ANALYTICAL MODELS FOR THE NONLINEAR SEISMIC RESPONSE OF REINFORCED CONCRETE FRAMES. A Thesis in. Architectural Engineering. Michael W.

Transcription

1 The Pennsylvania State University The Graduate School College of Engineering ANALYTICAL MODELS FOR THE NONLINEAR SEISMIC RESPONSE OF REINFORCED CONCRETE FRAMES A Thesis in Architectural Engineering by Michael W. Hopper 2009 Michael W. Hopper Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science December 2009

2 The thesis of Michael W. Hopper was reviewed and approved* by the following: Andres Lepage Assistant Professor of Architectural Engineering Thesis Advisor Ali Memari Associate Professor of Architectural Engineering M. Kevin Parfitt Associate Professor of Architectural Engineering Chimay Anumba Professor of Architectural Engineering Head of the Department of Architectural Engineering *Signatures are on file in the Graduate School.

3 iii ABSTRACT This thesis aims to identify the optimal combination of hysteresis-modeling and damping parameters implemented in nonlinear dynamic analysis to obtain satisfactory correlation between calculated and measured seismic response of reinforced concrete frames. A total of five parameters are included in this investigation: initial stiffness, bond-slip rotations, post-yield stiffness, unloading stiffness, and viscous damping. These parameters directly influence the seismic response of individual frame members. In this study, frame elements are modeled using lumped plasticity by means of an elastic middle portion bounded by nonlinear springs that connect each end of the member to a rigid segment representing the beam-column joints. Three small-scale shake-table multistory test structures and two orthogonal structural systems from an existing seven-story building with recorded seismic responses are used in this study. Each test structure was analyzed using three computer programs for two separate base accelerations and the analytical responses were compared to the measured responses using the Frequency Domain Error (FDE) index. The existing building was analyzed for a single recorded seismic event. After analyzing each structure with all possible combinations of modeling parameters, the optimal combinations of parameters leading to the best correlations between the calculated and measured response were identified. Simplified rules are given to derive the modeling parameters that give consistent low values of FDE for the various structures and structural analysis programs considered.

4 iv TABLE OF CONTENTS LIST OF TABLES... vi LIST OF FIGURES... viii ACKNOWLEDGMENTS... xii CHAPTER 1 INTRODUCTION Statement of the Problem Objectives and Scope Organization... 3 CHAPTER 2 ANALYTICAL TOOLS USED TO PERFORM NONLINEAR DYNAMIC ANALYSIS LARZ SAP PERFORM 3D CHAPTER 3 NONLINEAR MODELING PARAMETERS Initial Stiffness Bond-Slip Effect Post-Yield Stiffness Unloading Stiffness Viscous Damping CHAPTER 4 MULTISTORY TEST STRUCTURES SUBJECTED TO STRONG GROUND MOTIONS Test Structures MF1 and MF Test Structure FNW CHAPTER 5 SEVEN-STORY HOLIDAY INN BUILDING IN VAN NUYS, CALIFORNIA Building Description Earthquake Damage Modeling Assumptions CHAPTER 6 DISCUSSION OF RESULTS... 28

5 v 6.1 Calculated Responses for Test Structures MF1, MF2, and FNW Calculated Responses for the Holiday Inn Building CHAPTER 7 SUMMARY AND CONCLUSIONS APPENDIX FREQUENCY DOMAIN ERROR INDEX A.1 Combinations of Parameter Values A.2 FDE Clocks TABLES FIGURES LIST OF REFERENCES BIOGRAPHICAL SKETCH

6 vi LIST OF TABLES Table Viscous Damping Parameter, α and β Values Table Assumed Material Properties for Specimens MF1, MF2, and FNW Table Assumed Member Properties, Test Structure MF Table Assumed Member Properties, Test Structure MF Table Assumed Member Properties, Test Structure FNW Table Calculated First-Mode Periods of Vibration Table Yield Point Data, Test Structure MF Table Yield Point Data, Test Structure MF Table Yield Point Data, Test Structure FNW Table Specified Material Properties for the Holiday Inn Building Table Holiday Inn Building Assumed Beam/Slab Reinforcement Table Holiday Inn Building Column Reinforcement Table Column Bar Arrangements, Holiday Inn Building Table Yield Point Data, Structure HNS Table Yield Point Data, Structure HEW Table Number of Nonlinear Analysis Cases Considered, Test Structures Table Number of Nonlinear Analysis Cases Considered, Holiday Inn Building Table FDE Index Averages for Specimens MF1, MF2, and FNW, Using LARZ Table FDE Index Averages for Specimens MF1, MF2, and FNW, Using SAP

7 vii Table FDE Index Averages for Specimens MF1, MF2, and FNW, Using PERFORM 3D Table FDE Index Averages for Specimens MF1, MF2, and FNW, Using LARZ, SAP 2000, and PERFORM 3D Table FDE Index Averages for HEW and HNS, Using LARZ Table Summary of Best Models for Test Structures MF1, MF2, and FNW Table A.1 - Summary of Parameters Considered... 52

8 viii LIST OF FIGURES Figure Idealized Reinforced Concrete Frame Figure One-Component Model Figure Moment-Curvature Relationship Figure Moment-Rotation Relationship Including Bond-Slip Rotation Figure Flowchart of Program LARZ, After Saiidi and Sozen (1979) Figure Bilinear Moment-Rotation Relationship in SAP Figure Assumed Moment and Curvature Diagrams for Frame Members Figure Frame Compound Components Used in PERFORM 3D Figure Moment-Rotation Diagram for the Elastic Component in PERFORM 3D Figure Moment-Rotation Relationship of the Inelastic Component in PERFORM 3D Figure Moment-Curvature Relationship for Uncracked Case (U) Figure Moment-Curvature Relationship for Cracked Case (C) Figure Rotation Due to Bond Slip, After Saiidi and Sozen (1979) Figure Moment-Rotation Relationship with a Soft Post-Yield Stiffness (S) Figure Moment-Rotation Relationship with a Hard Post-Yield Stiffness (H) Figure Moment-Rotation Relationship with Non-Reducing (N) Unloading Stiffness Figure Moment-Rotation Relationship with Reducing (R) Unloading Stiffness Figure The University of Illinois Earthquake Simulator, After Lepage (1997).. 65

9 ix Figure Specimen MF1 Tested by Moehle and Sozen (1978) Figure Representative Reinforcement Details, After Moehle and Sozen (1980) Figure Frame Element Property Types, Test Structure MF Figure Frame Element Property Types, Test Structure MF Figure Base Acceleration Records, Test Structures MF1 and MF Figure Specimen FNW Tested by Moehle and Sozen (1980) Figure Frame Element Property Types, Test Structure FNW Figure Base Acceleration Records, Test Structure FNW Figure Plan and Elevations of the Holiday Inn Building, After Lepage (1997).. 74 Figure Exterior Frames, North-South Direction, T-Beam Reinforcement Assumptions Figure Interior Frames, North-South Direction, Slab Reinforcement Assumptions Figure Exterior Frames, East-West Direction, T-Beam Reinforcement Assumptions Figure Interior Frames, East-West Direction, Slab Reinforcement Assumptions Figure Base Acceleration Records, 1994 Holiday Inn Building in Van Nuys, California Figure Frame Element Property Types, Interior Frame, Structure HNS Figure Frame Element Property Types, Exterior Frame, Structure HNS Figure Frame Element Property Types, Interior Frame, Structure HEW Figure Frame Element Property Types, Exterior Frame, Structure HEW Figure FDE Clocks, Test Structure MF1, Run 1 and Run 2, LARZ Figure FDE Clocks, Test Structure MF1, Run 1 and Run 2, SAP Figure FDE Clocks, Test Structure MF1, Run 1 and Run 2, PERFORM 3D... 82

10 x Figure FDE Clocks, Test Structure MF2, Run 1 and Run 2, LARZ Figure FDE Clocks, Test Structure MF2, Run 1 and Run 2, SAP Figure FDE Clocks, Test Structure MF2, Run1 and Run 2, PERFORM 3D Figure FDE Clocks, Test Structure FNW, Run 1 and Run 2, LARZ Figure FDE Clocks, Test Structure FNW, Run 1 and Run 2, SAP Figure FDE Clocks, Test Structure FNW, Run 1 and Run 2, PERFORM 3D Figure Ratios of Calculated-to-Measured Maximum Roof Displacement, Average of Test Structures MF1, MF2, and FNW for Run 1, LARZ Figure Roof Displacement Histories, MF1 Run 1, LARZ Figure Roof Displacement Histories, MF2 Run 1, LARZ Figure Roof Displacement Histories, FNW Run 1, LARZ Figure Ratios of Calculated-to-Measured Maximum Roof Displacement, Average of Test Structures MF1, MF2, and FNW for Run 2, LARZ Figure Roof Displacement Histories, MF1 Run 2, LARZ Figure Roof Displacement Histories, MF2 Run 2, LARZ Figure Roof Displacement Histories, FNW Run 2, LARZ Figure Base Shear Histories, MF1, LARZ Figure Base Shear Histories, MF2, LARZ Figure Base Shear Histories, FNW, LARZ Figure Overturning Moment Histories, MF1, LARZ Figure Overturning Moment Histories, MF2, LARZ Figure Overturning Moment Histories, FNW, LARZ Figure Mean Drift Ratio and Story Drift Ratio Envelopes, MF1 Run 1, Model CHRT - α Figure Mean Drift Ratio and Story Drift Ratio Envelopes, MF1 Run 2, Model CHRL - α

11 xi Figure Mean Drift Ratio and Story Drift Ratio Envelopes, MF2 Run 1, Model CHRT - α Figure Mean Drift Ratio and Story Drift Ratio Envelopes, MF2 Run 2, Model CHRL - α Figure Mean Drift Ratio and Story Drift Ratio Envelopes, FNW Run 1, Model CHRT - α Figure Mean Drift Ratio and Story Drift Ratio Envelopes, FNW Run 2, Model CHRL - α Figure FDE Clocks, Structure HNS and HEW, LARZ Figure FDE Clocks, Structure HNS and HEW, SAP Figure FDE Clocks, Structure HNS and HEW, PERFORM 3D Figure Ratios of Calculated-to-Measured Maximum Roof Displacement, Average of Holiday Inn HNS and HEW, LARZ Figure Roof Displacement Histories, HNS, LARZ Figure Roof Displacement Histories, HEW, LARZ Figure A.1 - FDE Representation Figure A.2 - FDE Clock

12 xii ACKNOWLEDGMENTS The writer is grateful to his advisor, Andres Lepage, for the assistance and guidance provided throughout this project. He is also thankful to Sebastian Delgado for introductory sessions on the nonlinear modeling options of the computer programs used in this study. Appreciation is given to Jeff Dragovich for his support with the Frequency Domain Error program. He is also grateful to Jose Pincheira for providing a legible copy of the original drawings of the Holiday Inn building, located in Van Nuys, California.

