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Capacity-Demand Index Relationships for Performance-
Based Seismic Design
Structural Engineering Research Report
Department of Civil Engineering and Geological SciencesUniversity of Notre Dame
Notre Dame, Indiana
November 2001
Kenneth T. Farrow and Yahya C. Kurama Report #NDSE-01-02
2
0 0 T (sec) 2.0
MIV)S (T
a oˆ
MIVT )
µS (T →
a oˆ
= 1, 2, 4, 6, 8 (thin → thick lines)R
γ CO
V(∆ nl
in )
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
250 10
0
4
µr
5000
4
0 250
4
0 500
25
ny
0 100
50
0 40
50
µp
0 250
50
dataregression
d = 1.41, f = 0.72
d = 0.85, f = 1.17
d = 2.59, f = 1.23
d = 1.41, f = 0.72 d = 0.85, f = 1.17 d = 2.59, f = 1.23
g = 0.64, h = 1.75
g = 1.56, h = 0.57 g = 0.26, h = 0.34
g = 3.92, h = 2.93
g = 0.24, h = 0.70
g = 4.10, h = 1.42
ρ = 0.99
ρ = 0.97
ρ = 0.90
ρ = 0.96
ρ = 0.89 ρ = 0.95
ρ = 0.99 ρ = 0.97 ρ = 0.90
ρ = 0.89ρ = 0.96
ρ = 0.95
02468101214
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraDES spectrum
Capacity-Demand Index Relationships for Performance-
Based Seismic Design
Structural Engineering Research Report
Department of Civil Engineering and Geological SciencesUniversity of Notre Dame
Notre Dame, Indiana
November 2001
byKenneth T. Farrow
Former GraduateResearch Assistant
and
Yahya C. KuramaAssistant Professor
Report #NDSE-01-02
This report may be downloaded from http://www.nd.edu/~concrete/
ABSTRACT
Seismic design approaches in current U.S. building provisions advocate using severalstatic and dynamic analysis procedures. Among the static procedures, it is common to use linearand nonlinear methods that depend on capacity-demand index relationships (e.g., the relationshipbetween the design lateral strength and the maximum lateral displacement). The benefit of usingthese relationships comes from their simplicity and adaptability, however significant deficienciesexist in their development.
Foremost, previous research on the development of capacity-demand index relationshipsis based on linear-elastic single-degree-of-freedom acceleration response spectra, whereas currentdesign procedures are based on “smooth” design response spectra. For the design procedures to beconsistent, new relationships need to be developed using smooth design response spectra.
Furthermore, previous research on capacity-demand index relationships is limited to themaximum displacement ductility demand. However, other demand indices such as to quantifycumulative damage and residual displacement are needed for use in the framework of a perfor-mance-based design approach that allows the designer to specify and predict the performance of abuilding under an earthquake.
Finally, using nonlinear dynamic analysis procedures as part of a performance-baseddesign approach has become increasingly common. These procedures are often conducted usingground 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 meth-ods that adequately define the damage potential for given site conditions and structural character-istics, thus resulting in consistent prediction of the demand estimates by minimizing the scatter.
This research proposes new capacity-demand index relationships and ground motion scal-ing methods and shows that: (1) previous capacity-demand index relationships developed usinglinear-elastic ground motion spectra can lead to unconservative designs, particularly for survival-level, soft soil, and near-field conditions; (2) the correlation between the maximum displacementductility demand and other demand indices is relatively strong; and (3) scaling methods that workwell for ground motions recorded on stiff soil and far-field conditions lose their effectiveness forsoft soil and near-field conditions.
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CONTENTS
LIST OF TABLES ...........................................................................................................................v
LIST OF FIGURES ..................................................................................................................... viii
LIST OF SYMBOLS.....................................................................................................................xx
ACKNOWLEDGEMENTS ....................................................................................................... xxix
CHAPTER 1 INTRODUCTION .......................................................................................11.1 Problem Statement ..................................................................................................11.2 Research Objectives.................................................................................................41.3 Dissertation Scope and Organization.......................................................................41.4 Research Significance ..............................................................................................6
CHAPTER 2 BACKGROUND .........................................................................................72.1 Previous Research on Maximum Displacement Ductility Demand.........................7
2.1.1 Site soil characteristics.................................................................................92.1.2 Near-field ground motions .........................................................................102.1.3 Hysteretic lateral load-displacement behavior...........................................11
2.2 Previous Research on Other Demand Indices........................................................122.3 Current Capacity-Based Design Procedures..........................................................12
2.3.1 Equivalent lateral force (ELF) procedure ..................................................122.3.2 Capacity spectrum procedure.....................................................................14
2.4 Scaling of Ground Motion Records .......................................................................17
CHAPTER 3 DESCRIPTION OF THE RESEARCH PROGRAM................................193.1 Analytical Models..................................................................................................19
3.1.1 Single-degree-of-freedom (SDOF) models................................................193.1.2 Multi-degree-of-freedom (MDOF) models................................................25
3.2 Seismic Demand Levels.........................................................................................333.3 Ground Motion Records ........................................................................................34
3.3.1 Important properties of the ground motion records ...................................383.3.2 Strong motion duration ..............................................................................39
3.4 Ground Motion Scaling Methods...........................................................................40
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3.5 Reference Response Spectra ..................................................................................433.6 Nonlinear Dynamic Time-History Analyses..........................................................47
3.6.1 SDOF analyses...........................................................................................473.6.2 MDOF analyses .........................................................................................48
3.7 Statistical Evaluation of the Results ......................................................................513.7.1 SDOF demand estimates............................................................................523.7.2 MDOF demand estimates ..........................................................................56
CHAPTER 4 VALIDATION OF ANALYTICAL MODEL AND COMPARISONWITH PREVIOUS RESULTS ..................................................................58
4.1 Nonlinear Dynamic Time-History Analyses..........................................................584.2 Spectral Analyses...................................................................................................624.3 Comparison with Nassar and Krawinkler (1991) ..................................................62
4.3.1 Constant-R versus constant-µ approaches .................................................624.3.2 IND spectra versus smooth response spectra.............................................65
CHAPTER 5 EFFECT OF HYSTERETIC BEHAVIOR................................................675.1 Effect of Post-Yield Stiffness Ratio,α...................................................................675.2 EP Hysteresis Type versus SD Hysteresis Type ....................................................675.3 EP Hysteresis Type versus BE Hysteresis Type ....................................................705.4 EP Hysteresis Type versus BP Hysteresis Type.....................................................70
CHAPTER 6 EFFECT OF SITE CONDITIONS............................................................746.1 Site Soil Characteristics .........................................................................................746.2 Seismic Demand Level ..........................................................................................766.3 Site Seismicity .......................................................................................................796.4 Epicentral Distance ................................................................................................82
CHAPTER 7 EFFECT OF REFERENCE RESPONSE SPECTRA ...............................867.1 Low Seismicity (Boston), Stiff Soil Profile (SD) ...................................................86
7.2 Low Seismicity (Boston), Soft Soil Profile (SE) ....................................................86
7.3 High Seismicity (Los Angeles), Stiff Soil Profile (SD)..........................................88
7.4 High Seismicity (Los Angeles), Soft Soil Profile (SE) ..........................................88
7.5 High Seismicity (Los Angeles), Near-Field (NF), Stiff Soil Profile (SD)..............90
7.6 SD, BE, and BP Hysteresis Types .........................................................................91
CHAPTER 8 REGRESSION ANALYSES FORµ .........................................................938.1 Comparison ofR-µ-T Relationships with Previous Results ..................................938.2 R-µ-T Relationships Developed in this Study........................................................94
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CHAPTER 9 REGRESSION ANALYSES BETWEEN THE DEMAND INDICES...1049.1 Regression Relationships Developed Based on IND Spectra ..............................1049.2 Effect of Reference Response Spectra on the Relationships Between
the Demand Indices..............................................................................................123
CHAPTER 10 RECOMMENDED DESIGN PROCEDURE USING CAPACITY-DEMAND INDEX RELATIONSHIPS...................................................140
10.1 Inelastic Demand Spectra ....................................................................................14010.2 Design Example...................................................................................................140
CHAPTER 11 EFFECT OF GROUND MOTION SCALING METHOD .....................14711.1 EP Hysteresis Type ..............................................................................................14711.2 SD, BE, and BP Hysteresis Types .......................................................................15011.3 Site Soil Characteristics .......................................................................................15111.4 Epicentral Distance ..............................................................................................15511.5 Results for the MDOF Frame Structures .............................................................157
CHAPTER 12 SUMMARY, CONCLUSIONS, AND FUTURE RESEARCH..............16012.1 Summary..............................................................................................................16012.2 Conclusions..........................................................................................................16212.3 Future Research ...................................................................................................166
REFERENCES ............................................................................................................................167
APPENDIX A CHARACTERISTICS, TIME HISTORIES, ANDRESPONSE SPECTRA OF GROUND MOTION RECORDS ..............174
APPENDIX B CDSPEC (CAPACITY-DEMANDSPECTRA) PROGRAM LISTING ...220B.1 CDSPEC.M: Main Program ................................................................................220B.2 EQSCALE.M: Ground Motion Scaling Function................................................224B.3 NEHRPDES.M: Smooth Design Reference Spectrum Function.........................229B.4 LNLTHIST.M, LNLTHISTSD.M: Nonlinear Dynamic Time-History Analysis
Functions..............................................................................................................230B.5 CDSPECPOST.M: Post-Processing Function .....................................................235B.6 SDREG.M:R-µ-T Nonlinear Regression Program..............................................243B.7 DEMANDREG.M: Cross-Correlation Nonlinear Regression Program ..............252B.8 NKREGRESSION.M, NKREGRESSION_ST.M, NKREGRESSION_ME.M,
NKREGRESSION_SO.M, NKREGRESSION_NF.M: Regression Functions ...255B.9 EPDISP.M, BEDISP.M, SDDISP.M: Hysteretic Rule Functions ........................257
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LIST OF TABLES
Table 2.1: Values fora andb coefficients .................................................................................8
Table 2.2: Demand indices introduced by Mahin and Lin (1983). .........................................13
Table 3.1: Frame lateral-system properties.............................................................................27
Table 3.2: Gravity loads..........................................................................................................27
Table 3.3: Yield moment capacities for beam rotational springs............................................31
Table 3.4: Column base fiber element properties....................................................................32
Table 3.5: Site soil definitions (adapted fromIBC 2000 (ICC, 2000)) ...................................35
Table 3.6: Values fora andb coefficients developed in this study .........................................43
Table 3.7: Seismic coefficients for the smooth design (DES) response spectra .....................45
Table 3.8: SAC/N&K ground motion ensembles: parameters studied (shadedareas indicate the N&K ensemble) ........................................................................49
Table 3.9: UND/SAC ground motion ensembles: parameters studied (shadedareas indicate the SAC ensemble)..........................................................................50
Table 8.1: Regression coefficientsa andb for the N&K ground motion ensemble,EP hysteresis type,α = 0.10 ..................................................................................94
Table 8.2: Regression coefficientsa andb for the SAC ground motion ensemble.................95
Table 9.1: Regression coefficientsd andf: EP hysteresis type, IND spectra........................105
Table 9.2: Regression coefficientsd andf: SD, BE, and BP hysteresis types,IND spectra ..........................................................................................................106
Table 9.3: Regression coefficientsg andh: EP hysteresis type, IND spectra.......................107
Table 9.4: Regression coefficientsg andh: SD, BE, and BP hysteresis types,IND spectra ..........................................................................................................108
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Table 9.5: Correlation coefficient,ρ: EP hysteresis type, IND spectra ................................109
Table 9.6: Correlation coefficient,ρ: SD, BE, and BP hysteresis types, INDspectra ..................................................................................................................110
Table 9.7: Regression coefficientsd andf: AVG spectra ......................................................124
Table 9.8: Regression coefficientsd andf: DES spectra.......................................................125
Table 9.9: Regression coefficientsg andh: AVG spectra .....................................................126
Table 9.10: Regression coefficientsg andh: DES spectra......................................................127
Table 9.11: Correlation coefficient,ρ: AVG spectra...............................................................128
Table 9.12: Correlation coefficient,ρ: DES spectra ...............................................................129
Table 10.1: Structure properties and results for the design example ......................................145
Table 10.2: µp, µr, andny demands for the design example ...................................................146
Table A.1: University of Notre Dame (UND) very dense (SC) soil ensemble.......................174
Table A.2: University of Notre Dame (UND) stiff (SD) soil ensemble .................................175
Table A.3: University of Notre Dame (UND) soft (SE) soil ensemble..................................176
Table A.4: Nassar and Krawinkler (N&K) 15s very dense (SC) soil ensemble.....................177
Table A.5: SAC Boston design-level stiff (SD) soil ensemble...............................................178
Table A.6: SAC Boston survival-level stiff (SD) soil ensemble.............................................179
Table A.7: SAC Boston design-level soft (SE) soil ensemble................................................180
Table A.8: SAC Boston survival-level soft (SE) soil ensemble .............................................181
Table A.9: SAC Los Angeles design-level stiff (SD) soil ensemble ......................................182
Table A.10: SAC Los Angeles survival-level stiff (SD) soil ensemble....................................183
Table A.11: SAC Los Angeles design-level soft (SE) soil ensemble.......................................184
Table A.12: SAC Los Angeles survival-level soft (SE) soil ensemble.....................................185
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Table A.13: SAC Los Angeles near-field design-level stiff (SD) soil ensemble......................186
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LIST OF FIGURES
Figure 1.1: (a) Lateral force-displacement relationships; (b) Ground motionresponse spectra versus smooth design response spectra. .......................................2
Figure 2.1: R-µ-T relationships developed by Miranda (1993) for rock and soft soil sites. ....10
Figure 2.2: Hysteresis types used by Foutch and Shi (1998) ...................................................11
Figure 2.3: Bilinear elasto-plastic (EP) hysteresis type with definitions for demand indicesin Table 2.2.............................................................................................................13
Figure 2.4: Capacity spectrum procedures: (a) highly-damped linear-elasticdemand spectra; (b) inelastic demand spectra. ......................................................15
Figure 3.1: SDOF model properties. ........................................................................................20
Figure 3.2: Hysteresis types: (a) LE; (b) EP; (c) SD; (d) BE; (e) BP. ......................................21
Figure 3.3: Schematic of stiffness degrading (SD) hysteresis type. .........................................22
Figure 3.4: Construction of the bilinear-elastic/elasto-plastic (BP) hysteresis type.................23
Figure 3.5: BP hysteresis type with: (a)βs > βr, βr < 1; (b)βs = βr < 1. .................................24
Figure 3.6: EP hysteresis type versus BP hysteresis type: (a)βs > βr; (b) βs = βr (matchingyield point). ............................................................................................................25
Figure 3.7: Layout of structural system: (a) elevation; (b) four-story structure;(c) eight-story structure..........................................................................................26
Figure 3.8: MDOF models: (a) four-story elevation; (b) eight-story elevation;(c) close-up of analytical model.............................................................................29
Figure 3.9: Element models: (a) beam end rotational spring element; (b) columnbase fiber element ..................................................................................................30
Figure 3.10: Normalized cyclic base-shear-roof-drift behavior: (a) four-story frame;(b) eight-story frame. .............................................................................................32
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Figure 3.11: Equivalent linear model for the site response analyses: (a) assumed nonlinearhysteretic stress-strain behavior of the soil; (b) equivalentlinear model. ..........................................................................................................36
Figure 3.12: Soil properties for the site response analyses: (a) shear wave velocity;(b) shear modulus reduction factor and damping ratio. .........................................38
Figure 3.13: Acceleration time-history, cumulative RMSA function (CRF), and derivative ofthe CRF function (1940 El Centro (ELCN) x 2.0):(a) forward CRF; (b) reverse CRF. ........................................................................39
Figure 3.14: Scaling based on the spectral acceleration (UNDSC soil ground motionensemble): (a) at the structure period ( method); (b) over arange of structure periods ( method). ..............................................41
Figure 3.15: Design spectra: (a) general shape; (b) design-level; (c) survival-level. .................44
Figure 3.16: Smooth response spectra: (a) Boston, design-level,SD soil; (b) Boston, survival-level,SD soil...........................................................................................................45
Figure 3.17: Smooth response spectra: (a) Boston, design-level,SE soil; (b) Boston, survival-level,SE soil. ..........................................................................................................46
Figure 3.18: Smooth response spectra: (a) Los Angeles, design-level,SD soil;(b) Los Angeles, survival-level,SD soil. ................................................................46
Figure 3.19: Smooth response spectra: (a) Los Angeles, design-level,SE soil;(b) Los Angeles, survival-level,SE soil. ................................................................46
Figure 3.20: Smooth response spectra: (a) Los Angeles, design-level,SC soil;(b) Los Angeles, design-level,SD soil,NF. ...........................................................47
Figure 3.21: Flowchart describing parameters studied in the analytical procedure. ..................51
Figure 3.22: Average response spectra of the ground motions used in the MDOF analyses:(a) MIV scaling method; (b) scaling method. ................................52
Figure 3.23: Regression analysis (IND spectra, EP hysteresis type,α = 0.10):(a) first step,R-µ domain (T = 0.92 sec.); (b) second step,c-T domain. ...............53
Figure 3.24: Effect ofa andb coefficients onc coefficient: (a)c1 term; (b)c2 term;(c) c = c1 + c2. ........................................................................................................54
Figure 3.25: Effect ofa andb coefficients onµ: (a)b = 0.1; (b)b = 1.0; (c)b = 2.0. ...............55
Saˆ To( )
Saˆ To Tµ→( )
Saˆ To Tµ→( )
x
Figure 4.1: Comparison between CDSPEC and DRAIN-2DX (EP hysteresis type,α = 0.10,R = 8): (a-b) EQ09; (c-d) EQ15. ............................................................59
Figure 4.2: Comparison between CDSPEC and BISPEC (SD hysteresis type,α = 0.10,R =8): (a-b) LPPR; (c-d) PACH; (e) different reloading rules.....................................60
Figure 4.3: Comparison between CDSPEC and DRAIN-2DX (α = 0.10,R = 8):(a-d) BE hysteresis type; (e-h) BP hysteresis type,βs = βr = 1/3. .........................61
Figure 4.4: Comparison between CDSPEC and BISPEC for the N&K groundmotion ensemble (α = 0.10): (a) EP hysteresis type; (b) SD hysteresis type. .......62
Figure 4.5: R-µ relationship for the EP hysteresis type withα = 0.00 andT = 0.20 sec.under the 1983 Coalinga Parkfield Zone 16 ground motion (IND responsespectrum)................................................................................................................63
Figure 4.6: R-µ relationships for the EP hysteresis type withα = 0.00 (INDreference spectra): (a)T = 0.20 sec.; (b)T = 0.92 sec. ..........................................64
Figure 4.7: R-µ spectra (N&K ensemble, EP hysteresis type,α = 0.10): (a) AVG versusIND spectra; (b) DES versus IND spectra. ............................................................65
Figure 4.8: Smooth design (DES) response spectrum versus average (AVG)ground motion response spectrum. ........................................................................66
Figure 4.9: R-µ relationships using different reference response spectra (EPhysteresis type,α = 0.10,T = 3.0 sec.): (a) N&K ensemble; (b) N&K ensemblewithout EQ14. ........................................................................................................66
Figure 5.1: Effect of post-yield stiffness ratio,α (EP hysteresis type, SAC Los Angeles,survival-level,SD soil): (a)µ; (b) µp; (c) µr; (d) ny. ...............................................68
Figure 5.2: EP versus SD hysteresis types (α = 0.10, SAC Los Angeles,survival-level,SD soil): (a)µ; (b) µp; (c) µr; (d) ny................................................69
Figure 5.3: Comparison between force-displacement responses of the EP and SDhysteresis types: (a)α = 0.10; (b)α = 0.00. ..........................................................70
Figure 5.4: EP versus BE hysteresis types (α = 0.10, SAC Los Angeles,survival-level,SD soil): (a)µ; (b) µp; (c) µr; (d) ny................................................71
Figure 5.5: EP versus BP hysteresis types (α = 0.10, SAC Los Angeles,survival-level,SD soil): (a)µ; (b) µp; (c) µr; (d) ny................................................72
Figure 5.6: Hysteretic energy dissipation of the BP hysteresis type. .......................................73
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Figure 6.1: Effect of site soil characteristics (EP hysteresis type,α = 0.10, SACLos Angeles, design-level): (a)µ; (b) µp; (c) µr; (d) ny.........................................75
Figure 6.2: Effect of site soil characteristics (SD, BE, and BP hysteresis types,α = 0.10, SAC Los Angeles, design-level): (a)µ; (b) µp; (c) µr; (d) ny. ...............76
Figure 6.3: Response spectra: (a)SD soil ground motion spectrum versusSE soil groundmotion spectrum; (b) ground motion spectra versus smoothdesign spectra.........................................................................................................77
Figure 6.4: Effect of seismic demand level (EP hysteresis type,α = 0.10, SACLos Angeles,SD andSE soil): (a)µ; (b) µp; (c) µr; (d) ny. ....................................78
Figure 6.5: Effect of seismic demand level (SD, BE, and BP hysteresis types,α = 0.10,SAC Los Angeles,SD soil): (a)µ; (b) µp; (c) µr; (d) ny.........................................79
Figure 6.6: Average response spectra: (a) design-level versus survival-level;(b) Boston versus Los Angeles; (c) far-field versus near-field (NF)......................80
Figure 6.7: Effect of site seismicity (EP hysteresis type,α = 0.10, SACsurvival-level,SD andSE soil): (a)µ; (b) µp; (c) µr; (d) ny. ...................................81
Figure 6.8: Effect of site seismicity (EP hysteresis type,α = 0.10, SACdesign-level,SD andSE soil): (a)µ; (b) µp; (c) µr; (d) ny. .....................................83
Figure 6.9: Effect of site seismicity (SD, BE, and BP hysteresis types,α = 0.10,SAC survival-level,SD soil): (a)µ; (b) µp; (c) µr; (d) ny. ......................................84
Figure 6.10: Effect of epicentral distance (EP hysteresis type,α = 0.10, SAC Los Angeles,design-level,SD soil): (a)µ; (b) µp; (c) µr; (d) ny. .................................................85
