.2T (sec)

Σ∆δ i

/δy

median R=5

84th - R=5

median R=2

84th - R=2

Figure C.24 Variation of Cumulative Pre-Peak Excursion Ranges with Period,

Pinching Model, κ = 0.5

Norm. Cum. Primary Excursion Ranges vs. PeriodPinching Model;LA 10/50; ξ=5%;α=8%; P∆=-3%;κ=0.5

0

20

40

60

80

100

120

140

160

0 0.2 0.4 0.6 0.8 1 1.2T (sec)

Σ∆δ p

i/δy

median R=5

84th - R=5

median R=2

84th - R=2

Figure C.25 Variation of Cumulative Primary Excursion Ranges with Period,

Pinching Model, κ = 0.5

65

84th % Centered Def. Ranges, All Exc. T=0.5s, Pinching M. LA 10/50 ;R=5 (η=0.22) ; δy=1.398cm; ξ=5%; α=8%; P-∆= -3% ; κ=0.25

-15

-10

-5

0

5

10

15

∆δi/ δ

y

Figure C.26 84th Percentile Values of Ordered Deformation Ranges, All Excursions, Ordinary Records, T = 0.5 sec., Pinching Model, κ = 0.5

84th % Centered Def. Ranges Pre-Peak Ex. T=0.5s, Pinching M. LA 10/50;R=5 (η=0.22) ;δy=1.398cm;ξ=5%;α=8%;P∆=-3%;κ=0.25

-15

-10

-5

0

5

10

15

∆δi/ δ

y

Figure C.27 84th Percentile Values of Ordered Deformation Ranges, Pre-Peak Excursions, Ordinary Records, T = 0.5 sec., Pinching Model, κ = 0.5

66

Resp. Time History-All Exc.,NR94newh;T=0.5 s Pinching M.Nr-Flt, R=2 (η=0.563); δy=3.39cm; ζ=5%, α=8%, P∆=-3%, κ=0.5

-20

-15

-10

-5

0

5

10

15

20

25

0 20 40 60 80 100 120Time (sec)

Dis

pla

cem

ent

(cm

)

Response Time History-All Exc.,NR94rrs;T=0.5 s Pinching M.Nr-Flt, R=2 (η=0.563); δy=3.39cm; ζ=5%, α=8%, P∆=-3%, κ=0.5

-20

-10

0

10

20

30

40

0 20 40 60 80 100 120Time (sec)

Dis

pla

cem

ent

(cm

)

Response Time History-All Exc.,KB95kobj;T=0.5 s Pinching M.Nr-Flt, R=2 (η=0.563); δy=3.39cm; ζ=5%, α=8%, P∆=-3%, κ=0.5

-50

-40

-30

-20

-10

0

10

20

30

0 20 40 60 80 100 120Time (sec)

Dis

pla

cem

ent

(cm

)

Figure C.28 Response Time Histories for Pinching System (T = 0.5, η = 0.563, κ = 0.5) Subjected to Three Near-Fault Records

67

Force-Def. Response -All Exc.,NR94newh;T=0.5 s Pinching M.Nr-Flt, R=2 (η=0.563); δy=3.39cm; ζ=5%, α=8%, P∆=-3%, κ=0.5

-800

-600

-400

-200

0

200

400

600

800

-20 -10 0 10 20 30Displacement (cm)

Fo

rce

(N)

Figure C.29 Typical Force-Displacement Response to Near-Fault Time History

68

Median Def. Ranges (counted), All Exc. T=0.5s, Pinching M. Near-Fault (η=0.225) ; δy=1.356cm; ξ=5%; α=8%; P∆= -3%; κ=0.5

-40

-30

-20

-10

0

10

20

30

40

∆δi/ δ

y

Figure C.30 Median Values of Ordered Deformation Ranges, All Excursions,

Near-Fault Records, T = 0.5 sec., η = 0.225, Pinching Model, κ = 0.5

Median Def. Ranges(counted), Pre-Peak. T=0.5s, Pinching M. Near-Fault (η=0.225) ; δy=1.356cm; ξ=5%; α=8%; P∆= -3%; κ=0.5

-40

-30

-20

-10

0

10

20

30

40

∆δi/ δ

y

Figure C.31 Median Values of Ordered Deformation Ranges, Pre-Peak Excursions,

Near-Fault Records, T = 0.5 sec., η = 0.225, Pinching Model, κ = 0.5

69

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-1.5 -1 -0.5 0 0.5 1 1.5

Displacement

Lo

ad

Figure C.32 Simulation of Basic Loading History with Nondeteriorating Pinched Hysteresis

Model

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-1 -0.5 0 0.5 1 1.5

Displacement

Lo

ad

Figure C.33 Simulation of Near-Fault Loading History with Nondeteriorating Pinched

Hysteresis Model

70

Force Resp. Time Hist.;all Exc, NOR3; T=0.5 s.Pinching Mod. LA 10/50;R=5(η=0.225);δy=1.398cm;ξ=5%;α=8%;P∆=-3%;κ=0.25

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 20 40 60 80 100

Time(sec)

F/F

y

(a) All Excursions

Force Resp. Time Hist.;PrePeak Ex, NOR3; T=0.5 s.Pinch. M. LA 10/50;R=5(η=0.225);δy=1.398cm;ξ=5%;α=8%;P∆=-3%;κ=0.25

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 20 40 60 80 100

Time(sec)

F/F

y

(b) Pre-Peak Excursions

Figure C.34 Representative Force History for an Ordinary Record

71

0.1 0.2 0.3 0.4 0.7 1.0

1.0

0.8

0.6

0.4

0.2

DISPLACEMENT

FO

RC

E

Figure C.35 Estimation of Force Amplitudes from Deformation Amplitudes of Basic Loading History