13 1 CHAPTER 1 INTRODUCTION The structural engineering community is increasingly using nonlinear static or dynamic analysis to evaluate the response of a structure subjected to seismic events. When designing new buildings, structural engineers refer to ASCE 7-05 (ASCE, 2005) for loads and analysis procedures. Anytime designers choose a non-compliant system to resist lateral loads, sufficient analytical and test data must be submitted to assure the noncompliant system has adequate seismic response comparable to that of a conventional code-compliant system. These deviations from ASCE 7-05 typically trigger a two-step design process. First, a traditional code-level design is performed where engineers use a linear response analysis and apply a response modification factor, R, which is typically defined in relation to traditional lateral force-resisting systems. Second, the response of the building to the more severe 2500-year seismic event is evaluated using nonlinear dynamic analysis. Another case in which nonlinear dynamic analysis is required is in the rehabilitation of existing buildings. ASCE 31 (ASCE, 2003) and ASCE 41 (ASCE, 2006) require nonlinear dynamic analysis in a tier-3 type assessment of a structure. Nonlinear dynamic analysis is also the prevalent method of analysis for structures with damping systems and is mandatory for structures with base isolation. In both of these cases, nonlinear response is usually limited to a reduced number of elements and the definition of their hysteretic response generally follows simple rules. Although a nonlinear dynamic analysis is an excellent way to evaluate the performance of structures subjected to strong ground motions, the modeling complexities involved in characterizing the number and type of material nonlinearities often discourage designers from using this advanced option. This chapter defines the problem statement, followed by the objectives and scope of this study. The chapter ends with a brief description of how this thesis is organized.

14 2 1.1 Statement of the Problem Today, a nonlinear dynamic analysis for any structure is not an easy task. To make an attempt to reduce the effort involved, specialized software must be purchased that focuses on nonlinear analysis. Expensive training sessions must also be scheduled to educate engineers on how to use the specialized software. Even with employees who can efficiently operate a nonlinear analysis program, preparing data for analysis and interpreting output from the analysis can be very time consuming and therefore expensive in wages. This is partially due to the limited guidance offered by design specifications for engineers to use in nonlinear analysis, as well as to the limited guidelines towards modeling methods and assumptions provided by the available computer programs with nonlinear capabilities. This study is in search of practical computer models capable of representing realistic nonlinear seismic response of reinforced concrete frames. Simple rules are needed to identify and characterize the modeling parameters and how to implement them using standard structural analysis software. 1.2 Objectives and Scope The objective of this thesis is to develop a set of rules which consistently lead to realistic response of reinforced concrete moment frames when performing a nonlinear dynamic analysis. This study is limited to mid-rise reinforced concrete frames that are subjected to strong ground motions. Analysis of these reinforced concrete moment frames include material and geometric nonlinearities. The hysteretic response is represented by key modeling parameters and their optimal combination is identified by comparing calculated versus measured response histories of shake-table test structures and an instrumented building. Nonlinear dynamic analyses were performed using three computer programs: LARZ by Saiidi and Sozen (1979), SAP 2000 by CSI (2009), and PERFORM 3D by CSI (2006).

15 3 SAP 2000 and PERFORM 3D are commercially available structural analysis programs used commonly in engineering practice. In this study, reinforced concrete frames are modeled using a lumped plasticity model where nonlinear properties are concentrated in a spring at the beam-column interfaces. Beam-column joints are modeled as rigid segments and the members in the clear span are represented by a linear-elastic portion. Figure 1.1 displays the idealized reinforced concrete frame. Masses are assumed lumped at floor level and beams are assumed axially rigid. The optimal combinations of modeling parameters are identified by comparing the calculated response to experimental data obtained at the University of Illinois Earthquake Simulator (Sozen et al., 1969), as well as from the instrumented Holiday Inn building in Van Nuys, California, with recorded data during the Northridge earthquake in The information presented in this study provides practicing structural engineers with a set of modeling assumptions for performing nonlinear dynamic analysis capable of representing the actual response of reinforced concrete moment frames. This study is an extension of the work initiated by Lepage et al. (2008). The study expands on the number of structures and base motions. The Holiday Inn building computer model incorporates significant revisions in member properties. 1.3 Organization Chapter 2 introduces the three computer programs used in this study to perform the nonlinear dynamic analyses. The chapter includes a summary of the modeling assumptions and the required input information. Chapter 3 presents the five main nonlinear modeling parameters considered in this study. Four of these parameters directly influence the hysteretic response: initial stiffness, bond-slip effects, post-yield stiffness, and reducing stiffness. The fifth parameter, viscous damping, is assumed either mass proportional or stiffness proportional.

16 4 Chapter 4 describes the multistory test structures that were used in this study to correlate the measured and calculated response. Details are presented for their geometry, material properties, and base motions to which they were subjected to. Chapter 5 extends the applicability of the nonlinear models used in Chapter 4 to the instrumented Holiday Inn building in Van Nuys, California. This flat-plate reinforced concrete structure with perimeter moment frames was heavily damaged during the 1994 Northridge earthquake. Since the building response was recorded during this earthquake, it provides the opportunity to check if the modeling assumptions that best represent the small-scale test structures of Chapter 4 are also capable of representing the nonlinear response of an actual full-scale building. Chapter 6 discusses the calculated responses after the nonlinear dynamic analyses were performed in this study. In Chapter 7, the summary and conclusions include simple rules for defining the modeling parameters that consistently led to the best correlations between the measured and calculated nonlinear seismic responses. The Frequency Domain Error (FDE) index is described in an Appendix. With the large number of models and response data, the FDE index is used to help discern the set of parameters that minimizes the error in the calculated response.

17 5 CHAPTER 2 ANALYTICAL TOOLS USED TO PERFORM NONLINEAR DYNAMIC ANALYSIS There are several computer programs available today with nonlinear dynamic analysis capabilities. One academic program, LARZ, and two commercial programs, SAP 2000 and PERFORM 3D, are used in this study to perform nonlinear dynamic analyses of reinforced concrete frames. A brief program description and a simplified method to calculate yield moments and curvatures for LARZ input are introduced in Section 2.1. Section 2.2 describes the input data for SAP 2000 and presents a method to convert moment-curvature data to moment-rotation. Section 2.3 briefly describes PERFORM 3D. Geometric nonlinearities (P-delta effects) are taken into account by all of the programs considered. 2.1 LARZ Program LARZ is a specialized computer program developed for calculating the nonlinear seismic response of reinforced concrete frames. The original program was named SAKE and was written by Otani (1975) at the University of Illinois. The program was renamed LARZ after modifications by Saiidi (1979). Lopez (1988) further modified the program to study frame-wall structures. LARZ incorporates several hysteresis models to characterize the moment-rotation cyclical response of reinforced concrete members. The Takeda hysteresis model, the Sina hysteresis model, the Otani hysteresis model, the Simple Bilinear model, and the Q- hysteresis model all can be invoked by the program. If a nonlinear dynamic analysis for a steel frame is desired, LARZ can be used with the appropriate hysteresis model. In this study, the Takeda hysteresis model (Takeda et al., 1970) is used to represent the cyclic

18 6 response of reinforced concrete frame members. The unloading exponent parameter, γ, was set to 0.6 (as discussed in Chapter 3). Each member of the structure is modeled by the one-component model, originally developed by Giberson (1969) as represented in Figure 2.1. Beams and columns are considered massless line elements with their positions representing their centroidal axes. Each element consists of a linear elastic line that is bounded at each end by a nonlinear spring. Nonlinear properties of these springs account for the inelastic deformations of the structure. Rigid segments represent the beam-column joints. For each cross section, the moment-curvature relationship is specified to account for material nonlinearities. Cracking, yielding, and ultimate moment-curvature values at each end of the member are given to the program. From the moment-curvature data, LARZ calculates the moment-rotation relationship for each member. Rotations due to the slip of reinforcement within the beam-column joint are then added to the momentrotation curve calculated by LARZ. Dimensions of the planar structure are input by defining the joints with coordinates of beam-column intersections. Elements with assigned geometric and material properties were then defined between joints. More than one lateral-translation degree-of-freedom can be applied to each floor level if some beams are discontinued, which allows irregular structures to be analyzed using LARZ. However, this study was limited to one translational degree-of-freedom per level. Figure 2.2 shows a representative moment-curvature relationship that must be given to program LARZ for each element. Figure 2.3 displays the moment-rotation relationship calculated by LARZ from the input moment-curvature data. Bond-slip rotations, which are input by the user, are added to the LARZ-calculated rotation to obtain the final moment-rotation relationship to be used by the program when performing nonlinear dynamic analysis. A simplified procedure may be used to calculate the yield point for each frame element. This method uses interpolation between the balanced point and the purebending point of a P-M interaction diagram when calculating the yield moment and yield curvature for members with axial load. In this study, the method is only applied when

19 modeling the test structures described in Chapter 4. Several assumptions are implicit when using this method: Compressive and tensile steel are identical and in a single layer, Axial loads are below the balanced condition, Yield moment at zero axial load neglects the contribution of the compression steel, Yield moment at the balanced condition assumes yielding of the compression and tension reinforcement, and P-M and P-φ diagrams are assumed to vary linearly between pure bending and the balanced point. To determine the yield moment and curvature, the neutral axis depth,, is obtained using Equation 2.1 for the balanced condition and Equation 2.2 for the case of pure bending: where: 1 ε ε ε (2.1) (2.2) Equation 2.1 is used with Equation 2.3 to calculate the moment at the balanced condition:, 0.85 β 2 β (2.3) The moment at the pure bending condition is calculated using Equation 2.2 with Equation 2.4:, 3 (2.4) The curvature for both the balanced and pure-bending cases is determined using Equation 2.5: 7

20 8 One can then calculate the moment and curvature at yielding for a given member by knowing the axial force in that member and using Equation 2.6 and Equation 2.7: (2.5),,, (2.6),,, (2.7) where: P axial force in member Pb balanced point 0.85 β 0.85 After the frame geometry has been defined through joint coordinates, the member properties are assigned using moment and curvature values. LARZ calculates the elements moment-rotation stiffness matrix and then assembles a condensed structural stiffness matrix. The equation of motion is solved using the constant-averageacceleration method (Newmark, 1959). The instantaneous element and structure stiffness matrices are updated following the chosen hysteresis rules for cyclic loading. This process is repeated throughout the duration of the base motion. See Figure 2.4 for a flowchart diagram of the analysis steps implemented by LARZ. Gravity loads are optional, but definitions of hysteresis models and input base accelerations are required. 2.2 SAP 2000 Another program that was used in this study to perform nonlinear dynamic analyses for planar reinforced concrete frames under seismic loading is SAP 2000 (CSI, 2009). The program SAP was first developed by CSI in 1975 and several versions have been released since then. Today, its 3D graphical interface is very user-friendly and facilitates the input of structural models in a very efficient manner. SAP 2000 is easy to learn and can be used for simple or complex structural analysis and is used by design professionals in over 160 countries around the world.