Figure 7.1: R-µ spectra for low seismicity (EP hysteresis type,α = 0.10):(a-d) stiff soil profile,SD; (e-h) soft soil profile,SE...............................................87
Figure 7.2: R-µ spectra for high seismicity (EP hysteresis type,α = 0.10):(a-d) stiff soil profile,SD; (e-h) soft soil profile,SE...............................................89
Figure 7.3: R-µ spectra for near-field (Los Angeles design-levelSD soil, EPhysteresis type,α = 0.10: (a) AVG versus IND spectra; (b) DESversus IND spectra. ................................................................................................90
Figure 7.4: Effect of hysteresis type and reference response spectra on theµdemand (Los Angeles survival-levelSD soil, α = 0.10): (a-b) SD hysteresistype; (c-d) BE hysteresis type; (e-f) BP hysteresis type (βs = βr = 1/3). ...............92
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Figure 8.1: Regression curves (EP hysteresis type,α = 0.10): (a) constant-µversus constant-R approaches; (b) IND spectra versus AVG and DES spectra. ....93
Figure 8.2: Comparison between meanR-µ spectra and regression curves (EP hysteresistype,α = 0.10): (a-b) SAC Boston design-level; (c-d) SACLos AngelesSD soil; (e-f) SAC Los AngelesSE soil; (g) SAC Bostonsurvival-levelSE soil; (h) SAC Los Angeles design-levelSD soil,NF. .................96
Figure 8.3: Effect of hysteretic behavior on regression curves using IND spectra (SACLos Angeles, survival-level,SD soil): (a) post-yield stiffnessratio,α; (b) hysteresis type; (c)βs = βr..................................................................97
Figure 8.4: Effect of site soil characteristics on regression curves using INDspectra (SAC Los Angeles, design-level): (a) EP hysteresis type;(b) SD hysteresis type; (c) BE hysteresis type; (d) BP hysteresis type. ................98
Figure 8.5: Effect of seismic demand level on regression curves using INDspectra (SAC Los Angeles): (a-b) EP hysteresis type; (c) SDhysteresis type; (d) BE hysteresis type; (e) BP hysteresis type. ............................99
Figure 8.6: Effect of site seismicity on regression curves using IND spectra:(a-d) EP hysteresis type; (e) SD hysteresis type; (f) BE hysteresistype; (g) BP hysteresis type. ................................................................................100
Figure 8.7: Effect of epicentral distance on regression curves using IND spectra(SAC Los Angeles, design-level,SD soil)............................................................101
Figure 8.8: Effect of reference response spectra on regression curves (SACBoston, EP hysteresis type,α = 0.10): (a-b)SD soil; (c-d)SE soil. .....................101
Figure 8.9: Effect of reference response spectra on regression curves (SAC Los Angeles,EP hysteresis type,α = 0.10): (a-b)SD soil; (c-d)SE soil. ..................................102
Figure 8.10: Effect of reference spectra on regression curves (SAC Los Angeles,design-level,SD soil,NF, EP hysteresis type,α = 0.10)......................................102
Figure 8.11: Effect of reference response spectra on regression curves (SAC Los Angeles,survival-level,SD soil, α = 0.10): (a) SD hysteresis type;(b) BE hysteresis type; (c) BP hysteresis type (βs = βr. = 1/3). ...........................103
Figure 9.1: Matrix plots of cross-correlations: effect of post-yield stiffness ratio,α (EPhysteresis type, SAC Los Angeles, survival-level,SD soil): (a)α = 0.00;(b) α = 0.05; (c)α = 0.10.....................................................................................111
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Figure 9.2: Matrix plots of cross-correlations: effect of hysteresis type (α = 0.10, SACLos Angeles, survival-level,SD soil): (a) SD hysteresis type;(b) BE hysteresis type. .........................................................................................112
Figure 9.3: Matrix plots of cross-correlations: effect ofβs = βr (BP hysteresistype,α = 0.10, SAC Los Angeles, survival-level,SD soil): (a)βs = βr = 1/6;(b) βs = βr = 1/3; (c)βs = βr = 1/2. ......................................................................113
Figure 9.4: Matrix plots of cross-correlations: effect of site soil characteristics (EPhysteresis type,α = 0.10, SAC Los Angeles, design-level): (a)SD soil;(b) SE soil. ............................................................................................................114
Figure 9.5: Matrix plots of cross-correlations: effect of site soil characteristics (SDhysteresis type,α = 0.10, SAC Los Angeles, design-level): (a)SD soil;(b) SE soil. ............................................................................................................115
Figure 9.6: Matrix plots of cross-correlations: effect of site soil characteristics (BEhysteresis type,α = 0.10, SAC Los Angeles, design-level): (a)SD soil;(b) SE soil. ............................................................................................................115
Figure 9.7: Matrix plots of cross-correlations: effect of site soil characteristics (BPhysteresis type,α = 0.10,βs = βr = 1/3, SAC Los Angeles,design-level): (a)SD soil; (b)SE soil....................................................................116
Figure 9.8: Matrix plots of cross-correlations: effect of seismic demand level (EPhysteresis type,α = 0.10, SAC Los Angeles,SD soil):(a) survival-level; (b) design-level. ......................................................................116
Figure 9.9: Matrix plots of cross-correlations: effect of seismic demand level (EPhysteresis type,α = 0.10, SAC Los Angeles,SE soil):(a) survival-level; (b) design-level. ......................................................................117
Figure 9.10: Matrix plots of cross-correlations: effect of seismic demand level (SDhysteresis type,α = 0.10, SAC Los Angeles,SD soil):(a) survival-level; (b) design-level. ......................................................................117
Figure 9.11: Matrix plots of cross-correlations: effect of seismic demand level (BEhysteresis type,α = 0.10, SAC Los Angeles,SD soil):(a) survival-level; (b) design-level. ......................................................................118
Figure 9.12: Matrix plots of cross-correlations: effect of seismic demand level (BPhysteresis type,α = 0.10,βs = βr = 1/3, SAC Los Angeles,SD soil):(a) survival-level; (b) design-level. ......................................................................119
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Figure 9.13: Matrix plots of cross-correlations: effect of site seismicity (EPhysteresis type,α = 0.10, SAC survival-level,SD soil): (a) LosAngeles; (b) Boston. ............................................................................................119
Figure 9.14: Matrix plots of cross-correlations: effect of site seismicity (EPhysteresis type,α = 0.10, SAC survival-level,SE soil): (a) LosAngeles; (b) Boston. ............................................................................................120
Figure 9.15: Matrix plots of cross-correlations: effect of site seismicity (EPhysteresis type,α = 0.10, SAC design-level,SD soil): (a) Los Angeles;(b) Boston.............................................................................................................120
Figure 9.16: Matrix plots of cross-correlations: effect of site seismicity (EPhysteresis type,α = 0.10, SAC design-level,SE soil): (a) Los Angeles;(b) Boston.............................................................................................................121
Figure 9.17: Matrix plots of cross-correlations: effect of site seismicity (SDhysteresis type,α = 0.10, SAC survival-level,SD soil): (a) LosAngeles; (b) Boston. ............................................................................................121
Figure 9.18: Matrix plots of cross-correlations: effect of site seismicity (BEhysteresis type,α = 0.10, SAC survival-level,SD soil): (a) LosAngeles; (b) Boston. ............................................................................................122
Figure 9.19: Matrix plots of cross-correlations: effect of site seismicity (BPhysteresis type,α = 0.10,βs = βr = 1/3, SAC survival-level,SD soil):(a) Los Angeles; (b) Boston.................................................................................122
Figure 9.20: Matrix plots of cross-correlations: effect of epicentral distance (EP hysteresistype,α = 0.10, SAC Los Angeles, design-level,SD soil):(a) far-field; (b) near-field. ...................................................................................123
Figure 9.21: Matrix plots of cross-correlations: effect of reference responsespectrum (EP hysteresis type,α = 0.10, SAC Boston, design-level,SD soil):(a) AVG spectrum; (b) DES spectrum. ................................................................130
Figure 9.22: Matrix plots of cross-correlations: effect of reference responsespectrum (EP hysteresis type,α = 0.10, SAC Boston, survival-level,SD soil): (a) AVG spectrum; (b) DES spectrum...................................................130
Figure 9.23: Matrix plots of cross-correlations: effect of reference responsespectrum (EP hysteresis type,α = 0.10, SAC Boston, design-level,SE soil):(a) AVG spectrum; (b) DES spectrum. ................................................................131
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Figure 9.24: Matrix plots of cross-correlations: effect of reference responsespectrum (EP hysteresis type,α = 0.10, SAC Boston, survival-level,SE soil):(a) AVG spectrum; (b) DES spectrum. ................................................................131
Figure 9.25: Matrix plots of cross-correlations: effect of reference responsespectrum (EP hysteresis type,α = 0.10, SAC Los Angeles, design-level,SD soil): (a) AVG spectrum; (b) DES spectrum. ........................................132
Figure 9.26: Matrix plots of cross-correlations: effect of reference responsespectrum (EP hysteresis type,α = 0.10, SAC Los Angeles, survival-level,SD soil): (a) AVG spectrum; (b) DES spectrum .........................................132
Figure 9.27: Matrix plots of cross-correlations: effect of reference responsespectrum (EP hysteresis type,α = 0.10, SAC Los Angeles, design-level,SE soil): (a) AVG spectrum; (b) DES spectrum..........................................133
Figure 9.28: Matrix plots of cross-correlations: effect of reference responsespectrum (EP hysteresis type,α = 0.10, SAC Los Angeles, survival-level,SE soil): (a) AVG spectrum; (b) DES spectrum..........................................133
Figure 9.29: Matrix plots of cross-correlations: effect of reference responsespectrum (EP hysteresis type,α = 0.10, SAC Los Angeles, design-level,SD soil,NF): (a) AVG spectrum; (b) DES spectrum. .................................134
Figure 9.30: Matrix plots of cross-correlations: effect of reference responsespectrum (SD hysteresis type,α = 0.10, SAC Los Angeles, survival-level,SD soil): (a) AVG spectrum; (b) DES spectrum. ........................................134
Figure 9.31: Matrix plots of cross-correlations: effect of reference responsespectrum (BE hysteresis type,α = 0.10, SAC Los Angeles, survival-level,SD soil): (a) AVG spectrum; (b) DES spectrum. ........................................135
Figure 9.32: Matrix plots of cross-correlations: effect of reference responsespectrum (BP hysteresis type,α = 0.10,βs = βr = 1/3, SAC LosAngeles, survival-level,SD soil): (a) AVG spectrum; (b) DESspectrum...............................................................................................................135
Figure 9.33: Λ-µ regression curves using IND, AVG, and DES spectra (EPhysteresis type,α = 0.10, SAC Boston): (a-b)SD soil; (c-d)SE soil. ..................136
Figure 9.34: Λ-µ regression curves using IND, AVG, and DES spectra (EPhysteresis type,α = 0.10, SAC Los Angeles): (a-b)SD soil; (c-d)SE
soil........................................................................................................................137
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Figure 9.35: Λ-µ regression curves using IND, AVG, and DES spectra (EPhysteresis type,α = 0.10, SAC Los Angeles, design-level,SD soil,NF). ......................................................................................................................138
Figure 9.36: Λ-µ regression curves using IND, AVG, and DES spectra (α = 0.10,SAC Los Angeles, survival-level,SD soil): (a) SD hysteresis type;(b) BE hysteresis type; (c) BP hysteresis type,βs = βr = 1/3. .............................139
Figure 10.1: Inelastic demand spectra based on DES response spectra (EPhysteresis type,α = 0.10): (a-b) SAC Boston,SD soil; (c-d) SACBoston,SE soil. ....................................................................................................141
Figure 10.2: Inelastic demand spectra based on DES response spectra (EPhysteresis type,α = 0.10): (a-b) SAC Los Angeles,SD soil; (c-d) SACLos Angeles,SE soil.............................................................................................141
Figure 10.3: Inelastic demand spectra based on DES response spectra (EPhysteresis type,α = 0.10): SAC Los Angeles,SD soil,NF. .................................142
Figure 10.4: Inelastic demand spectra based on DES response spectra (SAC Los Angeles,survival-level,SD soil, α = 0.10): (a) SD hysteresis type;(b) BE hysteresis type; (c) BP hysteresis type,βs = βr = 1/3. .............................142
Figure 10.5: Capacity curve-demand spectra for design example: (a) using DES spectrum;(b) using IND spectra...........................................................................................144
Figure 11.1: Scatter in∆nlin for the UNDSD soil ensemble (EP hysteresis type,α = 0.10) -PGA method compared to: (a) EPA method; (b) EPV method;(c) MIV method; (d)A95 method; (e) method;(f) method.....................................................................................148
Figure 11.2: Scatter in∆nlin using the and methodscompared to the MIV method (EP hysteresis type,α = 0.10): (a) UNDSD soil; (b) UNDSC soil; (c) UNDSE soil; (d) SACSD soil,NF........................150
Figure 11.3: Scatter inµ (UND SD soil ensemble, EP hysteresis type,α = 0.10):(a) standard deviation,σ; (b) coefficient of variation, COV.................................151
Figure 11.4: Effect of hysteresis type on the scatter in∆nlin (UND SD soil ensemble,α = 0.10): (a) SD type; (b) BE type; (c) BP type (βs = βr = 1/3).........................152
Figure 11.5: Scatter in∆nlin for the UNDSC soil ensemble (EP hysteresis type,α = 0.10) -PGA method compared to: (a) EPA method; (b) EPV method;
Saˆ To( )
Saˆ To Tµ→( )
Saˆ To( ) Sa
ˆ To Tµ→( )
xvii
(c) MIV method; (d)A95 method; (e) method;(f) method.....................................................................................153
Figure 11.6: Scatter in∆nlin for the UNDSE soil ensemble (EP hysteresis type,α = 0.10) -PGA method compared to: (a) EPA method; (b) EPV method;(c) MIV method; (d)A95 method; (e) method;(f) method.....................................................................................154
Figure 11.7: Scatter in∆nlin for the SAC Los Angeles, design-level,SD soil,NF ensemble(EP hysteresis type,α = 0.10) - PGA method compared to:(a) EPA method; (b) EPV method; (c) MIV method; (d)A95 method;(e) method; (f) method. ...................................................156
Figure 11.8: Scatter in MDOF demands for the UNDSE soil ensemble using theMIV method and the method: (a) four-story structure;(b) eight-story structure........................................................................................158
Figure 11.9: Covariance in the MDOF demands for the UNDSE soil ensembleusing the MIV method and the method: (a) four-story structure;(b) eight-story structure........................................................................................159
Figure A.1: Ground motion records: University of Notre Dame (UND) very dense(SC) soil ensemble................................................................................................187
Figure A.2: Ground motion records: University of Notre Dame (UND) stiff (SD)soil ensemble........................................................................................................189
Figure A.3: Ground motion records: University of Notre Dame (UND) soft (SE)soil ensemble........................................................................................................191
Figure A.4: Ground motion records: Nassar and Krawinkler 15s very dense (SC)soil ensemble........................................................................................................193
Figure A.5: Ground motion records: SAC Boston design-level stiff (SD) soilensemble. .............................................................................................................195
Figure A.6: Ground motion records: SAC Boston survival-level stiff (SD) soilensemble ..............................................................................................................197
Figure A.7: Ground motion records: SAC Boston design-level soft (SE) soilensemble ..............................................................................................................199
Figure A.8: Ground motion records: SAC Boston survival-level soft (SE) soil ensemble(generated using EERA site response analysis program,Bardet et al., 2000)...............................................................................................201
Saˆ To( )
Saˆ To Tµ→( )
Saˆ To( )
Saˆ To Tµ→( )
Saˆ To( ) Sa
ˆ To Tµ→( )
Saˆ To Tµ→( )
Saˆ To Tµ→( )
xviii
Figure A.9: Ground motion records: SAC Los Angeles design-level stiff (SD) soilensemble. .............................................................................................................203
Figure A.10: Ground motion records: SAC Los Angeles survival-level stiff (SD) soilensemble. .............................................................................................................205
Figure A.11: Ground motion records: SAC Los Angeles design-level soft (SE) soilensemble. .............................................................................................................207
Figure A.12: Ground motion records: SAC Los Angeles survival-level soft (SE)soil ensemble (generated using EERA site response analysis program, Bardetet al., 2000). .........................................................................................................209
Figure A.13: Ground motion records: SAC Los Angeles design-level near-field(NF) ensemble......................................................................................................211
Figure A.14: Response spectra: University of Notre Dame (UND) very dense (SC)soil ensemble........................................................................................................213
Figure A.15: Response spectra: University of Notre Dame (UND) stiff (SD) soil ensemble. ..213
Figure A.16: Response spectra: University of Notre Dame (UND) soft (SE) soil ensemble. ...214
Figure A.17: Response spectra: Nassar and Krawinkler 15s very dense (SC) soil ensemble....214
Figure A.18: Response spectra: SAC Boston design-level stiff (SD) soil ensemble. ................215
Figure A.19: Response spectra: SAC Boston survival-level stiff (SD) soil ensemble. ..............215
Figure A.20: Response spectra: SAC Boston design-level soft (SE) soil ensemble..................216
Figure A.21: Response spectra: SAC Boston survival-level soft (SE) soil ensemble(generated using EERA site response analysis program, Bardet et al., 2000).....216
Figure A.22: Response spectra: SAC Los Angeles design-level stiff (SD) soil ensemble. .......217
Figure A.23: Response spectra: SAC Los Angeles survival-level stiff (SD) soil ensemble. .....217
Figure A.24: Response spectra: SAC Los Angeles design-level soft (SE) soil ensemble. ........218
Figure A.25: Response spectra: SAC Los Angeles survival-level soft (SE) soil ensemble(generated using EERA site response analysis program,Bardet et al., 2000)...............................................................................................218
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Figure A.26: Response spectra: SAC Los Angeles design-level stiff (SD) soil near-field (NF) ensemble..............................................................................................219
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LIST OF SYMBOLS
a = regression coefficient used forR-µ-T relationship
a* = acceleration of the idealized SDOF representation for the structure
A95 = Arias intensity-based parameter
AVG = average ground motion response spectrum based on the ground motion ensemble
a*y = acceleration capacity
b = regression coefficient used forR-µ-T relationship
BE = bilinear-elastic hysteresis type
BP = combined bilinear-elastic/elasto-plastic hysteresis type
c = coefficient used forR-µ-T relationship
C = sample covariance
c1 = 1st term in equation for thec coefficient
c2 = 2nd term in equation for thec coefficient
Ca = seismic coefficient tabulated inUBC 1997based on site seismicity and site soil characteris-
tics
cd = viscous damping coefficient
COV = coefficient of variation
Cs = seismic response coefficient determined using a smooth design response spectrum
xxi
Cv = seismic coefficient tabulated inUBC 1997based on site seismicity and site soil characteris-
tics
d = regression coefficient forΛ-T relationships
DES = smooth design response spectra from current U.S. seismic design provisions
DL = design dead load
Dsm = 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
Es = Arias Intensity
f = regression coefficient used forΛ-T relationships
f1 = design live load modification factor
Fbe = yield capacity for the BE component of the BP hysteresis type
Fdes = design lateral strength based on first “significant yield” of the structure
Felas = linear-elastic force demand
Fep = yield capacity for the EP component of the BP hysteresis type
Fi = lateral force applied at leveli
Fn = lateral force applied at roof leveln
Fnlin = maximum nonlinear force demand
xxii
Ft = portion of the design force concentrated at the top of the structure in addition toFn
Fy = design lateral strength based on a “global” yield point of the structure
= required nonlinear force capacity based on bedrock motion below the soft soil site
= linear-elastic force demand based on bedrock motion below the soft soil site
= required nonlinear (i.e., reduced) force capacity at the soft soil site
g = regression coefficient used for relationships betweenΛ demand indices
G = equivalent linear shear modulus
G/Gmax = shear modulus reduction factor
h = regression coefficient used for relationships betweenΛ demand indices
hc = column depth
hi = height from ground to leveli
hx = height from ground to levelx
I = seismic importance factor
Ie = 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
kbe = linear-elastic stiffness for the BE component of the BP hysteresis type
kep = linear-elastic stiffness for the EP component of the BP hysteresis type
ks1 = linear-elastic moment-rotation stiffness of the zero-length rotational spring model
ks2 = moment-rotation post-yield stiffness of the zero-length rotational spring model
Fyr µ( )
Fy e,r
Fys µ( )
xxiii
Ks2 = moment-curvature post-yield stiffness of the zero-length rotational spring model
ksh = “shooting” stiffness of the SD hysteresis type
ktot = total linear-elastic stiffness of the BP hysteresis type
kµ = secant stiffness
L = beam clear span length
LE = linear-elastic hysteresis type
LL = design live load
lp = equivalent plastic hinge length
m = structure mass
m* = equivalent mass of the idealized SDOF representation for the structure
MIV = maximum incremental velocity
Ms = yield moment capacity of the zero-length rotational spring model
N = standard penetration resistance
Na = near-field factor inUBC 1997based on the closest distance from the site to a known seismic
source
Nv = near-field factor inUBC 1997based on the closest distance from the site to a known seismic
source
neq = number of ground motion records
NF = near-field
ny = number of yield events
PGA = peak ground acceleration
PI = plasticity index
xxiv
px = probability of havingx occurrences during the design life,tdes
px ≠ 0 = probability of exceedance (or occurrence)
R = response modification coefficient
Rdes = response modification coefficient based on first “significant yield” of the structure
= response modification coefficient proposed by Miranda (1993)
S1 = rock soil profile classification perUBC 1994
S2 = stiff soil profile classification perUBC 1994
Sa = spectral acceleration
Sai = inelastic acceleration demand
= average spectral acceleration at the structure fundamental period
= average spectral acceleration over a range of structure periods
SC = very dense soil profile
Sd = spectral displacement
SD = stiffness-degrading hysteresis type
SD = stiff soil profile
Sd1 = seismic coefficient used to define smooth design response spectra
SD1 = seismic coefficient corresponding to the “design earthquake” inIBC 2000
Sde = linear-elastic displacement demand
Sdi = inelastic displacement demand
Sds = seismic coefficient used to define smooth design response spectra
SDS = seismic coefficient corresponding to the “design earthquake” inIBC 2000
SE = soft soil profile
R̂µ
Saˆ To( )
Saˆ To Tµ→( )
xxv
SM1 = seismic coefficient corresponding to the “maximum considered earthquake” inIBC 2000
SMS = seismic coefficient corresponding to the “maximum considered earthquake” inIBC 2000
= soft soil modification factor
su = soil unconfined shear strength
t = time
T = period
tdes = design life
tend = final cutoff point
Teq = equivalent structure period
tinit = initial cutoff point
tj = time at discretization pointj
To = linear-elastic structure fundamental period
Tr = return period
ts = time step used in CDSPEC program
Ts = soil predominant period of vibration
Tµ = elongated structure period
V = design structure base shear
vs = soil shear wave velocity
W = structure seismic weight
wi = seismic weight at floor or roof leveli
wx = seismic weight at floor or roof levelx
α = post-yield stiffness ratio
S Ts µ,( )
xxvi
αbe = post-yield stiffness ratio of the BE component for the BP hysteresis type
αep = post-yield stiffness ratio of the EP component for the BP hysteresis type (equal to zero)
αφ = moment-curvature post-yield stiffness ratio of the zero-length rotational spring model
βr = BP hysteresis type strength ratio
βs = BP hysteresis type stiffness ratio
γ = shear strain
Γ = modal participation factor
= shear strain rate
γc = shear strain amplitude
γCOV = 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
= ground motion acceleration
∆i = lateral displacement at floor or roof leveli
∆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
°γ
∆̇̇g
xxvii
∆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
θi = interstory drift at story leveli
θmax = maximum mean interstory drift over the height of the structure
θp = plastic rotation of the zero-length rotational spring model
θs = yield rotation of the zero-length rotational spring model
Λ = demand indices,µp, µr, andny
Λ’ = log of the demand indices,µp, µr, andny
µ = maximum displacement ductility demand
µ’ = log of the maximum displacement ductility demand,µ
µc = cyclic displacement ductility demand
µE = normalized hysteretic energy demand
µp = cumulative plastic deformation ductility demand
µr = residual displacement ductility demand
µrmax = maximum possible residual displacement ductility demand
µt = target displacement ductility
ξ = viscous damping ratio
ξeq = equivalent viscous damping ratio
ρ = correlation coefficient
xxviii
Ρ = coefficient of determination
σ = sample standard deviation
τ = shear stress
φp = curvature corresponding toθp
φs = curvatures corresponding toθs
ϕ = stiffness factor
ω = equivalent linear soil frequency
xxix
ACKNOWLEDGMENTS
The project was funded by the National Science Foundation (NSF) under Grant No. CMS98-74872 as part of the 1999 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 ofLehigh University for his comments and suggestions. The opinions, findings, and conclusionsexpressed in this report are those of the authors and do not necessarily reflect the views of theindividuals and organizations acknowledged above.