72

ELASTIC STRENGTH DEMAND SPECTRANR94cnp (Two Horizontal Components) ξ = 5%

0

1

2

3

0 0.5 1 1.5 2

Period (sec)

Fy

/ W

Smaller Component

Larger Component

LA NEHRP Design Spectrum

(a) Unscaled Record

ELASTIC STRENGTH DEMAND SPECTRANR94cnp (Scale Factor = 1.3, Two Horizontal Components) ξ = 5%

0

1

2

3

0 0.5 1 1.5 2

Period (sec)

Fy

/ W

Smaller Component

Larger Component

LA NEHRP Design Spectrum

(b) Scaled by Factor of 1.3

Figure C.36 Recommended Choice for Ordinary Record, Northridge 94, Canoga Park Record

73

ELASTIC STRENGTH DEMAND SPECTRALP89hda (Two Horizontal Components) ξ = 5%

0

1

2

3

0 0.5 1 1.5 2

Period (sec)

Fy

/ W

Larger Component

Smaller Component

LA NEHRP Design Spectrum

(a) Unscaled Record

ELASTIC STRENGTH DEMAND SPECTRALP89hda (Scale Factor = 1.3, Two Horizontal Components) ξ = 5%

0

1

2

3

0 0.5 1 1.5 2

Period (sec)

Fy

/ W

Larger Component

Smaller Component

LA NEHRP Design Spectrum

(b) Scaled by factor of 1.3

Figure C.37 Recommended Alternate Choice for Ordinary Record, Loma Prieta 89, Hollister

Differential Array Record

74

Elastic SDOF Strength DemandsNR94rrs, ξ = 5%

0

0.5

1

1.5

2

2.5

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2T (sec)

Sa

(g)

Fault-Normal

Fault-Parallel

Elastic SDOF Velocity DemandsNR94rrs, ξ = 5%

0

100

200

300

400

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2T (sec)

Sv (c

m/s

ec)

Fault-Normal

Fault-Parallel

Elastic SDOF Displacement DemandsNR94rrs, ξ = 5%

0

20

40

60

80

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2T (sec)

Sd

(cm

)

Fault-Normal

Fault-Parallel

Figure C.38 Recommended Choice for Near-Fault Record, Northridge 94, Rinaldi Receiving Station Record

75

Elastic SDOF Strength DemandsKB95tato, ξ = 5%

0

0.5

1

1.5

2

2.5

3

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2T (sec)

Sa

(g)

Fault-Normal

Fault-Parallel

Elastic SDOF Velocity Demands

KB95tato, ξ = 5%

0

100

200

300

400

500

600

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2T (sec)

Sv (c

m/s

ec)

Fault-Normal

Fault-Parallel

Elastic SDOF Displacement DemandsKB95tato, ξ = 5%

0

40

80

120

160

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2T (sec)

Sd

(cm

)

Fault-Normal

Fault-Parallel

Figure C.39 Recommended Alternate Choice for Near-Fault Record, Kobe 95, Takatori Station

76

Appendix A – Representative Results for Response to Ordinary Ground Motion

Case: T = 0.5 sec., Pinching Model with κ = 0.25, R = 5 (η = 0.225)

Def. Resp. Time Hist., all Exc. ,NOR3 ;T=0.5s ; Pinching Mod.LA 10/50;R=5(η=0.22); δy=1.398cm; ξ=5%; α=8%; P∆=−3%; κ=0.25

-20

-15

-10

-5

0

5

10

15

20

0 20 40 60 80 100

Time (sec)

δ i/ δ

y

Def. Resp.Time Hist,Pre-Peak Exc.,NOR3;T=0.5 s;Pinch. M.LA 10/50;R=5(η=0.22);δy=1.398cm;ξ=5%;α=8%;P∆=−3%;κ=0.25

-20

-15

-10

-5

0

5

10

15

20

0 20 40 60 80 100

Time (sec)

δ i/ δ

y

77

Force-Def. Response;All Exc., NOR3 ;T=0.5 s. Pinching Mod. LA 10/50; R=5 (η=0.22); δy=1.398cm; ξ=5%; α=8%; P∆=-3%; κ=0.25

-1.5

-1

-0.5

0

0.5

1

1.5

-10 -5 0 5 10

δi/δy

Fi/F

y

Force-Def. Response;PrePeak., NOR3 ;T=0.5 s. Pinching Mod. LA 10/50; R=5 (η=0.22); δy=1.398cm; ξ=5%; α=8%; P∆=-3%;κ=0.25

-1.5

-1

-0.5

0

0.5

1

1.5

-10 -5 0 5 10

δi/δy

Fi/F

y

78

Rainflow Excursions, all Exc., NOR3; T=0.5 s;Pinching ModelLA 10/50;R=5 (η=0.22);δy=1.398cm;ξ=5%;α=8%;P∆=-3%;κ=0.25

-20

-15

-10

-5

0

5

10

15

20

Dis

plac

emen

t (c

m)

Rainflow Excursions, Pre-Peak Ex., NOR3; T=0.5 s;Pinching M.LA 10/50;R=5 (η=0.22);δy=1.398cm;ξ=5%;α=8%;P∆=-3%;κ=0.25

-20

-15

-10

-5

0

5

10

15

20

Dis

plac

emen

t (c

m)

79

Median Deformation Ranges, All Exc. T=0.5s, Pinching M. LA 10/50; R=5 (η=0.22) ; δy=1.398cm ; ξ=5% ; α=8% ; P∆= -3%; κ=0.25