21 9 The user-friendly interface of SAP 2000, the numerous types of members, and the potential to model arbitrary geometries provide engineers with ample freedom when performing either a nonlinear static or dynamic analysis. Good judgment and experience is essential for obtaining meaningful output, as the results are very sensitive to the assumptions made. One way to perform a nonlinear dynamic analysis using SAP 2000 that leads to acceptable results when compared to experimental structures is to use a lumped plasticity model and force the program to behave similar to proven models in LARZ. A onecomponent model can be simulated using a flexurally rigid element bounded with nonlinear links (NL links) assigned at each member end. These NL links are then attached to rigid ends that simulate beam-column joints. NL links contain all linear and nonlinear flexural properties, given that the central segment is assumed flexurally rigid. Shear deformations may be accounted for by assigning shear area to the central linearelastic segment. Several types of hysteresis models are available. For reinforced concrete members, the Pivot hysteresis and Takeda hysteresis models are available. In this study, the Pivot hysteresis model (Dowell et al., 1998) was used instead of the Takeda hysteresis model because SAP 2000 implements Takeda with a fixed value for the unloading stiffness. The Pivot hysteresis model typically requires input values of three parameters: α pivot, β pivot, and η. For more information regarding these parameters, see Dowell et al., The range of values to represent behavior of reinforced concrete members may include 1 to 10 for α pivot, 0.25 to 0.75 for β pivot, and 0 to 10 for η. In this study, reducing unloading stiffness models (as discussed in Chapter 3) adopted values of α pivot = 1.0, β pivot = 0.3, and η = 10 (as discussed in Chapter 3). In SAP 2000, users must input moment-rotation relationships for each NL link. Two points of the moment-rotation curve are given to the program: yielding and ultimate. The hysteresis models in the current version of SAP 2000 do not allow a breakpoint before the yield point, and therefore the cracking point is not represented. The initial effective stiffness is then specified as the slope of the moment-rotation curve from the origin to the yield point. Bond-slip rotations can be accounted for by directly adding

22 10 them to the moment-rotation curves. See Figure 2.5 for the representative momentrotation curve when using SAP This is also known as a bilinear backbone curve. Although there are several methods to calculate the moment-rotation properties for a given cross section, a method similar to what LARZ implements was used in this study. A general expression was developed to calculate cracking, yielding, and ultimate rotations from known moments and curvatures. Figure 2.6 displays the momentcurvature diagram assumed for each frame member. The slope of line 1 of the curvature diagram in Figure 2.6 can be defined as: where: 2 λ (2.8) The slope of line 2 in Figure 2.6 can be defined as: where: 2 λ (2.9) Line 3 in Figure 2.6 can be defined using: A general formula for rotation is: 1 1 λ 2 λ (2.10) where: θ τ / 2 (2.11) τ /, tangential deviation from A measured at B, A the left support of the beam, and B the midspan of the beam. Integrating the curvature to obtain rotation, per Equation 2.11, gives:

23 11 θ 2λ 2 At cracking, Equation 2.12 may be used with: λ λ 1 resulting in Equation 2.13: At yielding, Equation 2.12 is used with: λ 1 resulting in Equation 2.14: At ultimate, Equation 2.12 is used with: 1 λ 2 λ 1 λ (2.12) θ 6 (2.13) θ 6 λ 1 λ (2.14) Bond-slip rotations, described in Chapter 3, are then added to the rotations calculated using Equation 2.12 through Equation PERFORM 3D PERFORM 3D (CSI, 2006) is the third program considered in this study for performing nonlinear dynamic analysis of reinforced concrete frames subjected to strong base motions. Also developed by CSI, PERFORM 3D is strictly a nonlinear analysis software program that is used for performance assessment of 3D structures subjected to seismic events. Structural models can be imported directly from SAP 2000, and therefore it is likely that engineers would migrate to PERFORM 3D, a more specialized program

24 12 for evaluation of buildings using performance-based design principles. Future versions of the program are likely to be integrated into SAP There are several options and parameters available with PERFORM 3D. A useful option for this study was to invoke the use of hysteresis loops with unloading and reloading stiffness reductions tied to a user-defined energy ratio in relation to energy dissipated by elastoplastic systems. Structural members are defined by means of compound components, which are shown in Figure 2.7. Elastic properties are assigned to a central portion and inelastic properties are assigned to rigid hinges at member ends. This information is given to PERFORM 3D by assigning cross sectional properties to the elastic component and by defining the basic post-yield force-displacement relationship for the inelastic component. The elastic component is typically defined by width, depth, modulus of elasticity, and Poisson's ratio. The program automatically calculates the cross-sectional properties from this information. In this study, the moment of inertia was modified so that the moment-rotation relationships account for rotations due to bond slip. Figure 2.8 displays the moment-rotation relationship implemented in PERFORM 3D for the elastic component. This figure shows the yield point as a function of the modified moment of inertia to include bond-slip rotations. If the gross section moment of inertia is used, the program would ignore the softening due to bond-slip effects. The modified moment of inertia,, is calculated using: 6 θ θ (2.15) where: member clear length, θ yield rotation, θ yield rotation due to bond-slip of the reinforcement (see Chapter 3), E modulus of elasticity of concrete, and moment at yielding. The inelastic component uses a rigid-plastic hinge. To specify the momentrotation relationship for the rigid-plastic hinge, three points (Y, U, and L) can be input as

25 13 displayed in Figure 2.9. These three points are the first-yield point (Y); the ultimate strength point (U) where the maximum strength is reached; and the ductile limit point (L) where strength loss begins. PERFORM 3D uses energy dissipation ratios for hysteresis loop rules. Users have the option to define the loop shape, as well as energy dissipation factors at different values of ductility. To represent elastoplastic behavior, users may input an energy dissipation ratio of 1.0. To represent systems with stiffness-reducing hysteresis, typical of reinforced concrete members, energy ratio values range between 0.1 and 0.4 (Otani, 1981). In this study, an energy ratio of 0.2 was adopted for models with unloading stiffness reductions (as discussed in Chapter 3).

26 14 CHAPTER 3 NONLINEAR MODELING PARAMETERS Due to the competitive environment in the business of consulting engineering, structural engineers rarely have the time to test their modeling techniques and compare the results with measured data. Nonlinear dynamic analysis can be complicated and time consuming and therefore, recommendations for input values on key modeling parameters are investigated in this study, with the intention of helping engineers create meaningful models and computer output. To give simple modeling recommendations for nonlinear seismic analysis of reinforced concrete frames, the key modeling parameters must be identified. Four parameters directly influence the primary backbone moment-rotation curve: initial stiffness, bond-slip deformations, post-yield stiffness, and unloading stiffness. Two relatively extreme values were chosen for each of these parameters to guide engineers toward the modeling assumptions leading to realistic response. A fifth parameter, viscous damping, also plays a role in predicting the response of structures subjected to strong ground motions. Four values of viscous damping are investigated. Each parameter is described in detail in Sections 3.1 through 3.5. The test structures described in Chapter 4 are modeled with all possible combinations of the modeling parameters listed in this chapter and are implemented with the three computer programs described in Chapter Initial Stiffness In this study, the initial slope of the moment-curvature diagrams is determined using cracked (C) or uncracked (U) section properties. For uncracked cases, the initial stiffness is based on gross section properties and the cracking point is based on the

27 15 modulus of rupture, taken as ½ [MPa]. For cases in which the cross section is cracked, the initial stiffness is defined using the secant stiffness to the yield point. Both of these cases share the same yield point. Concrete stress-strain relationships are based on Hognestad (1951) and steel reinforcement is taken as elastoplastic. Figure 3.1 and Figure 3.2 display the moment-curvature relationships for the uncracked and cracked section properties. 3.2 Bond-Slip Effect This parameter accounts for the additional rotation due to bond-slip effects inside the beam-column joints. Moment-rotation curves can be derived from moment-curvature relationships with the assumption that inflection points are at midspan. Once the primary moment-rotation relationship is determined, additional rotation due to bond-slip is added. Bond-slip rotation is a softening effect occurring inside the beam-column joint due to the elongation of the longitudinal reinforcement beyond the column face. In this study, bond-slip rotation is determined as a function of the development length, λdb: where: θ bond-slip rotation, curvature at yielding, θ 1 2 λ λ reinforcement bar diameter lengths, diameter of reinforcement bar, M moment at point of interest, yield moment, yield strength of reinforcement, and bond stress. (3.1) Equation 3.1 contains the following simplifying assumptions: the reinforcement is fully developed and pullout will not occur; the steel stress varies linearly from a

28 16 maximum at the beam-column interface to zero inside the joint; the rotation due to bond slip is measured with respect to the neutral axis; and the tensile stress in the reinforcement is proportional to the moment. These assumptions are shown in Figure 3.3. This study accounts for two cases of bond-slip rotation: a case of tight (T) bond where λ is taken as 20 bar diameter lengths, and a case of loose (L) bond where λ is taken as 40 bar diameter lengths. The values of λ closely correspond to practical values of f y and u. For instance, a case where f y = 410 MPa (60 ksi) and u = 5.1 MPa (0.75 ksi), leads to λ = 20, and a case where f y = 410 MPa (60 ksi) and u = 2.6 MPa (0.38 ksi), leads to λ = 40. The rotations due to bond slip may also be considered as a partial correction for the rigid joint assumption, especially for the L cases, where bond stress is assumed the lowest. 3.3 Post-Yield Stiffness The slope of the moment-rotation relationship after yielding of the reinforcement is known as the post-yield stiffness, K p. For this study, K p is expressed as a fraction of the secant-to-yield moment-rotation stiffness, K e. Two cases are considered: a soft case (S) in which K p = 0.02K e and a hard case (H) where K p = 0.10K e. To ensure these postyield stiffnesses are achieved, the ultimate point of the moment-rotation relationship is modified accordingly after incorporating bond-slip effects. Figure 3.4 displays the soft case and Figure 3.5 displays the hard case. Instead of determining the actual post-yield stiffness for a known cross section and material stress-strain relationships, the approach studied is to assign two extreme values and later identify the one leading to the best correlation with measured data. This approach gives an engineer the option of creating a model that mostly depends on the yield point.

29 Unloading Stiffness Past nonlinear analysis studies have been successful when using hysteresis rules based on Takeda (Saiidi and Sozen, 1979; Takeda at al., 1970). Therefore in this study, the unloading stiffness,, of the moment-rotation hysteresis model is controlled by the Takeda exponent parameter, γ: where: θ yield rotation θ maximum rotation θ θ Two values of γ are considered. A value of 0 represents a non-reducing (N) (3.2) unloading slope and a value of 0.6 represents a reducing (R) case. Figure 3.6 and Figure 3.7 display these cases. Equivalent cases were studied in SAP 2000 and PERFORM 3D, although these programs use different hysteresis models. Both N and R cases were reproduced by modifying the Pivot hysteresis input values in SAP 2000 and by modifying the energy dissipation factors in PERFORM 3D to behave similarly to a Takeda hysteresis model with γ equal to 0 and 0.6. In this study, the Pivot hysteresis model of SAP 2000 represents the N case using α pivot = 10, β pivot = 0.5, and η = 0. The R cases are represented with α pivot = 1.0, β pivot = 0.3, and η = 10. For the energy-ratio hysteresis of PERFORM 3D, this study assigns the N models a ratio of 0.5 and for the R models a ratio of 0.2 is adopted. 3.5 Viscous Damping Four individual cases of viscous damping are considered in this study. Two cases use mass-proportional damping at 2 and 5 percent of critical damping (α 2, α 5 ) and two cases use stiffness-proportional damping at 2 and 5 percent of critical damping (β 2, β 5 ).