1
CHAPTER 1
INTRODUCTION
1.1 Problem Statement
Seismic design and evaluation/rehabilitation approaches in current U.S. building provi-sions (e.g.,International Building Code 2000(ICC, 2000),FEMA 302(BSSC, 1998),UniformBuilding Code(ICBO, 1997),FEMA 356(ASCE, 2000)) advocate the use of several static anddynamic analysis procedures, such as the equivalent lateral force procedure, modal analysis pro-cedure, capacity spectrum procedure, and nonlinear dynamic analysis procedure. Among theseprocedures, it is common to use linear and nonlinear static methods that depend on capacity-demand index relationships such as the relationship between the design lateral strength and themaximum lateral displacement. Design approaches that use these relationships include the con-ventional “equivalent lateral force” procedure (ICC, 2000; BSSC, 1998; ICBO, 1997) and themore elaborate “capacity spectrum” procedure based on inelastic acceleration and displacementspectra (Reinhorn, 1997; Freeman, 1998; Chopra and Goel, 1999; Fajfar, 1999).
In current practice, the design lateral strength,Fdes, is often determined by dividing thedesign lateral force required to keep the structure linear-elastic during an earthquake,Felas, by aresponse modification coefficient,R = Rdes, as shown in Figure 1.1a. This force reduction isallowed provided that the resulting maximum nonlinear displacement demand,∆nlin, can beaccommodated. The maximum displacement,∆nlin, depends on theR coefficient used in designand can be estimated from simple capacity-demand index relationships based on an idealizedbilinear lateral force-displacement relationship of the structure as shown in Figure 1.1a. The ben-efit of using these capacity-demand index relationships comes from their simplicity, however sig-nificant deficiencies exist in their development, which are addressed by this research as follows.
Foremost, previous research on the development of seismic capacity-demand index rela-tionships is based onFelas values determined using linear-elastic single-degree-of-freedom(SDOF) acceleration response spectra from a selected ensemble of ground motion records. Whilethese relationships may be appropriate for the ground motion ensembles that were used, the seis-mic design of most building structures is based on “smooth” response spectra as specified bymodel building design and rehabilitation codes and provisions.
For the design procedures to be consistent, there is a need to develop capacity-demandindex relationships using smooth response spectra from current design provisions. This is particu-larly important for near-field ground motion records and for sites with soft soil conditions where
2
lateral displacement, ∆
late
ral f
orce
, FFORCE REDUCTION
CHANGE IN LATERAL DISPLACEMENT
nlinF
elasF
∆elas
2
1
∆des
3
∆y
K
αKy
F elasF
=
R
0
"significant yield" 1 Linear-Elastic Behavior
2 Nonlinear Behavior
3 Idealized Bilinear Behavior
desF = elas
F
Rdes
∆ = µ nlin
∆y
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
period, T (sec)
R = 2
Ground motion response spectrum (ξ = 5%)
Design response spectrum,IBC 2000 (ξ = 5%)
unconservative inconsistency in determining F / R
BOSTON DESIGN-LEVEL SOFT (S ) SOIL
Fel
as
R
1 W.
R = 1 (linear-elastic)
elas
E
Figure 1.1: (a) Lateral force-displacement relationships; (b) Ground motion response spec-tra versus smooth design response spectra.
(a)
(b)
3
the characteristics of the ground motion spectra may be significantly different than the character-istics of smooth design response spectra as shown in Figure 1.1b (whereW is the total seismicweight assigned to the lateral load-resisting system). As illustrated in Figure 1.1b, this inconsis-tency may result in different values for the design lateral force,Felas/R, for the same value of theRcoefficient. In particular, the design force using the smooth spectrum may be lower than thedesign force using the ground motion spectrum, leading to an unconservative design.
To the best of the authors’ knowledge, previous research on the development of capacity-demand index relationships using smooth design response spectra instead of linear-elastic groundmotion response spectra as the basis for calculatingFy = Felas/R has not been published in the lit-erature. This research shows that previous relationships developed using linear-elastic groundmotion response spectra can be significantly different than those developed using smooth designresponse 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 literatureare limited to the maximum displacement ductility demand,µ = ∆nlin/∆y. The ductility demand,µ,provides an estimate of the maximum nonlinear displacement demand of the structure corre-sponding to theR coefficient used in design. While these relationships may be adequate for thebasic seismic design of building structures, other demand indices such as to quantify cumulativedamage, hysteretic energy, and residual displacement are needed for use in the framework of aperformance-based design approach that allows the designer to specify and predict, with reason-able accuracy, the performance (degree of damage) of a building for a specified level of groundmotion intensity.
To the best of the authors’ knowledge, previous research on the development of closed-form relationships for a comprehensive set of seismic demand indices has not been published inthe literature. This research develops such relationships and shows that the correlation betweenµand the other demand indices is usually relatively strong. In some cases, the cross-correlationsbetween the demand indices show weak to no correlation, indicating that these demand indicescan carry independent measures of seismic demand.
As an alternative to capacity-demand index relationships, the use of nonlinear dynamictime-history procedures in design has become increasingly popular and cost-effective in recentyears due to the expanding availability of faster computers and better analysis models. Nonlineardynamic time-history analyses conducted as a part of a performance-based design approachrequire that the ground motion records are scaled to a specified level of seismic intensity. Currentseismic design provisions provide guidelines for the number of ground motion records to be usedin the nonlinear dynamic analysis procedure, however the manner in which these ground motionrecords are to be scaled is not explicitly addressed. Merely, the provisions require that the averageacceleration response spectrum of the ground motion ensemble is not less than 1.4 times a 5%-damped smooth design response spectrum, specified by the provisions, for periods between 0.2Tand 1.5T, whereT is the structure fundamental period (ICC, 2000; ASCE, 2000; ICBO, 1997).While the intent is straightforward (i.e., to match or exceed 1.4 times the smooth design responsespectrum), there are several ways that a ground motion ensemble can be scaled so as to producean average response spectrum that satisfies this requirement. However, recent research has dem-
4
onstrated that some scaling methods result in an excessive scatter in the estimated seismicdemands, indicating that these scaling methods may not be able to adequately define the damagepotential (i.e., seismic intensity) for given site conditions and structural characteristics.
For example, nonlinear dynamic analysis procedures have been often performed usingground motions scaled to constant peak ground motion characteristics (e.g., peak ground acceler-ation, PGA, and peak ground velocity, PGV). However, it has been shown that this method ofground motion scaling introduces a large scatter in the analysis results (Nau and Hall, 1984;Miranda, 1993; Vidic et al., 1994; Shome and Cornell, 1998). This indicates that the seismicdemand estimates may be biased, leading to designs with significant uncertainty and unknownmargins of safety, unless a relatively large ensemble of ground motions are used (Shome and Cor-nell, 1998). Therefore, it is imperative that nonlinear dynamic analysis procedures are performedusing ground motion records scaled based on methods that adequately define the damage potentialfor given site conditions and structural characteristics, thus resulting in consistent prediction ofthe demand estimates by minimizing the scatter.
To the best of the authors’ knowledge, an investigation of ground motion scaling methodsincorporating 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 notbeen previously published in the literature. The research shows that scaling methods that workwell for ground motion records on stiff soil profiles lose their effectiveness for ground motions onother site conditions, particularly on soft soil profiles and for largeR coefficients. A new scalingmethod is proposed for use under these conditions.
1.2 Research Objectives
The broad objective of this research is to address some of the research needs for the imple-mentation of performance-based procedures in current U.S. seismic design provisions. Theresearch has four specific objectives: (1) to develop new nonlinear SDOF capacity-demand indexrelationships based on linear-elastic smooth design response spectra; (2) to develop new relation-ships that quantify cumulative damage, hysteretic energy, and residual displacement demands; (3)to investigate the effects of the structure fundamental period of vibration, strength level, hystereticbehavior, and site conditions on the demand estimates; and (4) to investigate the effect of theground motion scaling method on the scatter in the demand estimates.
1.3 Scope and Organization
Chapter 2 presents a review of the previous research on capacity-demand index relation-ships, site conditions, hysteresis type, current seismic design procedures, and ground motion scal-ing methods.
Chapter 3 describes the research program in terms of the analytical models, groundmotion records, analysis procedures, demand indices, and statistical evaluation of results. Threemajor suites of ground motion records are used in this research: (1) ground motions compiled by
5
the authors at the University of Notre Dame; (2) ground motions compiled by the SAC steelproject (SAC, 1997; Somerville et al., 1997); and (3) ground motions compiled by Nassar andKrawinkler (1991).
The University of Notre Dame (UND) ground motions are used to investigate the effect ofthe ground motion scaling method on the scatter in the demand estimates. The SAC steel project(SAC) ground motions are primarily used to develop new capacity-demand index relationships,including relationships based on linear-elastic smooth design response spectra. Finally, the Nassarand Krawinkler (N&K) ground motions are used to validate the analytical procedure used in thisresearch by comparing the results with previous results. Within these major suites, the groundmotion records are further subdivided into ensembles with different site soil characteristics, siteseismicities, epicentral distances, and seismic demand levels (earthquake hazard level as definedin FEMA 356 (ASCE, 2000)).
Five single-degree-of-freedom (SDOF) hysteresis types are considered: (1) linear-elastic(LE); (2) bilinear elasto-plastic (EP); (3) stiffness degrading (SD); (4) bilinear-elastic (BE); and(5) combined bilinear-elastic/elasto-plastic (BP). The hysteretic load-displacement relationshipsof these systems are described later. The research provides possible correlations between the hys-teretic behavior and the seismic demand estimates for each system. The development of theserelationships will, in turn, provide design guidelines for structures with different types of hyster-etic behavior.
In addition to SDOF models, two multi-degree-of-freedom (MDOF) models representa-tive of four-story and eight-story cast-in-place reinforced-concrete special moment-resistingoffice building frame structures are considered. These MDOF models were used to reinforce thefindings from the SDOF models as described later.
Chapter 4 presents comparisons between results obtained in this research program andresults presented by Nassar and Krawinkler (1991) as validation for the analytical procedure. Inaddition, implications of some of the assumptions used by Nassar and Krawinkler are investi-gated.
Chapters 5 through 7 present a comprehensive examination of the capacity-demand indexrelationships for SDOF systems. The effects of variations in the following major types of parame-ters on the demand indices are examined: (1) input parameters such as site soil characteristics, siteseismicity, epicentral distance, and seismic demand level; and (2) structural parameters such asperiod of vibration, post-yield stiffness ratio, hysteresis type, and strength level (i.e., responsemodification coefficient,R, and reference response spectra).
Chapters 8 and 9 present the capacity-demand index relationships in terms of nonlinearregression curves for: (1) the maximum displacement ductility demand; and (2) demand indicesthat quantify cumulative damage, hysteretic energy, and residual displacement.
In Chapter 10, the results of the regression analyses in the previous two chapters are inter-preted for the purpose of producing viable structural designs using performance-based engineer-ing concepts (e.g., inelastic capacity-demand spectra methods).
6
Chapter 11 presents a comprehensive examination of the effect of several ground motionscaling methods on the scatter in the demand estimates. The effects of variations in the followingmajor types of parameters on the effectiveness of the scaling methods are examined: (1) inputparameters such as site soil characteristics and epicentral distance; and (2) structural parameterssuch as fundamental period of vibration, hysteresis type, and strength level (i.e., response modifi-cation coefficient,R). MDOF analyses are conducted to validate the findings from the SDOF anal-yses.
Chapter 12 summarizes the research and the major conclusions derived from this research.
It is emphasized that this research focuses almost solely on SDOF systems. A total of300,000 SDOF dynamic time-history analyses were conducted. A limited number of MDOF anal-yses (a total of 80 analyses) were performed to validate the SDOF analysis results on the scalingof ground motion records, however this was not done for the full range of analysis parameters(e.g., reference response spectra, demand indices).
For a wide range of structures, results derived from the SDOF analyses can be directlyapplied to current seismic design provisions based on simplifying assumptions for mode shapesand mass participation (SEAOC, 1996). Implicitly, these design provisions assume that the “glo-bal” seismic displacement response of the structure (e.g., roof-displacement response) is governedby the fundamental (first) mode of vibration, making a SDOF representation adequate.
Conversely, the results from this research may not be directly applicable to structures thatare expected to have significant multi-mode effects (e.g., flexible diaphragm structures (Fleis-chman and Farrow, 2001)) and modifications to the capacity-demand index relationships may benecessary. Furthermore, local force and deformation demands (i.e., demands in the structuralmembers and joints) are not addressed in this research.
1.4 Research Significance
Nonlinear static and dynamic procedures specified in current seismic design provisionsare popular because of their simplicity and cost-effectiveness. However, the use of these proce-dures in design may be compromised with unknown levels of protection imposed by the presentcode provisions. In some cases, these procedures may lead to significantly unconservativedesigns. This research represents an attempt to address some of the present needs for seismicdesign by developing capacity-demand index relationships and ground motion scaling methodsthat are consistent with current code provisions.
7
CHAPTER 2
BACKGROUND
This chapter provides background information on the following topics: (1) previousresearch on maximum displacement ductility demand relationships; (2) previous research on otherdemand indices; (3) current seismic design procedures that use capacity-demand index relation-ships; and (4) previous research on ground motion scaling methods.
2.1 Previous Research on Maximum Displacement Ductility Demand
Investigations formulating maximum displacement ductility demand relationships havebeen conducted by several researchers (e.g., Newmark and Hall, 1973; Riddell and Newmark,1979; Riddell et al., 1989; Nassar and Krawinkler, 1991; Nassar et al., 1992; Miranda, 1993;Vidic et al., 1994; Borzi and Elnashai, 2000).
Nassar and Krawinkler (1991) investigated both single-degree-of-freedom (SDOF) andmulti-degree-of-freedom (MDOF) systems using an ensemble of fifteen ground motion recordsrepresentative of records on soil profileS1 (rock, perUBC 1994(ICBO, 1994)) from the westernUnited States. Through a statistical evaluation of the dynamic response of SDOF systems withidealized bilinear behavior as shown in Figure 1.1a, theRcoefficient was determined as a functionof theµ demand, the post-yield stiffness ratio,α (see Fig. 1.1a), and period of vibration,T, as:
2.1
where,
, 2.2
2.3
Values for thea andb coefficients in Equation 2.3 can be found in Table 2.1 for a given value ofα(Nassar and Krawinkler, 1991).
R µ T α, ,( ) c µ 1–( ) 1+[ ]1 c/=
RFelas
Fy------------= µ
∆nlin
∆y-----------=
c T α,( ) Ta
Ta 1+--------------- b
T---+=
8
It should be noted that the definition of theR coefficient in current code provisions (e.g.,IBC 2000) is based on a point of first “significant yield” in the lateral load resisting system (i.e.,Fdes, ∆desin Fig. 1.1a) which is different from theRcoefficient defined in Equation 2.2 based on a“global” yield point for the structure (i.e.,Fy, ∆y). The R coefficient defined in Equation 2.2 isused in the remainder of this research since it takes into account the overstrength in the structureand is considered to result in a better measure of the maximum displacement ductility demandthroughµ.
In the development of the aboveR-µ-T relationship, the lateral strength,Fy = Felas/R wasdetermined from the 5%-damped linear-elastic SDOF acceleration response spectrum for eachground motion. As mentioned earlier, this is different from current code provisions where thedesign lateral strength is calculated using a linear-elastic smooth design response spectrum.
It can be verified that Equation 2.1 satisfies the following well-established limits (Chopra,1995): (1) asT → 0, R → 1 (linear-elastic response); and (2) asT → ∞, R → µ (equal displace-ment assumption). The first limit recognizes that for very short-period (i.e.,T < 0.25 sec.) struc-tures, the maximum displacement ductility demand becomes extremely excessive andunattainable for relatively small reductions in the lateral force capacity. Therefore, these struc-tures should be designed to remain linear-elastic (i.e.,R = 1). The second limit implies that forlong-period structures, the maximum nonlinear displacement demand,∆nlin, approaches the max-imum linear-elastic displacement demand,∆elas. This result is expected since for very long-period(i.e., flexible) structures, the maximum displacement demand is similar to the peak ground dis-placement, independent ofR.
In addition, Equation 2.1 can be used to deduce the often-used relationship,, for short- to medium-period (i.e., 0.25≤ T < 0.70 sec.) structures which is based
on the assumption that the linear-elastic system and the nonlinear system absorb the same amountof energy (i.e., equal energy assumption) (Newmark and Hall, 1973; Miranda and Bertero, 1994).This relationship implies thatµ tends to be greater than or equal toR for short- to medium-periodstructures.
It is noted that theR-µ-T relationship defined in Equations 2.1-2.3 was developed usingfar-field ground motions recorded at sites with stiff soil, making it likely that the results may not
Table 2.1: Values fora andb coefficients
α (%) a b
0 1.00 0.42
2 1.01 0.37
10 0.80 0.29
R 2µ 1–=
9
be directly applicable for sites with medium and soft soil or near-field conditions. This isdescribed next.
2.1.1 Site soil characteristics
The effects of site soil characteristics on theR-µ-T relationship have been reported byElghadamsi and Morhaz (1987), Peng et al. (1988), Krawinkler and Rahnama (1992), Miranda(1993), Rahnama and Krawinkler (1993), and Miranda and Bertero (1994). The studies byKrawinkler and Rahnama (1992), Miranda (1993), and Miranda and Bertero (1994) suggest thatlocal site soil characteristics may have a significant effect on theR-µ-T relationship, especially forsites with soft soil.
Krawinkler and Rahnama (1992) recommend a modification to be used for sites with softsoil in the form:
2.4
where is the required nonlinear (i.e., reduced) lateral force capacity at the soft soil site,is the required nonlinear force capacity based on the bedrock motion below the soft soil
site, is the linear-elastic force demand based on the bedrock motion below the soft soil site,is a soft soil modification factor, andTs is the predominant period of vibration for the
soft soil site. Although an expression is not provided for the soft soil modification factor,, it was observed that is approximately 5 for linear-elastic structures (R = 1)
and 3-4 for nonlinear structures with fundamental periods close toTs.
Miranda (1993) conducted a regression analysis to determine an approximate responsemodification coefficient, , in the form:
2.5
whereΦ is a function ofT, µ, and site soil characteristics. Expressions forΦ are given by Miranda(1993) for rock, alluvium, and soft soil sites as:
for rock sites 2.6
for alluvium sites 2.7
for soft sites 2.8
Fys µ( ) Fy
r µ( )S Ts µ,( )Fy e,
r
R µ( )------------S Ts µ,( )= =
Fys µ( )
Fyr µ( )
Fy e,r
S Ts µ,( )
S Ts µ,( ) S Ts µ,( )
R̂µ
R̂µµ 1–
Φ------------ 1 1≥+=
Φ 1 110T µT–-----------------------
12T------- 1.5 T( )ln 0.6–( )2–( )exp–+=
Φ 1 112T µT–-----------------------
25T------- 2 T( )ln 0.2–( )2–( )exp–+=
Φ 1Ts
3T-------
3Ts
4T--------- 3
TTs-----
ln 0.25– 2
– exp–+=
10
Figure 2.1 shows theR-µ-T relationships for rock and soft soil sites using Equations 2.5,2.6, and 2.8. From this study it was determined that: (1) the use ofR coefficients developed forsites with rock or alluvium can lead to unconservative designs (due to largerR coefficients, andthus, smaller lateral strengths) for short-period (i.e., shorter than two-thirds of the site soil pre-dominant period,Ts) structures located on sites with soft soil; and (2) theRcoefficients developedfor sites with soft soil are much larger than the displacement ductility ratio,µ, for structures withperiods near the site soil predominant period (0.75Ts ≤ T ≤ 1.5Ts). The second observation above,which implies that smaller lateral strengths are needed for structures with periods near the sitepredominant period, is probably because the lateral force capacities,Fy = Felas/R, of the systemsin Miranda’s study were determined based on the linear-elastic acceleration response spectrum foreach ground motion. As shown in Figure 1.1b and discussed in more detail in Chapter 6, thisresults in increasedFy values for structures with periods close to the site predominant period,leading to the reduced seismic demands for soft soil profiles.
2.1.2 Near-field ground motions
The occurrence of recent earthquakes close to heavily urbanized areas (e.g., 1994Nothridge, California; 1995 Kobe, Japan; 1999 Izmit, Turkey; 1999 Chi-Chi, Taiwan; 2001Gujarat, India) has shown that severe damage can occur to the built environment due to near-fieldground motions. The effects of near-field ground motions on seismic demands have been exam-ined by several researchers (e.g., Hall et al., 1995; Naeim, 1995; Makris, 1997; Attalla et al.,1998; Bozorgnia and Mahin, 1998; Malthora, 1999). These studies suggest that near-field groundmotions may impose large ductility demands and residual displacements (Attalla et al., 1998),particularly for flexible and base-isolated (i.e., long-period) structures (Hall et al., 1995; Makris,1997). However, closed-formR-µ-T relationships are not available in the literature.
T (sec)0 1 2 3
1
2
3
4
5
6
7
8
9
R
T s
µ = 2 ,3, 4, 5, 6 (thin → thick lines) rock site soft soil site
2/3T s
Figure 2.1:R-µ-T relationships developed by Miranda (1993) for rock and soft soil sites.
11
2.1.3 Hysteretic lateral load-displacement behavior
The differences between the hysteretic lateral load-displacement behavior of differenttypes of structures may dramatically affect the seismic capacity-demand index relationships.Foutch and Shi (1998) performed a study on eight hysteresis types as shown in Figure 2.2, rangingfrom a bilinear elasto-plastic hysteresis type (Type 1) to a bilinear-elastic hysteresis type (Type 8).Structures that have pinched lateral load-displacement relationships (e.g., monolithic cast-in-place reinforced concrete structures) resemble Hysteresis Types 5 and 6; and self-centered sys-tems (e.g., unbonded post-tensioned precast concrete structures (Kurama et al., 1997)) have non-linear but nearly elastic behavior similar to Hysteresis Type 8.
Foutch and Shi performed a parametric study on MDOF models of three-, six-, and nine-story structures which were designed for Los Angeles using the 1994 NEHRP provisions (BSSC,1994). They concluded that the hysteretic behavior has a small effect on the maximum displace-ment demand (defined as the maximum story displacement in the structure) with the exceptionthat the maximum displacement of systems with pinched (i.e., Types 5 and 6) and bilinear-elastic(i.e., Type 8) hysteresis types can be up to 40 percent larger than the maximum displacement ofsystems with a bilinear elasto-plastic hysteresis type (i.e., Type 1). Note that the results in Foutchand Shi’s study are presented for anR value of 8 only.
Nassar and Krawinkler (1991) conducted SDOF nonlinear dynamic time-history analysesusing two hysteresis types, namely, the bilinear elasto-plastic (EP) hysteresis type and the stiff-ness degrading (SD) hysteresis type (i.e., Types 1 and 3 in Fig. 2.2), however, theR-µ-T relation-ship in Equation 2.1 was developed only for the EP hysteresis type. They observed that, except forvery short-period systems, the SD hysteresis type allows largerR coefficients (i.e., larger forcereduction) for a givenµ than the EP hysteresis type for systems with no post-yield stiffness (i.e.,α= 0.0). This interesting behavior can be attributed to the larger self-centering capability (i.e., abil-ity of the system to return toward zero displacement upon unloading from a nonlinear displace-ment) of the SD hysteresis type as compared with the EP hysteresis type forα = 0.0. It is alsonoted that even though the inelastic energy dissipation capacity of the EP hysteresis type is largerthan that of the SD hysteresis type during large displacement cycles, the SD hysteresis type con-tinues to dissipate energy during small displacement cycles whereas the EP hysteresis type doesnot dissipate any energy unless yielding occurs.
F
∆
F
∆
F
∆
F
∆
F
∆
F
∆
F
∆
F
∆
Hysteresis Type 1 Hysteresis Type 2 Hysteresis Type 3 Hysteresis Type 4
Hysteresis Type 8Hysteresis Type 7Hysteresis Type 6Hysteresis Type 5
10% 10%
10%
10%
10%
40%
Figure 2.2: Hysteresis types used by Foutch and Shi (1998)
12
Similar studies by Gupta and Krawinkler (1998) and Gupta and Kunnath (1998) usedground motions recorded on stiff soil sites from the western United States to examine the effect ofthe hysteretic lateral load-displacement behavior on the maximum displacement demand of SDOFstructures. It was observed that for structure periods longer than 0.5 seconds, the effect of hyster-etic behavior on the maximum displacement demand is small except for systems with a negativepost-yield stiffness ratio,α (which may occur due to P-∆ effects).