-15

-10

-5

0

5

10

15

∆δi/ δ

y

Median Deformation Ranges Pre-Peak Exc. T=0.5s, Pinching M. LA 10/50;R=5 (η=0.22) ;δy=1.398cm;ξ=5%;α=8%;P∆=-3%;κ=0.25

-15

-10

-5

0

5

10

15

∆δi/ δ

y

80

Median Centered Def. Ranges, All Exc.; T=0.5s; M. Pinching LA 10/50 ;R=5 (η=0.22) ; δy=1.398cm; ξ=5%; α=8%; P∆= -3%; κ=0.25

-15

-10

-5

0

5

10

15

∆δi/ δ

y

Median Centered Def. Ranges Pre-Peak Ex. T=0.5s, M. Pinching LA 10/50 ;R=5 (η=0.22) ;δy=1.398cm ; ξ=5% ; α=8%; P-∆= -3%; κ=0.25

-15

-10

-5

0

5

10

15

∆δi/ δ

y

81

84th % Centered Def. Ranges, All Exc. T=0.5s, Pinching M. LA 10/50 ;R=5 (η=0.22) ; δy=1.398cm; ξ=5%; α=8%; P-∆= -3% ; κ=0.25

-15

-10

-5

0

5

10

15

∆δi/ δ

y

84th % Centered Def. Ranges Pre-Peak Ex. T=0.5s, Pinching M. LA 10/50;R=5 (η=0.22) ;δy=1.398cm;ξ=5%;α=8%;P∆=-3%;κ=0.25

-15

-10

-5

0

5

10

15

∆δi/ δ

y

82

CDF of Deform. Ranges, all Exc. T=0.5s Pinching ModelLA 10/50;R=5 (η=0.22);δy=1.398cm;ξ=5%;α=8%;P∆=-3%;κ=0.25

0.00.10.20.30.40.50.60.70.80.91.0

0 5 10 15 20 25

∆δ/δy

CD

F Real Data

CDF of Deform. Ranges, Pre-Peak Ex. T=0.5s Pinching ModelLA 10/50;R=5 (η=0.22);δy=1.398cm;ξ=5%;α=8%;P∆=-3%;κ=0.25

0.00.10.20.30.40.50.60.70.80.91.0

0 5 10 15 20 25

∆δ/δy

CD

F Real Data

83

Pinching Model -T=0.5 s. -All Exc.LA 10/50; κ=0.25; P-∆=-3%; α=8%; R=5 (η=0.22); δy=1.398 cmNormalized Deformation Ranges

