Capacity-Demand Index Relationships for Performance- Based Seismic Design

Transcription

1 Capacity-Demand Index Relationships for Performance- Based Seismic Design November Kenneth T. Farrow and Yahya C. Kurama Report #NDSE R =,,, 6, 8 (thin thick lines) IND spectra DES spectrum ρ =.99 d =., f =.7 ρ =.97 d =.85, f = ρ =.99 ρ =.97 ρ =.9 d =., f =.7 d =.85, f =.7 d =.59, f =.3 p ρ =.96 g =.6, h = ρ =.96 ρ =.89 g =.56, h =.57 g =.6, h =.3 5 ρ =.95 g =., h =.7 γ COV( nlin ) R =,,, 6, 8 (thin thick lines) MIV S ˆ (T ) a o MIV S ˆ (T T ) a o. ρ = d =.59, f = r ρ =.89 ρ =.95 g = 3.9, h =.93 g =., h =. 5 5 n y data regression Structural Engineering Research Report Department of Civil Engineering and Geological Sciences University of Notre Dame Notre Dame, Indiana

2 Capacity-Demand Index Relationships for Performance- Based Seismic Design November Report #NDSE-- by Kenneth T. Farrow Former Graduate Research Assistant and Yahya C. Kurama Assistant Professor Structural Engineering Research Report Department of Civil Engineering and Geological Sciences University of Notre Dame Notre Dame, Indiana

3 ABSTRACT Seismic design approaches in current U.S. building provisions advocate using several static and dynamic analysis procedures. Among the static procedures, it is common to use linear and nonlinear methods that dep on capacity-demand index relationships (e.g., the relationship between the design lateral strength and the maximum lateral displacement). The benefit of using these relationships comes from their simplicity and adaptability, however significant deficiencies exist in their development. Foremost, previous research on the development of capacity-demand index relationships is based on linear-elastic single-degree-of-freedom acceleration response spectra, whereas current design procedures are based on smooth design response spectra. For the design procedures to be consistent, new relationships need to be developed using smooth design response spectra. Furthermore, previous research on capacity-demand index relationships is limited to the maximum displacement ductility demand. However, other demand indices such as to quantify cumulative damage and residual displacement are needed for use in the framework of a performance-based design approach that allows the designer to specify and predict the performance of a building under an earthquake. Finally, using nonlinear dynamic analysis procedures as part of a performance-based design approach has become increasingly common. These procedures are often conducted using ground motions scaled to constant peak ground motion characteristics (e.g., peak acceleration) resulting in a large scatter in the analysis results. Ground motions should be scaled based on methods that adequately define the damage potential for given site conditions and structural characteristics, thus resulting in consistent prediction of the demand estimates by minimizing the scatter. This research proposes new capacity-demand index relationships and ground motion scaling methods and shows that: () previous capacity-demand index relationships developed using linear-elastic ground motion spectra can lead to unconservative designs, particularly for survivallevel, soft soil, and near-field conditions; () the correlation between the maximum displacement ductility demand and other demand indices is relatively strong; and (3) scaling methods that work well for ground motions recorded on stiff soil and far-field conditions lose their effectiveness for soft soil and near-field conditions. This report may be downloaded from

4 CONTENTS LIST OF TABLES...v LIST OF FIGURES... viii LIST OF SYMBOLS...xx ACKNOWLEDGEMENTS... xxix CHAPTER INTRODUCTION.... Problem Statement.... Research Objectives....3 Dissertation Scope and Organization.... Research Significance...6 CHAPTER BACKGROUND...7. Previous Research on Maximum Displacement Ductility Demand Site soil characteristics Near-field ground motions Hysteretic lateral load-displacement behavior.... Previous Research on Other Demand Indices....3 Current Capacity-Based Design Procedures Equivalent lateral force (ELF) procedure Capacity spectrum procedure.... Scaling of Ground Motion Records...7 CHAPTER 3 DESCRIPTION OF THE RESEARCH PROGRAM Analytical Models Single-degree-of-freedom (SDOF) models Multi-degree-of-freedom (MDOF) models Seismic Demand Levels Ground Motion Records Important properties of the ground motion records Strong motion duration Ground Motion Scaling Methods... ii

5 3.5 Reference Response Spectra Nonlinear Dynamic Time-History Analyses SDOF analyses MDOF analyses Statistical Evaluation of the Results SDOF demand estimates MDOF demand estimates...56 CHAPTER VALIDATION OF ANALYTICAL MODEL AND COMPARISON WITH PREVIOUS RESULTS Nonlinear Dynamic Time-History Analyses Spectral Analyses Comparison with Nassar and Krawinkler (99) Constant-R versus constant- approaches IND spectra versus smooth response spectra...65 CHAPTER 5 EFFECT OF HYSTERETIC BEHAVIOR Effect of Post-Yield Stiffness Ratio, α EP Hysteresis Type versus SD Hysteresis Type EP Hysteresis Type versus BE Hysteresis Type EP Hysteresis Type versus BP Hysteresis Type...7 CHAPTER 6 EFFECT OF SITE CONDITIONS Site Soil Characteristics Seismic Demand Level Site Seismicity Epicentral Distance...8 CHAPTER 7 EFFECT OF REFERENCE RESPONSE SPECTRA Low Seismicity (Boston), Stiff Soil Profile (S D ) Low Seismicity (Boston), Soft Soil Profile (S E ) High Seismicity (Los Angeles), Stiff Soil Profile (S D ) High Seismicity (Los Angeles), Soft Soil Profile (S E ) High Seismicity (Los Angeles), Near-Field (NF), Stiff Soil Profile (S D ) SD, BE, and BP Hysteresis Types...9 CHAPTER 8 REGRESSION ANALYSES FOR Comparison of R--T Relationships with Previous Results R--T Relationships Developed in this Study...9 iii