2.2 Previous Research on Other Demand Indices
As described above, a significant amount of previous research is available on the maxi-mum displacement ductility demand,µ. In comparison, a limited amount of research exists onother demand indices. Mahin and Lin (1983) performed a study on SDOF systems with EP andSD hysteresis types. This study is important because the demand indices listed in Table 2.2 andillustrated in Figure 2.3 were introduced. Similar demand indices are discussed by Ghobarah et al.(1999) and Loh and Ho (1990). Their research acknowledges that the displacement ductilitydemand,µ, alone is inadequate to describe the amount of damage that a structure may sustain dur-ing an earthquake. Loh and Ho (1990) performed a parametric study on a number of SDOF hys-teresis types (e.g., EP and SD types) in an attempt to identify correlations betweenµ, µc, µp, andµE. It was found that the demand indicesµc andµp are highly correlated withµ. The demand indi-cesµ andµE were found to have weak correlation with each other, implying that they carry inde-pendent measures of seismic demand for the structures. Capacity-demand index relationshipswere not developed in these studies.
MacRae and Kawashima (1997) performed a study on the permanent (i.e., residual) dis-placement demand,∆r, for elasto-plastic SDOF systems with varying post-yield stiffness ratios,α(see Fig. 1.1a). This demand index, which is shown in Figure 2.3, is important because even if astructure survives a severe seismic event without collapse, large residual displacements may ren-der the structure irreparable, leading to eventual demolition. MacRae and Kawashima (1997) con-cluded that: (1) the residual displacement demand is almost completely independent ofearthquake magnitude, epicentral distance, and site soil conditions; and (2) the residual displace-ment demand depends significantly on the post-yield stiffness ratio,α.
2.3 Current Capacity-Based Design Procedures
Current seismic design procedures that use capacity-demand index relationships include:(1) the equivalent lateral force procedure; and (2) the capacity spectrum procedure. These proce-dures are described below.
2.3.1 Equivalent lateral force (ELF) procedure
According to theIBC 2000(ICC, 2000) design provisions, all “regular” structures up to 70m high may be designed using the ELF procedure. The design structure base shear,V (which isequal toFdes in Fig. 1.1a), is specified as:
13
Table 2.2: Demand indices introduced by Mahin and Lin (1983).
Demand Index Description
maximum displacement ductility
cyclic displacement ductility
cumulative plastic deformation ductility
normalized hysteretic energy
ny number of yield events
nyrev number of complete yield reversals
nzero number of zero crossings
∆p,1
∆p,2
yield event
zero crossing
complete yield reversal
F
∆
(F , ∆ )i+1 i+1(F , ∆ )
i i
time, t
∆
∆max
∆min
∆r
unloading
F
∆0 0
0
(F , ∆ )max max
(F , ∆ )min min
(F , ∆ )y y
(F , ∆ )y y
Figure 2.3: Bilinear elasto-plastic (EP) hysteresis type with definitions for demand indicesin Table 2.2.
µ∆nlin
∆y-----------
max ∆min ∆max,( )∆y
-----------------------------------------------= =
µc
∆– min ∆max+
∆y-------------------------------- 1– 1.0≥=
µp
∆p i,∑∆y
-----------------=
µE
Fi Fi 1++
2------------------------
∆i 1+ ∆i–( )∑Fy∆y
----------------------------------------------------------------- 1+=
14
2.9
whereCs is the seismic response coefficient determined using a smooth design response spectrumand a response modification coefficient,R, andW is the seismic weight. The value of theRcoeffi-cient ranges from a low of 1.25 for structural systems with a limited amount of ductility andredundancy (e.g., ordinary steel moment frames in inverted pendulum systems) to a high of 8 forstructural systems with a high level of ductility and redundancy (e.g., special reinforced-concretemoment resisting frames) (ICC, 2000).
As noted earlier, the calculation ofV is based on a point of significant yield in the lateralload resisting system. The term “significant yield” is defined as that level causing complete plasti-fication of at least the most critical region of the structure (ICC, 2000, e.g., formation of a firstplastic hinge in the structure). In the case of a reinforced concrete frame, the level of significantyield is reached when at least one of the sections of its most highly stressed member reaches itsstrength.
The base shear,V, is used to prescribe an equivalent static lateral force distribution alongthe height of the structure. The ELF procedure is a “force-based” design procedure where a linear-elastic analysis of the lateral load resisting system is conducted under the equivalent lateral forces.The elastic deformations calculated at this reduced force level are then amplified to account forthe expected nonlinear behavior of the structure. The structural members are designed anddetailed for the reduced forces from the linear-elastic analysis and the amplified nonlinear defor-mations.
2.3.2 Capacity spectrum procedure
The ELF procedure does not explicitly consider the inherent relationship between theRcoefficient and theµ demand. Recognizing that nonlinear-inelastic procedures may provide a bet-ter basis for seismic design, the capacity spectrum procedure, which was first proposed by Free-man et al. (1975), is specified in theATC 40 (ATC, 1996) andFEMA 356 (ASCE, 2000)provisions for the seismic evaluation and rehabilitation of existing structures.
By graphical implementation, the capacity spectrum procedure compares the lateral forceand displacement capacity of a structure with estimated demands from an earthquake groundmotion (Freeman, 1998; Reinhorn, 1997; Chopra and Goel, 1999). The design seismic demandcan be represented using two different approaches: (1) highly-damped linear-elastic responsespectra; and (2) inelastic response spectra. As described below, the first approach does not requireanR-µ-T relationship, whereas anR-µ-T relationship is necessary in the second approach.
(1) In the first approach (Fig. 2.4a), the ground motion demand spectra are constructed byplotting the linear-elastic SDOF acceleration response spectra,Sa, versus the displacementresponse spectra,Sd, for different values of viscous damping ratio,ξ. Radial lines (originatingfrom “0”) in Figure 2.4a represent constant values ofSa/Sd which are related to the period,T, asfollows:
V CsW=
15
2.10
The lateral force and displacement capacity of the structure is represented using a globalforce-displacement (F-∆) relationship obtained from a nonlinear-static pushover analysis (curve“2” in Fig. 1.1a). Assuming that the global seismic displacement response of the structure is gov-erned by the fundamental (first) mode of vibration, the pushover curve is converted into an ideal-ized equivalent SDOF acceleration versus displacement (a*-∆*) relationship as follows:
, 2.11
wherem* is the mass of the equivalent SDOF system andΓ is the modal participation factor(Chopra, 1995).
The a*-∆* relationship (i.e., capacity curve) is plotted together with theSa-Sd demandspectra as shown in Fig. 2.4a. The design approach requires an iterative procedure based on anassumed initial value of the maximum nonlinear displacement,∆*
nlin, as described by Freeman(1998). The assumed displacement is used to estimate an equivalent viscous damping ratio,ξeq,based on the expected inelastic energy dissipation of the structure. Then, a new value of∆*
nlin isestimated from the intersection point of the capacity curve and the demand spectrum correspond-ing to ξeq. The procedure is repeated iteratively until the∆*
nlin value from the intersection of thecapacity curve and the demand spectrum (forξeq) matches the∆*
nlin value used to estimateξeq.
Sa
Sd----- 4π2
T2
---------≅
0
Highly-damped linear-elastic demand spectrum (ξ = ξ )
Linear-elastic demand spectrum (ξ = 5%)
Capacity curve
Idealized capacity curve
Inelastic acceleration and displacement demands
ya
T eq
eq
*
∆ y* ∆
nlin*
acc
eler
atio
n, S
o
ra
a*
displacement, S or d ∆ *
0
Inelastic demand spectrum (µ = µ )
Linear-elastic demand spectrum (µ = 1)
Capacity curve
Idealized capacity curve
Linear-elastic acceleration and displacement demands
Inelastic acceleration and displacement demands
acc
eler
atio
n, S
a
displacement, S d
t
2
1
aS
aiS
Sde
Sd
Sdi
A
A’ A’’
Figure 2.4: Capacity spectrum procedures: (a) highly-damped linear-elastic demand spec-tra; (b) inelastic demand spectra.
(a) (b)
a* F
m* Γ
----------= ∆* ∆Γ---=
16
The intersection point between the capacity curve and the demand spectrum represents the periodof an equivalent linear-elastic SDOF system,Teq, as shown in Figure 2.4a.
It is noted that the use of highly-damped linear-elastic response spectra to estimate∆*nlin
poses some difficulties as follows: (1) there is no significant evidence that justifies the existence ofa stable relationship betweenξeq and the inelastic energy dissipation corresponding to a maxi-mum nonlinear displacement,∆*
nlin, especially for highly nonlinear systems; and (2)Teq mayhave little to do with the dynamic response of the nonlinear system (Krawinkler, 1994).
(2) To overcome the difficulties associated with the approach above, a second approach(Fig. 2.4b) was recommended by Reinhorn (1997) and Chopra and Goel (1999). This approachuses inelastic demand spectra developed from linear-elastic smooth design response spectra withestablishedR-µ-T relationships. Essentially, the same procedure as above is used to develop thecapacity curve for the structure, however in contrast, the demand spectra are constructed frominelastic SDOF acceleration and displacement spectra for different values ofR or µ. In general,inelastic demand spectra are expected to provide better design estimates than highly-damped lin-ear-elastic demand spectra, especially for high levels of ductility demand,µ (Reinhorn, 1997).
The inelastic demand spectra are constructed by first dividing the linear-elastic accelera-tion demand,Sa, with theR coefficient to determine the inelastic acceleration demand (see line“1” in Fig. 2.4b) as:
2.12
Then, the inelastic displacement demand,Sdi, is determined by multiplying the linear-elastic dis-placement demand,Sde(corresponding toSai), with the displacement ductility demand,µ (see line“2” in Fig. 2.4b) as:
2.13
Thus, point A on the linear-elastic demand spectrum in Figure 2.4b is transformed to point A” onthe inelastic demand spectrum through path A-A’-A”.
EitherR or µ can be specified to construct the inelastic demand spectra for use within theframework of a “force-based” or “displacement-based” design procedure. In the displacement-based design procedure, a target displacement ductility,µ = µt, is specified for Equation 2.13 andthe correspondingR coefficient to be used in Equation 2.12 is calculated using an assumedR-µ-Trelationship with the linear-elastic structure period,T, and the specified target displacement ductil-ity, µt. The inelastic demand spectrum shown in Figure 2.4b is constructed using this procedure.Inversely, in the force-based procedure, anRcoefficient is specified for Equation 2.12 and the cor-responding displacement ductility demand,µ, for Equation 2.13 is calculated using anR-µ-T rela-tionship based on the linear-elastic structure period,T, and the specifiedR coefficient.
Sai
Sa
R-----=
Sdi µSde µ T2π------
2Sai= =
17
The point at which the capacity curve and a demand spectrum intersect in Figure 2.4bdefines the seismic demand on the structure, as in the first approach. If the acceleration demand,Sai, is less than or equal to the acceleration capacity,a*
y, and the displacement demand,Sdi, is lessthan or equal to the displacement capacity,∆*
ult, then the current design of the structure is ade-quate. Otherwise, redesign is required. Note that in this approach neither an equivalent viscousdamping ratio,ξeq, nor an equivalent period,Teq, of a linear-elastic SDOF system is specified.
2.4 Scaling of Ground Motion Records
As an alternative to using capacity-demand index relationships, nonlinear dynamic analy-sis procedures can be used to estimate seismic demands for design. In this case, current seismicdesign provisions require that a series of nonlinear time-history analyses are conducted with pairsof horizontal ground motion components selected from not less than three events. If three pairs ofground motion records are used, then the maximum value of the parameter of interest (e.g., maxi-mum nonlinear displacement) is taken for design. If seven or more pairs of records are used, thenthe average value of the response parameter of interest may be taken for design.
Nonlinear dynamic time-history analyses conducted as a part of a performance-baseddesign approach require that the ground motion records are scaled to a specified level of seismicintensity. With the small number of ground motion records required by current seismic designprocedures, the manner in which these records are scaled to describe the expected seismic inten-sity is of concern (Shome and Cornell, 1998). Scaling of ground motions to a specified level ofintensity (e.g., PGA or PGV) is relatively simple, however with this simplicity come inconsistentpredictions in demand estimates, which can be measured by the amount of scatter. This observa-tion is particularly significant for structures that are designed to undergo large nonlinear displace-ments, since, it has been shown that scatter in the demand estimates tends to increase at higherlevels of nonlinear behavior (Lam et al., 1998).
There are a number of papers that mention or explicitly address the scaling of groundmotion records for seismic design and analysis (Nassar and Krawinkler, 1991; Miranda, 1993;Kennedy, 1984; Arias, 1969; Carballo and Cornell, 1998; Shome and Cornell, 1998; Shome et al.,1998; Martinez-Rueda, 1998; Nau and Hall, 1984; Vidic et al., 1994). Scaling, in general, is per-formed: (1) to match a parameter associated with the expected damage potential for a given site;and/or (2) to match smooth design response spectra (Nassar and Krawinkler, 1991). In the studyby Nassar and Krawinkler (1991) to determine capacity-demand index relationships between theresponse modification coefficient,R, the maximum displacement ductility demand,µ, the post-yield stiffness ratio,α, and the period of vibration,T, the ground motions were scaled to a peakground acceleration (PGA) of 0.4g. A large scatter in the results was observed, particularly forlarge values of theR coefficient. Miranda (1993) performed a similar study and observed thatusing acceleration parameters (i.e., PGA and “effective” peak acceleration, EPA) to scale theground motions increases the scatter in the inelastic acceleration demand spectra (calculated forconstant values ofµ) for longer-period structures.
Shome and Cornell (1998) and Shome et al. (1998) found that seismic demand estimatesare strongly correlated with the linear-elastic spectral response acceleration at the structure funda-
18
mental period,To, also called the spectral intensity, . This observation was made for twostructures (To = 0.25 and 1.05 sec.) using ground motions representative of records on soil profileS2 (stiff, per UBC 1994(ICBO, 1994)) in California with similar magnitudes (measured as theamount of strain energy released at the source of the earthquake) and similar epicentral distancesfrom the source of the earthquake. Shome and Cornell (1998) demonstrated that the scatter in thedemand estimates can be significantly reduced by scaling the ground motion records in an ensem-ble up or down so that their spectral intensities at the structure linear-elastic fundamental periodare equal (i.e., scaling method). Scatter can be further reduced by: (1) scaling the groundmotion records based on the average spectral intensity of the ensemble over a range of periods(Kennedy et al., 1984; Shome and Cornell, 1998; Shome et al., 1998; Martinez-Rueda, 1998); and(2) scaling at higher levels of damping (typically around 5% to 20% of critical damping)(Kennedy et al., 1984; Shome and Cornell, 1998). These methods of scaling were found to be bet-ter methods to define the damage potential (i.e., seismic intensity) for given site conditions andstructural characteristics by reducing the scatter in the demand estimates.
Additionally, for scaling methods based on peak ground motion characteristics (e.g.,PGA), scatter was found to be significantly large (e.g., the scatter,δ, calculated as the standarddeviation of the log demand estimates, which is approximately equal to the coefficient of varia-tion, COV, was as much as 92% of the mean demand) even for a relatively large sample size (20ground motions), implying that the demand estimates are subject to significant uncertainty(Shome et al., 1998). Shome et al. concluded that to obtain an estimate of the mean response,X,within a certain range,X (e.g.,X ± 0.10X or X = 0.10X), with 95% confidence, the number ofground motions required,neq, can be determined using the following:
2.14
Thus, the required number of ground motion records to obtain a reasonably good estimate of themean response can be significantly reduced by reducing the scatter in the demand estimates.
Saˆ To( )
Saˆ To( )
neq 4δ2
X2
------=
19
CHAPTER 3
DESCRIPTION OF THE RESEARCH PROGRAM
This chapter describes the research program as follows: (1) analytical models; (2) seismicdemand levels; (3) earthquake ground motion ensembles; (4) ground motion scaling methods; (5)reference response spectra; (6) nonlinear dynamic time-history analysis procedure; and (7) statis-tical evaluation of the results.
3.1 Analytical Models
Single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) models areinvestigated as described below.
3.1.1 Single-degree-of-freedom (SDOF) models
SDOF models with idealized bilinear force-displacement relationships as shown in Figure3.1 are used. The linear-elastic period,T, is varied by varying the mass,m. The linear-elastic stiff-ness,k, is set constant at 175,000 kN/m. The viscous damping ratio is assumed to be equal to aconstant value ofξ = 5%. Note that the period,T, can be varied either by varying the mass,m,while keeping the stiffness,k, constant (as is done in this research) or vice versa without affectingthe results. This follows since the governing equilibrium equation for the dynamic time-historyresponse of a SDOF system:
3.1
can be rewritten in terms of the structure period,T, as:
3.2
where∆(t) is the time-history displacement response,cd is the damping coefficient, is theground motion acceleration time-history,t is time, and the overdots represent derivatives withrespect to time,t. Equation 3.2 shows that the time-history response is independent of whichparameter (m or k) is varied or kept constant to varyT.
m∆̇̇ t( ) cd∆̇ t( ) k∆ t( )+ + m∆̇̇g t( )–=
∆̇̇ t( ) 4πT------ξ∆̇ t( ) 4π2
T2
---------∆ t( )+ + ∆̇̇g t( )–=
∆̇̇g t( )
20
Five hysteresis types are considered as illustrated in Figure 3.2. The importance of eachhysteresis type is briefly described as follows:
(1) The linear-elastic (LE) hysteresis type is used to determine the force demand,Felas, forthe structure to remain linear-elastic under a ground motion. These analyses correspond to aresponse modification coefficient,R, of 1;
(2) The bilinear elasto-plastic (EP) hysteresis type represents a structure with a high levelof inelastic energy dissipation and small self-centering capability (e.g., a steel structure);
(3) The stiffness-degrading (SD) hysteresis type represents a structure with a modest levelof inelastic energy dissipation and stiffness degradation due to cyclic damage (e.g., a monolithiccast-in-place reinforced-concrete structure).
The SD hysteresis type is used to represent the behavior of a structure that has a memorylimited to the largest positive and negative displacements from past yield excursions and may berepresentative of reinforced-concrete structures in which opening and closing of cracks lead topinching in the hysteretic behavior (Mahin and Lin, 1983; Hadidi-Tamjed, 1988; Nassar andKrawinkler, 1991). Figure 3.3 shows a detailed schematic of the SD type.
The initial linear-elastic stiffness of the SD hysteresis type is equal tok. Yielding duringfirst loading in the positive direction occurs whenFy is reached (point a in Fig. 3.3). Upon follow-ing a “yielding” branch with stiffnessαk, unloading occurs with a stiffness equal to the initial lin-ear-elastic stiffness,k, similar to the EP type (line bc). Loading in the reverse (i.e., negative)direction from point c requires that cracks be closed before yielding can occur at point d. This
Figure 3.1: SDOF model properties.
lateral displacement, ∆
late
ral f
orce
, F
FORCE REDUCTION
CHANGE IN LATERAL DISPLACEMENT
nlinF
elasF
∆elas
∆y
Linear-ElasticBehavior
k
αky
F elasF
=
R
F (1 - ) elas
1
R
m
k
∆
∆ = µ nlin
∆y0
IdealizedBilinearBehavior
kµ
21
results in a reduction in stiffness (i.e., stiffness degradation) upon crossing the zero force axis(point c), and, loading in the negative direction occurs along a “shooting” branch cd (with a stiff-ness ofksh) towards the yield point d. If yielding in the current loading direction has occurred dur-ing previous cycles, then the “shooting” branch ends at the previous unloading point with thelargest displacement (e.g., lines fb, jb, and he in Fig. 3.3).
(4) The bilinear-elastic (BE) hysteresis type represents a structure with no inelastic energydissipation and a high level of self-centering capability (e.g., a post-tensioned precast concretestructure as described by Kurama (2001)); and
(5) The combined bilinear-elastic/elasto-plastic (BP) hysteresis type represents a structurewith a modest level of inelastic energy dissipation and modest self-centering capability (e.g., apost-tensioned precast concrete structure with supplemental energy dissipation as described byKurama (2001)).
The BP hysteresis type is constructed by placing the BE and EP hysteresis types in paral-lel, as illustrated in Figure 3.4. The BE hysteresis type represents the primary system (e.g., precastconcrete structure) and the EP hysteresis type represents the secondary system (e.g., supplementalenergy dissipation system). The structural properties of the BP hysteresis type are defined by set-
F
∆
F
∆
F
∆
Bilinear Elasto-Plastic (EP)
Combined Bilinear Elastic/Elasto-Plastic (BP)
Bilinear-Elastic (BE)
F
∆
Linear-Elastic (LE)
F
∆
Stiffness-Degrading (SD)
0 0 0
0 0
"yielding" branch
"shooting" branch
"yielding" branch
Figure 3.2: Hysteresis types: (a) LE; (b) EP; (c) SD; (d) BE; (e) BP.
(a) (b) (c)
(d) (e)
22
ting the combined linear-elastic stiffness,ktot, to 175,000 kN/m. The linear-elastic force demand,Felas, is based on this linear-elastic stiffness. The reduced force,Fy, is likewise based on the result-ing linear-elastic force demand,Felas (see Fig 3.4). Thus, the BP hysteresis type lateral stiffnessand strength are:
3.3
3.4
wherekbeandkepare the linear-elastic stiffnesses of the BE and EP hysteresis types, respectively,andFbe andFep are the yield strengths of the BE and EP hysteresis types, respectively.
The relationships between the lateral stiffnesses and strengths of the BE and EP hysteresistypes are:
3.5
3.6
a
b
c
d
e
f
g
h
i
j
F
∆o
k
αk
k
k =sh,cd ∆ + ∆y c
Fy
αk
k =sh,fb ∆ - ∆b f
Fb
k
(F , ∆ )y y
(-F , -∆ )y y
"yielding" branch
(F , ∆ )b b
∆c
"shooting" branch
∆f
Figure 3.3: Schematic of stiffness degrading (SD) hysteresis type.
ktot kbe kep+=
Fy Fbe Fep+Felas
R------------= =
kep βskbe=
Fep βrFbe=
23
whereβs andβr are the BP type stiffness and strength ratios, respectively. The inelastic hystereticenergy dissipation of the BP type can be increased by increasingβr .
From Equations 3.3 through 3.6, the BE type stiffness and strength are:
3.7
3.8
The EP type post-yield stiffness ratio,αep, is set equal to zero and the BE type post-yieldstiffness ratio,αbe, is prescribed to result in an overall combined system post-yield stiffness ratioof α, i.e.:
elasF
yF elas
F=
R
epF
beF
ep2F
kbe
αbe
kbe
kep
αep
k = 0ep
kbe
αbe
kbe
k +be
kep
kep
αbe
k +be
∆ep
∆rmax
∆be
kbe
kep
m
Bili
near
-Ela
stic
Ela
sto-
Pla
stic
Bili
near
-Ela
stic
/Ela
sto-
Pla
stic
F
∆
+
=
Figure 3.4: Construction of the bilinear-elastic/elasto-plastic (BP) hysteresis type.
kbe
ktot
1 βs+( )-------------------=
Fbe
Felas
R 1 βr+( )-----------------------=
24
3.9
thus,
3.10
The increase in strength due to the supplemental energy dissipation system is typicallysmaller than the yield strength of the primary system (i.e.,Fep < Fbe ⇒ βr < 1). Assuming thatβs> βr, the load-displacement relationship for the BP hysteresis type can be seen in Figure 3.5a. Thebehavior of a similar system withβs = βr < 1 is shown in Figure 3.5b.