30 The damping matrix,, is defined as a linear combination of the mass matrix,, and the stiffness matrix,, both defined at global structural degrees-of-freedom: and where: α β (3.3) ξ 1 2ω α βω (3.4) ξ the fraction of critical damping for mode of frequency ω. Values of α and β are determined based on the first mode of vibration corresponding to uncracked section properties. For the case of mass-proportional damping, use Equation 3.5: α 2 2π and β 0 (3.5) For the case of stiffness-proportional damping, use Equation 3.6: where: ξ 0.02 or 0.05 and β π and α 0 (3.6) the first-mode period of vibration. Assumed values of α and β for the test structures presented in Chapter 4 and the case study presented in Chapter 5 are shown in Table

31 19 CHAPTER 4 MULTISTORY TEST STRUCTURES SUBJECTED TO STRONG GROUND MOTIONS This chapter describes the test structures considered in this study to help identify the combinations of modeling parameters that best represent measured seismic response. Three small-scale experimental structures previously tested will be modeled using the nonlinear modeling parameters described in Chapter 3 and implemented with the three computer programs presented in Chapter 2. Results of the runs are first reported through the use of the Frequency Domain Error index (Appendix) and followed by a more detailed report (Chapter 6) of the global and local structural response of the best models. Section 4.1 provides the geometry, material properties, reinforcement data, and base acceleration information for test structures MF1 (Healey, 1978) and MF2 (Moehle, 1978). Section 4.2 provides similar information for test structure FNW (Moehle, 1980). 4.1 Test Structures MF1 and MF2 Healey (1978) tested a ten story, three-bay frame named MF1 and Moehle (1978) tested a similar structure named MF2. Each test structure had two frames in parallel and was subjected to unidirectional base motions. These tests were performed at the University of Illinois Earthquake Simulator. This facility was designed to test smallscale structures subjected to base motion in one horizontal direction (see Figure 4.1) and is described in detail by Sozen et al. (1969). Both structures have a first and tenth story height of 279 mm (11 in.), while all other levels are 229 mm (9 in.). Frame MF2 was identical to frame MF1, shown in Figure 4.2, except that the first level had a discontinued beam in one exterior bay.

32 20 Each story had a 4.45 kn (1000 lbs) mass attached between the planar frames so that their center-of-mass coincided with the center-of-mass of the frames. Gravity loads were evenly distributed directly to the columns, and therefore all plastic hinges formed in the vicinity of the beam-column interface. Connections were designed so that each element at a given level experienced the same amount of horizontal displacement. The mass at level one of MF2 was decreased by approximately one-third due to the discontinued beam. Each frame had a 254 mm (10 in.) girder at the base to replicate a rigid foundation. Bays were 305 mm (12 in.) for each structure. Column dimensions were 38 x 51 mm (1.5 x 2 in.) and beams were 38 x 38 mm (1.5 x 1.5 in.). At the top story, the columns extended 83 mm (3.3 in.) and at each floor level the beams extended 76 mm (3 in.) beyond the end bays to develop the reinforcing wire. Number 13 gage annealed wire was used in each member for flexural reinforcement. Transverse reinforcement consisted of number 16 gage wire bent into rectangular spirals. Beam-column joints were reinforced using number 16 gage wire spirals, as well as metal tubing to prevent deterioration of the concrete at the connection of the masses to the frames (see Figure 4.3). Completed reinforcement cages were sprayed with 10 percent hydrochloric acid to rust the steel to improve bond. This rusting procedure had negligible effects on the steel force-strain properties. Loose rust scales were removed with a wire brush before the concrete was cast. Measured material properties are summarized in Table 4.1. Moment-curvature values for the cracking and yielding points of each nonlinear spring are provided in Table 4.2 for MF1 and Table 4.3 for MF2. Figure 4.4 and Figure 4.5 show the assignments of the element property types for MF1 and MF2. Moment-rotation yield-point data for test structures MF1 and MF2 is given in Table 4.6 and Table 4.7. Each test structure was subjected to a total of three runs, where each run had an increased intensity of base acceleration. During test runs, horizontal displacements and accelerations were recorded at each level. After the response was recorded for each run, displacements and accelerations were reset to zero for the next run.

33 21 In this study, Run 1 and Run 2 were modeled. Run 1 represents a structure that has never been subjected to strong ground motions and therefore has no previous structural damage. Run 2 represents a structure that has previously been subjected to strong ground motions and therefore has some structural damage. The base motions applied to frames MF1 and MF2 were patterned after the North- South component of the 1940 Imperial Valley earthquake at El Centro Station, California. The peak base acceleration was 0.40g for MF1 and 0.38g for MF2 in Run 1. In Run 2, the peak base acceleration was 0.93g for MF1 and 0.83g for MF2. Base acceleration records for MF1 and MF2 are presented in Figure 4.6. The time scale was compressed by 2.5 to account for the reduced scale of the experiments. Displacement and acceleration response was recorded at steps of seconds. Table 4.5 contains the calculated initial period for test structures MF1 and MF Test Structure FNW Moehle (1980) also tested a combination of frames and walls. Two nine story planar frames and one wall acting in parallel were mounted to the Earthquake Simulator at the University of Illinois. Frame FNW, shown in Figure 4.7, had two frames in parallel without walls and was a parent model to his other tests with walls. Each frame had a large first story that was twice the height of the typical stories. Column crosssectional dimensions are identical to MF1 and MF2. Longitudinal reinforcement in beams and columns were number 13 gage wire and transverse reinforcement was number 16 gage wires bent into rectangular spirals. A controlled procedure, similar to MF1 and MF2, was used to rust the reinforcement cages to improve bonding kn (1000 lbs) was distributed equally to each parallel frame. They were placed so that the center-of-masses of the weights were vertically aligned with the centerof-mass of each level of the frames. Connections between the masses and frames were designed so that each element at a given level experienced the same amount of horizontal displacement.

34 22 Measured material properties are summarized in Table 4.1. Moment-curvature values for the cracking and yielding points of the nonlinear springs of FNW are provided in Table 4.4. Figure 4.8 displays the assumed locations of the nonlinear springs for FNW. Moment-rotation yield-point data for test structure FNW is given in Table 4.8. Test structure FNW was subjected to 2 incremental runs, where Run 2 had increased base acceleration intensities in relation to Run 1. After the horizontal displacement and acceleration responses were recorded at each level for Run 1, the displacements and accelerations were set to zero and Run 2 was implemented. In this study, both Run 1 and Run 2 were modeled. The base motions that were applied to frame FNW were patterned after the North- South component of the 1940 Imperial Valley earthquake at El Centro Station, California. The peak acceleration of the applied base motion during Run 1 was 0.39g and during Run 2 was 0.78g. Base acceleration records for FNW are shown in Figure 4.9. The time scale was compressed by 2.5 to account for the reduced scale of the experiment. Displacement and acceleration response was recorded at a time step of seconds. Table 4.5 contains the calculated initial period for test structure FNW.

35 23 CHAPTER 5 SEVEN-STORY HOLIDAY INN BUILDING IN VAN NUYS, CALIFORNIA This chapter investigates the applicability of nonlinear models, similar to those described in Chapter 4 with the nonlinear parameters presented in Chapter 3, to represent the measured response of an instrumented full-scale building subjected to a strong seismic event. For this purpose, the seven-story Holiday Inn building located in Van Nuys, California provides an interesting case study. A total of 16 accelerometers located at the roof, fifth, second, first, and ground levels recorded motions in the East-West, North-South, and vertical directions. The recorded peak ground acceleration at the building site during the 1994 Northridge earthquake was 0.42g and 0.45g in the North- South and East-West directions, respectively. Section 5.1 gives a brief description of the building followed by Section 5.2 with a summary of the structural damage due to the seismic event. Section 5.3 describes the analytical models used in this study to identify the modeling assumptions that best correlate with the measured response. 5.1 Building Description The building was designed in 1965 and was built on the North-East side of the Los Angeles basin in The overall dimensions of the building are 19 m (62 ft) wide, 46 m (150 ft) in length, and 20 m (65 ft) in height. Perimeter spandrel beams create reinforced concrete frames to resist lateral forces while interior framing consists of a flat slab. The building is symmetrical about both axes, while the stairwell in the southwest corner is supported by light framing members. Typical bays are approximately 6.1 x 5.72 m (20 x ft) and floor-to-floor heights are 4.11 m (13.5 ft) for the first level, 2.64 m (8.67 ft) for the top story, and 2.65 m (8.7 ft) for all other levels. East-West frames have

36 24 eight bays and the North-South frames have three bays. The foundation systems used for this structure consist of groups of cast-in-place reinforced concrete friction piles with pile caps connected by grade beams. A typical framing plan and elevations are shown in Figure 5.1. Note that level 1 refers to the first elevated floor. The original design included both the exterior beam-column frames and the interior slab-column frames as part of the lateral force-resisting system. The interior columns are 510 x 510 mm (20 x 20 in.) at level one and 460 x 460 mm (18 x 18 in.) above level one. The exterior columns are 360 x 510 mm (14 x 20 in.) at all levels, with the 510 mm dimension along the North-South direction. The spandrel beam sizes vary throughout the height of the building. In the North-South frames, beams are 360 x 760 mm (14 x 30 in.) at level one, 360 x 570 mm (14 x 22.5 in.) at levels two through six, and 360 x 560 mm (14 x 22 in.) at the roof level. Beams of the East-West perimeter frames have the same depth dimensions as those of the North-South perimeter frames, but the width dimensions are 410 mm (16 in.). Slab thicknesses also vary throughout the height of the structure. Slab thicknesses are 250 mm (10 in.) at the first level, 220 mm (8.5 in.) at levels two through six, and 200 mm (8 in.) at the roof level. Specified material properties for the building are summarized in Table Earthquake Damage The two largest earthquakes experienced in the greater Los Angeles metropolitan area in recent past are the 1971 San Fernando earthquake and the 1994 Northridge earthquake. During both of these seismic events, the Holiday Inn building in Van Nuys was the closest instrumented building to the epicenter (21 km in 1971 and 6.4 km in 1994). The 1971 San Fernando earthquake caused minor structural damage. This damage came in the form of minor spalling and cracking of concrete. These damages were repaired by patching the first level beam-column joints in the northeast corner of the building and by using epoxy to repair concrete which had spalled (Blume, 1973). Although the structural damage was minor, the nonstructural damage was extensive.