6 CHAPTER 9 REGRESSION ANALYSES BETWEEN THE DEMAND INDICES Regression Relationships Developed Based on IND Spectra Effect of Reference Response Spectra on the Relationships Between the Demand Indices...3 CHAPTER RECOMMENDED DESIGN PROCEDURE USING CAPACITY- DEMAND INDEX RELATIONSHIPS.... Inelastic Demand Spectra.... Design Example... CHAPTER EFFECT OF GROUND MOTION SCALING METHOD...7. EP Hysteresis Type...7. SD, BE, and BP Hysteresis Types Site Soil Characteristics...5. Epicentral Distance Results for the MDOF Frame Structures...57 CHAPTER SUMMARY, CONCLUSIONS, AND FUTURE RESEARCH...6. Summary...6. Conclusions Future Research...66 REFERENCES...67 APPENDIX A CHARACTERISTICS, TIME HISTORIES, AND RESPONSE SPECTRA OF GROUND MOTION RECORDS...7 APPENDIX B CDSPEC (CAPACITY-DEMAND SPECTRA) PROGRAM LISTING... B. CDSPEC.M: Main Program... B. EQSCALE.M: Ground Motion Scaling Function... B.3 NEHRPDES.M: Smooth Design Reference Spectrum Function...9 B. LNLTHIST.M, LNLTHISTSD.M: Nonlinear Dynamic Time-History Analysis Functions...3 B.5 CDSPECPOST.M: Post-Processing Function...35 B.6 SDREG.M: R--T Nonlinear Regression Program...3 B.7 DEMANDREG.M: Cross-Correlation Nonlinear Regression Program...5 B.8 NKREGRESSION.M, NKREGRESSION_ST.M, NKREGRESSION_ME.M, NKREGRESSION_SO.M, NKREGRESSION_NF.M: Regression Functions...55 B.9 EPDISP.M, BEDISP.M, SDDISP.M: Hysteretic Rule Functions...57 iv

7 LIST OF TABLES Table.: Values for a and b coefficients...8 Table.: Demand indices introduced by Mahin and Lin (983)....3 Table 3.: Table 3.: Table 3.3: Table 3.: Frame lateral-system properties...7 Gravity loads...7 Yield moment capacities for beam rotational springs...3 Column base fiber element properties...3 Table 3.5: Site soil definitions (adapted from IBC (ICC, ))...35 Table 3.6: Values for a and b coefficients developed in this study...3 Table 3.7: Seismic coefficients for the smooth design (DES) response spectra...5 Table 3.8: Table 3.9: Table 8.: Table 8.: Table 9.: Table 9.: Table 9.3: Table 9.: SAC/N&K ground motion ensembles: parameters studied (shaded areas indicate the N&K ensemble)...9 UND/SAC ground motion ensembles: parameters studied (shaded areas indicate the SAC ensemble)...5 Regression coefficients a and b for the N&K ground motion ensemble, EP hysteresis type, α =....9 Regression coefficients a and b for the SAC ground motion ensemble...95 Regression coefficients d and f: EP hysteresis type, IND spectra...5 Regression coefficients d and f: SD, BE, and BP hysteresis types, IND spectra...6 Regression coefficients g and h: EP hysteresis type, IND spectra...7 Regression coefficients g and h: SD, BE, and BP hysteresis types, IND spectra...8 v

8 Table 9.5: Correlation coefficient, ρ: EP hysteresis type, IND spectra...9 Table 9.6: Correlation coefficient, ρ: SD, BE, and BP hysteresis types, IND spectra... Table 9.7: Regression coefficients d and f: AVG spectra... Table 9.8: Regression coefficients d and f: DES spectra...5 Table 9.9: Regression coefficients g and h: AVG spectra...6 Table 9.: Table 9.: Regression coefficients g and h: DES spectra...7 Correlation coefficient, ρ: AVG spectra...8 Table 9.: Correlation coefficient, ρ: DES spectra...9 Table.: Structure properties and results for the design example...5 Table.: p, r, and n y demands for the design example...6 Table A.: University of Notre Dame (UND) very dense (S C ) soil ensemble...7 Table A.: University of Notre Dame (UND) stiff (S D ) soil ensemble...75 Table A.3: Table A.: Table A.5: Table A.6: Table A.7: University of Notre Dame (UND) soft (S E ) soil ensemble...76 Nassar and Krawinkler (N&K) 5s very dense (S C ) soil ensemble...77 SAC Boston design-level stiff (S D ) soil ensemble...78 SAC Boston survival-level stiff (S D ) soil ensemble...79 SAC Boston design-level soft (S E ) soil ensemble...8 Table A.8: SAC Boston survival-level soft (S E ) soil ensemble...8 Table A.9: SAC Los Angeles design-level stiff (S D ) soil ensemble...8 Table A.: Table A.: Table A.: SAC Los Angeles survival-level stiff (S D ) soil ensemble...83 SAC Los Angeles design-level soft (S E ) soil ensemble...8 SAC Los Angeles survival-level soft (S E ) soil ensemble...85 vi

9 Table A.3: SAC Los Angeles near-field design-level stiff (S D ) soil ensemble...86 vii

10 LIST OF FIGURES Figure.: (a) Lateral force-displacement relationships; (b) Ground motion response spectra versus smooth design response spectra.... Figure.: R--T relationships developed by Miranda (993) for rock and soft soil sites.... Figure.: Hysteresis types used by Foutch and Shi (998)... Figure.3: Figure.: Bilinear elasto-plastic (EP) hysteresis type with definitions for demand indices in Table...3 Capacity spectrum procedures: (a) highly-damped linear-elastic demand spectra; (b) inelastic demand spectra....5 Figure 3.: SDOF model properties.... Figure 3.: Figure 3.3: Figure 3.: Hysteresis types: (a) LE; (b) EP; (c) SD; (d) BE; (e) BP... Schematic of stiffness degrading (SD) hysteresis type... Construction of the bilinear-elastic/elasto-plastic (BP) hysteresis type...3 Figure 3.5: BP hysteresis type with: (a) β s > β r, β r < ; (b) β s = β r <.... Figure 3.6: Figure 3.7: Figure 3.8: Figure 3.9: Figure 3.: EP hysteresis type versus BP hysteresis type: (a) β s > β r ; (b) β s = β r (matching yield point)...5 Layout of structural system: (a) elevation; (b) four-story structure; (c) eight-story structure...6 MDOF models: (a) four-story elevation; (b) eight-story elevation; (c) close-up of analytical model...9 Element models: (a) beam rotational spring element; (b) column base fiber element...3 Normalized cyclic base-shear-roof-drift behavior: (a) four-story frame; (b) eight-story frame....3 viii