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 203.988 5.253 5.681 6.269 6.360 7.812 9.104 9.732 9.793 9.954 11.231 12.266 14.289 14.510 14.716 15.272 15.375 16.093 23.621 26.2773.617 5.248 5.539 5.578 5.639 7.069 8.533 8.554 9.298 9.795 10.530 12.004 13.488 13.929 14.468 14.792 15.062 15.198 23.140 24.3473.513 4.887 5.017 5.239 5.392 6.853 7.981 8.096 8.353 8.477 9.367 11.268 12.289 13.368 13.591 13.822 14.207 15.131 18.150 23.6733.124 4.554 4.700 5.017 5.146 5.380 6.373 7.042 7.906 8.034 8.915 9.691 11.138 11.859 12.396 13.643 14.088 14.207 18.150 19.5312.994 3.978 4.541 4.987 5.146 5.365 6.373 7.042 7.145 7.584 7.810 8.227 9.395 9.676 10.457 13.594 13.643 14.180 15.258 18.2262.994 3.922 4.294 4.876 4.968 4.992 5.010 5.034 6.920 7.056 7.466 7.810 7.876 8.170 10.457 10.637 13.542 13.593 14.635 17.4272.758 2.991 3.927 4.294 4.625 4.792 4.992 5.010 5.470 6.920 7.056 7.069 7.408 8.170 9.760 10.637 10.769 12.783 13.542 15.8972.684 2.991 3.711 3.927 4.221 4.515 4.625 4.673 4.835 6.063 6.115 6.261 6.681 6.711 8.883 9.232 10.376 10.659 12.962 15.8972.684 2.755 3.498 3.647 3.711 4.221 4.409 4.522 4.835 5.062 6.115 6.261 6.360 6.681 8.094 8.173 8.642 9.292 12.962 15.4932.489 2.580 2.720 3.317 3.465 4.121 4.128 4.374 4.550 4.810 5.062 5.891 5.972 6.300 6.636 6.681 8.094 8.549 10.734 15.4932.148 2.580 2.720 3.317 3.465 3.956 3.984 4.030 4.197 4.333 4.482 4.810 5.160 5.972 6.249 6.636 7.355 7.771 10.734 13.6112.148 2.485 2.649 2.702 3.247 3.321 3.984 3.995 4.029 4.333 4.482 4.735 5.122 5.149 5.377 6.358 6.609 7.771 7.938 12.6852.137 2.410 2.649 2.702 2.755 2.833 3.321 3.916 3.946 3.995 4.202 4.635 4.684 4.735 5.377 6.078 6.218 6.331 7.234 11.8652.081 2.137 2.489 2.597 2.619 2.755 3.265 3.348 3.379 3.470 3.969 4.202 4.203 4.635 4.684 4.926 5.732 6.218 6.477 10.2871.996 2.081 2.421 2.580 2.597 2.643 2.951 3.265 3.348 3.379 3.470 3.683 3.954 3.969 4.203 4.425 5.732 5.739 6.086 10.2871.996 2.016 2.350 2.537 2.580 2.643 2.830 2.951 3.095 3.172 3.242 3.612 3.683 3.954 3.961 4.117 4.523 5.316 5.384 9.1651.973 2.016 2.067 2.485 2.489 2.537 2.626 2.830 2.837 3.036 3.172 3.242 3.336 3.401 3.930 3.961 4.523 5.169 5.180 7.9971.870 1.974 2.014 2.067 2.170 2.489 2.626 2.655 2.738 2.837 2.949 3.027 3.155 3.379 3.598 3.859 4.129 5.169 5.180 7.9971.782 1.870 1.947 1.974 2.014 2.170 2.474 2.586 2.655 2.699 2.949 3.027 3.155 3.379 3.433 3.454 3.672 4.464 4.878 7.6361.782 1.844 1.847 1.911 1.947 2.059 2.067 2.485 2.580 2.580 2.582 2.699 2.753 2.909 3.433 3.454 3.672 4.067 4.464 7.5461.775 1.784 1.844 1.893 1.911 2.059 2.067 2.363 2.508 2.580 2.580 2.580 2.580 2.582 2.909 3.036 3.170 3.234 4.067 7.5461.606 1.772 1.775 1.782 1.807 1.835 1.996 2.363 2.485 2.487 2.489 2.508 2.580 2.580 2.582 2.747 2.902 3.170 3.234 7.1551.606 1.677 1.760 1.782 1.807 1.835 1.996 2.093 2.211 2.264 2.485 2.485 2.485 2.510 2.582 2.628 2.867 2.902 3.039 7.1551.577 1.585 1.677 1.713 1.720 1.746 1.780 1.782 2.065 2.093 2.211 2.233 2.308 2.510 2.570 2.628 2.747 2.867 3.021 6.4231.555 1.563 1.603 1.649 1.720 1.746 1.780 1.782 1.977 2.021 2.042 2.065 2.308 2.474 2.474 2.485 2.570 2.778 2.944 5.0641.555 1.563 1.583 1.585 1.603 1.603 1.613 1.710 1.867 1.977 2.021 2.042 2.108 2.156 2.352 2.361 2.489 2.582 2.937 4.8861.362 1.424 1.561 1.563 1.585 1.585 1.603 1.613 1.780 1.780 1.827 1.867 1.995 2.015 2.108 2.156 2.289 2.582 2.937 4.8861.362 1.424 1.561 1.563 1.563 1.565 1.583 1.585 1.765 1.780 1.780 1.827 1.872 1.995 2.015 2.059 2.289 2.554 2.574 4.4571.304 1.367 1.367 1.536 1.563 1.563 1.565 1.569 1.583 1.600 1.607 1.678 1.780 1.841 1.872 2.059 2.187 2.554 2.574 4.4571.304 1.367 1.367 1.367 1.536 1.561 1.563 1.563 1.569 1.600 1.601 1.678 1.780 1.834 1.841 1.885 2.187 2.474 2.489 4.2941.236 1.238 1.238 1.367 1.367 1.524 1.561 1.561 1.563 1.583 1.583 1.583 1.596 1.819 1.834 1.885 1.997 2.113 2.250 4.2431.236 1.238 1.238 1.238 1.367 1.367 1.496 1.524 1.561 1.561 1.575 1.576 1.596 1.600 1.606 1.819 1.997 2.113 2.250 3.8781.205 1.209 1.209 1.238 1.238 1.365 1.365 1.367 1.496 1.561 1.575 1.576 1.583 1.600 1.606 1.778 1.976 1.990 2.080 3.8781.205 1.209 1.209 1.214 1.238 1.238 1.365 1.365 1.365 1.365 1.561 1.561 1.561 1.577 1.583 1.778 1.787 1.976 2.080 3.4011.085 1.148 1.153 1.209 1.209 1.238 1.239 1.350 1.365 1.365 1.555 1.561 1.561 1.561 1.561 1.693 1.787 1.877 2.012 3.4011.085 1.148 1.153 1.209 1.209 1.209 1.239 1.239 1.239 1.350 1.365 1.365 1.459 1.555 1.561 1.586 1.693 1.877 2.012 3.1781.065 1.071 1.071 1.096 1.170 1.208 1.209 1.239 1.239 1.343 1.365 1.365 1.402 1.459 1.461 1.585 1.586 1.781 1.915 3.178

1.065 1.071 1.071 1.104 1.170 1.208 1.208 1.208 1.242 1.340 1.363 1.402 1.408 1.461 1.577 1.584 1.781 1.831 3.1751.058 1.070 1.071 1.104 1.144 1.208 1.208 1.242 1.340 1.362 1.363 1.408 1.416 1.555 1.584 1.706 1.831 3.1021.058 1.069 1.069 1.070 1.071 1.071 1.207 1.239 1.239 1.313 1.362 1.365 1.416 1.555 1.563 1.706 1.782 3.1021.007 1.028 1.046 1.069 1.069 1.071 1.207 1.236 1.239 1.239 1.313 1.365 1.365 1.543 1.563 1.577 1.782 2.9261.007 1.028 1.039 1.043 1.046 1.066 1.171 1.208 1.209 1.236 1.239 1.339 1.365 1.513 1.543 1.576 1.758 2.926

1.039 1.043 1.066 1.086 1.208 1.209 1.226 1.239 1.239 1.339 1.418 1.513 1.576 1.758 2.7471.007 1.026 1.086 1.128 1.205 1.208 1.215 1.239 1.239 1.418 1.475 1.555 1.707 2.675

1.026 1.069 1.128 1.205 1.208 1.208 1.215 1.239 1.390 1.475 1.555 1.707 2.6751.022 1.069 1.069 1.069 1.202 1.208 1.208 1.208 1.386 1.390 1.493 1.585 2.474