Notice that for the system withβs > βr (Fig. 3.5a), the combined load-displacement curveis trilinear. This behavior is typical of a structure with an “early-yield” supplemental energy dissi-pation system (Kasai et al., 1998). Although the early yielding of the supplemental energy dissi-pation system is not expected to significantly affect the overall dynamic response of the combined(i.e., primary plus secondary) system, it does make comparison with the other hysteresis types dif-ficult since their respective yield points do not match. This is illustrated in Fig. 3.6a. To make aclearer comparison possible, the alternate system in Figure 3.5b which has a bilinear load-dis-placement relationship is evaluated. In this case,∆epmust equal∆be (see Fig. 3.4) and, thus, it can
αbekbe αktot α 1 βs+( )kbe= =
αbe α 1 βs+( )=
Bilinear-Elastic (BE) Elasto-Plastic (EP)
Bilinear-Elastic/ Elasto-Plastic (BP)
+ =k
be
β kbes
F
∆
F F
0 0 0
(F , ∆ )be be
(1 + β )k s be
((1 + β )F , ∆ ) r be be
∆ ∆
α(1 + β )k s be
(β F , ∆ )be ber β
s
βr
(β =s
β < 1)r
α(1 + β )k s be
+ =k
be
(F , ∆ )be be
α(1 + β )k s be
β kbes
(β F , ∆ )be ber β
s
βr
(1 + β )k s be
kbe
((1 + β )F , ∆ ) r be be
α(1 + β )k s be
Bilinear-Elastic (BE) Elasto-Plastic (EP)
Bilinear-Elastic/ Elasto-Plastic (BP)
β < 1)r
F
∆
F
∆
F
∆
(β s > β ,
0 0 0
r
Figure 3.5: BP hysteresis type with: (a)βs> βr, βr < 1; (b) βs= βr < 1.
(a)
(b)
25
be shown thatβs must equalβr . This constraint requiresβs to be less than 1.0 sinceβr is assumedto be less than 1.0. The resulting system is compared to the EP hysteresis type in Figure 3.6b.
The BP hysteresis type improves upon the BE hysteresis type by increasing the inelastichysteretic energy dissipation of the system. However, this increase in the inelastic energy dissipa-tion is obtained at the expense of the system’s inherent self-centering capability. It can be shownthat the resulting maximum possible residual displacement,∆rmax (see Fig. 3.4), of the BP hyster-esis type is equal toβr∆be. As recommended by Kurama (2001),βr is limited to ≤ 0.5 in thisresearch to ensure that the residual displacements after a ground motion event are small.
3.1.2 Multi-degree-of-freedom (MDOF) models
Two MDOF models are considered as shown in Figure 3.7. These models represent four-story and eight-story cast-in-place reinforced concrete special moment-resisting office buildingframes, designed according to theUBC 1997equivalent lateral force procedure for a region withhigh seismicity (Los Angeles) and for theSE (soft) soil profile. Both structures have identical floorplans. For each structure, an interior frame in the E-W direction is analyzed assuming that thefloor diaphragms are sufficiently rigid under in-plane forces (Fig. 3.7).
Beam and column member sizes are shown in Figure 3.7. The concrete unconfined com-pressive strength is 41.4 MPa. The assumed yield strength of the reinforcing steel is 414 MPa.Structural properties for the four-story and eight-story frames are provided in Table 3.1. Gravityloads, including the weight of partitions, architectural finishes, utilities, structural members, andlive loads, are provided in Table 3.2. Note that the differences (due to different column tributary
F
∆
EP type
BP type, β > β ,s
β < 1r
EP type
BP type, β =s
βr< 1
F
∆
r
Figure 3.6: EP hysteresis type versus BP hysteresis type: (a)βs> βr; (b) βs= βr (matchingyield point).
(a) (b)
26
PLAN
4 bays at 5m = 20m
6 ba
ys a
t 5.5
m =
33m
ELEVATION
4 bays at 5m = 20m 4 bays at 5m = 20m
4 st
orie
s at
3.3
5m =
13.
4m
7 st
orie
s at
3.3
5m4m
Interior Frame Analyzed
N
4 bays at 5m = 20m
6 ba
ys a
t 5.5
m =
33m
Interior Frame Analyzed
600mm x 600mm (stories 1-4)525mm x 525mm (stories 5-8)
500mm x 500mm 400mm x 750mm (levels 1-4)400mm x 650mm (levels 5-8)
500mm x 500mm
525mm x 525mm
400mm x 650mm
120mm c.i.p. slab 120mm c.i.p. slab
2nd floor(level 1)
roof(level 4)
roof(level 8)
2nd floor(level 1)
Figure 3.7: Layout of structural system: (a) elevation; (b) four-story structure; (c) eight-story structure.
(b) (c)
(a)
27
heights) between the seismic weights,wx, of the intermediate floors and the seismic weights of the2nd floor and roof are assumed to be negligible.
The required design base shear capacities of the frames,Fdes(denoted asV in UBC 1997),defined at first “significant yield” (see Fig. 1.1a) were determined based on theUBC 1997linear-elastic smooth design response spectrum withR = 8.5 (ICBO, 1997) using:
3.11
whereW is the total frame seismic weight (including tributary self-weight and partitions),I is theseismic importance factor (taken as 1.0), andCv andCa are seismic coefficients tabulated in UBC1997 based on site seismicity and site soil characteristics. For Los Angeles (seismic zone 4 inUBC 1997) and theSE soil profile, theCv andCa coefficients are given as:
, 3.12
whereNv and Na are near-field factors based on the closest distance from the site to aknown seismic source. The structures are assumed to be farther than 10 km from a known seismicsource, thusNv = Na = 1.0 (ICBO 1997).
Table 3.1: Frame lateral-system properties
Structure Height
(m)
Length
(m)
Frame seismicweight per floor or
roof, wx (kN)
Total frameseismic weight,W
(kN)
Fundamentalperiod,T
(sec.)
Frame designbase shear,Fdes
(kN)
4-story 13.4 20 756.2 3024.8 0.49 332.7
8-story 27.5 20 Levels 1-4: 790.7Levels 5-8: 756.2
6187.5 0.87 681.0
Table 3.2: Gravity loads
Load description Gravity load
Normal weight reinforced concrete (i.e., floor slab,columns, and beams)
24 kN/m3
Floor finish, ceiling, utilities, and movable partitions 1.20 kPa
Curtain walls, glazing, etc., supported by peripherybeams only, extending over floor height of 3.35 m
0.50 kPa
Live load on all floors and the roof 2.50 kPa
2.5CaI
R-----------------W Fdes≥
CvI
RT---------W 0.11CaIW≥=
Cv 0.36Na= Ca 0.96Nv=
28
The design base shear force,Fdes, is distributed over the height of each structure as:
3.13
wheren is the number of floor and roof levels,Ft is the portion of the design force concentrated atthe top of the structure in addition toFn (for the roof), andFi is the lateral force applied at leveli.The concentrated force at the roof,Ft, is calculated as:
3.14
The value ofFt need not exceed 0.25Fdesand may be considered zero whenT is less than or equalto 0.7 sec. (which is the case for the four-story structure). The remaining portion ofFdesis distrib-uted over the height of the structure as follows:
3.15
wherewx andwi are equal to the seismic weights at levelx and leveli, respectively, andhx andhiare the heights from ground to levelx and leveli, respectively.
The beams and columns were proportioned based on theUBC 1997load combinations,including both gravity and earthquake loads. The load combinations considered for this designare:
3.16
and
3.17
whereDL is the design dead load,LL is the design live load,f1 is the live load modification factor,andEL is the design earthquake load. The live load modification factor,f1, depends on the type oflive load and was taken as 0.5 (for miscellaneous live loads (ICBO, 1997)). The load combina-tions in Equations 3.16-3.17 were applied to each structure to determine the design member forcedemands, as illustrated in Figures 3.8a-b. The gravity loads on the beams were assumed to act at adistance of 1/3L from the column faces, whereL is the beam clear span length.
A simple analytical model is developed to conduct nonlinear static and nonlinear dynamictime-history analyses of each frame. The structures are assumed to be fixed at the column bases.The initial flexural stiffnesses of the interior columns are assumed to be 100% of the gross-section
Fdes Ft Fii 1=
n
∑+=
Ft 0.07TFdes=
Fx
Fdes Ft–( )wxhx
wihii 1=
n
∑---------------------------------------=
U1 1.2DL f 1LL 1.0EL+ +=
U2 0.9DL 1.0EL±=
29
linear-elastic flexural stiffnesses. The initial “effective” flexural stiffnesses of the exterior columnsand floor and roof beams are assumed to be 80% and 30% of the gross-section linear-elastic flex-ural stiffnesses, respectively, to account for cracking in the members. An effective T-beam flangewidth of 625mm (1/8 the center-to-center span length of the beam (Paulay and Priestley, 1992)) isincluded in the calculation of the gross-section flexural stiffness for the beams. The beams andcolumns are modeled using linear-elastic beam-column elements in the structural analysis pro-gram, DRAIN-2DX (Prakash et al., 1993). Shear deformations in the members are included. Fulldepth rigid end zones are assumed at the beam-column joints (Fig. 3.8c).
Nonlinear behavior at the beam ends and in the beam-column joints is modeled as concen-trated plastic hinges at the beam-column joints using zero-length rotational spring elements withstiffness-degrading properties, as shown in Figures 3.8c and 3.9a (Wu, 1995). The moment-rota-tion behavior of the springs under positive and negative bending are assumed to be the same. The
Figure 3.8c
node
linear-elasticbeam-column element
zero-length rotationalspring element
rigid end zone
fiber beam-columnelement1.5hc
hc
Figure 3.8: MDOF models: (a) four-story elevation; (b) eight-story elevation; (c) close-upof analytical model.
(a) (b)
(c)
30
yield moment capacities of the springs,Ms, are set equal to the design beam moment demands,ignoring any overstrength. It is assumed that the interior and exterior joint springs used in everyfour levels of each structure are the same. The largest design beam moment demand is used as theMs value for each set of interior and exterior joint springs in the structure as shown in Table 3.3.
The initial stiffness of a beam rotational spring,ks1, is set greater than the flexural stiffnessof the adjacent linear-elastic beam element such that little deformation occurs in the spring beforeyielding. This is achieved by assigning a linear-elastic stiffness to the spring as follows:
3.18
where ϕ is a stiffness factor,E is the concrete Young’s modulus,Ie is the assumed effective(including the effect of cracking) moment of inertia of the adjacent beam element (i.e., 30% of thegross-section T-beam moment of inertia), andL is the adjacent clear span length. For this study,ϕis taken as 5, resulting in a maximum spring rotation of 0.4% relative to the beam-column joint(i.e., node) rotation before yielding occurs.
The moment-rotation post-yield stiffness of the zero-length springs is:
3.19
whereMp, θp, andθs are defined in Figure 3.9a. The plastic rotation, (θp - θs), assuming an equiv-alent plastic hinge length oflp (Paulay and Priestley, 1992), can be calculated as:
ks1
ϕ4EIe
L----------------=
ks2
M p Ms–( )θp θs–( )
--------------------------=
θM
M
θ0
ks1
ks2
P
M
0 0.060
700
Str
ess
(MP
a)
Strain (mm/mm)
−4 0 4x 10−3−5
0
45
Str
ess
(MP
a)
Strain (mm/mm)
unconfined concrete stress-strain relationship
steel stress-strain relationship
fiber
sM , s
θ
tension
compression
pM , p
θ
axis of bending
M-θ
crushing ofconcrete notmodeled
Figure 3.9: Element models: (a) beam end rotational spring element; (b) column base fiberelement.
(a) (b)
31
3.20
whereφp andφs are the curvatures corresponding toθp andθs, respectively.
With the linear-elastic beam elements and the beam rotational springs in series, the follow-ing expression can be written:
3.21
whereαφ is the moment-curvature post-yield stiffness ratio andKs2 is the moment-curvature post-yield stiffness for the equivalent plastic hinge length as:
3.22
Equation 3.21 can be rearranged to determineKs2 as:
3.23
Thus, using Equations 3.19, 3.20, 3.22, and 3.23, the moment-rotation post-yield stiffness of thesprings,ks2, can be calculated as:
3.24
The equivalent plastic hinge length,lp, is assumed to be 1.5 times the beam depth and themoment-curvature post-yield stiffness ratio,αφ, is set at 2%.
Table 3.3: Yield moment capacities for beam rotational springs
Frame Floor or roof level Beam end locationYield moment capacity,
Ms (kN-m)
4-story 1-4interior joint 136.4
exterior joint 128.7
8-story1-4
interior joint 376.1
exterior joint 299.9
5-8interior joint 231.7
exterior joint 206.1
θp θs–( ) φp φs–( )l p=
1αφEIe--------------- 1
EIe-------- 1
Ks2--------+=
Ks2
M p Ms–( )φp φs–( )
--------------------------=
Ks2
αφEIe
1 αφ–( )--------------------=
ks2
Ks2
l p--------
αφEIe
1 αφ–( )l p-------------------------= =
32
Yielding in the columns other than at the bases is prevented. Nonlinear behavior at the col-umn bases is modeled using fiber beam-column elements to account for axial-flexural interaction(Prakash et al., 1993). The cross-section of each column at the base is modeled with an arrange-ment of longitudinal “fibers”, as shown in Figure 3.9b. The length of the fiber beam-column ele-ments is assumed to be 1.5 times the column depth,hc (see Fig. 3.8c). The stress-strain behaviorof each fiber is assigned based on the assumed material properties. The assumed stress-strain rela-tionships of the reinforcing steel and unconfined concrete used in this study are provided in Figure3.9b. The tensile strength of the concrete is included, considering the effect of tension stiffening.Crushing of concrete in compression is not modeled assuming that a sufficient amount of confine-ment reinforcement is used in the critical regions of the members to ensure ductile behavior. Thenumber of fibers and the amount of steel reinforcement used in the columns at the base are listedin Table 3.4.
The normalized (with respect to the seismic weight,W) base-shear-roof-drift behavior ofthe two frames under combined gravity and cyclic lateral loading is shown in Figure 3.10. Theroof drift is equal to the roof lateral displacement divided by the structure height. The frames areassumed to be sufficiently detailed to achieve ductile behavior. For the analyses of the frames
Table 3.4: Column base fiber element properties
Frame Column base location Number of fibersaxial/flexural steel
reinforcementa
aThe steel reinforcement is arranged symmetrically about the axis of bending, as shown in Fig. 3.9b.
4-storyinterior 24 4-#7 bars
exterior 22 6-#6 bars
8-storyinterior 26 6-#8 bars
exterior 22 4-#8 bars
−0.03 −0.02 −0.01 0 0.01 0.02 0.03−0.3
−0.2
−0.1
0
0.1
0.2
0.3
roof drift (rad)
base
she
ar/s
eism
ic w
eigh
t (g)
−0.03 −0.02 −0.01 0 0.01 0.02 0.03−0.3
−0.2
−0.1
0
0.1
0.2
0.3
roof drift (rad)
base
she
ar/s
eism
ic w
eigh
t (g)
Figure 3.10: Normalized cyclic base-shear-roof-drift behavior: (a) four-story frame; (b) eight-story frame.
(a) (b)
33
other than the analyses used in design, the gravity load is assumed to be 100% of the design deadload,DL, plus 25% of the design live load,LL, with no live load reduction. The lateral load distri-bution over the height of the frames follows theUBC 1997equivalent lateral force pattern given inEquation 3.15 (ICBO, 1997). P-∆ effects are considered in the analyses of the frames.
It is noted that it may be possible to develop more accurate analytical models for the struc-tures, but this was not done to keep the computational time manageable for the large number ofground motion records considered in the research.
3.2 Seismic Demand Levels
The primary objective of the recent seismic design provisions inIBC 2000is to presentcriteria for the seismic design of building structures to minimize the risk to life (ICC, 2000;BSSC, 1998). To satisfy this objective, two seismic demand levels are specified in these provi-sions.
The seismic demand levels inIBC 2000are defined by three characteristics (Hart andWong, 2000): (1) design life; (2) probability of exceedance; and (3) return period. Design life, orexposure time, of the structure is a best estimate at the time of the design of the structure of howlong the building will be operational before demolition or seismic upgrading. Probability ofexceedance is the chance of a particular level of earthquake being exceeded during the design lifeof the structure. Return period is the expected number of years between occurrences of a particu-lar level of earthquake.
Predictions for possible future earthquake occurrences are often modeled with a Poissondistribution (Ang and Tang, 1975):
, x = 0, 1, 2, . . . 3.25
whereTr is the return period,tdes is the design life, andpx is the probability of havingx occur-rences during the design life,tdes(Hart and Wong, 2000). Using Equation 3.25, the probability ofnon-occurrence can be found by settingx equal to 0:
3.26
Inversely, the probability of exceedance (or occurrence) is:
3.27
px
tdes
Tr--------
xe
tdes
Tr-------
–
x!--------------------------------=
px 0= e
tdes
Tr-------
–
=
px 0≠ 1 px 0=– 1 e
tdes
Tr-------
–
–= =
34
Thus, for a given expected design life and probability of exceedance, the return period of a partic-ular earthquake can be calculated by rearranging Equation 3.27:
3.28
The first seismic demand level inIBC 2000 is defined using a “maximum consideredearthquake” ground motion which has a 2 percent probability of being exceeded in 50 years (cor-responding to a return period,Tr, of approximately 2500 years, by substitutingtdes= 50 andpx ≠ 0= 0.02 in Eq. 3.28). The maximum considered earthquake ground motion is referred to as the“survival-level” ground motion herein.
The second seismic demand level specified in the currentIBC 2000provisions is definedusing a “design-level” ground motion. The acceleration response spectrum for the design-levelground motion is determined by multiplying the spectrum for the survival-level ground motion bya factor of 2/3 (referred to as the “seismic margin” in the NEHRP provisions (BSSC, 1998)). Thisdemand level roughly corresponds to an earthquake with a 10 percent probability of beingexceeded in 50 years, or a return period of approximately 500 years, for coastal California and alower probability of occurrence (a return period of approximately 1400 years) for the easternUnited States (BSSC, 1998).
In anticipation of performance-based seismic design provisions in future design codes,with explicit objectives for allowable damage within a structure, the capacity-demand index rela-tionships in this research are developed using both design-level ground motions and survival-levelground motions as described below.
3.3 Ground Motion Records
To derive clear conclusions and unbiased results from this research, it is necessary toappropriately select and group ground motion records considering the following parameters: (1)site soil characteristics; (2) seismic demand level; (3) site seismicity; and (4) epicentral distance.The ground motion records used in this research are listed in Tables A.1 through A.13 and theiracceleration time histories are shown in Figures A.1 through A.13 of Appendix A. The groundmotion records include both historic and generated records. A large number of records are used inorder to generate a large statistical data set.
Three major ground motion suites are used in the research: (1) ground motions compiledat the University of Notre Dame; (2) ground motions compiled by Nassar and Krawinkler (1991);and (3) ground motions compiled by the SAC* steel project (Somerville et al., 1997).
* SAC is a joint venture funded by the Federal Emergency Management Agency (FEMA) comprised of threeorganizations: Structural Engineers Association of California (SEAOC), Applied Technology Council(ATC), and California Universities for Research in Earthquake Engineering (CUREE).
Tr
tdes
1 px 0≠–( )ln--------------------------------–=
35
Ground motion records for sites with very dense (SC), stiff (SD), and soft (SE) soil profileswere collected at the University of Notre Dame (UND) to investigate the effect of ground motionscaling method on the scatter in the seismic demand indices, including the effect of site soil char-acteristics (Tables A.1 through A.3). TheSC, SD, andSE soil profiles correspond to site classes C,D, and E inIBC 2000(ICC, 200), respectively. Twenty ground motion records were selected foreach soil profile. Detailed site soil properties (e.g., shear wave velocities) were not available formany recording stations, thus the classification of the ground motion records was based on gen-eral soil descriptions (e.g., rock, alluvium, soft clay, etc.), which are available for all stations.Each ground motion record was classified by matching, as close as possible, the general soildescriptions with the definitions for the site soil characteristics provided in Table 3.5 (ICC, 2000).
The ground motions compiled by Nassar and Krawinkler (N&K) are used as a baselineensemble to validate the results in this research and to provide additional results for the very dense(SC) soil profile. These ground motions include fifteen records (Table A.4) representative of theS1soil profile (equivalent to theSC soil profile inIBC 2000) from the western United States.
The ground motion records compiled by the SAC steel project (Tables A.5 through A.13)are primarily used to investigate the effect of the reference response spectra on theR-µ-T relation-ships and to generate new relationships for other seismic demand indices. These ground motionsare categorized by: (1) site soil characteristics (stiff soil and soft soil, i.e.,SD andSE soil profilesin IBC 2000); (2) seismic demand level (design-level and survival-level); (3) site seismicity (lowseismicity and high seismicity, i.e., Boston and Los Angeles); and (4) epicentral distance (near-field, NF, and far-field).
The ground motion records in Tables A.5, A.6, A.9, and A.10 were scaled by the SACsteel project to match (at selected periods of 0.3, 1, 2, and 4 sec.) linear-elastic smooth designacceleration response spectra representative of each ground motion ensemble in the tables listedabove (BSSC, 1998; ICC, 2000; Somerville et al., 1997). The shapes of the acceleration responsespectra of the individual ground motion records were not modified in the scaling procedure.Instead, this scaling was done by finding a single factor for each ground motion record that mini-mized the weighted sum of the square error between the linear-elastic ground motion response
Table 3.5: Site soil definitions (adapted fromIBC 2000 (ICC, 2000))
Site soil
profilea
aTheSC, SD, andSE soil profiles correspond to site classes C, D, and E inIBC 2000, respectively.
Description
Average properties in top 30.5 m of site soil
Soil shear wavevelocity,vs (m/s)
Standardpenetration
resistance,N
Soil unconfinedshear strength,su,
(kPa)
SC Very dense soil 366 <vs ≤ 762 N > 50 su ≥ 96
SD Stiff soil 183≤ vs ≤ 366 15≤ N ≤ 50 48≤ su ≤ 96
SE Soft soil vs < 183 N < 15 su < 48
36
spectrum and the corresponding smooth design response spectrum for 5% damping. The weightsused were 0.1, 0.3, 0.3, and 0.3 forT = 0.3, 1, 2, and 4 sec., respectively.
The design-level soft (SE) soil ground motions in Tables A.7 and A.11 were generated bythe SAC steel project using site response analyses conducted on different soil profiles broadly rep-resentative ofSE soil conditions in Boston and Los Angeles (Somerville et al., 1997). The siteresponse analyses were conducted using the equivalent linear analysis computer programSHAKE91 (Idriss and Sun, 1992). The design-level ground motions for soft (SE) soil were gener-ated using the SAC Boston and Los Angeles design-level ensembles for stiff (SD) soil as the bed-rock motions.
The ground motions in Table A.13 were selected by the SAC steel project to representnear-field (NF) ground motions from earthquakes having a variety of faulting mechanisms (strike-slip, oblique, and thrust) with magnitudes ranging from 6 3/4 to 7 1/2 and epicentral distancesranging from 0 to 18 km (Somerville et al., 1997). The ground motions were not scaled to match asmooth design response spectrum.
The SAC ground motion ensembles compiled by Somerville et al. (1997) do not includesurvival-level ground motions on soft (SE) soil for sites with low and high seismicities. Thus,these ground motions were augmented with two ground motion ensembles (Tables A.8 and A.12)generated at the University of Notre Dame following the same procedure used by Somerville et al.(1997). The additional ensembles are intended to be representative of survival-level groundmotions onSE soil for Boston and Los Angeles.