11 Figure 3.: Figure 3.: Figure 3.3: Figure 3.: Equivalent linear model for the site response analyses: (a) assumed nonlinear hysteretic stress-strain behavior of the soil; (b) equivalent linear model Soil properties for the site response analyses: (a) shear wave velocity; (b) shear modulus reduction factor and damping ratio Acceleration time-history, cumulative RMSA function (CRF), and derivative of the CRF function (9 El Centro (ELCN) x.): (a) forward CRF; (b) reverse CRF Scaling based on the spectral acceleration (UND S C soil ground motion ensemble): (a) at the structure period ( Sˆ a ( T o ) method); (b) over a range of structure periods ( Sˆ a ( T o T ) method).... Figure 3.5: Design spectra: (a) general shape; (b) design-level; (c) survival-level.... Figure 3.6: Figure 3.7: Figure 3.8: Figure 3.9: Figure 3.: Smooth response spectra: (a) Boston, design-level, S D soil; (b) Boston, survivallevel, S D soil...5 Smooth response spectra: (a) Boston, design-level, S E soil; (b) Boston, survivallevel, S E soil....6 Smooth response spectra: (a) Los Angeles, design-level, S D soil; (b) Los Angeles, survival-level, S D soil....6 Smooth response spectra: (a) Los Angeles, design-level, S E soil; (b) Los Angeles, survival-level, S E soil....6 Smooth response spectra: (a) Los Angeles, design-level, S C soil; (b) Los Angeles, design-level, S D soil, NF....7 Figure 3.: Flowchart describing parameters studied in the analytical procedure....5 Figure 3.: Average response spectra of the ground motions used in the MDOF analyses: (a) MIV scaling method; (b) Sˆ a ( T o T ) scaling method....5 Figure 3.3: Regression analysis (IND spectra, EP hysteresis type, α =.): (a) first step, R- domain (T =.9 sec.); (b) second step, c-t domain Figure 3.: Effect of a and b coefficients on c coefficient: (a) c term; (b) c term; (c) c = c + c....5 Figure 3.5: Effect of a and b coefficients on : (a) b =.; (b) b =.; (c) b = ix

12 Figure.: Comparison between CDSPEC and DRAIN-DX (EP hysteresis type, α =., R = 8): (a-b) EQ9; (c-d) EQ Figure.: Comparison between CDSPEC and BISPEC (SD hysteresis type, α =., R = 8): (a-b) LPPR; (c-d) PACH; (e) different reloading rules...6 Figure.3: Comparison between CDSPEC and DRAIN-DX (α =., R = 8): (a-d) BE hysteresis type; (e-h) BP hysteresis type, β s = β r = / Figure.: Figure.5: Figure.6: Figure.7: Figure.8: Figure.9: Figure 5.: Figure 5.: Figure 5.3: Figure 5.: Figure 5.5: Comparison between CDSPEC and BISPEC for the N&K ground motion ensemble (α =.): (a) EP hysteresis type; (b) SD hysteresis type....6 R- relationship for the EP hysteresis type with α =. and T =. sec. under the 983 Coalinga Parkfield Zone 6 ground motion (IND response spectrum)...63 R- relationships for the EP hysteresis type with α =. (IND reference spectra): (a) T =. sec.; (b) T =.9 sec....6 R- spectra (N&K ensemble, EP hysteresis type, α =.): (a) AVG versus IND spectra; (b) DES versus IND spectra Smooth design (DES) response spectrum versus average (AVG) ground motion response spectrum R- relationships using different reference response spectra (EP hysteresis type, α =., T = 3. sec.): (a) N&K ensemble; (b) N&K ensemble without EQ...66 Effect of post-yield stiffness ratio, α (EP hysteresis type, SAC Los Angeles, survival-level, S D soil): (a) ; (b) p ; (c) r ; (d) n y...68 EP versus SD hysteresis types (α =., SAC Los Angeles, survival-level, S D soil): (a) ; (b) p ; (c) r ; (d) n y...69 Comparison between force-displacement responses of the EP and SD hysteresis types: (a) α =.; (b) α = EP versus BE hysteresis types (α =., SAC Los Angeles, survival-level, S D soil): (a) ; (b) p ; (c) r ; (d) n y...7 EP versus BP hysteresis types (α =., SAC Los Angeles, survival-level, S D soil): (a) ; (b) p ; (c) r ; (d) n y...7 Figure 5.6: Hysteretic energy dissipation of the BP hysteresis type x

13 Figure 6.: Figure 6.: Figure 6.3: Figure 6.: Effect of site soil characteristics (EP hysteresis type, α =., SAC Los Angeles, design-level): (a) ; (b) p ; (c) r ; (d) n y...75 Effect of site soil characteristics (SD, BE, and BP hysteresis types, α =., SAC Los Angeles, design-level): (a) ; (b) p ; (c) r ; (d) n y Response spectra: (a) S D soil ground motion spectrum versus S E soil ground motion spectrum; (b) ground motion spectra versus smooth design spectra...77 Effect of seismic demand level (EP hysteresis type, α =., SAC Los Angeles, S D and S E soil): (a) ; (b) p ; (c) r ; (d) n y Figure 6.5: Effect of seismic demand level (SD, BE, and BP hysteresis types, α =., SAC Los Angeles, S D soil): (a) ; (b) p ; (c) r ; (d) n y...79 Figure 6.6: Figure 6.7: Figure 6.8: Average response spectra: (a) design-level versus survival-level; (b) Boston versus Los Angeles; (c) far-field versus near-field (NF)...8 Effect of site seismicity (EP hysteresis type, α =., SAC survival-level, S D and S E soil): (a) ; (b) p ; (c) r ; (d) n y...8 Effect of site seismicity (EP hysteresis type, α =., SAC design-level, S D and S E soil): (a) ; (b) p ; (c) r ; (d) n y Figure 6.9: Effect of site seismicity (SD, BE, and BP hysteresis types, α =., SAC survival-level, S D soil): (a) ; (b) p ; (c) r ; (d) n y...8 Figure 6.: Effect of epicentral distance (EP hysteresis type, α =., SAC Los Angeles, design-level, S D soil): (a) ; (b) p ; (c) r ; (d) n y...85 Figure 7.: R- spectra for low seismicity (EP hysteresis type, α =.): (a-d) stiff soil profile, S D ; (e-h) soft soil profile, S E...87 Figure 7.: R- spectra for high seismicity (EP hysteresis type, α =.): (a-d) stiff soil profile, S D ; (e-h) soft soil profile, S E...89 Figure 7.3: R- spectra for near-field (Los Angeles design-level S D soil, EP hysteresis type, α =.: (a) AVG versus IND spectra; (b) DES versus IND spectra...9 Figure 7.: Effect of hysteresis type and reference response spectra on the demand (Los Angeles survival-level S D soil, α =.): (a-b) SD hysteresis type; (c-d) BE hysteresis type; (e-f) BP hysteresis type (β s = β r = /3)....9 xi