1.022 1.069 1.069 1.136 1.202 1.208 1.208 1.362 1.362 1.386 1.563 2.378 Pinching Model -T=0.5 s. -All Exc.LA 10/50; k=0.25; P-D=-3%; a=8%; R=5 (h=0.22); dy=1.398 cm

Number of excursionscm1 cm2 lan1 lan2 lp1 lp2 lp3 lp4 lp5 lp6 nor2 nor3 nor4 nor5 nor6 nor9 nor10 sup1 sup2 sup3 Median 84% th

>6dy 15 6 24 14 11 1 1 5 10 8 3 9 9 13 9 13 5 8.500 13.000>2 dy 29 21 56 32 36 18 21 14 18 23 26 24 17 30 26 27 25 28 26 19 25.495 34.576>dy 58 42 85 56 65 46 45 38 42 42 47 47 44 54 52 52 46 51 59 47 47.000 56.480

Sum of Normalized Deformation Ranges

cm1 cm2 lan1 lan2 lp1 lp2 lp3 lp4 lp5 lp6 nor2 nor3 nor4 nor5 nor6 nor9 nor10 sup1 sup2 sup3 Median 84% th>6dy 173.595 62.499 319.617 147.419 130.085 6.360 6.269 40.413 150.355 81.990 28.226 70.819 68.686 128.282 77.130 118.281 34.481 69.753 147.419>2 dy 221.236 117.305 420.698 206.394 207.753 63.953 78.385 48.985 87.537 196.278 140.832 103.433 46.608 142.932 124.000 180.829 137.207 155.712 104.030 69.240 130.436 226.658>dy 259.092 143.887 461.996 242.378 250.364 102.809 111.316 85.509 120.671 221.598 170.650 134.590 86.253 178.787 159.414 216.148 166.131 187.449 153.302 109.936 162.738 247.305

84

Pinching Model Modified -T=0.5 s. - Pre-Peak Exc.LA 10/50; κ=0.25; P-∆=-3%; α=8%; R=5 (η=0.22); δy=1.398 cmNormalized Deformation Ranges

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 203.988 5.253 5.681 6.269 6.360 7.812 9.104 9.732 9.793 9.954 11.231 12.266 14.289 14.510 14.716 15.272 15.375 16.093 23.621 26.2773.513 3.978 5.017 5.248 5.380 5.891 7.069 7.981 9.298 9.367 9.795 11.138 12.004 12.289 13.929 14.468 14.792 15.131 23.140 24.3472.755 2.758 3.265 4.374 5.017 5.146 5.239 5.365 5.370 5.377 5.470 8.353 9.037 11.859 13.591 13.594 13.822 14.207 18.150 23.6732.489 2.580 3.265 4.030 4.095 4.197 4.541 4.550 4.968 5.146 5.377 7.145 7.906 10.457 12.396 12.783 13.643 14.207 18.150 19.5312.137 2.580 2.582 2.755 2.833 2.951 3.612 3.711 4.084 4.294 4.876 6.920 7.584 8.883 9.676 10.457 10.769 13.542 13.643 18.2262.137 2.485 2.582 2.597 2.755 2.949 2.951 3.401 3.711 3.927 4.625 4.635 6.920 7.056 7.408 8.173 9.760 10.637 13.542 17.4271.782 1.911 2.489 2.570 2.597 2.626 2.702 2.753 2.949 3.927 4.625 4.635 5.062 6.300 6.711 7.056 9.573 10.637 12.962 15.8971.782 1.911 2.170 2.485 2.537 2.570 2.580 2.626 2.649 2.702 3.036 3.465 3.598 5.062 5.657 6.681 8.094 8.642 12.962 15.8971.585 1.807 2.065 2.170 2.474 2.489 2.537 2.580 2.580 2.649 2.778 3.027 3.242 4.333 4.810 6.681 7.771 7.938 8.094 15.4931.563 1.807 2.059 2.065 2.067 2.317 2.485 2.489 2.489 2.580 2.582 2.747 3.027 3.242 4.333 4.810 6.636 7.355 7.771 15.4931.563 1.600 1.720 1.780 1.782 1.841 2.059 2.067 2.067 2.485 2.580 2.582 2.582 2.628 3.916 4.203 6.609 6.636 7.234 12.6851.424 1.600 1.603 1.606 1.720 1.780 1.782 1.841 2.014 2.067 2.554 2.580 2.582 2.628 2.837 4.203 6.078 6.358 6.477 10.2871.424 1.583 1.585 1.596 1.603 1.603 1.606 1.661 1.819 2.014 2.474 2.485 2.489 2.554 2.837 3.379 4.117 4.523 5.732 10.2871.367 1.561 1.563 1.577 1.583 1.585 1.596 1.603 1.782 1.819 1.844 1.977 2.059 2.489 2.699 3.379 3.859 4.523 5.732 9.1651.367 1.555 1.561 1.561 1.563 1.563 1.583 1.585 1.693 1.782 1.844 1.977 2.012 2.059 2.699 2.830 3.672 4.129 5.180 7.9971.238 1.365 1.367 1.555 1.561 1.561 1.563 1.563 1.585 1.693 1.780 1.835 1.885 2.012 2.580 2.830 3.433 3.672 5.180 7.9971.238 1.362 1.365 1.367 1.367 1.524 1.561 1.563 1.563 1.585 1.780 1.782 1.835 1.885 2.580 2.655 3.036 3.433 4.464 7.6361.209 1.238 1.239 1.362 1.365 1.367 1.367 1.524 1.563 1.583 1.584 1.606 1.780 1.782 2.485 2.655 2.747 3.234 4.464 7.5461.209 1.236 1.238 1.238 1.239 1.365 1.365 1.367 1.367 1.561 1.584 1.606 1.758 1.780 1.780 2.474 2.485 2.580 3.234 7.5461.071 1.208 1.209 1.236 1.238 1.238 1.239 1.365 1.367 1.561 1.563 1.577 1.613 1.758 1.780 2.289 2.363 2.580 3.021 6.4231.071 1.205 1.208 1.209 1.209 1.238 1.238 1.239 1.239 1.496 1.555 1.563 1.583 1.613 1.707 2.289 2.363 2.485 2.574 5.064