37 25 Most of the severe damage was located on the first and second floors, and the majority of the repair costs were spent on drywall, bathroom tile, and plumbing fixtures. During the 1994 Northridge earthquake, the structure experienced severe damage. The structural damage was located primarily in the East-West perimeter moment frames. Damage to the South perimeter frame was severe between the third and fourth floor levels, where shear failure of the columns was followed by buckling of the column longitudinal reinforcement. The frames in the North-South direction experienced minor damage, such as flexural cracks (EERI, 1995). The nonstructural damage caused by the 1994 Northridge Earthquake was more severe than that of the 1971 San Fernando earthquake. Most of the damage was concentrated in the lower 4 floors, where many bathroom tiles and mirrors were completely shattered. 5.3 Modeling Assumptions In this study, the nonlinear dynamic analysis of the Holiday Inn building is performed by modeling the East-West and North-South frames separately. Given the symmetrical plan of the Holiday Inn building and the presence of perimeter moment frames, the merits of a two-dimensional model is investigated. This analytical model uses massless line elements bounded by rigid segments to represent beam-column joints. Springs containing all nonlinear properties for each member were placed between the massless line elements and the rigid end segments. The analysis included flexural, axial, and shear deformations in all frame members, except axial deformations in beams due to the rigid diaphragm assumption. In each direction of analysis, the model incorporates interior slab-column frames and exterior beam-column frames. The interior frame slab width in the North-South direction (structure HNS) uses the column strip width of 2.87 m (113 in.) and in the East- West direction (structure HEW) uses the column strip width of 3.12 m (123 in.). Beam depths at perimeter framing (for both HNS and HEW) are 0.76 m (30 in.) at level 1, 0.57 m (22.5 in.) at levels 2 to 6, and 0.56 m (22 in.) at the roof. The stiffnesses of the exterior spandrel beams included the composite beam and slab section. In the North-South

38 26 direction, the assumed effective width of the slab contributing to the beams is 1.60 m (63 in.) at all stories. In the East-West direction the assumed effective width of the slab contributing to the beams is 1.67 m (66 in.) at typical stories and 1.78 m (70 in.) at the first and roof levels. All reinforcement was taken from the original structural drawings by Rissman and Rissman Associates. The beam and slab reinforcement considered in the models are shown in Figure 5.2 to Figure 5.5. Each seven-story frame was modeled with three different levels of beam and slab reinforcement: the first-level reinforcement, four levels of typical-level reinforcement, and the roof-level reinforcement. The reinforcement in the members was simplified so that a typical reinforcement is assigned to beams and slabs at levels 2 through 6 using the reinforcement indicated in the drawings for level 2 (level 2 as indicated in Figure 5.1). Assignments of element property types are shown in Figure 5.7 through Figure 5.10 for structures HNS and HEW. Moment-rotation yieldpoint data for structures HNS and HEW are given in Table 5.5 and Table 5.6. For a given beam or slab element, the reinforcement was averaged to account for the positive and negative bending moments. In other words, the top reinforcing steel at the left support was added to the bottom reinforcing steel at the right support and the bottom reinforcing steel at the left support was added to the top reinforcing steel at the right support. The average of these sums is used to characterize identical momentrotation relationships at each end of each bay. The areas of reinforcement used to derive the moment-rotation relationships are presented in Table 5.2. Values of d were assumed to be equal to 1.1 inches in the North-South direction slabs and 1.9 inches in the East- West direction slabs. The column reinforcement schedule is presented in Table 5.3. Expected, rather than specified, material properties were used to derive member moment-rotation relationships. The expected values of the reinforcing steel yield strength were taken as 1.2 f y (Mirza and MacGregor, 1979). Expected values of the concrete compressive strength were taken as 1.2 f c. The modulus of elasticity of the concrete, in units of psi, was based on Equation 5.1: 57, (5.1)

39 27 Steel was assumed to have an elastoplastic stress-strain relationship when calculating the primary trilinear moment-rotation curve for each member. The stressstrain curve of the concrete was characterized by Hognestad (1951), where a parabolic and linear function is used. Concrete usable compressive strains were limited to and the tensile strength of the concrete was neglected. Bond-slip rotations were computed assuming the slab reinforcement was #6 bars and the beam reinforcement was #7 bars for d b in Equation 4.1. Column reinforcing bar diameters were taken from the column schedule in Table 5.3. The ultimate point of the moment-rotation curve was a function of the Hard (H) and Soft (S) cases as defined in Chapter 4.2. The effect of axial load was taken into account when defining the moment-rotation relationship for each column. The recorded peak base acceleration at the Holiday Inn building during the 1994 Northridge earthquake in the North-South direction was 0.42g. In the East-West direction, the recorded peak base acceleration was 0.45g. These seismic records are shown in Figure 5.6.

40 28 CHAPTER 6 DISCUSSION OF RESULTS This chapter contains the results from the nonlinear dynamic analyses performed in this study. These results were determined after using three computer programs to represent five nonlinear modeling parameters on three test structures (MF1, MF2, and FNW) subjected to two base motion intensities. Additional data is presented for the calculated response of the two full-scale structures (HNS and HEW), composing the Holiday Inn building in Van Nuys, California. This amounted to a total of 928 calculated FDE index values, as derived from Table 6.1 and Table 6.2. To identify the best combinations of modeling parameters that led to realistic response, the calculated FDE index values were sorted based on the type of damping and structure. Section 6.1 identifies trends in the data determined after the calculated responses for the test structures (MF1, MF2, and FNW) using programs LARZ, SAP 2000, and PERFORM 3D. The Holiday Inn building was modeled to verify if the combinations that led to satisfactory calculated-to-measured correlations for the test structures also work for the full-scale building, as discussed in Section Calculated Responses for Test Structures MF1, MF2, and FNW FDE clocks (see Appendix) for the calculated roof displacement response for test structures MF1, MF2, and FNW are shown in Figure 6.1 to Figure 6.9. The data in the FDE clocks represent not only the FDE indexes, but may also show + or x to identify the cases where the amplitude of the maximum measured displacement exceeds the maximum calculated displacement by more than 25%. For program LARZ, the average of the ratios of calculated-to-measured maximum roof displacement for all test structures when subjected to Run 1 are shown in Figure This figure emphasizes how the

41 29 models with 5% mass-proportional damping underestimate the roof displacement by 20% or more. Table 6.3 presents the FDE index averages for the test structures using LARZ. For Run 1, the best models for each damping include: CHRT α 2, CHRL α 5, UHRT β 2, and UHRL β 5. The roof displacement histories for these models applied to the test structures exhibit satisfactory correlations in Figure 6.11 to Figure For Run 2, LARZ models with 5% mass-proportional damping continue to underestimate the roof displacement by 20% or more, as indicated in Figure From Table 6.3, model CHRL α 2 was the only model with FDE indexes less than 25% for all three test structures considered. The roof displacement histories for LARZ models applied to the test structures are shown in Figure 6.15 to Figure 6.17, which exhibit satisfactory correlations. All of the models mentioned as having satisfactory correlations for the roof displacements, also gave satisfactory correlations for the base-shear histories and overturning-moment histories. Representative plots are shown in Figure 6.18 to Figure The measured base shear and overturning moment response histories were derived from the known masses and elevations and from the recorded accelerations at each level. The models with the lowest FDE index values were able to represent satisfactorily the global response of the test structures as measured by roof displacement, base shear, and overturning moment. The ability of the models to capture localized rotation demands may be measured by the story drift ratios (story displacement divided by story height). Representative plots of story drift ratios, shown in Figure 6.24 to Figure 6.29, indicate that the analytical models tend to do better in representing the maximum measured story response in Run 1 than Run 2. The calculated responses for Run 2 show larger deviations from the measured data, an indication that the analytical models do not properly account for high-mode effects in cases where the structures had previously yielded. When running programs SAP 2000 and PERFORM 3D, U models were not included due to limitations in the programs of only allowing bilinear primary momentrotation relationships. Both SAP 2000 and PERFORM 3D assume plastic response occurs after the first breakpoint of the primary force-displacement curves.

42 30 FDE clocks corresponding to the calculated response using SAP 2000 and PERFORM 3D for test structure MF1, subjected to Run 1 and Run 2, are shown in Figure 6.2 and Figure 6.3. FDE clocks for test structure MF2, subjected to Run 1 and Run 2, are shown in Figure 6.5 and Figure 6.6, while those for test structure FNW are shown in Figure 6.8 and Figure 6.9. Average tabulated FDE index values for SAP 2000 and PERFORM 3D are presented in Table 6.4 and Table 6.5. The trends for SAP 2000 are consistent with those of PERFORM 3D. In general, models CHRT α 2 and CHRL α 5 were among the best for Run 1, and model CHRL α 2 was among the best for Run 2, which is consistent with the best models identified after LARZ. Table 6.6 contains the overall averages of the FDE index values for all three test structures and all three programs considered. The table shows that models CHRT for Run 1 and CHRL for Run 2 were the best models regardless of the type of damping, excluding α 5 models (which consistently underestimated the roof displacement by more than 20%). 6.2 Calculated Responses for the Holiday Inn Building FDE clocks (see Appendix) for the calculated roof displacement response of structures HNS and HEW are shown in Figure 6.30 to Figure The FDE clocks suggest that the C models in LARZ have a poor correlation with the measured response when compared to the U models. The FDE clocks, obtained for α 2 damping after programs SAP 2000 and PERFORM 3D (Figure 6.31 and Figure 6.32) also testify on the poor correlation of the C models and that the C model with α 2 damping points to CHRT as the best model. For program LARZ, the average of the ratios of calculated-to-measured maximum roof displacement for structures HNS and HEW are shown in Figure The figure indicates that with the exception of α 5 models, the C models overestimated the roof displacement by as much as 60%. This may be an indication that the nonstructural components of the building affected the building response either by providing an initial stiffening effect or additional damping.

43 31 In Table 6.7, the FDE index averages show that ten models were capable of attaining an FDE index of 25% or less for both HNS and HEW. Noteworthy are models UHRT and USRT, which regardless of damping, discarding α 5 models, gave consistently low FDE index values. This finding is somewhat analogous to the selection in Table 6.6 of the CHRT models (Run 1), only that the full-scale structure calls for a U model over the C model. This is possibly due to the stiffening effect of the nonstructural components in HNS and HEW. When comparing Table 6.3 with Table 6.7, only two models coincidentally give an FDE index below 25% for all structures: models UHRT β 2 and UHRL β 5. The goodness-of-fit of models UHRT are shown in Figure 6.34 and Figure Note that except for the case of α 5 damping, models UHRT attained an FDE index below 25%, which is an indication of satisfactory correlation with the measured roof displacement response. Base shear and overturning moment histories are not presented for HNS and HEW because recorded data is not available for each floor. For this same reason, envelopes of story drift ratios are not presented.