14 Figure 8.: Figure 8.: Figure 8.3: Figure 8.: Figure 8.5: Figure 8.6: Figure 8.7: Figure 8.8: Figure 8.9: Figure 8.: Figure 8.: Figure 9.: Regression curves (EP hysteresis type, α =.): (a) constant- versus constant-r approaches; (b) IND spectra versus AVG and DES spectra Comparison between mean R- spectra and regression curves (EP hysteresis type, α =.): (a-b) SAC Boston design-level; (c-d) SAC Los Angeles S D soil; (e-f) SAC Los Angeles S E soil; (g) SAC Boston survival-level S E soil; (h) SAC Los Angeles design-level S D soil, NF Effect of hysteretic behavior on regression curves using IND spectra (SAC Los Angeles, survival-level, S D soil): (a) post-yield stiffness ratio, α; (b) hysteresis type; (c) β s = β r...97 Effect of site soil characteristics on regression curves using IND spectra (SAC Los Angeles, design-level): (a) EP hysteresis type; (b) SD hysteresis type; (c) BE hysteresis type; (d) BP hysteresis type Effect of seismic demand level on regression curves using IND spectra (SAC Los Angeles): (a-b) EP hysteresis type; (c) SD hysteresis type; (d) BE hysteresis type; (e) BP hysteresis type Effect of site seismicity on regression curves using IND spectra: (a-d) EP hysteresis type; (e) SD hysteresis type; (f) BE hysteresis type; (g) BP hysteresis type.... Effect of epicentral distance on regression curves using IND spectra (SAC Los Angeles, design-level, S D soil)... Effect of reference response spectra on regression curves (SAC Boston, EP hysteresis type, α =.): (a-b) S D soil; (c-d) S E soil.... Effect of reference response spectra on regression curves (SAC Los Angeles, EP hysteresis type, α =.): (a-b) S D soil; (c-d) S E soil.... Effect of reference spectra on regression curves (SAC Los Angeles, design-level, S D soil, NF, EP hysteresis type, α =.)... Effect of reference response spectra on regression curves (SAC Los Angeles, survival-level, S D soil, α =.): (a) SD hysteresis type; (b) BE hysteresis type; (c) BP hysteresis type (β s = β r. = /3)....3 Matrix plots of cross-correlations: effect of post-yield stiffness ratio, α (EP hysteresis type, SAC Los Angeles, survival-level, S D soil): (a) α =.; (b) α =.5; (c) α =... xii

15 Figure 9.: Figure 9.3: Figure 9.: Figure 9.5: Figure 9.6: Figure 9.7: Figure 9.8: Figure 9.9: Figure 9.: Figure 9.: Figure 9.: Matrix plots of cross-correlations: effect of hysteresis type (α =., SAC Los Angeles, survival-level, S D soil): (a) SD hysteresis type; (b) BE hysteresis type.... Matrix plots of cross-correlations: effect of β s = β r (BP hysteresis type, α =., SAC Los Angeles, survival-level, S D soil): (a) β s = β r = /6; (b) β s = β r = /3; (c) β s = β r = /....3 Matrix plots of cross-correlations: effect of site soil characteristics (EP hysteresis type, α =., SAC Los Angeles, design-level): (a) S D soil; (b) S E soil.... Matrix plots of cross-correlations: effect of site soil characteristics (SD hysteresis type, α =., SAC Los Angeles, design-level): (a) S D soil; (b) S E soil....5 Matrix plots of cross-correlations: effect of site soil characteristics (BE hysteresis type, α =., SAC Los Angeles, design-level): (a) S D soil; (b) S E soil....5 Matrix plots of cross-correlations: effect of site soil characteristics (BP hysteresis type, α =., β s = β r = /3, SAC Los Angeles, design-level): (a) S D soil; (b) S E soil...6 Matrix plots of cross-correlations: effect of seismic demand level (EP hysteresis type, α =., SAC Los Angeles, S D soil): (a) survival-level; (b) design-level....6 Matrix plots of cross-correlations: effect of seismic demand level (EP hysteresis type, α =., SAC Los Angeles, S E soil): (a) survival-level; (b) design-level....7 Matrix plots of cross-correlations: effect of seismic demand level (SD hysteresis type, α =., SAC Los Angeles, S D soil): (a) survival-level; (b) design-level....7 Matrix plots of cross-correlations: effect of seismic demand level (BE hysteresis type, α =., SAC Los Angeles, S D soil): (a) survival-level; (b) design-level....8 Matrix plots of cross-correlations: effect of seismic demand level (BP hysteresis type, α =., β s = β r = /3, SAC Los Angeles, S D soil): (a) survival-level; (b) design-level....9 xiii

16 Figure 9.3: Figure 9.: Figure 9.5: Figure 9.6: Figure 9.7: Figure 9.8: Figure 9.9: Figure 9.: Figure 9.: Figure 9.: Figure 9.3: Matrix plots of cross-correlations: effect of site seismicity (EP hysteresis type, α =., SAC survival-level, S D soil): (a) Los Angeles; (b) Boston....9 Matrix plots of cross-correlations: effect of site seismicity (EP hysteresis type, α =., SAC survival-level, S E soil): (a) Los Angeles; (b) Boston.... Matrix plots of cross-correlations: effect of site seismicity (EP hysteresis type, α =., SAC design-level, S D soil): (a) Los Angeles; (b) Boston... Matrix plots of cross-correlations: effect of site seismicity (EP hysteresis type, α =., SAC design-level, S E soil): (a) Los Angeles; (b) Boston... Matrix plots of cross-correlations: effect of site seismicity (SD hysteresis type, α =., SAC survival-level, S D soil): (a) Los Angeles; (b) Boston.... Matrix plots of cross-correlations: effect of site seismicity (BE hysteresis type, α =., SAC survival-level, S D soil): (a) Los Angeles; (b) Boston.... Matrix plots of cross-correlations: effect of site seismicity (BP hysteresis type, α =., β s = β r = /3, SAC survival-level, S D soil): (a) Los Angeles; (b) Boston... Matrix plots of cross-correlations: effect of epicentral distance (EP hysteresis type, α =., SAC Los Angeles, design-level, S D soil): (a) far-field; (b) near-field....3 Matrix plots of cross-correlations: effect of reference response spectrum (EP hysteresis type, α =., SAC Boston, design-level, S D soil): (a) AVG spectrum; (b) DES spectrum....3 Matrix plots of cross-correlations: effect of reference response spectrum (EP hysteresis type, α =., SAC Boston, survival-level, S D soil): (a) AVG spectrum; (b) DES spectrum...3 Matrix plots of cross-correlations: effect of reference response spectrum (EP hysteresis type, α =., SAC Boston, design-level, S E soil): (a) AVG spectrum; (b) DES spectrum....3 xiv