1.069 1.153 1.205 1.208 1.209 1.209 1.238 1.239 1.496 1.513 1.555 1.576 1.583 1.707 1.997 2.211 2.308 2.574 4.8861.065 1.069 1.071 1.153 1.208 1.208 1.209 1.209 1.362 1.365 1.513 1.561 1.576 1.585 1.997 2.211 2.308 2.474 4.886

1.065 1.069 1.071 1.071 1.071 1.208 1.209 1.362 1.365 1.386 1.561 1.561 1.563 1.787 1.976 2.021 2.108 4.4571.069 1.069 1.071 1.071 1.170 1.236 1.239 1.365 1.386 1.561 1.563 1.787 1.976 2.021 2.108 4.457

1.061 1.069 1.170 1.236 1.239 1.365 1.365 1.367 1.448 1.586 1.872 1.877 2.015 3.8781.061 1.071 1.205 1.208 1.350 1.365 1.367 1.448 1.586 1.872 1.877 2.015 3.878

1.071 1.205 1.208 1.239 1.251 1.350 1.372 1.577 1.678 1.781 1.834 3.4011.069 1.202 1.239 1.242 1.251 1.372 1.555 1.678 1.781 1.834 3.4011.069 1.202 1.208 1.238 1.242 1.367 1.555 1.577 1.583 1.600 3.175

1.197 1.208 1.238 1.239 1.367 1.418 1.575 1.576 1.600 3.1021.069 1.197 1.209 1.238 1.239 1.418 1.575 1.576 1.583 3.1021.069 1.098 1.209 1.209 1.238 1.362 1.555 1.561 1.561 2.926

Pinching Model -T=0.5 s. -Pre-Peak Exc.LA 10/50; k=0.25; P-D=-3%; a=8%; R=5 (h=0.22); dy=1.398 cm

Number of excursionscm1 cm2 lan1 lan2 lp1 lp2 lp3 lp4 lp5 lp6 nor2 nor3 nor4 nor5 nor6 nor9 nor10 sup1 sup2 sup3 Median 84% th

>6dy 9 2 20 12 7 1 1 2 5 5 2 7 1 11 2 12 2 2.000 11.000>2 dy 15 11 44 23 16 12 11 6 10 13 17 12 6 20 9 24 9 27 9 12 12.000 19.799>dy 36 24 63 44 39 27 24 21 25 28 32 29 25 39 24 49 22 50 34 37 30.463 41.450

Sum of Normalized deformation ranges

cm1 cm2 lan1 lan2 lp1 lp2 lp3 lp4 lp5 lp6 nor2 nor3 nor4 nor5 nor6 nor9 nor10 sup1 sup2 sup3 Median 84% th>6dy 119.424 24.270 279.830 128.494 80.957 6.360 6.269 17.774 93.830 47.497 19.749 56.993 9.104 105.562 20.597 111.950 14.882 22.433 111.950>2 dy 134.019 52.037 357.928 165.646 104.117 44.348 37.913 19.764 41.690 119.842 88.924 58.297 17.906 103.175 36.349 152.793 44.122 149.381 34.205 40.924 55.078 119.939>dy 162.419 69.671 385.514 196.725 136.734 64.155 55.598 40.656 62.041 140.225 110.104 83.272 46.301 131.174 57.570 188.112 61.791 181.117 70.857 77.139 80.147 144.508

85

Normalized Deformation Ranges from Primary Excursions for each History; T=0.5 s.; Pinching Model

LA 10/50; R=5 (η=0.22), δy=1.398 cm; ξ=5%; α=8%; P∆=-3%; κ=0.25

cm1 cm2 lan1 lan2 lp1 lp2 lp3 lp4 lp5 lp6 nor2 nor3 nor4 nor5 nor6 nor9 nor10 sup1 sup2 sup31 15.275 12.268 26.280 14.291 16.095 5.253 6.361 6.270 9.795 23.624 15.377 9.956 3.988 9.734 9.105 14.512 11.232 14.718 7.813 5.6822 14.794 12.006 24.350 13.931 15.133 5.248 4.969 3.979 7.982 23.143 11.139 9.796 3.514 9.299 5.892 12.291 9.368 14.470 7.070 5.3803 7.939 5.471 23.676 13.824 13.596 5.240 2.833 2.755 4.375 10.770 7.146 4.877 2.759 8.355 4.198 11.861 3.402 13.593 5.3664 2.748 4.550 19.534 8.643 12.785 4.542 2.755 4.030 3.037 3.917 7.908 3.613 9.761 2.754 12.3985 18.229 7.235 8.884 4.294 2.755 7.585 7.356 9.6776 17.430 6.478 8.174 7.409 6.610 6.3597 12.687 4.129 6.301 6.712 6.0798 9.166 3.021 3.599 4.1189 7.637 2.779 3.860

10 6.424 3.03711 5.065 2.74812 3.17513 2.74814151617181920

n 4 4 13 8 9 5 4 3 5 4 4 3 3 11 4 7 4 6 2 3Sum norm 40.755 34.294 176.401 71.553 87.346 24.578 16.919 13.004 28.937 60.575 37.578 24.629 10.261 70.763 22.807 68.470 26.756 71.214 14.884 16.428Sum 56.984 47.950 246.644 100.045 122.127 34.365 23.656 18.182 40.460 84.696 52.542 34.436 14.347 98.941 31.889 95.735 37.410 99.572 20.810 22.970