44 32 CHAPTER 7 SUMMARY AND CONCLUSIONS To identify the modeling assumptions that lead to the best correlation between calculated and measured seismic response, a series of nonlinear dynamic analyses were performed on three small-scale shake-table test structures (MF1, MF2, and FNW) and on the orthogonal structural systems (HNS and HEW) of an instrumented seven-story building located in Van Nuys, California. The structures represent multibay multistory reinforced concrete moment frames. The nonlinear dynamic analyses were implemented using three programs (LARZ, SAP 2000, and PERFORM 3D). Each program was used to represent the influence of five modeling parameters: initial stiffness, bond-slip rotations, post-yield stiffness, unloading stiffness, and viscous damping. In all, a total of 928 analysis cases (see Table 6.1 and Table 6.2) were considered. The calculated responses were processed using the Frequency Domain Error (FDE) clocks, which is an effective tool for visualizing the influence of multiple parameter values on the correlation between the calculated response histories and the measured responses. The FDE index helped identify the analytical models having the best and most consistent correlation with the measured data. From the limited number of structures, modeling parameters, parameter range of values, and computer programs considered, the following are concluded (refer to Table A.1 for parameter identifications): (1) For the test structures (MF1, MF2, and FNW) subjected to Run 1, the best models for each type of damping were: CHRT α 2 CHRL α 5 UHRT β 2 UHRL β 5 These models are derived by the following simple rules:

45 33 Use H and R; Use C models with α damping and U models with β damping. The softer C models are compensated for by using constant damping (mass proportional); Use T models with 2% damping and L models with 5% damping. Higher damping is used on the softer L models. (2) For the test structures (MF1, MF2, and FNW) subjected to Run 2, one model outperformed the others: CHRL α 2 This model may be viewed as a logical outcome from the rules given for Run 1. Since the structures had previously yielded when subjected to Run 1, the use of a C model is sensible. The change from T to L is justified because the structure is effectively softer due to concrete spalling. (3) For the full-scale Holiday Inn building (structures HNS and HEW) two of the best models were consistent with the rules given in conclusion (1): UHRT β 2 UHRL β 5 (4) In general, these models had satisfactory accuracy when representing the global measured response of the test structures for both Run 1 and Run 2. Calculated roof displacement, base shear, and overturning moment histories successfully tracked the measured response. (5) Although the models for the test structures in Run 1 were able to satisfactorily represent the local measured response, indicated by story drift ratios, the models for Run 2 were not as accurate. This is possibly due to limitations in the analytical models to properly account for high-mode effects. The findings from this thesis suggest that a valuable contribution to practicing engineers would be to have developers of structural software incorporate moment frame model templates with pre-assigned nonlinear springs. The spring definition may be prepopulated with default property values based on the rules defined in conclusions (1) through (3).

46 34 APPENDIX FREQUENCY DOMAIN ERROR INDEX The large number of models and output information make this study impractical to determine the combinations of the nonlinear modeling parameters (Chapter 3) that give the best correlation between measured and calculated response for each of the structures considered. This Appendix describes the Frequency Domain Error (FDE) method (Dragovich and Lepage, 2009) for calculating the error between two signals. The FDE index lends itself to graphical analysis to identify trends associated with the values assigned to the five parameters used in this study (Table A.1). A.1 Combinations of Parameter Values When all combinations of the parameters described in Chapter 3 are accounted for, a total of 64 models are created for each structure. Table A.1 shows a summary of the parameters considered in this study. Due to the large amount of models analyzed, the FDE index was used to determine the combinations of parameters that led to the best correlation between the calculated and measured response for reinforced concrete moment frames. A.2 FDE Clocks The FDE index is a method for measuring the correlation between two waveforms. In this study, the index is used to determine the combinations of parameters leading to the best correlation between measured and calculated roof displacement responses for the structures considered.

47 35 The FDE index quantifies amplitude and phase deviations between two waveforms. A number between 0 and 1 is calculated, where 0 represents a perfect correlation and 1 indicates a waveform that is 180 degrees out of phase. To determine this number, the Fourier Transforms of the calculated and measured signals must be computed so that the signals can be represented in the frequency domain. The real and imaginary components of the complex number can be thought of as x and y components. The x-axis is represented by the real component and the y-axis is represented by the imaginary component (Argand diagram). The FDE index is based on the error vector between the calculated and measured vectors. Figure A.1 displays the x-y error representation for a given frequency for the calculated and measured response signals. The FDE index is based on the error vector (Figure A.1) normalized by the sum of the vector magnitudes of the measured and calculated signals: (A.1) where:,, real and imaginary component of the measured signal at frequency i real and imaginary component of the calculated signal at frequency i, starting and ending frequency for summation To visually discern the parameters which lead to the best correlation between measured and calculated results, the FDE index for each structure was plotted using FDE clocks (Figure A.2). These clocks are a graphical representation consisting of 16 circular sectors, with each sector representing one of the models resulting from the combination of the first four parameters of Table A.1. The perimeter of the circle marks an FDE index value of 0.75 and the center marks an FDE index value of 0. Uncracked (U) models are shown in the right half of each FDE clock and cracked (C) models are shown in the left half. Every sector has an opposite sector located at 180 degrees with the alternative set of parameters. For example, UHNT is the model most resistant to deformation and on its opposite side is model CSRL, which is the least resistant to deformation. FDE clocks for

48 36 each type and value of damping are presented in Chapter 6 for each of the structures considered.

49 TABLES 37

50 Table Viscous Damping Parameter, α and β Values Structure T 1 α 2 α 5 β 2 β 5 MF MF FNW HNS HEW Note: Subscript 2 corresponds to 2% of critical damping and subscript 5 to 5% of critical damping.

51 39 Table Assumed Material Properties for Specimens MF1, MF2, and FNW Specimen Property MF1 (MPa) MF2 (MPa) FNW (MPa) f c E c 22,000 21,000 20,000 f y Table Assumed Member Properties, Test Structure MF1 Element a Dimension Long. Reinf. EI M c φ c M y φ y (mm) (gage wire/face) (kn-m 2 ) (kn-m) (rad/m) (kn-m) (rad/m) Beam 1 38 x 38 (2) # Beam 2 38 x 38 (3) # Column 1 38 x 51 (2) # Column 2 38 x 51 (2) # Column 3 38 x 51 (3) # a See Figure 4.4 for property assignments.

52 40 Table Assumed Member Properties, Test Structure MF2 Element a Dimension Long. Reinf. EI M c φ c M y φ y (mm) (gage wire/face) (kn-m 2 ) (kn-m) (rad/m) (kn-m) (rad/m) Beam 1 38 x 38 (2) # Beam 2 38 x 38 (3) # Column 1 38 x 51 (2) # Column 2 38 x 51 (2) # Column 3 38 x 51 (4) # a See Figure 4.5 for property assignments.

53 41 Table Assumed Member Properties, Test Structure FNW Element a Dimension Long. Reinf. EI M c φ c M y φ y (mm) (gage wire/face) (kn-m 2 ) (kn-m) (rad/m) (kn-m) (rad/m) Beam 1 38 x 38 (3) # Beam 2 38 x 38 (2) # Column 1 38 x 51 (4) # Column 2 38 x 51 (4) # Column 3 38 x 51 (2) # Column 4 38 x 51 (2) # Column 5 38 x 51 (2) # a See Figure 4.8 for property assignments. Table Calculated First-Mode Periods of Vibration Structure a T 1 (Seconds) MF MF FNW a Based on gross-section properties

54 42 Table Yield Point Data, Test Structure MF1 Element a P φ y M y θ y θ y b (kn) (rad) (kn-m) (rad) (rad) E E E E E E E E E E E E E E E E E E E E-03 a See Figure 4.4 for element locations. b Values based on L case. For T case, divide by 2 per Section 3.2. Table Yield Point Data, Test Structure MF2 Element a P φ y M y θ y θ y b (kn) (rad) (kn-m) (rad) (rad) E E E E E E E E E E E E E E E E E E E E-03 a See Figure 4.5 for element locations. b Values based on L case. For T case, divide by 2 per Section 3.2. Table Yield Point Data, Test Structure FNW Element a P φ y M y θ y θ y b (kn) (rad) (kn-m) (rad) (rad) E E E E E E E E E E E E E E E E E E E E E E E E E E E E-03 a See Figure 4.8 for element locations. b Values based on L case. For T case, divide by 2 per Section 3.2.

55 5 43 Table Specified Material Properties for the Holiday Inn Building Element Level f c f y (ksi) (ksi) L 3 Roof 3 40 Slabs & Beams L L L 3 Roof 3 60 Columns L L Note: 1 ksi = MPa Level Table Holiday Inn Building Assumed Beam/Slab Reinforcement North-South Direction (HNS) East-West Direction (HEW) Exterior Frame Interior Frame Exterior Frame Interior Frame Exterior Bay A s (in 2 ) Interior Bay A s (in 2 ) Exterior Bay A s (in 2 ) Interior Bay A s (in 2 ) Exterior Bay A s (in 2 ) Interior Bay A s (in 2 ) Exterior Bay A s (in 2 ) Interior Bay A s (in 2 ) Roof Typical Note: in = 25.4 mm

56 44 Level Table Holiday Inn Building Column Reinforcement North-South Direction (HNS) East-West Direction (HEW) Exterior Frame Interior Frame Exterior Frame Interior Frame Edge Interior Edge Interior Edge Interior Edge Interior Column Column Column Column Column Column Column Column 14 x20 14 x20 14 x20 18 x18 20 x14 20 x14 20 x14 18 x18 Roof 6#7 6#7 6#7 6#7 6#7 6#7 6#7 6#7 6 6#7 6#7 6#7 6#7 6#7 6#7 6#7 6#7 5 6#7 6#7 6#7 6#7 6#7 6#7 6#7 6#7 4 6#7 6#9 6#7 6#8 6#7 6#7 6#9 6#8 3 6#7 8#9 6#9 8#9 6#7 6#9 8#9 8#9 2 6#7 8#9 6#9 10#9 6#7 6#9 8#9 10#9 1 8#9 12#9 10#9 10#9* 8#9 10#9 12#9 10#9* * Column is 20 x 20 Note: 1 in = 25.4 mm Table Column Bar Arrangements, Holiday Inn Building Reinforcement a 14 x20 18 x18 and 20 x20 6 Bars 8 Bars 10 Bars 12 Bars a Cover to centroid of longitudinal reinforcement taken as 2.5. Note: 1 in = 25.4 mm

57 45 Table Yield Point Data, Structure HNS Element a P φ y M y θ y θ y b (kn) (rad) (kn-m) (rad) (rad) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-03 a See Figure 5.7 and Figure 5.8 for element locations. b Values based on L case. For T case, divide by 2 per Section 3.2. Note: 1 kip = kn 1 in = 25.4 mm

58 46 Table Yield Point Data, Structure HEW Element a P φ y M y θ y θ y b (kn) (rad) (kn-m) (rad) (rad) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-03 a See Figure 5.9 and Figure 5.10 for element locations. b Values based on L case. For T case, divide by 2 per Section 3.2. Note: 1 kip = kn 1 in = 25.4 mm

59 6 47 Table Number of Nonlinear Analysis Cases Considered, Test Structures Program Cases Total U/C, H/S, N/R, T/L = 16 α LARZ 2, α 5, β 2, β 5 = MF1, MF2, FNW = 3 Run 1, Run 2 = 2 SAP 2000 PERFORM 3D C, H/S, N/R, T/L α 2, α 5, β 2, β 5 MF1, MF2, FNW Run 1, Run 2 C, H/S, N/R, T/L α 2, α 5, β 2, β 5 MF1, MF2, FNW Run 1, Run 2 = 8 = 4 = 3 = 2 = 8 = 4 = 3 = Table Number of Nonlinear Analysis Cases Considered, Holiday Inn Building Program Cases Total U/C, H/S, N/R, T/L = 16 α LARZ 2, α 5, β 2, β 5 = HEW, HNS = Record = 1 SAP 2000 PERFORM 3D C, H/S, N/R, T/L α 2 HEW, HNS 1994 Record C, H/S, N/R, T/L α 2 HEW, HNS 1994 Record = 8 = 1 = 2 = 1 = 8 = 1 = 2 =

60 Table FDE Index Averages for Specimens MF1, MF2, and FNW, Using LARZ Model Average FDE Index RUN 1, % Average FDE Index RUN 2, % α 2 α 5 β 2 β 5 α 2 α 5 β 2 β 5 CHNL CHRL CSNL CSRL CHNT CHRT CSNT CSRT UHNL UHRL USNL USRL UHNT UHRT USNT USRT Note: Bolded values identify lowest average FDE index for the given damping and are shaded if values are below 25% for all test structures considered (MF1, MF2, and FNW). 48 Table FDE Index Averages for Specimens MF1, MF2, and FNW, Using SAP 2000 Model Average FDE Index RUN 1, % Average FDE Index RUN 2, % α 2 α 5 β 2 β 5 α 2 α 5 β 2 β 5 CHNL CHRL CSNL CSRL CHNT CHRT CSNT CSRT Note: Bolded values identify lowest average FDE index for the given damping.