17 Figure 9.: Figure 9.5: Figure 9.6: Figure 9.7: Figure 9.8: Figure 9.9: Figure 9.3: Figure 9.3: Figure 9.3: Figure 9.33: Figure 9.3: Matrix plots of cross-correlations: effect of reference response spectrum (EP hysteresis type, α =., SAC Boston, survival-level, S E soil): (a) AVG spectrum; (b) DES spectrum....3 Matrix plots of cross-correlations: effect of reference response spectrum (EP hysteresis type, α =., SAC Los Angeles, designlevel, S D soil): (a) AVG spectrum; (b) DES spectrum....3 Matrix plots of cross-correlations: effect of reference response spectrum (EP hysteresis type, α =., SAC Los Angeles, survivallevel, S D soil): (a) AVG spectrum; (b) DES spectrum...3 Matrix plots of cross-correlations: effect of reference response spectrum (EP hysteresis type, α =., SAC Los Angeles, designlevel, S E soil): (a) AVG spectrum; (b) DES spectrum...33 Matrix plots of cross-correlations: effect of reference response spectrum (EP hysteresis type, α =., SAC Los Angeles, survivallevel, S E soil): (a) AVG spectrum; (b) DES spectrum...33 Matrix plots of cross-correlations: effect of reference response spectrum (EP hysteresis type, α =., SAC Los Angeles, designlevel, S D soil, NF): (a) AVG spectrum; (b) DES spectrum....3 Matrix plots of cross-correlations: effect of reference response spectrum (SD hysteresis type, α =., SAC Los Angeles, survivallevel, S D soil): (a) AVG spectrum; (b) DES spectrum....3 Matrix plots of cross-correlations: effect of reference response spectrum (BE hysteresis type, α =., SAC Los Angeles, survivallevel, S D soil): (a) AVG spectrum; (b) DES spectrum Matrix plots of cross-correlations: effect of reference response spectrum (BP hysteresis type, α =., β s = β r = /3, SAC Los Angeles, survival-level, S D soil): (a) AVG spectrum; (b) DES spectrum...35 Λ- regression curves using IND, AVG, and DES spectra (EP hysteresis type, α =., SAC Boston): (a-b) S D soil; (c-d) S E soil Λ- regression curves using IND, AVG, and DES spectra (EP hysteresis type, α =., SAC Los Angeles): (a-b) S D soil; (c-d) S E soil...37 xv

18 Figure 9.35: Λ- regression curves using IND, AVG, and DES spectra (EP hysteresis type, α =., SAC Los Angeles, design-level, S D soil, NF) Figure 9.36: Λ- regression curves using IND, AVG, and DES spectra (α =., SAC Los Angeles, survival-level, S D soil): (a) SD hysteresis type; (b) BE hysteresis type; (c) BP hysteresis type, β s = β r = / Figure.: Figure.: Figure.3: Figure.: Figure.5: Inelastic demand spectra based on DES response spectra (EP hysteresis type, α =.): (a-b) SAC Boston, S D soil; (c-d) SAC Boston, S E soil.... Inelastic demand spectra based on DES response spectra (EP hysteresis type, α =.): (a-b) SAC Los Angeles, S D soil; (c-d) SAC Los Angeles, S E soil... Inelastic demand spectra based on DES response spectra (EP hysteresis type, α =.): SAC Los Angeles, S D soil, NF.... Inelastic demand spectra based on DES response spectra (SAC Los Angeles, survival-level, S D soil, α =.): (a) SD hysteresis type; (b) BE hysteresis type; (c) BP hysteresis type, β s = β r = /3.... Capacity curve-demand spectra for design example: (a) using DES spectrum; (b) using IND spectra... Figure.: Scatter in nlin for the UND S D soil ensemble (EP hysteresis type, α =.) - PGA method compared to: (a) EPA method; (b) EPV method; (c) MIV method; (d) A 95 method; (e) Sˆ a ( T o ) method; (f) Sˆ a ( T o T ) method...8 Figure.: Scatter in nlin using the Sˆ a ( T o ) and Sˆ a ( T o T ) methods compared to the MIV method (EP hysteresis type, α =.): (a) UND S D soil; (b) UND S C soil; (c) UND S E soil; (d) SAC S D soil, NF...5 Figure.3: Scatter in (UND S D soil ensemble, EP hysteresis type, α =.): (a) standard deviation, σ; (b) coefficient of variation, COV...5 Figure.: Effect of hysteresis type on the scatter in nlin (UND S D soil ensemble, α =.): (a) SD type; (b) BE type; (c) BP type (β s = β r = /3)...5 Figure.5: Scatter in nlin for the UND S C soil ensemble (EP hysteresis type, α =.) - PGA method compared to: (a) EPA method; (b) EPV method; xvi