Mean-ln Median STDlog 84% thn 1.549 4.709 0.482 7.622 n = number of primary excursions in response time historySum norm 3.552 34.889 0.747 73.636 Sum Norm = sum of normalized primary excursion ranges Sum 3.887 48.782 0.747 102.958 Sum = sum of primary excursion ranges in a response time history

Deformation Ranges of Primary Excursions; T=0.5 s.; Pinching Model

LA 10/50; R=5 (η=0.22), δy=1.398 cm; ξ=5%; α=8%; P∆=-3%; κ=0.25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201 5.576 7.344 7.943 8.766 8.893 10.923 12.729 13.608 13.693 13.918 15.703 17.150 19.979 20.288 20.575 21.354 21.497 22.501 33.027 36.7402 4.913 5.562 6.947 7.337 7.522 8.237 9.884 11.159 13.000 13.097 13.695 15.573 16.784 17.183 19.475 20.229 20.682 21.156 32.355 34.0423 3.852 3.857 3.961 4.756 5.869 6.116 6.818 7.325 7.502 7.648 9.990 11.098 11.680 15.057 16.582 19.002 19.008 19.325 33.1004 3.841 3.850 3.852 4.245 5.051 5.476 5.634 6.350 6.361 11.055 12.083 13.646 17.332 17.874 27.3085 3.852 6.004 10.115 10.284 10.603 12.420 13.528 25.4846 8.890 9.057 9.240 10.358 11.427 24.3677 5.773 8.498 8.809 9.384 17.7378 4.224 5.031 5.756 12.8149 3.885 5.396 10.676

10 4.245 8.98011 3.841 7.08112 4.43913 3.841141516171819

Normalized Deformation Ranges of Primary Excursions; T=0.5 s.; Pinching Model

LA 10/50; R=5 (η=0.22), δy=1.398 cm; ξ=5%; α=8%; P∆=-3%; κ=0.25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201 3.988 5.253 5.681 6.269 6.360 7.812 9.104 9.732 9.793 9.954 11.231 12.266 14.289 14.510 14.716 15.272 15.375 16.092 23.621 26.2772 3.513 3.978 4.969 5.248 5.380 5.891 7.069 7.981 9.298 9.367 9.795 11.138 12.004 12.289 13.929 14.468 14.792 15.131 23.140 24.3473 2.755 2.758 2.833 3.401 4.197 4.374 4.876 5.239 5.365 5.470 7.145 7.938 8.353 10.769 11.859 13.591 13.594 13.822 23.6734 2.747 2.753 2.755 3.036 3.612 3.916 4.030 4.541 4.550 7.906 8.642 9.760 12.396 12.783 19.5315 2.755 4.294 7.234 7.355 7.584 8.883 9.676 18.2266 6.358 6.477 6.609 7.408 8.173 17.4277 4.129 6.078 6.300 6.711 12.6858 3.021 3.598 4.117 9.1659 2.778 3.859 7.636

10 3.036 6.42311 2.747 5.06412 3.17513 2.747141516171819

86

Deformation Ranges for RainFlow Cycle Counting in Pinching Model

LA 10/50; R=5 (η=0.22), δy=1.398 cm; ξ=5%; α=8%; P∆=-3%; κ=0.25

Median and 84th of Deformation Range Median and 84th of Deformation Range All Excursions Pre-Peak Excursions

Mean-ln Median σ ln 84thMedian 84th

Mean-ln Median σ ln 84thMedian 84th

(cm) (cm) Normaliz Normaliz (cm) (cm) Normaliz Normaliz2.695 14.809 0.499 24.404 10.591 17.454 2.695 14.809 0.499 24.404 10.591 17.4542.639 13.995 0.513 23.384 10.009 16.724 2.565 13.003 0.549 22.519 9.3 16.1062.564 12.99 0.503 21.483 9.291 15.365 2.32 10.175 0.636 19.217 7.277 13.7442.472 11.845 0.511 19.748 8.472 14.124 2.226 9.263 0.643 17.629 6.625 12.6082.397 10.989 0.49 17.93 7.859 12.824 2.04 7.691 0.668 15.002 5.5 10.7292.312 10.092 0.487 16.419 7.218 11.743 1.926 6.862 0.627 12.851 4.908 9.1912.227 9.273 0.496 15.23 6.632 10.893 1.825 6.202 0.651 11.894 4.436 8.5062.138 8.48 0.484 13.766 6.065 9.846 1.692 5.429 0.641 10.303 3.883 7.3692.073 7.951 0.479 12.837 5.687 9.181 1.604 4.971 0.609 9.135 3.555 6.5331.978 7.231 0.478 11.658 5.172 8.338 1.52 4.571 0.588 8.234 3.269 5.8891.908 6.741 0.465 10.732 4.821 7.676 1.396 4.04 0.595 7.321 2.889 5.2361.838 6.285 0.442 9.778 4.495 6.993 1.317 3.732 0.573 6.616 2.669 4.732

1.768 5.862 0.429 9.005 4.193 6.44 1.23 3.42 0.525 5.78 2.446 4.134

1.679 5.362 0.409 8.072 3.835 5.773 1.168 3.215 0.519 5.4 2.299 3.862

1.622 5.061 0.402 7.568 3.62 5.413 1.123 3.073 0.479 4.963 2.198 3.551.561 4.763 0.367 6.873 3.407 4.916 1.076 2.932 0.49 4.787 2.097 3.4241.505 4.505 0.356 6.429 3.222 4.598 1.029 2.798 0.471 4.48 2.001 3.2041.454 4.281 0.373 6.217 3.062 4.446 0.979 2.661 0.483 4.312 1.903 3.0841.400 4.055 0.366 5.847 2.9 4.182 0.902 2.466 0.447 3.856 1.763 2.758