The site response analyses were conducted using the equivalent linear analysis computerprogram EERA, which has been shown to compare well with the SHAKE91 program used by theSAC steel project (Bardet et al., 2000). The equivalent linear approach consists of performing lin-ear dynamic time-history analyses by approximating the expected nonlinear hysteretic shearstress-strain behavior of the soil, shown in Figure 3.11a, as an equivalent linear system, shown in
τ
G
η
Gγ
ηγ°
γ°γ
shear stress, τ
shear strain, γ
Gmax
G
0
(τ , γ )c c
Figure 3.11: Equivalent linear model for the site response analyses: (a) assumed nonlinearhysteretic stress-strain behavior of the soil; (b) equivalent linear model.
(a) (b)
37
Figure 3.11b. In this procedure, the shear stress,τ, depends on the shear strain,γ, and the shearstrain rate, , as follows:
3.29
whereG is the equivalent linear shear modulus andη is the viscosity (Fig. 3.11b). The equivalentlinear shear modulus,G, is taken as the secant shear modulus as shown in Figure 3.11a. The vis-cosity,η, is related toG and the equivalent linear damping ratio,ξ:
3.30
whereω is the equivalent linear soil frequency (Bardet et al., 2000). Both the equivalent linearshear modulus,G, and the equivalent linear damping ratio,ξ, are dependent on the shear strainamplitude,γc (Fig. 3.11a). The shear strain amplitude,γc, is determined as the maximum shearstrain from equivalent linear site response analyses that are conducted iteratively as described inBardet et al. (2000).
The site response analyses were performed based on similar soil conditions as used bySomerville et al. (1997) in the development of the SAC design-levelSE soil ground motionrecords for Boston and Los Angeles. A soft soil profile with a unit weight of 19.66 kN/m3, anaverage shear wave velocity of 136 m/sec., and a total depth of 30.5 m was used. The shear wavevelocities as a function of depth (Fig. 3.12a) were taken from Somerville et al. (1997), and theshear modulus reduction factor,G/Gmax, and damping ratio,ξ, as a function of the shear strainamplitude,γc, for soil with a plasticity index, PI†, of 30 (Fig. 3.12b) were taken from empiricalcurves proposed by Vucetic and Dobry (1991). This soil profile is broadly representative of softsoil conditions encountered in Boston and Los Angeles. The survival-level ground motions forsoft (SE) soil were generated using the SAC Boston and Los Angeles survival-level ensembles forstiff (SD) soil as the bedrock motions. Tables A.8 and A.12 list the ground motions generated fromthe site response analyses using the EERA program.
In total, 255 ground motion records are used in this research. Apart from the groundmotions generated using the EERA program, the acceleration time-history records of the groundmotions were obtained from the National Geophysical Data Center (NGDC) data archive, theMultidisciplinary Center for Earthquake Engineering Research (MCEER) database, groundmotion records collected by Kurama et al. (1997), and the SAC steel project (Somerville et al.,1997).
†The plasticity index, PI, is a measure of the consistency, or degree of firmness, of the soil based on theplastic state, which is between the semisolid and liquid states (Cernica, 1995). PI = 0 corresponds to grav-els and sands (cohesionless) and PI≅ 50 corresponds to high plasticity clays (Vucetic and Dobry, 1991).
°γ
τ Gγ η°γ+=
η 2ξGω
-----------=
38
3.3.1 Important properties of the ground motion records
Properties of the ground motion records, including peak ground acceleration, PGA, andmaximum incremental velocity, MIV, are given in Tables A.1 through A.13. Incremental velocity,IV, is the area under the acceleration time-history of a ground motion between two consecutivezero acceleration crossings. The impulsive loading during a ground motion can be calculated bymultiplying the mass,m, with the IV. The maximum IV (i.e., MIV, which is the maximum areaunder the acceleration time-history of a ground motion between two consecutive zero accelerationcrossings) may be a better indicator of the damage potential of a ground motion than PGA since itcaptures the impulsive characteristics of the ground motion (Kurama et al., 1997).
For a ground motion with a large PGA and a small MIV (short-duration pulse, called anacceleration spike), most of the applied impulse is absorbed by the structural inertia and minordeformation, thus damage, occurs in the structure. A ground motion with a moderate PGA and alarge MIV (long-duration pulse) can produce significant structural deformation especially if theduration of the acceleration pulse is long compared to the fundamental period of the structure. Ascan be seen in Tables A.3, A.7-A.9, and A.11-A.13, the ground motions with large accelerationpulses (i.e., larger values of MIV) are more likely to occur on soft soil and near-field sites (Singh,1985; Naeim and Anderson, 1993).
The linear-elastic SDOF pseudo-acceleration and pseudo-velocity response spectra of theground motions with 5 percent viscous damping are shown in Figures A.14 through A.26 ofAppendix A. These response spectra are calculated from the acceleration time-history records ofthe ground motions using 301 period values varying exponentially between 0.1 and 3.0 seconds.
10−4
10−3
10−2
10−1
100
1010
15
30D
amping R
atio, ξ (%)
10−4
10−3
10−2
10−1
100
101
0
0.5
1
Shear Strain, γ (%)S
hear
Mod
ulus
Red
uctio
n F
acto
r (G
/Gm
ax)
ξ
G/Gmax
c
100 120 140 160 180
−30
−25
−20
−15
−10
−5
0
Shear Wave Velocity (m/sec)
Soi
l Dep
th (
m)
Figure 3.12: Soil properties for the site response analyses: (a) shear wave velocity; (b)shear modulus reduction factor and damping ratio.
(a) (b)
39
The periods are exponentially spaced because the variation in the seismic response of SDOF sys-tems tend to be large at short periods and small at long periods.
3.3.2 Strong motion duration
The strong motion duration for the ground motions,Dsm, was determined by cutoff meth-ods and by the root mean square acceleration (RMSA) method (McCann and Shah, 1979). Thecutoff methods defineDsm as the maximum duration for which the acceleration remains largerthan a specified value. In this research, ten percent of PGA and five percent of the gravitationalconstant, g, were used as cutoff values. The RMSA method consistently predicted longer strongmotion durations for the ground motion records, and thus was adopted.
The RMSA method makes use of a cumulative root mean square function (CRF), definedin discrete form as:
3.31
wherea(tj) is the acceleration record,tj is the time at discretization pointj, n is the current discret-ization point, andl is the total number of discretization points. Figure 3.13 shows a plot of theCRF function for the 1940 El Centro ground motion record (ELCN) scaled by a factor of 2.0. Thederivatives of the CRF and reverse CRF (where the acceleration time-history is reversed) aretaken as a function oft and the times at which these derivatives become and remain negative aretaken as the final and initial cutoff points,tend and tinit, respectively (see Fig. 3.13). The cutoff
CRF
a2 t j( )j 1=
n
∑n 1–
------------------------ n 2 … l, ,= =
0 10 20 30 40 50 60−1
−0.5
0
0.5
1
0 10 20 30 40 50 600
100
200
300
0 10 20 30 40 50 60−10
−5
0
5
10
tend
0 10 20 30 40 50 60−1
−0.5
0
0.5
1
0 10 20 30 40 50 600
50
100
0 10 20 30 40 50 60−10
−5
0
5
10
tinitD = t - t
sm end init
acce
lera
tion
(g)
CR
F (
cm/s
ec )2
dCR
F/d
t (cm
/sec
)
3
t (sec) t (sec)
Figure 3.13: Acceleration time-history, cumulative RMSA function (CRF), and derivativeof the CRF function (1940 El Centro (ELCN) x 2.0): (a) forward CRF; (b) reverse CRF.
(a) (b)
40
times for the strong motion duration are taken at these points because these points indicate that therate of input energy is decreasing (the slope of the CRF function remains negative), implying thatthe remainder of the ground motion record does not contain any high energy pulses.
3.4 Ground Motion Scaling Methods
Seven different ground motion scaling methods are used to investigate the scatter in theestimated demands from the dynamic analyses. These scaling methods are described below:
(1) Peak ground acceleration (PGA): Each ground motion record is scaled to the averagePGA of the ground motion ensemble. As described earlier, most of the previous research usingnonlinear dynamic analysis procedures is based on this scaling method.
(2) Effective peak acceleration (EPA): Each ground motion record is scaled to the averageEPA of the ground motion ensemble. According to the 1994 NEHRP provisions (BSSC, 1994),EPA is calculated as the average linear-elastic 5%-damped spectral acceleration for the periodrange of 0.1 and 0.5 seconds divided by 2.5. The 2.5 coefficient relates back to the formulation ofthe smooth design response spectra inATC 3-06 (ATC, 1978).
(3) Effective peak velocity (EPV): Each ground motion record is scaled to the averageEPV of the ground motion ensemble. According to the 1994 NEHRP provisions (BSSC, 1994),EPV is equal to the linear-elastic 5%-damped spectral pseudo-velocity at a period of 1 second. Inthis research, the EPV values of the ground motions are calculated as the average spectral pseudo-velocity for periods between 0.8 and 1.2 seconds as recommended by Kurama et al. (1997).
(4) Maximum incremental velocity (MIV): Each ground motion record is scaled to theaverage MIV of the ground motion ensemble.
(5) Arias intensity-based parameter (A95): Each ground motion record is scaled to theaverageA95 parameter of the ground motion ensemble. TheA95 parameter is defined as the levelof acceleration which contains up to 95 percent of the Arias Intensity,Es (Sarma and Yang, 1987),which is defined as (Arias, 1969):
3.32
wherea(t) is the acceleration time-history of the ground motion record. TheA95 parameter, simi-lar to MIV, has been shown to be a good measure of the damage potential of a ground motionrecord (Sarma and Yang, 1987). Based on a linear regression study performed by Sarma and Yang(1987), the relationship between theA95 parameter andEs can be approximated as:
3.33
or,
Es a2 t( ) td0
t
∫=
A95log 0.438 Eslog 0.117–=
41
3.34
(6) Spectral acceleration at the structure fundamental period ( ): Each groundmotion record is scaled to the average linear-elastic 5%-damped spectral acceleration of theground motion ensemble at the linear-elastic fundamental period of the structure being analyzed,To. Different from the scaling methods described above, the method depends on the struc-ture properties (i.e.,To) as well as the ground motion characteristics. For example, if the structurehas a period ofTo = 0.18 sec., each ground motion is scaled such that the spectral acceleration at0.18 sec. is equal to the average spectral acceleration, , of the ground motion ensemble at0.18 sec. as illustrated in Figure 3.14a. The parameter is sometimes referred to as thestructure-specific ground motion spectral intensity (Shome and Cornell, 1998).
(7) Spectral acceleration over a range of structure periods ( ): Each groundmotion record is scaled to the average linear-elastic 5%-damped spectral acceleration of theground motion ensemble over a range of structure periods. First, the average spectral acceleration,
A95 0.764Es0.438=
Saˆ To( )
Saˆ To( )
Saˆ To( )
ξ = 0.05To
0 0.5 1 1.5 2 2.5 3 3.50
1
2
T (sec)
Sa (
g)
S (T )a o
ˆ
Average Response Spectrum
Scaled Acceleration Response Spectra
T (sec)
TµTo
µ
0 0.5 1 1.5 2 2.5 30
1
2
S (T →T )a o
ˆ
Scaled Acceleration Response Spectra
3.5
Sa (
g)
ξ = 0.05
Average Response Spectrum
Figure 3.14: Scaling based on the spectral acceleration (UNDSC soil ground motionensemble): (a) at the structure period ( method); (b) over a range of structureperiods ( method).
Saˆ To( )
Saˆ To Tµ→( )
(a)
(b)
Saˆ To( )
Saˆ To Tµ→( )
42
, of the ground motion ensemble over the period range is calculated.Then, the ground motions are scaled such that the average spectral acceleration of each groundmotion over the period range is equal to , as illustrated in Figure 3.14b.This scaling method takes into account the elongation of the structure period due to nonlinearbehavior. Thus, the period range depends on the amount of nonlinear deformation expected in thestructure. In this research, the elongated period,Tµ, is calculated based on the secant stiffness,kµ,corresponding to the maximum displacement demand,∆nlin, as (Fig. 3.1):
3.35
Since the structure period is defined as:
3.36
the ratio betweenTµ andTo can be calculated as:
3.37
Finally, Equation 3.37 is rearranged to findTµ:
3.38
The maximum displacement ductility demand,µ, can be estimated as a function ofR, T,and α using existing relationships. Equations 2.1-2.3, as developed by Nassar and Krawinkler(1991), are used in this research.
Note that thea andb coefficients in Equation 2.3 were developed by Nassar and Krawin-kler (1991) using nonlinear regression analyses based on far-field ground motions recorded atsites representative of theSC soil profile for the EP hysteresis type (Table 2.1). Thus, these coeffi-cients may not be applicable for sites withSD and SE soil profiles and near-field conditions(Miranda, 1993; Krawinkler and Rahnama, 1992) or for different hysteresis types. Newa andbcoefficients were developed in this study for the UNDSC, SD, andSE soil ground motions and theSAC Los Angeles near-field (NF) ground motions (Tables A.1-A.3 and A.13), and for the EP, SD,BE, and BP hysteresis types using regression analyses similar to those used by Nassar andKrawinkler (1991). These new coefficients are given in Table 3.6.
Saˆ To Tµ→( ) To Tµ→
To Tµ→ Saˆ To Tµ→( )
kµFnlin
∆nlin-----------
k∆y αk ∆nlin ∆y–( )+
∆nlin---------------------------------------------------= =
T 2π mk----=
Tµ2
To2
------ kkµ-----
∆nlin
∆y α ∆nlin ∆y–( )+---------------------------------------------
∆nlin
∆y 1 α∆nlin
∆y----------- 1–
+
--------------------------------------------------- µαµ 1 α–+--------------------------= = = =
Tµ Toµ
αµ 1 α–+--------------------------=
43
3.5 Reference Response Spectra
In order to investigate the effect of reference response spectra on theR-µ-T relationships,three types of linear-elastic acceleration response spectra were used to calculateFy = Felas/R,namely: (1) response spectra based on the individual ground motion records (IND); (2) averageground motion response spectra based on the ground motion ensembles (AVG); and (3) smoothdesign response spectra from current seismic design provisions (DES).
In the case of the IND spectra,Fy is calculated as the mass,m, times the linear-elasticspectral acceleration (at the structure period) of the individual ground motion record divided byR.The IND spectra for the ground motion records are shown by the thin solid lines in Figures A.14-A.26.
Similarly, in the case of the AVG spectra,Fy is calculated as the mass,m, times the aver-age linear-elastic spectral acceleration (at the structure period) of the ground motion ensembledivided byR. The AVG spectra for the ground motion ensembles are shown by the thick dashedlines in Figures A.14-A.26.
In the case of the DES spectra,Fy is calculated as the mass,m, times the spectral accelera-tion (at the structure period) from the design response spectrum divided byR. The general shapeof the design response spectra inIBC 2000andUBC 1997is used as shown in Figure 3.15a. Theseismic coefficientsSd1, Sds, Cv, andCa are mapped or tabulated based on site seismicity, seismicdemand level, site soil characteristics, and epicentral distance. Note thatSd1 andSds for the sur-vival-level ground motion correspond toSM1 andSMS for the “maximum considered earthquake”in IBC 2000, and similarly,Sd1 andSds for the design-level ground motion correspond toSD1 andSDS for the “design earthquake” inIBC 2000. Additionally, it is noted that the seismic coefficientsCv andCa in UBC 1997 are specified for the design-level ground motion.
Figures 3.15b and c show the smooth design (DES) response spectra used in this researchfor sites with low seismicity and high seismicity (i.e., Boston and Los Angeles, respectively) and
Table 3.6: Values fora andb coefficients developed in this study
EP typeα = 0.10
SD typeα = 0.10
BE typeα = 0.10
BP typeα = 0.10,βr = βs = 1/3
Ground motion ensemble a b a b a b a b
UND SC soil 1.46 0.58 -- -- -- -- -- --
UND SD soil 1.49 0.46 1.60 0.50 2.74 0.76 2.33 0.66
UND SE soil -0.41 0.95 -- -- -- -- -- --
SAC Los Angeles,NF 0.35 0.89 -- -- -- -- -- --
44
with design-level and survival-level seismic demands, respectively. The seismic coefficients thatdefine these response spectra are obtained from theIBC 2000provisions and are given in Table3.7. It is noted that theIBC 2000provisions do not provide design spectra for soft (SE) soil sites inregions with high seismicity. Thus, theSE soil design spectra for Los Angeles are based on theUBC 1997provisions (ICBO, 1997) instead as shown in Figure 3.15 and Table 3.7. Also, the IBC2000 provisions do not take near-field conditions into account. Thus, the Los Angeles design-level SD soil DES spectrum in Figure 3.15a and Table 3.7 is used for both far-field and near-field(NF) conditions.
Comparisons between the average (AVG) response spectra and the design (DES) responsespectra used in this research are provided in Figures 3.16-3.20.
0 0.5 1 1.5 2 2.5 3T (sec)
Sa (
g)
0 0.5 1 1.5 2 2.5 3T (sec)
Sa (
g)
0
0.5
1
1.5
2
2.5
0
0.5
1
1.5
2
2.5ξ = 5% ξ = 5%S
DS
EBoston
Los Angeles
SC
SD
SE
SD
SE
Boston
Los AngelesS
DS
E
T
Sa
ξ = 5%S (IBC 2000)
ds
2.5C (UBC 1997)a
0.4S (IBC 2000)ds
C (UBC 1997)a
S /T (IBC 2000)d1
C /T (UBC 1997)v
T = 0.2To sT = S /S (IBC 2000)
d1 dss
T = 0.4C /C (UBC 1997)v as
Figure 3.15: Design spectra: (a) general shape; (b) design-level; (c) survival-level.
(c)(b)
(a)
45
Table 3.7: Seismic coefficients for the smooth design (DES) response spectra
Site Seismicity Demand Level Site Soil Sds Sd1
Boston
DesignSD 0.31 0.13
SE 0.47 0.19
SurvivalSD 0.47 0.19
SE 0.70 0.28
Los Angeles
Design
SC 1.37 0.70
SD 1.37a 0.81a
SE 0.90b 0.96b
SurvivalSD 2.05 1.22
SE 1.35c 1.44c
aThese coefficients are used for both far-field and near-field (NF) conditions.b2.5Ca andCv from UBC 1997 (ICBO, 1997) are used for theSE soil design-levelresponse spectrum for Los Angeles
cThe seismic coefficients for survival-level are determined by multiplying the seis-mic coefficients for design-level by 3/2
0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
T (sec)
Sa
(g)
ξ = 0.05
Boston, SD soil, design−level
AVG spectrum (SAC) DES spectrum (IBC 2000)
0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
T (sec)
Sa
(g)
ξ = 0.05
Boston, SD soil, survival−level
AVG spectrum (SAC) DES spectrum (IBC 2000)
Figure 3.16: Smooth response spectra: (a) Boston, design-level,SD soil; (b) Boston, sur-vival-level,SD soil.
(b)(a)
46
0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
T (sec)
Sa
(g)
ξ = 0.05
Boston, SE soil, survival−level
AVG spectrum (SAC) DES spectrum (IBC 2000)
0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
T (sec)
Sa
(g)
ξ = 0.05
Boston, SE soil, design−level
AVG spectrum (SAC) DES spectrum (IBC 2000)
Figure 3.17: Smooth response spectra: (a) Boston, design-level,SE soil; (b) Boston, sur-vival-level,SE soil.
(b)(a)
0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
T (sec)
Sa
(g)
ξ = 0.05
Los Angeles, SD soil, design−level
AVG spectrum (SAC) DES spectrum (IBC 2000)
0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
T (sec)
Sa
(g)
ξ = 0.05
Los Angeles, SD soil, survival−level
AVG spectrum (SAC) DES spectrum (IBC 2000)
Figure 3.18: Smooth response spectra: (a) Los Angeles, design-level,SD soil; (b) LosAngeles, survival-level,SD soil.
(b)(a)
0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
T (sec)
Sa
(g)
ξ = 0.05
Los Angeles, SE soil, design−level
AVG spectrum (SAC) DES spectrum (UBC 1997)
0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
T (sec)
Sa
(g)
ξ = 0.05
Los Angeles, SE soil, survival−level
AVG spectrum (SAC) DES spectrum (UBC 1997)
Figure 3.19: Smooth response spectra: (a) Los Angeles, design-level,SE soil; (b) LosAngeles, survival-level,SE soil.
(b)(a)
47
3.6 Nonlinear Dynamic Time-History Analyses
The SDOF and MDOF dynamic time-history analyses conducted in this research aredescribed in the following sections.
3.6.1 SDOF analyses
A MATLAB (2000) algorithm,CDSPEC(Capacity-Demand SPECtra), was developed toconduct nonlinear dynamic time-history analyses of the SDOF models described previously (Far-row, 2001). An incremental step-by-step formulation is used for the solution of the nonlinear equi-librium equation of motion assuming that the response acceleration varies linearly between thetime discretization points (Clough and Penzien, 1993). Time step,ts, and error tolerance aredefined by the user (defaults are set for and 1% error in displacement, respectively).The CDSPEC program is listed in Appendix B.
The SDOF analyses were conducted for the following parameters:
(1) Thirteen ground motion ensembles as described earlier;
(2) Five response modification coefficients,R = 1(elastic), 2, 4, 6, and 8;
(3) Five hysteresis types described earlier (LE, EP, SD, BE, and BP);
(4) Three post-yield stiffness ratios,α = 0, 5, and 10%;
(5) Three BP stiffness and strength ratios,βs = βr = 1/6, 1/3, and 1/2;
(6) Thirty structure periods, exponentially spaced, ranging fromT = 0.1 to 3.0 seconds;
0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
T (sec)
Sa
(g)
ξ = 0.05
Los Angeles, SD soil, design−level, NF
AVG spectrum (SAC) DES spectrum (IBC 2000)
0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
T (sec)
Sa
(g)
ξ = 0.05
Los Angeles, SC soil, design−level
AVG spectrum (N&K) DES spectrum (IBC 2000)
Figure 3.20: Smooth response spectra: (a) Los Angeles, design-level,SC soil; (b) LosAngeles, design-level,SD soil,NF.
(b)(a)
ts T⁄ 1 50⁄≤
48
(7) Seven ground motion scaling methods as described earlier; and
(8) Three linear-elastic acceleration reference response spectra for calculatingFy asdescribed earlier.
In all, 300,000 SDOF nonlinear dynamic time-history analyses were conducted from com-binations of the parameters described above. The parameters that are included in the analyticalprocedure are provided in Tables 3.8 and 3.9 and shown in Figure 3.21. As shown in Table 3.8, theSAC ground motion records were primarily used to investigate the effect of the reference responsespectra on theR-µ-T relationships. The N&K ground motion records were used to validate theCDSPEC program and provide results for the very dense (SC) soil profile. Similarly, as shown inTable 3.9, the UND ground motion records were used to investigate the effect of ground motionscaling method on the scatter in the demand indices, including the effect of site soil characteris-tics. In addition, the SAC near field (NF) ground motion records were used to investigate theeffect of epicentral distance on the scatter in the demand indices. The IND reference responsespectra were used for the analyses listed in Table 3.9.