61 Table FDE Index Averages for Specimens MF1, MF2, and FNW, Using PERFORM 3D Model Average FDE Index RUN 1, % Average FDE Index RUN 2, % α 2 α 5 β 2 β 5 α 2 α 5 β 2 β 5 CHNL CHRL CSNL CSRL CHNT CHRT CSNT CSRT Note: Bolded values identify lowest average FDE index for the given damping. 49 Table FDE Index Averages for Specimens MF1, MF2, and FNW, Using LARZ, SAP 2000, and PERFORM 3D Model Average FDE Index RUN 1, % Average FDE Index RUN 2, % α 2 α 5 β 2 β 5 α 2 α 5 β 2 β 5 CHNL CHRL CSNL CSRL CHNT CHRT CSNT CSRT Note: Bolded values identify lowest average FDE index for the given damping. Boxed cells identify models with consistent low FDE index values regardless of damping type (excluding α 5 ).

62 7 50 Table FDE Index Averages for HEW and HNS, Using LARZ Note: Model Average FDE Index, % α 2 α 5 β 2 β 5 CHNL CHRL CSNL CSRL CHNT CHRT CSNT CSRT UHNL UHRL USNL USRL UHNT UHRT USNT USRT Shaded values identify models where FDE index values are below 25% for HEW and HNS. Boxed cells identify models with consistent low FDE index values regardless of damping type (excluding α 5 ).

63 51 Table Summary of Best Models for Test Structures MF1, MF2, and FNW Damping Parameter Run 1 Run 2 α 2 CHRT a CHRL a α β 2 UHRT b - β 5 UHRL b - a Models consistently led to low FDE index values in LARZ, SAP 2000, and PERFORM 3D, see Table 6.6. b Models consistently led to low FDE index values in test structures and full-scale structures, see Table 6.3 and Table 6.7.

64 52 Table A.1 - Summary of Parameters Considered Parameter Identification Characteristic Symbol Initial Stiffness Bond-Slip Effect Post-Yield Stiffness Unloading Stiffness Uncracked f r = ½ f' c [MPa] U Cracked f r 0 C Tight λ = 20 T Loose λ = 40 L Hard K p = 0.10K e H Soft K p = 0.02K e S Non Reducing γ = 0 N Reducing γ = 0.6 R 2% Mass Proportional ξ = 0.02 α 2 Viscous Damping 5% Mass Proportional ξ = 0.05 α 5 2% Stiffness Proportional ξ = 0.02 β 2 5% Stiffness Proportional ξ = 0.05 β 5

65 FIGURES 53

66 Figure Idealized Reinforced Concrete Frame

67 55 Linear-elastic segment Rigid ends Nonlinear rotational springs Figure One-Component Model M u M y Moment Secant to yield M c φ c φ y φ u Curvature Figure Moment-Curvature Relationship

68 56 Bond-slip at yielding, θ y M u Moment M y M c Rotation due to flexure Rotation with bond-slip θ c + θ c θ y + θ y θ u + θ u Rotation Figure Moment-Rotation Relationship Including Bond-Slip Rotation

69 57 Start Time = ΔT Calculate element codes Hysteresis Models Takeda Sina Otani Bilinear Calculate instantaneous element and structural stiffness matrices Calculate elastic element stiffnesses and structural stiffness matrices Q-hyst Condense structural stiffness matrix and calculate damping matrix No Time > Duration? Time = Time + ΔT Calculate member end forces Solve differential equation of motion Yes ΔT = time interval of integration Stop Figure Flowchart of Program LARZ, After Saiidi and Sozen (1979)

70 58 M u M y Moment K e θ y + θ y θ u + θ u Rotation Figure Bilinear Moment-Rotation Relationship in SAP 2000

71 59 /2 /2 x A B Moment Diagram, M M c M u M y Member Length, L c (λ 1 /λ 2 ) y A 1 B Curvature Diagram, y 2 C L 3 u Figure Assumed Moment and Curvature Diagrams for Frame Members Elastic component Inelastic component Rigid ends Figure Frame Compound Components Used in PERFORM 3D

72 60 Bond-slip at yielding, θ y Moment M y M c θ ys = θ y + θ y Neglecting bond-slip effects Using modified moment of inertia θ c θ y θ ys Rotation Figure Moment-Rotation Diagram for the Elastic Component in PERFORM 3D M u U L Moment M y Y YU: Post yield UL: Peak strength θ U Rotation Figure Moment-Rotation Relationship of the Inelastic Component in PERFORM 3D θ L

73 3 61 M u M y Moment M c φ c φ y φ u Curvature Figure Moment-Curvature Relationship for Uncracked Case (U) M u M y Moment K e φ y Curvature Figure Moment-Curvature Relationship for Cracked Case (C) φ u

74 62 For beams, kd d Figure Rotation Due to Bond Slip, After Saiidi and Sozen (1979)

75 63 M u M y K p = 0.02 K e Moment K e φ y Rotation Figure Moment-Rotation Relationship with a Soft Post-Yield Stiffness (S) φ u M u K p = 0.10 K e M y Moment K e φ y Rotation Figure Moment-Rotation Relationship with a Hard Post-Yield Stiffness (H) φ u

76 64 M u M y Moment K r = K e K e θ y Rotation Figure Moment-Rotation Relationship with Non-Reducing (N) Unloading Stiffness θ u M u M y Moment K r K e θ y Rotation Figure Moment-Rotation Relationship with Reducing (R) Unloading Stiffness θ u

77 4 65 Figure The University of Illinois Earthquake Simulator, After Lepage (1997)

78 66 Elevation Section through beams (All dimensions are in millimeters) Figure Specimen MF1 Tested by Moehle and Sozen (1978)

79 Figure Representative Reinforcement Details, After Moehle and Sozen (1980) 67

80 Note: For yield point data, see Table 4.6. Figure Frame Element Property Types, Test Structure MF1

81 Note: For yield point data, see Table 4.7. Figure Frame Element Property Types, Test Structure MF2

82 MF1 Run MF1 Run Ground Acceleration, g MF2 Run Time, s MF2 Run Figure Base Acceleration Records, Test Structures MF1 and MF2

83 71 Elevation Section through beams (All dimensions are in millimeters) Figure Specimen FNW Tested by Moehle and Sozen (1980)

84 Note: For yield point data, see Table 4.8. Figure Frame Element Property Types, Test Structure FNW

85 FNW Run Ground Acceleration, g FNW Run Time, s Figure Base Acceleration Records, Test Structure FNW

86 5 74 Figure Plan and Elevations of the Holiday Inn Building, After Lepage (1997)

87 75 2#9 + 10#5 (Slab) 2#6 + 4#5 (Slab) 2#9 + 10#5 (Slab) 2#9 + 10#5 (Slab) Roof Level 2#7 2#7 2#7 2#7 2#10 + 1#9 + 13#6 (Slab) 2#10 + 1#9 + 13#6 (Slab) 2#10 + 1#7 + 4#5(Slab) 2#10 + 1#9 + 13#6 (Slab) Typ. Level 2#9 2#9 2#9 2#9 3# #6 (Slab) 3# #6 (Slab) 2#9 + 1#10 + 4#5 (Slab) 3# #6 (Slab) Level 1 2#10 2#9 2#10 2#9 Exterior Bay Interior Bay Figure Exterior Frames, North-South Direction, T-Beam Reinforcement Assumptions

88 76 6#5 4#5 17#6 10#6 17#6 10#6 17#6 10#6 Roof Level 21#6 21#6 5#6 21#6 Typ. Level 4#5 8#6 + 2#6 8#6 + 2#6 8#6 + 2#6 16#7 16#7 7#6 16#7 Level 1 5#5 10#5 + 2#6 10#5 + 2#6 10#5 + 2#6 Exterior Bay Interior Bay Figure Interior Frames, North-South Direction, Slab Reinforcement Assumptions

89 77 2#8 + 10#5 (Slab) 2#6 + 4#5 (Slab) 2#8 + 10#5 (Slab) 2#8 + 10#5 (Slab) Roof Level 2#7 2#7 2#6 2#6 3#9 + 17#6 (Slab) 3#8 + 4#5 (Slab) 3#9 + 17#6 (Slab) 3#9 + 17#6 (Slab) Typ. Level 2#8 2#8 2#6 2#6 2#9 + 16#6 (Slab) 2#9 + 4#5 (Slab) 2#9 + 16#6 (Slab) 2#9 + 16#6 (Slab) Level 1 2#8 2#8 2#6 2#6 Exterior Bay Interior Bay Figure Exterior Frames, East-West Direction, T-Beam Reinforcement Assumptions

90 78 7#5 19#6 19#6 19#6 Roof Level 4#5 (10#5 + 10#6)/2 (10#5 + 10#6)/2 (10#5 + 10#6)/2 19#6 19#6 7#5 19#6 Typ. Level 4#5 (8#6 + 7#6)/2 + 2#6 (8#6 + 7#6)/2 + 2#6 (8#6 + 7#6)/2 + 2#6 16#7 16#7 6#5 16#7 Level 1 4#5 (8#6 + 7#6)/2 + 2#6 (8#6 + 7#6)/2 + 2#6 (8#6 + 7#6)/2 + 2#6 Exterior Bay Interior Bay Note: Bottom reinforcement is based on average within column-strip on either side of column gridline. Figure Interior Frames, East-West Direction, Slab Reinforcement Assumptions

91 NS 0.00 Ground Acceleration, g EW Time, s Figure Base Acceleration Records, 1994 Holiday Inn Building in Van Nuys, California

92 Note: For yield point data, see Table 5.5. Figure Frame Element Property Types, Interior Frame, Structure HNS Note: For yield point data, see Table 5.5. Figure Frame Element Property Types, Exterior Frame, Structure HNS

93 Note: For yield point data, see Table 5.6. Figure Frame Element Property Types, Interior Frame, Structure HEW Note: For yield point data, see Table 5.6. Figure Frame Element Property Types, Exterior Frame, Structure HEW