19 (c) MIV method; (d) A 95 method; (e) Sˆ a ( T o ) method; (f) Sˆ a ( T o T ) method...53 Figure.6: Scatter in nlin for the UND S E soil ensemble (EP hysteresis type, α =.) - PGA method compared to: (a) EPA method; (b) EPV method; (c) MIV method; (d) A 95 method; (e) Sˆ a ( T o ) method; (f) Sˆ a ( T o T ) method...5 Figure.7: Figure.8: Figure.9: Figure A.: Scatter in nlin for the SAC Los Angeles, design-level, S D soil, NF ensemble (EP hysteresis type, α =.) - PGA method compared to: (a) EPA method; (b) EPV method; (c) MIV method; (d) A 95 method; (e) Sˆ a ( T o ) method; (f) Sˆ a ( T o T ) method...56 Scatter in MDOF demands for the UND S E soil ensemble using the MIV method and the Sˆ a ( T o T ) method: (a) four-story structure; (b) eight-story structure...58 Covariance in the MDOF demands for the UND S E soil ensemble using the MIV method and the Sˆ a ( T o T ) method: (a) four-story structure; (b) eight-story structure...59 Ground motion records: University of Notre Dame (UND) very dense (S C ) soil ensemble...87 Figure A.: Ground motion records: University of Notre Dame (UND) stiff (S D ) soil ensemble...89 Figure A.3: Ground motion records: University of Notre Dame (UND) soft (S E ) soil ensemble...9 Figure A.: Ground motion records: Nassar and Krawinkler 5s very dense (S C ) soil ensemble...93 Figure A.5: Figure A.6: Figure A.7: Figure A.8: Ground motion records: SAC Boston design-level stiff (S D ) soil ensemble Ground motion records: SAC Boston survival-level stiff (S D ) soil ensemble...97 Ground motion records: SAC Boston design-level soft (S E ) soil ensemble...99 Ground motion records: SAC Boston survival-level soft (S E ) soil ensemble (generated using EERA site response analysis program, Bardet et al., )... xvii

20 Figure A.9: Figure A.: Figure A.: Ground motion records: SAC Los Angeles design-level stiff (S D ) soil ensemble....3 Ground motion records: SAC Los Angeles survival-level stiff (S D ) soil ensemble....5 Ground motion records: SAC Los Angeles design-level soft (S E ) soil ensemble....7 Figure A.: Ground motion records: SAC Los Angeles survival-level soft (S E ) soil ensemble (generated using EERA site response analysis program, Bardet et al., )....9 Figure A.3: Ground motion records: SAC Los Angeles design-level near-field (NF) ensemble... Figure A.: Response spectra: University of Notre Dame (UND) very dense (S C ) soil ensemble...3 Figure A.5: Response spectra: University of Notre Dame (UND) stiff (S D ) soil ensemble...3 Figure A.6: Response spectra: University of Notre Dame (UND) soft (S E ) soil ensemble.... Figure A.7: Response spectra: Nassar and Krawinkler 5s very dense (S C ) soil ensemble... Figure A.8: Response spectra: SAC Boston design-level stiff (S D ) soil ensemble....5 Figure A.9: Figure A.: Figure A.: Response spectra: SAC Boston survival-level stiff (S D ) soil ensemble...5 Response spectra: SAC Boston design-level soft (S E ) soil ensemble...6 Response spectra: SAC Boston survival-level soft (S E ) soil ensemble (generated using EERA site response analysis program, Bardet et al., )...6 Figure A.: Response spectra: SAC Los Angeles design-level stiff (S D ) soil ensemble....7 Figure A.3: Response spectra: SAC Los Angeles survival-level stiff (S D ) soil ensemble....7 Figure A.: Response spectra: SAC Los Angeles design-level soft (S E ) soil ensemble....8 Figure A.5: Response spectra: SAC Los Angeles survival-level soft (S E ) soil ensemble (generated using EERA site response analysis program, Bardet et al., )...8 xviii

21 Figure A.6: Response spectra: SAC Los Angeles design-level stiff (S D ) soil nearfield (NF) ensemble...9 xix

22 LIST OF SYMBOLS a = regression coefficient used for R--T relationship a * = acceleration of the idealized SDOF representation for the structure A 95 = Arias intensity-based parameter AVG = average ground motion response spectrum based on the ground motion ensemble a * y = acceleration capacity b = regression coefficient used for R--T relationship BE = bilinear-elastic hysteresis type BP = combined bilinear-elastic/elasto-plastic hysteresis type c = coefficient used for R--T relationship C = sample covariance c = st term in equation for the c coefficient c = nd term in equation for the c coefficient C a = seismic coefficient tabulated in UBC 997 based on site seismicity and site soil characteristics c d = viscous damping coefficient COV = coefficient of variation C s = seismic response coefficient determined using a smooth design response spectrum xx

23 C v = seismic coefficient tabulated in UBC 997 based on site seismicity and site soil characteristics d = regression coefficient for Λ-T relationships DES = smooth design response spectra from current U.S. seismic design provisions DL = design dead load D sm = strong motion duration E = concrete Young s modulus EL = design earthquake load EH = normalized energy dissipation EP = bilinear elasto-plastic hysteresis type EPA = effective peak acceleration EPV = effective peak velocity E s = Arias Intensity f = regression coefficient used for Λ-T relationships f = design live load modification factor F be = yield capacity for the BE component of the BP hysteresis type F des = design lateral strength based on first significant yield of the structure F elas = linear-elastic force demand F ep = yield capacity for the EP component of the BP hysteresis type F i = lateral force applied at level i F n = lateral force applied at roof level n F nlin = maximum nonlinear force demand xxi

24 F t = portion of the design force concentrated at the top of the structure in addition to F n F y = design lateral strength based on a global yield point of the structure F r y ( ) = required nonlinear force capacity based on bedrock motion below the soft soil site F r ye, = linear-elastic force demand based on bedrock motion below the soft soil site F s y ( ) = required nonlinear (i.e., reduced) force capacity at the soft soil site g = regression coefficient used for relationships between Λ demand indices G = equivalent linear shear modulus G/G max = shear modulus reduction factor h = regression coefficient used for relationships between Λ demand indices h c = column depth h i = height from ground to level i h x = height from ground to level x I = seismic importance factor I e = effective beam moment of inertia IND = response spectra based on the individual ground motion records k = linear-elastic stiffness of the idealized SDOF representation for the structure K = linear-elastic stiffness of the structure k be = linear-elastic stiffness for the BE component of the BP hysteresis type k ep = linear-elastic stiffness for the EP component of the BP hysteresis type k s = linear-elastic moment-rotation stiffness of the zero-length rotational spring model k s = moment-rotation post-yield stiffness of the zero-length rotational spring model xxii