3.6.2 MDOF analyses
The MDOF nonlinear dynamic time-history analyses were conducted using the DRAIN-2DX program (Prakash et al., 1993). A total of 80 analyses were conducted to reinforce the find-ings from the SDOF analyses to investigate the effect of the scaling method on the scatter in thedemand indices. The UND soft (SE) soil ground motion ensemble was used to excite the MDOFstructures shown in Figure 3.7. These ground motions were first scaled using the MIV method orthe method. Then, the entire ground motion ensemble was scaled so that the aver-age linear-elastic acceleration response spectrum of the ensemble is not less than 1.4 times the5%-dampedUBC 1997Los Angeles design-levelSE soil design response spectrum for periodsbetween 0.2To and 1.5To, whereTo is the structure fundamental period (see Table 3.1), as shown inFigure 3.22 and required byUBC 1997(ICBO, 1997). As described in Section 3.1.2, the MDOFframe structures were designed using this design response spectrum. In the scaling of the entireground motion ensemble, the scaling factors required for the eight-story structure are larger thanthe scaling factors required for the four-story structure. For simplicity, the scaling factors for theeight-story structure were used for both structures (scaling factors of 3.19 and 4.11 for the MIV-based and -based ensembles, respectively).
Note that the shapes of the acceleration response spectra of the individual ground motionrecords were not modified in the scaling procedure described above. The average response spec-trum for the UND soft (SE) soil ground motion ensemble shown in Figure 3.22a is different fromthe average response spectrum shown in Figure 3.22b since the method was used inthe latter case. While Figure 3.22a shows the average response spectrum for the ground motionensemble scaled to one constant factor using the MIV method, Figure 3.22b shows the averageresponse spectrum for the ground motion ensemble scaled by a constant factor at each structureperiod using the method, resulting in average response spectra that are different(see definition of the scaling method in Section 3.4).
Saˆ To Tµ→( )
Saˆ To Tµ→( )
Saˆ To Tµ→( )
Saˆ To Tµ→( )
Saˆ To Tµ→( )
49
Table 3.8: SAC/N&K ground motion ensembles: parameters studied (shaded areas indicate the N&K ensemble)
Variable Classification
Ref. Response Spectrum Individual Ground Motion Spectra (IND) Average Spectrum (AVG) Smooth Design Spectrum (DES)
Hysteresis Type EP SD BE BP EP SD BE BP EP SD BE BP
α (%) 0 5 10 10 10 10 10 10 10 10 10 10 10 10
βs = βr (%) - - - - - 17 33 50 - - - 33 - - - 33
Site Seismicitya
aL = Los Angeles; B = Boston
L L L B L B L B L L B L L B L L L L B L L L
Seismic Demand Levelb
bD = design-level; S = survival-level
D S S D S D S D S S D S S S D S S S D S D S S S S D S D S S S S
Site Soil Characteristicc
c C = very dense (SC) soil; D = stiff (SD) soil; E = soft (SE) soil
C D D C D E D E D E D E D E D D D E D D D D E D D D C D E D E D E D E D D D C D E D E D E D E D D D
Epicentral Distanced
dN = near-field (NF); F = far-field
F F F F N F F F F F F F F F F F F F F F F F F F F F F F N F F F F F F F F F F F F N F F F F F F F F F F F
50
Table 3.9: UND/SAC ground motion ensembles: parameters studied (shaded areas indicate the SAC ensemble)
Variable Classification
Scaling Method PGA EPA EPV MIV A95
Hysteresis Type EP SD BE BP EP SD BE BP EP SD BE BP EP SD BE BP EP SD BE BP EP SD BE BP EP SD BE BP
α (%) 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
βs = βr (%) 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33
Site Seismicitya
aL = Los Angeles; B = Boston
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
Seismic Demand Levelb
bD = design-level; S = survival-level
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
Site Soil Characteristicc
c C = very dense (SC) soil; D = stiff (SD) soil; E = soft (SE) soil
C D E D D D C D E D D D C D E D D D C D E D D D C D E D D D C D E D D D C D E D D D
Epicentral Distanced
dN = near-field (NF); F = far-field
F N F F F F F F N F F F F F F N F F F F F F N F F F F F F N F F F F F F N F F F F F F N F F F F F
Saˆ To( ) Sa
ˆ To Tµ→( )
51
The seismic mass of each frame was lumped at the beam-column joint nodal degrees-of-freedom in the horizontal direction based on tributary areas. Gravity loading of 100% of thedesign dead load,DL, plus 25% of the design live load,LL, was applied. P-∆ effects were consid-ered.
Mass and stiffness proportional Rayleigh damping ofξ = 5% was specified for the 1st and2nd vibrational modes for the four-story structure and the 1st and 3rd modes for the eight-storystructure. As described earlier in Section 3.1.2, the linear-elastic stiffnesses of the beam rotationalsprings were set equal to 5 times the linear-elastic stiffnesses of the adjacent beam elements (i.e.,stiffness factor,ϕ = 5). This introduced a significant increase in the stiffness proportional dampingfor the beam rotational springs since the stiffness proportional damping of an element in DRAIN-2DX is calculated based on the linear-elastic stiffness of the element even after yielding (Prakashet al., 1993). To reduce this effect, the stiffness proportional damping coefficient for the beamrotational springs was multiplied by the inverse of the stiffness factor, i.e., 1/ϕ = 0.2.
3.7 Statistical Evaluation of the Results
The statistical evaluation of the demand estimates from the SDOF and MDOF analyses aredescribed as follows.
255 ground motion records(soil profile, demand level, seismicity, distance)
270 SDOF systems (period, hysteresis type, α, β , β )
Nonlinear dynamic time-history analyses (mean spectra, regression curves)
CDSPEC
7 scaling methods(PGA, EPA, EPV, MIV, A , S (T ),S (T →T ))
aa oo µ95ˆ ˆ
5 R coefficients and3 reference spectra (IND, AVG, DES)
Input Parameters Structure Parameters
Demand Indices
s r
Figure 3.21: Flowchart describing parameters studied in the analytical procedure.
52
3.7.1 SDOF demand estimates
Results are presented as mean demand spectra of the maximum displacement ductility,µ,cumulative plastic deformation ductility,µp, residual displacement ductility,µr = ∆r/∆y, and thenumber of yield events,ny, corresponding to differentR coefficients. These demand indices areillustrated in Figure 2.3.
A two-step nonlinear regression analysis, similar to the procedure used by Nassar andKrawinkler (1991), is performed on the results to develop relationships betweenR, µ, andT asgiven by Equations 2.1-2.3. The first step regression is carried out in theR-µ domain, using theconstant-R procedure as described later in Chapter 4, to relate the period and other parameterdependencies (i.e., reference response spectra, post-yield stiffness, hysteresis type, and site condi-tions) to thec coefficient (e.g., see Fig. 3.23a). The second step regression is carried out in thec-Tdomain to relate the parameter dependencies to thea andb coefficients (e.g., see Fig. 3.23b).
To better understand the sensitivity of the regression equations to the variation in thea andb coefficients,c (Eq. 2.3) andµ (Eq. 2.1) are plotted in Figures 3.24 and 3.25, respectively, foraandb coefficients of 0.1, 1.0, and 2.0.
In Figure 3.24, thec coefficient is plotted as the sum of two separate terms as given inEquation 2.3:
3.39
0 3.50
2.5
T (sec)
S
(g)
a
To,4stry To,8stry
ξ = 5%
0 3.50
2.5
T (sec)
S
(g)
a
To,4stry To,8stry
1.4x design response spectrum
3.19x average response spectrum
UBC 1997 Los Angeles design-level S soil design response spectrum
UND S soil ground motion ensembleavg. response spectrum (MIV)
0.2T ≤ T ≤ 1.5To,4stry o,4stry
0.2T ≤ T ≤ 1.5To,8stry o,8stry
E
E
ξ = 5%
1.4x design response spectrum
4.11x average response spectrum
UBC 1997 Los Angeles design-levelS soil design response spectrum
UND S soil ground motion ensembleavg. response spectrum (S (T → T ))
0.2T ≤ T ≤ 1.5To,4stry o,4stry
0.2T ≤ T ≤ 1.5To,8stry o,8stry
E
Ea o µ
Figure 3.22: Average response spectra of the ground motions used in the MDOF analyses:(a) MIV scaling method; (b) scaling method.Sa
ˆ To Tµ→( )
(a) (b)
c1T
a
Ta
1+---------------=
53
3.40
3.41
In Figure 3.25,µ is plotted by rearranging Equation 2.1 as:
3.42
The values for thec1 term in Figure 3.24a are bounded and have lower and upper limits.The limit value forc1 asT → ∞ is 1.0, regardless ofa. As T → 0, c1 → 0, regardless ofa. T = 1sec. represents a transition point in the equation: forT < 1 sec.,c1 is inversely proportional toa;for T > 1 sec.,c1 is directly proportional toa. The coefficienta controls the rate at whichc1approaches 1.0 forT → ∞ and the rate at whichc1 approaches 0 forT → 0.
The coefficientc2 in Figure 3.24b has no upper bound asT → 0 and has a lower bound of0 asT → ∞. Relative to the relationship betweenc1 anda, c2 is significantly more sensitive tochanges inb especially asT decreases. When the two terms are combined, as in Figure 3.24c, itcan be seen thatc is highly sensitive to small changes inb and slightly sensitive to changes ina.
In terms ofµ, increasinga while keepingb constant decreases the ductility demand forT <1 sec. and increases the ductility demand forT >1 sec.;µ is more sensitive to changes ina forlarger values of theR coefficient (Fig. 3.25). Increasingb while keepinga constant results in sig-nificantly large increases inµ for the entire period range. It can be inferred thatb controls the rate
c2bT---=
c c1 c2+ Ta
Ta
1+--------------- b
T---+= =
0 5 10 150
1
2
3
4
5
6
7
8
9
10
R
µ0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
(sec)
c
T
data point, constant-Rmeanregression
data pointregression
T = 0.92 sec, IND spectra, EP type, α = 0.10
a = 1.35, b = 0.44
N&K Ground Motion EnsembleN&K Ground Motion Ensemble
c = 0.94
T = 0.92 sec, c = 0.94
IND spectra, EP type, α = 0.10
Figure 3.23: Regression analysis (IND spectra, EP hysteresis type,α = 0.10): (a) first step,R-µ domain (T = 0.92 sec.); (b) second step,c-T domain.
(a) (b)
µ Rc
1–c
-------------- 1+=
54
at whichµ → ∞ for shorter periods. From these observations, intuitive predictions based solely oncomparinga andb values for different parameters can be achieved.
A one-step nonlinear regression analysis is performed on the results to determine relation-ships betweenµ and the other demand indices,Λ = µp, µr, andny. For the regression analyses intheΛ-µ domain, a nonlinear least-squares fit on the data is performed using a simple equation:
3.43
whered and f are the regression coefficients. The regression for cross-correlations between thedemand indices other thanµ (i.e., Λ = µp, µr, andny) is performed using an equation of similarform:
3.44
0 1 2 30
1
2
3
4
T (sec)
c =
c1 +
c2
a = 0.1a = 1.0
a = 2.0
b = 0.1
0 1 2 30
0.2
0.4
0.6
0.8
1
c 1 = T
a /(T
a + 1
)
a = 0.1a = 1.0a = 2.0
0 1 2 30
2
4
6
8
10c 2 =
b/T
b = 0.1b = 1.0b = 2.0
T (sec)
T (sec)
b = 1.0
b = 2.0
Figure 3.24: Effect ofa andb coefficients onc coefficient: (a)c1 term; (b)c2 term; (c)c = c1 + c2.
(a)
(b)
(c)
Λ d µ 1–( )1 f⁄=
Λ j gΛi1 h⁄
=
55
The regression analyses relate the parameter dependencies (e.g., reference response spec-tra, post-yield stiffness, hysteresis type, and site conditions) to thea, b, d, f, g,andh coefficients.The correlation between the demand indices is measured by calculating the correlation coeffi-cient,ρ (Bendat and Piersol, 1986), defined as:
3.45
for the correlation betweenµ and the other demand indices,Λ, and:
3.46
for the cross-correlations between the demand indices,Λ, other thanµ. In these equations,σ is thestandard deviation andC is the covariance. This normalized quantity lies between -1 and +1,
0 1 2 30
2
4
6
8
10
µ
a = 0.1a = 1.0a = 2.0
R = 2R = 4
0 1 2 30
2
4
6
8
10µ
a = 0.1a = 1.0
a = 2.0R = 2R = 4
0 1 2 30
2
4
6
8
10
T (sec)
µ
a = 0.1a = 1.0a = 2.0
R = 2R = 4
b = 2.0
b = 1.0
b = 0.1
T (sec)
T (sec)
(a)
Figure 3.25: Effect ofa andb coefficients onµ: (a)b = 0.1; (b)b = 1.0; (c)b = 2.0.
(b)
(c)
ρΛµC Λ µ,( )
σ Λ( )σ µ( )-------------------------=
ρΛ ij,C Λi Λ j,( )
σ Λi( )σ Λ j( )-----------------------------=
56
inclusively. A correlation coefficient magnitude of 1 implies full correlation and a magnitude of 0implies no correlation.
It is noted that the square of the correlation coefficient,ρ, is commonly referred to as thecoefficient of determination,Ρ, which is used as a measure for the adequacy of the assumedregression relationship to represent the variability in the data (Montgomery and Peck, 1991).Sinceρ has a range of -1 to +1, inclusively, it follows thatΡ ranges from 0 to 1, inclusively. Forexample, values ofΡ that are close to 1 imply that most of the variability in the data is explainedby the regression relationship.
Note that the correlation coefficient,ρ, is only suitable for measuring the amount of thelinear correlation between two variables. Since the relationships between the demand indices areassumed to be exponential in nature (i.e., Eqs. 3.43 and 3.44), they can be transformed into linearrelationships by taking the log of the equations (Bates and Watts, 1988; Montgomery and Peck,1991):
3.47
3.48
Thus, the correlation coefficient,ρ, is calculated on the log of the data, i.e.,µ’ = log(µ-1) andΛ’ =log Λ. As shown in Equations 3.47-3.48, the correlation coefficient,ρ, depends on the assumedregression relationship. Using more accurate regression relationships than the relationships usedin this research (which are of simple exponential form) may increaseρ, implying better correla-tion.
The effect of ground motion scaling method on the scatter in the demand estimates fromthe SDOF analyses is investigated using the dimensional (i.e., unit dependent) maximum dis-placement demand,∆nlin, and the non-dimensional maximum displacement ductility demand,µ =∆nlin/∆y. The results are presented as dispersion spectra corresponding to differentR coefficientsby calculating the coefficient of variation, COV, defined as the ratio between the sample standarddeviation,σ, and the sample mean. This measure is used to assess the effectiveness of the differentscaling methods in reducing the scatter in the demand estimates.
3.7.2 MDOF demand estimates
The evaluation of scatter in the demand estimates from the MDOF analyses is presentedas response profiles of: (1) mean(∆i/∆max) ± σ(∆i/∆max), where∆i is the lateral displacement atfloor or roof leveli, and∆maxis the maximum mean floor or roof displacement, calculated by tak-ing the mean lateral displacement at each floor or roof level and then taking the maximum valueof the mean lateral displacements over the height of the structure; and (2) mean(θi/θmax) ± σ(θi/θmax), whereθi is the interstory drift at storyi:
Λlog dlog1f--- µ 1–( )log+=
Λlog j glog1h--- Λilog+=
57
, hi = height at leveli from ground level 3.49
andθmax is the maximum mean interstory drift, calculated by taking the mean interstory drift ateach story and then taking the maximum value of the mean interstory drifts over the height of thestructure.
Additionally, results are presented as dispersions by calculating the COV of∆i/∆max andθi/θmax for the MIV and scaling methods.
θi
∆i ∆i 1––
hi hi 1––-----------------------=
Saˆ To Tµ→( )
58
CHAPTER 4
VALIDATION OF ANALYTICAL MODEL AND
COMPARISON WITH PREVIOUS RESULTS
This chapter provides a verification of the CDSPEC program, which was developed toconduct nonlinear dynamic time-history analyses as well as pre-processing of the variables andpost-processing of the results. The results from CDSPEC are compared with results using twoanalysis packages as follows: (1) nonlinear-dynamic time-history analyses using DRAIN-2DX(Prakash et al., 1993); and (2) spectral analyses using BISPEC (Hachem, 2000). Furthermore,comparisons are made with results from Nassar and Krawinkler (1991). Unless otherwise noted,the lateral force capacities,Fy = Felas/R, of the systems described in this chapter are determinedbased on the linear-elastic acceleration response spectrum for each ground motion (i.e., IND spec-tra).
4.1 Nonlinear Dynamic Time-History Analyses
Figure 4.1 compares the nonlinear dynamic time-history response of two SDOF systemsobtained using CDSPEC and DRAIN-2DX. As examples of representative behavior, the resultsare presented for the EP hysteresis type withα = 0.10,R = 8, andT = 0.92 and 3.0 sec.
Figures 4.1a-b compare the responses under the 1965 Puget Sound, Olympia WashingtonHighway ground motion (EQ09) and Figures 4.1c-d compare the responses under the 1979 Impe-rial Valley, Calexico Fire Station ground motion (EQ15). Both ground motions are from the N&Kensemble. The responses obtained using the two analysis packages are nearly identical, validatingthe nonlinear dynamic time-history analysis algorithm used in CDSPEC.
Similarly, Figure 4.2 compares the nonlinear dynamic time-history response for the SDhysteresis type withα = 0.10,R = 8, andT = 0.50 and 3.0 sec. obtained using CDSPEC andBISPEC. Figures 4.2a-b compare the responses under the 1989 Loma Prieta, San Francisco-Pre-sidio ground motion (LPPR) and Figures 4.2c-d compare the responses under the 1966 Parkfield,Parkfield-Cholame Shandon #2 ground motion (PACH). Both ground motions are from the UNDensemble. The responses obtained using the two analysis packages are nearly identical, except forthe responses of theT = 3.0 sec. structure subjected to the PACH ground motion aftert = 8 sec.(Fig. 4.2d). The reason for the differences in the two responses is described below.
59
The SD type hysteretic reloading rules in the BISPEC and CDSPEC analysis packages areslightly different, as illustrated by the force-displacement responses in Figure 4.2e. After unload-ing with the linear-elastic stiffness from the “shooting” branch at point a, the structure reloadsback in the negative direction from point b. While the SD type in the CDSPEC program reloadsalong the linear-elastic branch towards point a, and then towards the largest displacement in thenegative direction along the shooting branch (line ac), the SD type in the BISPEC programreloads directly towards the largest displacement along line bc, resulting in a less realistic loop(Mahin and Lin, 1983).
Comparisons between the DRAIN-2DX and CDSPEC analysis packages for the BE hys-teresis type withα = 0.10 and the BP hysteresis type withα = 0.10 andβs = βr = 1/3 are shown inFigure 4.3. Again, the responses obtained using the two analysis packages are nearly identical,validating the nonlinear dynamic time-history analysis algorithm used in CDSPEC.
0 5 10 15 20 25
0
time, t (sec)
∆ (c
m)
N&K: EQ09
0 5 10 15 20 25−12.5
0
12.5
time, t (sec)
∆ (c
m)
0 5 10 15 20 25−12.5
0
12.5
0 5 10 15 20 25
0
DRAIN-2DXCDSPEC
T = 0.92 sec
N&K: EQ09
T =3.0 sec
N&K: EQ15
T = 0.92 sec
N&K: EQ15
T = 3.0 sec
DRAIN-2DXCDSPEC
DRAIN-2DXCDSPEC
DRAIN-2DXCDSPEC
−12.5
12.5
−12.5
12.5
time, t (sec)time, t (sec)
∆ (c
m)
∆ (c
m)
EP TYPE EP TYPE
EP TYPE EP TYPE
(a)
(c)
Figure 4.1: Comparison between CDSPEC and DRAIN-2DX (EP hysteresis type,α =0.10,R = 8): (a-b) EQ09; (c-d) EQ15.
(b)
(d)
60
25 25
0 5 10 15 20 25
0
time, t (sec)
∆ (c
m)
UND: LPPR
0 5 10 15 20 25
0
time, t (sec)
∆ (c
m)
0 5 10 15 20 25
0
0 5 10 15 20 25
0
BISPECCDSPEC
T = 0.50 sec
UND: LPPR
T =3.0 sec
UND: PACH
T = 0.50 sec
UND: PACH
T = 3.0 sec
BISPECCDSPEC
time, t (sec)time, t (sec)
∆ (c
m)
∆ (c
m)
BISPECCDSPEC
25
-25
-25 -25
25
-25
SD TYPE SD TYPE
SD TYPE SD TYPE
BISPECCDSPEC
−25 0 25
UND: PACH
T = 3.0 sec
BISPECCDSPEC
SD TYPE
F (
kN)
0
12000
-12000
∆ (cm)
different reloading rules
a
c
b
(a)
(c)
Figure 4.2: Comparison between CDSPEC and BISPEC (SD hysteresis type,α = 0.10,R = 8): (a-b) LPPR; (c-d) PACH; (e) different reloading rules.
(b)
(d)
(e)
61
0 5 10 15 20 25
0
time, t (sec)
∆ (c
m)
UND: LPPR
0 5 10 15 20 25
0
time, t (sec)
∆ (c
m)
0 5 10 15 20 25
0
0 5 10 15 20 25
0
DRAIN-2DXCDSPEC
T = 0.50 sec
UND: LPPR
T =3.0 sec
UND: PACH
T = 0.50 sec
UND: PACH
T = 3.0 sec
DRAIN-2DXCDSPEC
time, t (sec)time, t (sec)
∆ (c
m)
∆ (c
m)
DRAIN-2DXCDSPEC
30
-30
30
-30
30
-30
30
-30
BE TYPE BE TYPE
BE TYPE BE TYPE
DRAIN-2DXCDSPEC
0 5 10 15 20 25
0
time, t (sec)
∆ (c
m)
UND: LPPR
0 5 10 15 20 25
0
time, t (sec)
∆ (c
m)
0 5 10 15 20 25
0
0 5 10 15 20 25
0
DRAIN-2DXCDSPEC
T = 0.50 sec
UND: LPPR
T =3.0 sec
UND: PACH
T = 0.50 sec
UND: PACH
T = 3.0 sec
DRAIN-2DXCDSPEC
time, t (sec)time, t (sec)
∆ (c
m)
∆ (c
m)
DRAIN-2DXCDSPEC
DRAIN-2DXCDSPEC
30
-30
30
-30
30
-30
30
-30
BP TYPE BP TYPE
BP TYPE BP TYPE
(a)
Figure 4.3: Comparison between CDSPEC and DRAIN-2DX (α = 0.10,R = 8): (a-d)BE hysteresis type; (e-h) BP hysteresis type,βs = βr = 1/3.
(b)
(c) (d)
(e) (f)
(g) (h)
62
4.2 Spectral Analyses
The previous section provides a validation of CDSPEC for a limited number of parametersonly. To this end, results from a spectral analysis using CDSPEC under the entire N&K groundmotion ensemble are compared with results obtained using another nonlinear spectral analysisprogram, BISPEC (Hachem, 2000).
The results are compared in terms of meanR-µ spectra for the EP hysteresis type withα =0.10 as shown in Figure 4.4a. The thin to thick lines represent increasing values ofR = 1, 2, 4, 6,and 8. TheR-µ spectra generated by CDSPEC are almost identical to the spectra generated byBISPEC. Similar results are shown for the SD hysteresis type withα = 0.10 in Figure 4.4b. Thus,CDSPEC is satisfactorily validated.
4.3 Comparison with Nassar and Krawinkler (1991)
The results are compared with results previously obtained by Nassar and Krawinkler(1991). Comparisons are made in terms of: (1) constant-R versus constant-µ approaches; and (2)IND spectra versus smooth response spectra.
4.3.1 Constant-R versus constant-µ approaches
Two approaches can be used to obtain relationships betweenRandµ. In the first approach,the R coefficient is set to predetermined constant values and theµ demand for each structuremodel is determined from nonlinear dynamic time-history analyses. In the second approach,µ isset to predetermined constant values and the dynamic analyses are conducted to calculate the
Figure 4.4: Comparison between CDSPEC and BISPEC for the N&K ground motionensemble (α = 0.10): (a) EP hysteresis type; (b) SD hysteresis type.