94 CHRT CHNT USRT 0.50 USNT CHRT CHNT USRT 0.50 USNT CSNT UHRT CSNT UHRT CSRT UHNT CSRT UHNT CSRL UHNL CSRL UHNL CSNL UHRL CSNL UHRL CHRL CHNL USRL USNL α 2 β 2 CHRL CHNL USRL USNL CHRT CHNT USRT 0.50 USNT α 5 β 5 CHRT CHNT USRT 0.50 USNT CSNT UHRT CSNT UHRT CSRT UHNT CSRT UHNT CSRL UHNL CSRL UHNL CSNL UHRL CSNL UHRL CHRL CHNL USRL USNL CHRL CHNL USRL USNL R1 RUN 1 R2 RUN 2 Figure FDE Clocks, Test Structure MF1, Run 1 and Run 2, LARZ α 2 CHRT CHNT α 2 CHRT CHNT CSNT CSNT CSRT CSRT CSRL CSRL CSNL CSNL CHRL CHNL MF1 RUN 1 MF2 RUN 2 CHRL CHNL R1 RUN 1 R2 RUN 2 Figure FDE Clocks, Test Structure MF1, Run 1 and Run 2, SAP 2000 Figure FDE Clocks, Test Structure MF1, Run 1 and Run 2, PERFORM 3D

95 CHRT CHNT USRT 0.50 USNT CHRT CHNT USRT 0.50 USNT CSNT UHRT CSNT UHRT CSRT UHNT CSRT UHNT CSRL UHNL CSRL UHNL CSNL UHRL CSNL UHRL CHRL CHNL USRL USNL α 2 β 2 CHRL CHNL USRL USNL CHRT CHNT USRT 0.50 USNT α 5 β 5 CHRT CHNT USRT 0.50 USNT CSNT UHRT CSNT UHRT CSRT UHNT CSRT UHNT CSRL UHNL CSRL UHNL CSNL UHRL CSNL UHRL CHRL CHNL USRL USNL CHRL CHNL USRL USNL R1 RUN 1 R2 RUN 2 Figure FDE Clocks, Test Structure MF2, Run 1 and Run 2, LARZ α 2 CHRT CHNT α 2 CHRT CHNT CSNT CSNT CSRT CSRT CSRL CSRL CSNL CSNL CHRL CHNL MF1 RUN 1 MF2 RUN 2 CHRL CHNL R1 RUN 1 R2 RUN 2 Figure FDE Clocks, Test Structure MF2, Run 1 and Run 2, SAP 2000 Figure FDE Clocks, Test Structure MF2, Run1 and Run 2, PERFORM 3D

96 CHRT CHNT USRT 0.50 USNT CHRT CHNT USRT 0.50 USNT CSNT UHRT CSNT UHRT CSRT UHNT CSRT UHNT CSRL UHNL CSRL UHNL CSNL UHRL CSNL UHRL CHRL CHNL USRL USNL α 2 β 2 CHRL CHNL USRL USNL CHRT CHNT USRT 0.50 USNT α 5 β 5 CHRT CHNT USRT 0.50 USNT CSNT UHRT CSNT UHRT CSRT UHNT CSRT UHNT CSRL UHNL CSRL UHNL CSNL UHRL CSNL UHRL CHRL CHNL USRL USNL CHRL CHNL USRL USNL R1 RUN 1 R2 RUN 2 Figure FDE Clocks, Test Structure FNW, Run 1 and Run 2, LARZ α 2 CHRT CHNT α 2 CHRT CHNT CSNT CSNT CSRT CSRT CSRL CSRL CSNL CSNL CHRL CHNL MF1 RUN 1 MF2 RUN 2 CHRL CHNL R1 RUN 1 R2 RUN 2 Figure FDE Clocks, Test Structure FNW, Run 1 and Run 2, SAP 2000 Figure FDE Clocks, Test Structure FNW, Run 1 and Run 2, PERFORM 3D

97 Stiffer Softer α 2 UHNT UHRT USNT USRT UHNL UHRL USNL USRL CHNT CHRT CSNT CSRT CHNL CHRL CSNL CSRL α 5 UHNT UHRT USNT USRT UHNL UHRL USNL USRL CHNT CHRT CSNT CSRT CHNL CHRL CSNL CSRL β 2 UHNT UHRT USNT USRT UHNL UHRL USNL USRL CHNT CHRT CSNT CSRT CHNL CHRL CSNL CSRL β 5 UHNT UHRT USNT USRT UHNL UHRL USNL USRL CHNT CHRT CSNT CSRT CHNL CHRL CSNL CSRL Figure Ratios of Calculated-to-Measured Maximum Roof Displacement, Average of Test Structures MF1, MF2, and FNW for Run 1, LARZ

98 FDE = 22 CHRT α FDE = 21 CHRL α 5 Displacement, mm FDE = 16 UHRT β FDE = 20 UHRL β 5 30 Measured Calculated Time, s Figure Roof Displacement Histories, MF1 Run 1, LARZ

99 FDE = 16 CHRT α FDE = 28 CHRL α 5 Displacement, mm FDE = 17 UHRT β FDE = 17 UHRL β Measured Calculated Time, s Figure Roof Displacement Histories, MF2 Run 1, LARZ

100 FDE = 22 CHRT α FDE = 28 CHRL α 5 Displacement, mm FDE = 24 UHRT β FDE = 22 UHRL β Time, s Figure Roof Displacement Histories, FNW Run 1, LARZ

101 Stiffer Softer α 2 UHNT UHRT USNT USRT UHNL UHRL USNL USRL CHNT CHRT CSNT CSRT CHNL CHRL CSNL CSRL α 5 UHNT UHRT USNT USRT UHNL UHRL USNL USRL CHNT CHRT CSNT CSRT CHNL CHRL CSNL CSRL β 2 UHNT UHRT USNT USRT UHNL UHRL USNL USRL CHNT CHRT CSNT CSRT CHNL CHRL CSNL CSRL β 5 UHNT UHRT USNT USRT UHNL UHRL USNL USRL CHNT CHRT CSNT CSRT CHNL CHRL CSNL CSRL Figure Ratios of Calculated-to-Measured Maximum Roof Displacement, Average of Test Structures MF1, MF2, and FNW for Run 2, LARZ

102 90 Displacement, mm FDE = 19 CHRL α 2 Measured Calculated Time, s Figure Roof Displacement Histories, MF1 Run 2, LARZ Displacement, mm FDE = 23 CHRL α 2 Measured Calculated Time, s Figure Roof Displacement Histories, MF2 Run 2, LARZ Displacement, mm FDE = 23 CHRL α 2 Measured Calculated Time, s Figure Roof Displacement Histories, FNW Run 2, LARZ

103 Run 1: CHRT α Base Shear, kn Run 2: CHRL α Measured Calculated Time, s Figure Base Shear Histories, MF1, LARZ Run 1: CHRT α 2 Base Shear, kn Run 2: CHRL α Measured Calculated Time, s Figure Base Shear Histories, MF2, LARZ

104 Run 1: CHRT α Base Shear, kn Run 2: CHRL α Measured Calculated Time, s Figure Base Shear Histories, FNW, LARZ Overturning Moment, kn - m Run 1: CHRT α Run 2: CHRL α 2 Measured Calculated Time, s Figure Overturning Moment Histories, MF1, LARZ

105 93 Overturning Moment, kn - m Run 1: CHRT α Run 2: CHRT α 2 Measured Calculated Time, s Figure Overturning Moment Histories, MF2, LARZ Overturning Moment, kn - m Run 1: CHRT α Run 2: CHRL α 2 Measured Calculated Time, s Figure Overturning Moment Histories, FNW, LARZ

106 Measured LARZ SAP 2000 PERFORM 3D Level Mean Drift Ratio, % Story Drift Ratio, % Figure Mean Drift Ratio and Story Drift Ratio Envelopes, MF1 Run 1, Model CHRT - α Measured LARZ SAP 2000 PERFORM 3D Level Mean Drift Ratio, % Story Drift Ratio, % Figure Mean Drift Ratio and Story Drift Ratio Envelopes, MF1 Run 2, Model CHRL - α 2

107 Measured LARZ SAP 2000 PERFORM 3D Level Mean Drift Ratio, % Story Drift Ratio, % Figure Mean Drift Ratio and Story Drift Ratio Envelopes, MF2 Run 1, Model CHRT - α Measured LARZ SAP 2000 PERFORM 3D Level Mean Drift Ratio, % Story Drift Ratio, % Figure Mean Drift Ratio and Story Drift Ratio Envelopes, MF2 Run 2, Model CHRL - α 2

108 Measured LARZ SAP 2000 PERFORM 3D Level Mean Drift Ratio, % Story Drift Ratio, % Figure Mean Drift Ratio and Story Drift Ratio Envelopes, FNW Run 1, Model CHRT - α Measured LARZ SAP 2000 PERFORM 3D Level Mean Drift Ratio, % Story Drift Ratio, % Figure Mean Drift Ratio and Story Drift Ratio Envelopes, FNW Run 2, Model CHRL - α 2

109 CHRT CHNT USRT 0.50 USNT CHRT CHNT USRT 0.50 USNT CSNT UHRT CSNT UHRT CSRT UHNT CSRT UHNT CSRL UHNL CSRL UHNL CSNL UHRL CSNL UHRL CHRL CHNL USRL USNL α 2 β 2 CHRL CHNL USRL USNL CHRT CHNT USRT 0.50 USNT α 5 β 5 CHRT CHNT USRT 0.50 USNT CSNT UHRT CSNT UHRT CSRT UHNT CSRT UHNT CSRL UHNL CSRL UHNL CSNL UHRL CSNL UHRL CHRL CHNL USRL USNL CHRL CHNL USRL USNL H HNS H HEW Figure FDE Clocks, Structure HNS and HEW, LARZ α α 2 CHNT CHRT 0.50 CSNT CSNT CHRT CHNT CSRT CSRT CSRL CSRL CSNL CSNL CHRL CHNL S HNS S HEW CHRL CHNL H HNS H HEW Figure FDE Clocks, Structure HNS and HEW, SAP 2000 Figure FDE Clocks, Structure HNS and HEW, PERFORM 3D

110 Stiffer Softer α 2 UHNT UHRT USNT USRT UHNL UHRL USNL USRL CHNT CHRT CSNT CSRT CHNL CHRL CSNL CSRL α 5 UHNT UHRT USNT USRT UHNL UHRL USNL USRL CHNT CHRT CSNT CSRT CHNL CHRL CSNL CSRL β 2 UHNT UHRT USNT USRT UHNL UHRL USNL USRL CHNT CHRT CSNT CSRT CHNL CHRL CSNL CSRL β 5 UHNT UHRT USNT USRT UHNL UHRL USNL USRL CHNT CHRT CSNT CSRT CHNL CHRL CSNL CSRL Figure Ratios of Calculated-to-Measured Maximum Roof Displacement, Average of Holiday Inn HNS and HEW, LARZ

111 FDE = 23 UHRT α FDE = 31 UHRT α 5 Displacement, mm FDE = 20 UHRT β FDE = 20 UHRT β Measured Calculated Time, s Figure Roof Displacement Histories, HNS, LARZ

112 FDE = 25 UHRT α FDE = 33 UHRT α 5 Displacement, mm FDE = 25 UHRT β FDE =21 UHRT β Time, s Figure Roof Displacement Histories, HEW, LARZ

113 101 Figure A.1 - FDE Representation Figure A.2 - FDE Clock