25 K s = moment-curvature post-yield stiffness of the zero-length rotational spring model k sh = shooting stiffness of the SD hysteresis type k tot = total linear-elastic stiffness of the BP hysteresis type k = secant stiffness L = beam clear span length LE = linear-elastic hysteresis type LL = design live load l p = equivalent plastic hinge length m = structure mass m * = equivalent mass of the idealized SDOF representation for the structure MIV = maximum incremental velocity M s = yield moment capacity of the zero-length rotational spring model N = standard penetration resistance N a = near-field factor in UBC 997 based on the closest distance from the site to a known seismic source N v = near-field factor in UBC 997 based on the closest distance from the site to a known seismic source n eq = number of ground motion records NF = near-field n y = number of yield events PGA = peak ground acceleration PI = plasticity index xxiii

26 p x = probability of having x occurrences during the design life, t des p x = probability of exceedance (or occurrence) R = response modification coefficient R des = response modification coefficient based on first significant yield of the structure Rˆ = response modification coefficient proposed by Miranda (993) S = rock soil profile classification per UBC 99 S = stiff soil profile classification per UBC 99 S a = spectral acceleration S ai = inelastic acceleration demand Sˆ a ( ) T o = average spectral acceleration at the structure fundamental period ˆ ( T o T ) = average spectral acceleration over a range of structure periods S a S C = very dense soil profile S d = spectral displacement SD = stiffness-degrading hysteresis type S D = stiff soil profile S d = seismic coefficient used to define smooth design response spectra S D = seismic coefficient corresponding to the design earthquake in IBC S de = linear-elastic displacement demand S di = inelastic displacement demand S ds = seismic coefficient used to define smooth design response spectra S DS = seismic coefficient corresponding to the design earthquake in IBC S E = soft soil profile xxiv

27 S M = seismic coefficient corresponding to the maximum considered earthquake in IBC S MS = seismic coefficient corresponding to the maximum considered earthquake in IBC ST ( s, ) = soft soil modification factor s u = soil unconfined shear strength t = time T = period t des = design life t = final cutoff point T eq = equivalent structure period t init = initial cutoff point t j = time at discretization point j T o = linear-elastic structure fundamental period T r = return period t s = time step used in CDSPEC program T s = soil predominant period of vibration T = elongated structure period V = design structure base shear v s = soil shear wave velocity W = structure seismic weight w i = seismic weight at floor or roof level i w x = seismic weight at floor or roof level x α = post-yield stiffness ratio xxv

28 α be = post-yield stiffness ratio of the BE component for the BP hysteresis type α ep = post-yield stiffness ratio of the EP component for the BP hysteresis type (equal to zero) α φ = moment-curvature post-yield stiffness ratio of the zero-length rotational spring model β r = BP hysteresis type strength ratio β s = BP hysteresis type stiffness ratio γ = shear strain Γ = modal participation factor γ = shear strain rate γ c = shear strain amplitude γ COV = ratio of the COV-spectra for two different sets of parameters δ = dispersion * = displacement of the idealized SDOF representation for the structure be = yield displacement of the BE component for the BP hysteresis type des = yield displacement based on first significant yield of the structure elas = linear-elastic displacement demand ep = yield displacement of the EP component for the BP hysteresis type g = ground motion acceleration i = lateral displacement at floor or roof level i max = maximum mean floor or roof lateral displacement over the height of the structure nlin = maximum nonlinear displacement demand * nlin = maximum nonlinear displacement demand of the idealized SDOF representation for the structure xxvi

29 r = residual displacement demand rmax = maximum possible residual displacement * ult = displacement capacity of the idealized SDOF representation for the structure y = yield displacement based on a global yield point of the structure η = viscosity θ i = interstory drift at story level i θ max = maximum mean interstory drift over the height of the structure θ p = plastic rotation of the zero-length rotational spring model θ s = yield rotation of the zero-length rotational spring model Λ = demand indices, p, r, and n y Λ = log of the demand indices, p, r, and n y = maximum displacement ductility demand = log of the maximum displacement ductility demand, c = cyclic displacement ductility demand E = normalized hysteretic energy demand p = cumulative plastic deformation ductility demand r = residual displacement ductility demand rmax = maximum possible residual displacement ductility demand t = target displacement ductility ξ = viscous damping ratio ξ eq = equivalent viscous damping ratio ρ = correlation coefficient xxvii

30 Ρ = coefficient of determination σ = sample standard deviation τ = shear stress φ p = curvature corresponding to θ p φ s = curvatures corresponding to θ s ϕ = stiffness factor ω = equivalent linear soil frequency xxviii

31 ACKNOWLEDGMENTS The project was funded by the National Science Foundation (NSF) under Grant No. CMS as part of the 999 CAREER Program. The support of the NSF Program Directors Dr. S. C. Liu and Dr. P. Chang is gratefully acknowledged. The authors thank Professor R. Sause of Lehigh University for his comments and suggestions. The opinions, findings, and conclusions expressed in this report are those of the authors and do not necessarily reflect the views of the individuals and organizations acknowledged above. xxix

32 CHAPTER INTRODUCTION. Problem Statement Seismic design and evaluation/rehabilitation approaches in current U.S. building provisions (e.g., International Building Code (ICC, ), FEMA 3 (BSSC, 998), Uniform Building Code (ICBO, 997), FEMA 356 (ASCE, )) advocate the use of several static and dynamic analysis procedures, such as the equivalent lateral force procedure, modal analysis procedure, capacity spectrum procedure, and nonlinear dynamic analysis procedure. Among these procedures, it is common to use linear and nonlinear static methods that dep on capacitydemand index relationships such as the relationship between the design lateral strength and the maximum lateral displacement. Design approaches that use these relationships include the conventional equivalent lateral force procedure (ICC, ; BSSC, 998; ICBO, 997) and the more elaborate capacity spectrum procedure based on inelastic acceleration and displacement spectra (Reinhorn, 997; Freeman, 998; Chopra and Goel, 999; Fajfar, 999). In current practice, the design lateral strength, F des, is often determined by dividing the design lateral force required to keep the structure linear-elastic during an earthquake, F elas,bya response modification coefficient, R = R des, as shown in Figure.a. This force reduction is allowed provided that the resulting maximum nonlinear displacement demand, nlin, can be accommodated. The maximum displacement, nlin, deps on the R coefficient used in design and can be estimated from simple capacity-demand index relationships based on an idealized bilinear lateral force-displacement relationship of the structure as shown in Figure.a. The benefit of using these capacity-demand index relationships comes from their simplicity, however significant deficiencies exist in their development, which are addressed by this research as follows. Foremost, previous research on the development of seismic capacity-demand index relationships is based on F elas values determined using linear-elastic single-degree-of-freedom (SDOF) acceleration response spectra from a selected ensemble of ground motion records. While these relationships may be appropriate for the ground motion ensembles that were used, the seismic design of most building structures is based on smooth response spectra as specified by model building design and rehabilitation codes and provisions. For the design procedures to be consistent, there is a need to develop capacity-demand index relationships using smooth response spectra from current design provisions. This is particularly important for near-field ground motion records and for sites with soft soil conditions where