Deformation Ranges of Primary Excursions, T=0.5 s.; Pinching Model

LA 10/50; R=5 (η=0.22), δy=1.398 cm; ξ=5%; α=8%; P∆=-3%; κ=0.25

Median and 84th of Deformation Ranges Median and 84th of Deformation Ranges Counted Computed

Median 84thMedian 84th

Mean-ln Median σ ln 84thMedian 84th

(cm) (cm) Normaliz Normaliz (cm) (cm) Normaliz Normaliz1 14.811 21.497 10.593 15.375 1 2.695 14.809 0.499 24.404 10.591 17.4542 13.396 20.682 9.581 14.792 2 2.565 12.997 0.550 22.521 9.296 16.1073 7.575 19.002 5.418 13.5914 5.263 13.646 3.764 9.7605 10.603 7.584

6 9.240 6.609

7 8.498 6.0788 4.224 3.0219101112131415161718

19

87

Force Statistics (level crossings); All exc. T=0.5 s Pinching Model LA 10/50;R=5(η=0.22);δy=1.398cm;ξ=5%;α=8%; P-∆=-3%; κ=0.25;Fy= 220.79 (kg-cm/sec2/kg)

Force AmplitudeNumber of Peaks

cm1 cm2 lan1 lan2 lp1 lp2 lp3 lp4 lp5 lp6 nor2 nor3 nor4 nor5 nor6 nor9 nor10 sup1 sup2 sup3 Median 84% thFy Pos 4 3 10 5 4 3 2 1 1 3 5 4 1 7 3 4 2 3 3 4 3.061 5.628

Neg 7 3 9 7 6 3 4 2 3 6 4 3 2 7 3 6 3 6 2 2 3.923 6.440Max 11 6 19 12 10 6 6 3 4 9 9 7 3 14 6 10 5 9 5 6 7.160 11.649

0.75 fy Pos 10 6 16 10 8 6 5 3 6 6 9 7 4 9 8 9 5 9 6 7 6.987 10.116Neg 9 7 15 12 10 7 6 4 6 8 8 6 5 13 9 9 7 12 8 8 8.030 11.156Max 19 13 31 22 18 13 11 7 12 14 17 13 9 22 17 18 12 21 14 15 15.062 21.160

0.5 fy Pos 19 12 27 17 16 10 12 9 10 10 14 13 12 17 13 17 13 18 16 14 13.949 18.226Neg 15 12 24 21 21 14 13 10 14 15 16 17 15 20 15 17 12 17 18 17 15.805 19.575Max 34 24 51 38 37 24 25 19 24 25 30 30 27 37 28 34 25 35 34 31 29.852 37.422

Force ValueMax Absolute294.60 311.47 370.17 305.21 313.55 242.08 240.87 242.64 276.60 360.57 305.98 276.99 229.61 267.51 264.24 312.08 284.47 298.55 265.67 249.73 283.307 322.733Force αFy 1.33 1.41 1.68 1.38 1.42 1.10 1.09 1.10 1.25 1.63 1.39 1.25 1.04 1.21 1.20 1.41 1.29 1.35 1.20 1.13 1.283 1.462

Force Statistics (level crossings); Pre-Peak. T=0.5 s Pinching Model LA 10/50;R=5(h=0.22);dy=1.398cm;x=5%;a=8%; P-D=-3%; k=0.25;Fy= 220.79 (kg-cm/sec2/kg)

Force AmplitudeNumber of Peaks

cm1 cm2 lan1 lan2 lp1 lp2 lp3 lp4 lp5 lp6 nor2 nor3 nor4 nor5 nor6 nor9 nor10 sup1 sup2 sup3 Median 84% thFy Pos 4 2 9 5 4 3 2 1 1 3 4 4 1 7 3 4 2 3 3 3 2.909 5.266

Neg 7 3 9 7 5 3 4 2 3 5 4 3 2 7 3 6 3 6 2 2 3.852 6.239Max 11 5 18 12 9 6 6 3 4 8 8 7 3 14 6 10 5 9 5 5 6.893 11.169

0.75 fy Pos 8 4 13 10 7 6 5 3 6 6 8 7 4 9 5 9 4 9 6 6 6.342 9.144Neg 9 7 15 12 8 7 6 4 6 7 8 6 5 13 6 9 7 12 7 7 7.627 10.619Max 17 11 28 22 15 13 11 7 12 13 16 13 9 22 11 18 11 21 13 13 14.018 19.603

0.5 fy Pos 13 8 20 17 15 10 9 8 10 10 12 13 10 15 9 17 8 18 10 11 11.662 15.584Neg 15 12 24 18 13 12 12 9 11 12 15 13 13 20 12 16 12 17 17 15 14.032 17.652Max 28 20 44 35 28 22 21 17 21 22 27 26 23 35 21 33 20 35 27 26 25.784 32.930

Force ValueMax Abs 294.60 311.47 370.17 305.21 313.55 242.08 240.87 242.64 276.60 360.57 305.98 276.99 229.61 267.51 264.24 312.08 284.47 298.55 265.67 249.73 283.307 322.733Force αΦψ 1.33 1.41 1.68 1.38 1.42 1.10 1.09 1.10 1.25 1.63 1.39 1.25 1.04 1.21 1.20 1.41 1.29 1.35 1.20 1.13 1.283 1.462Amplit