33 F elas CHANGE IN LATERAL DISPLACEMENT FORCE REDUCTION lateral force, F F y F = des F nlin F = elas R F elas R des 3 αk K des y "significant yield" Linear-Elastic Behavior elas lateral displacement, (a) Nonlinear Behavior 3 Idealized Bilinear Behavior = nlin y W. F elas R BOSTON DESIGN-LEVEL SOFT (S E) SOIL Ground motion response spectrum (ξ = 5%) Design response spectrum, IBC (ξ = 5%) R = (linear-elastic). unconservative inconsistency in determining F elas/ R R = period, (b) Figure.: (a) Lateral force-displacement relationships; (b) Ground motion response spectra versus smooth design response spectra.

34 the characteristics of the ground motion spectra may be significantly different than the characteristics of smooth design response spectra as shown in Figure.b (where W is the total seismic weight assigned to the lateral load-resisting system). As illustrated in Figure.b, this inconsistency may result in different values for the design lateral force, F elas /R, for the same value of the R coefficient. In particular, the design force using the smooth spectrum may be lower than the design force using the ground motion spectrum, leading to an unconservative design. To the best of the authors knowledge, previous research on the development of capacitydemand index relationships using smooth design response spectra instead of linear-elastic ground motion response spectra as the basis for calculating F y = F elas /R has not been published in the literature. This research shows that previous relationships developed using linear-elastic ground motion response spectra can be significantly different than those developed using smooth design response spectra and can lead to unconservative designs, particularly for survival-level, soft soil, and near-field ground motions. Furthermore, the seismic capacity-demand index relationships available in the literature are limited to the maximum displacement ductility demand, = nlin / y. The ductility demand,, provides an estimate of the maximum nonlinear displacement demand of the structure corresponding to the R coefficient used in design. While these relationships may be adequate for the basic seismic design of building structures, other demand indices such as to quantify cumulative damage, hysteretic energy, and residual displacement are needed for use in the framework of a performance-based design approach that allows the designer to specify and predict, with reasonable accuracy, the performance (degree of damage) of a building for a specified level of ground motion intensity. To the best of the authors knowledge, previous research on the development of closedform relationships for a comprehensive set of seismic demand indices has not been published in the literature. This research develops such relationships and shows that the correlation between and the other demand indices is usually relatively strong. In some cases, the cross-correlations between the demand indices show weak to no correlation, indicating that these demand indices can carry indepent measures of seismic demand. As an alternative to capacity-demand index relationships, the use of nonlinear dynamic time-history procedures in design has become increasingly popular and cost-effective in recent years due to the expanding availability of faster computers and better analysis models. Nonlinear dynamic time-history analyses conducted as a part of a performance-based design approach require that the ground motion records are scaled to a specified level of seismic intensity. Current seismic design provisions provide guidelines for the number of ground motion records to be used in the nonlinear dynamic analysis procedure, however the manner in which these ground motion records are to be scaled is not explicitly addressed. Merely, the provisions require that the average acceleration response spectrum of the ground motion ensemble is not less than. times a 5%- damped smooth design response spectrum, specified by the provisions, for periods between.t and.5t, where T is the structure fundamental period (ICC, ; ASCE, ; ICBO, 997). While the intent is straightforward (i.e., to match or exceed. times the smooth design response spectrum), there are several ways that a ground motion ensemble can be scaled so as to produce an average response spectrum that satisfies this requirement. However, recent research has dem- 3

35 onstrated that some scaling methods result in an excessive scatter in the estimated seismic demands, indicating that these scaling methods may not be able to adequately define the damage potential (i.e., seismic intensity) for given site conditions and structural characteristics. For example, nonlinear dynamic analysis procedures have been often performed using ground motions scaled to constant peak ground motion characteristics (e.g., peak ground acceleration, PGA, and peak ground velocity, PGV). However, it has been shown that this method of ground motion scaling introduces a large scatter in the analysis results (Nau and Hall, 98; Miranda, 993; Vidic et al., 99; Shome and Cornell, 998). This indicates that the seismic demand estimates may be biased, leading to designs with significant uncertainty and unknown margins of safety, unless a relatively large ensemble of ground motions are used (Shome and Cornell, 998). Therefore, it is imperative that nonlinear dynamic analysis procedures are performed using ground motion records scaled based on methods that adequately define the damage potential for given site conditions and structural characteristics, thus resulting in consistent prediction of the demand estimates by minimizing the scatter. To the best of the authors knowledge, an investigation of ground motion scaling methods incorporating different site conditions and structure characteristics (e.g., site soil characteristics, epicentral distance, hysteretic behavior), which is one of the main focuses of this research, has not been previously published in the literature. The research shows that scaling methods that work well for ground motion records on stiff soil profiles lose their effectiveness for ground motions on other site conditions, particularly on soft soil profiles and for large R coefficients. A new scaling method is proposed for use under these conditions.. Research Objectives The broad objective of this research is to address some of the research needs for the implementation of performance-based procedures in current U.S. seismic design provisions. The research has four specific objectives: () to develop new nonlinear SDOF capacity-demand index relationships based on linear-elastic smooth design response spectra; () to develop new relationships that quantify cumulative damage, hysteretic energy, and residual displacement demands; (3) to investigate the effects of the structure fundamental period of vibration, strength level, hysteretic behavior, and site conditions on the demand estimates; and () to investigate the effect of the ground motion scaling method on the scatter in the demand estimates..3 Scope and Organization Chapter presents a review of the previous research on capacity-demand index relationships, site conditions, hysteresis type, current seismic design procedures, and ground motion scaling methods. Chapter 3 describes the research program in terms of the analytical models, ground motion records, analysis procedures, demand indices, and statistical evaluation of results. Three major suites of ground motion records are used in this research: () ground motions compiled by