(a) (b)
0 0.5 1 1.5 2 2.5 30
2
4
6
8
10
12
14
T (sec)
µ
R = 1,2,4,6,8 (thin → thick lines)
3.5
BISPEC
CDSPEC
0 0.5 1 1.5 2 2.5 30
2
4
6
8
10
12
14
T (sec)
µ
R = 1,2,4,6,8 (thin → thick lines)
3.5
EP type
BISPEC
CDSPEC
SD type
63
requiredR coefficient by using iteration. These two approaches are referred to as “constant-R”and “constant-µ” approaches, respectively. The constant-R approach is adopted in this researchfor reasons described below.
In previous research on the development ofR-µ-T relationships, the constant-µ approachis almost always used. A handful of researchers have used the alternate constant-R approach(Osteraas and Krawinkler, 1990; Astarlioglu et al., 1998; Lam et al., 1998). To illustrate the differ-ences between the two approaches, results obtained from CDSPEC for the N&K ground motionensemble are compared below to previous results by Nassar and Krawinkler (1991). The compar-isons are based on the EP hysteresis type withα = 0.00.
Figure 4.5 shows theR-µ relationship forT = 0.20 sec. under the 1983 Coalinga, ParkfieldZone 16 ground motion (EQ20). The relationship is determined by using the constant-R proce-dure for 25 linearly-spacedR coefficients ranging from 1 to 6 (represented by the solid circles).For this record, it can be seen that theµ demand does not always increase monotonically as theRcoefficient increases. In particular, there is a range where threeR coefficients can be found foreachµ (shaded region in Fig. 4.5).
This phenomenon has been reported by several researchers (Newmark and Hall, 1973;Nassar and Krawinkler, 1991; Miranda, 1993; Chopra, 1995) and must be considered when devel-oping constant-µ spectra. The smallestR coefficient is taken in these instances, since it corre-sponds to a larger strength, and thus, is more conservative (e.g., point a, instead of points b or c,for µ = 5 in Fig. 4.5). However, it can be shown that more conservative results are obtained usingthe constant-R approach as follows.
The thick lines in Figures 4.6a and b show the meanR-µ relationships under the N&Kground motion ensemble (15 ground motions as represented by the thin lines) forT = 0.2 sec. and0.92 sec., respectively. The thick dashed lines represent the constant-µ relationships with iterationto determine the smallest requiredRcoefficient, as is done by Nassar and Krawinkler (1991). Theempty circles show the meanR values corresponding toµ = 1, 2, 3, 4, 5, 6, and 8. Upon compari-
0 1 2 3 4 5 6 7 8 90
1
2
3
4
5
6
R
µ
constant-Rconstant-µ
1983 Coalinga Parkfield Zone 16 (N&K: EQ20)
a
b
c
T = 0.20 sec.
Figure 4.5:R-µ relationship for the EP hysteresis type withα = 0.00 andT = 0.20 sec.under the 1983 Coalinga Parkfield Zone 16 ground motion (IND response spectrum).
64
son with Nassar and Krawinkler’s results (Nassar and Krawinkler, 1991), represented by the dia-mond markers in Figure 4.6, a good match is observed.
The thick solid lines in Figure 4.6 show the mean constant-R relationships using 25 lin-early-spacedR coefficients ranging from 1 to 6. For a givenR coefficient, the mean constant-Rrelationship consistently predicts largerµ values than the mean constant-µ relationship. Theresults indicate that the mean constant-R relationship is highly nonlinear, especially for short-period structures. In other words, small incremental increases inR can lead to significantly largeincreases inµ, especially for large values ofR. This large (and non-constant) variance weighs themean distribution towards larger values ofµ (Lam et al., 1998). On the other hand, the mean con-stant-µ relationship has relatively less variance, resulting in little or no shift in the distribution.Therefore, comparison of the mean values with respect to each “variable” (eitherR or µ) leads tothe constant-R approach as being more conservative.
As an example, point 1 in Figure 4.6a shows that the requiredR coefficient to limitµ to atarget value of 7 for a structure withT = 0.20 sec. is approximately 3.4. However, if a structuredesigned usingR = 3.4 is subjected to the same ensemble of ground motions, an average value ofapproximatelyµ = 9 would be observed (point 2). Thus, the constant-R approach is more conser-vative than the constant-µ approach even when an iteration procedure that selects the smallestRcoefficient is used.
The constant-R approach is adopted in this study since: (1) current design provisions arebased on constantR coefficients; (2) developing constant-R spectra is relatively simple andrequires less computational effort since there is no need for an iterative procedure; and (3) theconstant-R approach is more conservative than the constant-µ approach.
µ0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
R
N&K Ground Motion Ensemble
0 1 2 3 4 5 6 7 8 90
1
2
3
4
5
6
R
N&K Ground Motion Ensemble
µ
EQ20
T = 0.20 sec.
T = 0.92 sec.
1 2
mean, constant-µmean, constant-RN&K 1991
mean, constant-µmean, constant-RN&K 1991
Figure 4.6:R-µ relationships for the EP hysteresis type withα = 0.00 (IND referencespectra): (a)T = 0.20 sec.; (b)T = 0.92 sec.
(a) (b)
65
4.3.2 IND spectra versus smooth response spectra
Figure 4.7 shows the meanR-µ spectra for the N&K ground motion ensemble (with verydense,SC, soil profile) using the constant-R approach with the IND, AVG, and DES spectra todetermineFy = Felas/R. There is a non-negligible increase inµ when the results from the INDspectra are compared with the results from the two smooth response spectra, especially the AVGspectrum for the ensemble at long periods (Fig. 4.7a). This increase inµ, together with theincrease inµ due to the constant-Rapproach, indicates that the use of previously-developedR-µ-Trelationships in design may be unconservative.
Figure 4.7b shows that the increase inµ is not as significant when the DES spectrum forthe ensemble (i.e.,IBC 2000Los Angeles, design-level,SC soil) is used, particularly forT > 1.25sec. This is because the AVG spectrum has spectral accelerations (thus, results inFy values) lowerthan the DES spectrum for this period range, as shown in Figure 4.8. It seems that a slightdecrease inFy results in a large increase in the meanµ demand. This is particularly true for the1979 Imperial Valley Holtville P.O. ground motion (EQ14) of the N&K ground motion ensemble,as shown by theR-µ relationships in Figure 4.9a. While the DES spectrum (represented by thedashed line with diamond markers) results in a large increase inµ for EQ14 as compared to usingIND spectra (represented by the dashed line with circle markers), a significantly larger increase inµ occurs when using the AVG spectrum (represented by the dashed line with square markers).Thus, the meanµ demand using the AVG spectrum (represented by the solid line with squaremarkers) is larger than the meanµ demand using the DES spectrum (represented by the solid linewith diamond markers). Once theµ demand for EQ14 is removed from the ensemble, as shown inFigure 4.9b, a slight decrease in the meanµ demand is observed when using the DES spectrum(represented by the solid line with diamond markers) and a significant decrease in the meanµdemand is observed when using the AVG spectrum (represented by the solid line with squaremarkers). Notice that the meanµ demand for the AVG spectrum is now smaller than the meanµdemand for the DES spectrum. Furthermore, the differences between the meanµ demands for the
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraAVG spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraDES spectrum
Figure 4.7: R-µ spectra (N&K ensemble, EP hysteresis type,α = 0.10): (a) AVG versusIND spectra; (b) DES versus IND spectra.
(a) (b)
66
AVG and IND spectra are smaller than the differences between the meanµ demands for the DESand IND spectra. This finding, in general, is expected since the AVG spectrum represents themean response spectrum of the ground motion ensemble, and thus, is considered to provide a bet-ter representation of the IND spectra.
0 3.50
2.0
T (sec)
N&K AVG spectrum, ξ = 5%
S
(g)
a
AVG spectrum results in lower F valuesthan DES spectrum
DES spectrum, ξ = 5%
R = 1 (linear-elastic)
LOS ANGELES DESIGN-LEVEL S SOIL C
y
Figure 4.8: Smooth design (DES) response spectrum versus average (AVG) groundmotion response spectrum.
0 5 10 15 20 25 30 351
2
3
4
5
6
7
8
EQ14
IND
N&K Ground Motion Ensemble
T = 3.0 sec.
R
µ
DES
AVG
IND spectraAVG spectrumDES spectrummeanEQ14
0 5 101
2
3
4
5
6
7
8
IND
N&K Ground Motion Ensemble
T = 3.0 sec.
R
µ
DESAVG
IND spectra, meanAVG spectrum, meanDES spectrum, mean
Figure 4.9:R-µ relationships using different reference response spectra (EP hysteresistype,α = 0.10,T = 3.0 sec.): (a) N&K ensemble; (b) N&K ensemble without EQ14.
(a) (b)
67
CHAPTER 5
EFFECT OF HYSTERETIC BEHAVIOR
In this chapter, the effect of hysteretic behavior on the capacity-demand index relation-ships is investigated in terms of mean spectral values of the demand indicesµ, µp, µr, andny. Theeffect of the post-yield stiffness ratio,α, is discussed first. Then, comparisons are given betweenthe EP hysteresis type and the SD, BE, and BP types. The results are presented for the SAC LosAngeles survival-levelSD (stiff) soil ground motion ensemble. The lateral force capacities,Fy =Felas/R, of the systems described in this chapter are determined based on the linear-elastic acceler-ation response spectrum for each ground motion (i.e., IND spectra).
5.1 Effect of Post-Yield Stiffness Ratio,α
Figure 5.1 compares the mean demand spectra for the EP hysteresis type withα = 0.10(solid lines) andα = 0.05 and 0.00 (dashed lines). The thin to thick lines represent results forincreasingR coefficients. Bothµ andµr demands increase withR. Figure 5.1a indicates that thereis a slight decrease in theµ demand asα increases, more for medium- to short-period structures (T< 1.0 sec.). Unlike theµ demand, Figure 5.1b shows that theµp demand increases asα increasesfor nearly the entire range of periods (T > 0.25 sec). This increase is possibly due to the ability ofthe system with a largerα to yield and accumulate plastic deformation in the opposite directionupon unloading (as a result of the larger amount of elastic energy stored during loading).
In general, the residual displacement ductility demand,µr, significantly decreases with asmall increase inα (Fig. 5.1c). Theµr demand is highly erratic at shorter periods (T < 0.5 sec.),tends to be more uniform at longer periods, and seems to increase asR increases. Figure 5.1cshows that the dependency of theµr demand on theR coefficient increases asα decreases; thisdependency is smaller at higher levels ofR. The number of yield events,ny, increases withR andis independent ofα, as shown in Figure 5.1d.
5.2 EP Hysteresis Type versus SD Hysteresis Type
Figure 5.2 compares the mean demand spectra for the EP (solid lines) and SD (dashedlines) hysteresis types withα = 0.10. Figure 5.2a indicates that there is a small increase inµ forthe SD hysteresis type. As shown in Figure 3.2, the SD hysteresis type dissipates less energy thanthe EP type during large displacement cycles. The increase inµ due to the smaller hysteretic
68
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10EP type, α = 0.00
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10EP type, α = 0.00
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10EP type, α = 0.00
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10EP type, α = 0.00
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10EP type, α = 0.05
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10EP type, α = 0.05
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10EP type, α = 0.05
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10EP type, α = 0.05
Figure 5.1: Effect of post-yield stiffness ratio,α (EP hysteresis type, SAC Los Angeles,survival-level,SD soil): (a)µ; (b) µp; (c) µr; (d) ny.
(a)
(b)
(c)
(d)
69
energy dissipation of the SD type during large displacement cycles, however, is mitigated by theadditional energy dissipation which occurs in the SD type during small displacement cycles (e.g.,see cycle g-h-i-j-b in Figure 3.3). As a result, the differences between theµ demands of the EPand SD hysteresis types are not large.
Theµp-spectra (Fig. 5.2b) exhibit considerably large increases for the SD hysteresis type,especially for smaller values of theRcoefficient. This increase inµp occurs as a result of the accu-mulation of “damage” during the shooting branch of the SD type (Fig. 3.2c). The ratio betweentheµp demands for the SD and EP types decreases as theR coefficient increases, in particular, forsystems with large values ofα. This is explained as follows. As shown in Figure 5.3a, the behav-ior of the SD type under large displacement cycles approaches the behavior of the EP type forlarge values ofα andR, and thus, the SD type spends “less time” in the shooting branch and moretime in the yielding branch during an earthquake. Figure 5.3b shows that the differences betweenthe SD and EP types are more prominent for smaller values ofα. Thus, for small values ofα, thedifferences between theµp demands for the two hysteresis types are expected to be larger.
The differences in theµr-spectra for the SD and EP hysteresis types are quite erratic butthe two systems follow a similar general trend as shown in Fig. 5.2c. The number of yield events,ny, for the SD type is smaller and seems to be slightly less dependent onR as compared to the EP
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10SD type, α = 0.10
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10SD type, α = 0.10
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10SD type, α = 0.10
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10SD type, α = 0.10
Figure 5.2: EP versus SD hysteresis types (α = 0.10, SAC Los Angeles, survival-level,SDsoil): (a)µ; (b) µp; (c) µr; (d) ny.
(a) (b)
(c) (d)
70
type for the entire period range (Fig. 5.2d). The SD type spends more time in the shooting branchduring smaller displacement cycles, and thus, experiences fewer number of yield events.
5.3 EP Hysteresis Type versus BE Hysteresis Type
Figure 5.4 compares the mean demand spectra for the EP and BE hysteresis types withα =0.10. For the BE hysteresis type, all demand indices, exceptµr, are significantly larger than thosefor the EP type: theµ demand can be 1.75 times as large (Fig. 5.4a), theµp demand can be morethan 4 times as large (Fig. 5.4b), and theny demand can be 2.25 times as large (Fig. 5.4d). The dif-ferences in theµ demands for the two hysteresis types remain relatively constant while the differ-ences in theµp andny demands decrease asR increases. These results demonstrate the effect ofhysteretic energy dissipation in reducing the seismic demands. The BE type, however, is superiorto the EP type in the residual displacement ductility demand,µr, since the system always returnsto the zero displacement, or plumb, position at the end of the earthquake event, regardless of theRcoefficient or structure period (Fig. 5.4c).
5.4 EP Hysteresis Type versus BP Hysteresis Type
Figure 5.5 compares the mean demand spectra for the EP hysteresis type (solid lines) andBP hysteresis type withβs = βr = 1/6, 1/3, and 1/2 (dashed lines). As compared to the BE type dis-cussed in the previous section (Figs. 5.4a and b), theµ andµp demands for the BP type (Figs. 5.5aand b) are smaller as a result of the increased hysteretic energy dissipation. There is a smallincrease in theµr demand (Fig. 5.5c) as compared to the BE type (Fig. 5.4c), however the BP typeis still far more effective in reducing the residual displacements to a negligible level than the EPtype (especially for short periods, i.e.,T < 0.5 sec.). Theny demand is slightly decreased for theBP type (Figure 5.5d), more for smallerR values, as compared to the BE type (Figure 5.4d).
−100 0 100−17800
0
17800F
(kN
)
EP TypeSD Type
−100 0 100
0
F (
kN)
T = 0.92 sec
R = 8
α = 0.10
∆ (cm) ∆ (cm)
−17800
17800
T = 0.92 sec
R = 8
α = 0.00
EP TypeSD Type
SAC Los Angeles, survival-level, S soil: Palos Verdes, LA40D SAC Los Angeles, survival-level, S soil: Palos Verdes, LA40
D
yielding branch
yielding branch
shooting branch
Figure 5.3: Comparison between force-displacement responses of the EP and SD hysteresistypes: (a)α = 0.10; (b)α = 0.00.
(a) (b)
71
As expected, the decrease in theµ, µp, andny demands for the BP hysteresis type is largerfor larger values ofβr due to the increased hysteretic energy dissipation asβr is increased (Fig.5.6). The amount of hysteretic energy dissipated by the BP type normalized with the amount ofhysteretic energy dissipated by the EP type during a displacement cycle to± ∆nlin = ± µ∆y can beexpressed as follows:
5.1
The above equation can be simplified forβs = βr as:
5.2
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BE type, α = 0.10
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ = 1, 2, 4, 6, 8 (thin → thick lines)R
EP type, α = 0.10BE type, α = 0.10
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BE type, α = 0.10
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BE type, α = 0.10
Figure 5.4: EP versus BE hysteresis types (α = 0.10, SAC Los Angeles, survival-level,SDsoil): (a)µ; (b) µp; (c) µr; (d) ny.
(a) (b)
(c) (d)
EHβr
1 βs+( ) 1 α–( )-------------------------------------
µβr
βs-----–
1 βs+
1 βr+--------------
µ 1–-----------------------------------------=
EH βs βr=( )βr
1 βr+( ) 1 α–( )-------------------------------------=
72
The normalized energy dissipation,EH, of the BP hysteresis type with differentβs = βr valuesused in this research are provided in Figure 5.6. For example, the BP type withβs = βr = 1/2 dissi-pates 37% of the energy dissipated by the EP type during the same displacement cycle.
For βr = 1/2, theµ andny demands are moderately decreased (by about 14 to 25 percent,Figs. 5.5a and d) and theµp demands are significantly decreased (by about 50 percent, Fig. 5.5b)as compared to the BE type, while theµr demands still remain negligible (µr ≅ 0.10, on average,Fig. 5.5c). Thus, the objective of reducing theµ, µp, andny demands while keeping theµr demandwithin allowable limits can be reasonably achieved with the BP hysteresis type.
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ = 1, 2, 4, 6, 8 (thin → thick lines)R
EP type, α = 0.10BP type, α = 0.10, β = β = 1/6s r
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/6s r
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/6s r
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/6s r
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/3s r
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/3s r
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/3s r
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/3s r
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/2s r
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/2s r
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/2s r
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/2s r
Figure 5.5: EP versus BP hysteresis types (α = 0.10, SAC Los Angeles, survival-level,SDsoil): (a)µ; (b) µp; (c) µr; (d) ny.
(a)
(b)
(c)
(d)
73
As described in Section 3.1.1, the maximum possible residual displacement of the BP hys-teresis type is limited to∆rmax= βr∆be= βr∆y. Thus, the maximum possible residual displacementductility demand isµrmax = ∆rmax/∆y = βr. The maximumµr values for the BP hysteresis type inFigures 5.5a-c are smaller than the correspondingβr values, especially forβr = 1/2. These resultsindicate that it may be possible to decrease theµ, µp, andny demands further without significantlyincreasing theµr demand by usingβr values larger than 1/2, however this is not investigated bythe research.
F
∆
EP type
BP type, βs = β
r = 1/6, EH = 0.16
BP type, βs = β
r = 1/3, EH = 0.28
BP type, βs = β
r = 1/2, EH = 0.37
EP type
EH = hysteretic energy dissipated, one cycle, BP type
hysteretic energy dissipated, one cycle, EP type
∆nlin
-∆nlin
∆y
Figure 5.6: Hysteretic energy dissipation of the BP hysteresis type.
74
CHAPTER 6
EFFECT OF SITE CONDITIONS
In this chapter, the effect of site conditions on the capacity-demand index relationships isinvestigated. The effects of site soil characteristics are discussed first. Then, the effects of seismicdemand level, site seismicity, and epicentral distance are described. The lateral force capacities,Fy = Felas/R, of the systems described in this chapter are determined based on the linear-elasticacceleration response spectrum for each ground motion (i.e., IND spectra). The results are pre-sented for the SAC ground motion ensemble.
6.1 Site Soil Characteristics
Figure 6.1 compares the mean demand spectra for the EP hysteresis type withα = 0.10 forstiff (SD) and soft (SE) soil profiles (solid and dashed lines, respectively). The results are presentedfor the SAC Los Angeles design-level ground motion ensemble. For structures withT < ~1.25-1.50 sec., theµ andµp spectra for theSE soil are on average larger than the spectra for theSD soil(up to 4 times as large, Figs. 6.1a and b). TheSE soil profile tends to result in largerµr demandsthan theSD soil profile for T < 1 sec. and smaller values ofR (Fig. 6.1c). For the same periodrange, theny demands are on average slightly less for theSE soil profile than for theSD soil profile(Fig. 6.1d). Similar observations can be made for the mean demand spectra of the SD, BE, and BP(with βs = βr = 1/3) hysteresis types in Figure 6.2, except thatµr for the BE type is zero andµr forthe BP type is negligible regardless of soil profile, andny for the SD type are more or less thesame for the two soil profiles.
For longer-period structures (T > ~1.0-1.5 sec.), theµ, µp, andµr demands are smaller andny is virtually unchanged for theSE soil profile as compared to theSD soil profile. Similar obser-vations can be made for the meanµ andµp demand spectra of the SD, BE, and BP hysteresistypes and for the meanµr demand spectra of the SD and BP hysteresis types. This phenomenonwas previously reported by Miranda (1993). Demands at shorter periods (relative to the predomi-nant ground motion period) are in general larger for soft soil profiles since period elongation dueto yielding runs the structure into energy-rich regions of the ground motion spectra as shown inFigures A.22 and A.24 (e.g.,To,sto Teff,sin Fig. 6.3a, dashed versus solid lines). In contrast, yield-ing in longer-period structures with initial linear-elastic periods around the predominant groundmotion period of the soft soil shifts the “effective” structure period,Teff, to the regions of theground motion spectra with significantly less energy, reducing the seismic demands (e.g.,To,l toTeff,l in Fig. 6.3a). As shown in Figure 6.1, the period at which the demands for theSE soil profile
75
become smaller than the demands for theSD soil profile decreases asR increases. This is expectedsince the amount of nonlinear behavior increases asR increases.
It is noted that the lateral force capacities,Fy = Felas/R, of the systems described in thischapter are determined based on the linear-elastic acceleration response spectrum for each groundmotion, referred to as IND reference spectra (e.g., thin solid line in Fig. 6.3b). This results inincreasedFy values for structures with linear-elastic periods near the predominant ground motionperiod, particularly for soft soil profiles, leading to the reducedµ, µp, andµr demands in Figures6.1 and 6.2. However, as previously discussed in Chapters 1 and 4, smooth design response spec-tra (referred to as DES reference spectra) are used in current design practice (ICC, 2000; BSSC,1998; ICBO, 1997), resulting in significantly smallerFy values for structures near the predomi-nant ground motion period (e.g., thin dashed line in Fig. 6.3b). Thus, the reducedµ, µp, andµrdemands forSE soil in Figures 6.1a-c and 6.2a-c for longer periods may be unconservative forstructures where smooth design response spectra are used to determineFy (see shaded region inFig. 6.3b forR = 2). This is discussed in more detail later in Chapter 7.
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ = 1, 2, 4, 6, 8 (thin → thick lines)R
EP type, α = 0.10, S soilEP type, α = 0.10, S soil
D
E
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, SEP type, α = 0.10, S
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, S soilEP type, α = 0.10, S soil
D
E
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/3r s
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, S soilEP type, α = 0.10, S soil
D
E
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/3r s
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, S soilEP type, α = 0.10, S soil
D
E
Figure 6.1: Effect of site soil characteristics (EP hysteresis type,α = 0.10, SAC Los Ange-les, design-level): (a)µ; (b) µp; (c) µr; (d) ny.
(a) (b)
(c) (d)
76
6.2 Seismic Demand Level
Figure 6.4 compares the mean demand spectra using the EP hysteresis type withα = 0.10under the SAC Los Angeles design-level and survival-level ground motion ensembles for siteswith SD andSE soil profiles. ForT < ~1.25 sec., theµ andµp demands are smaller (by as much as60%) for theSD andSE soil design-level ensembles as compared to theSD andSE soil survival-level ensembles, respectively (Figs. 6.4a and b). ForT > ~1.25 sec., theµ andµp demands for theSD andSE soil design-level ensembles are larger than the demands for theSD andSE soil survival-level ensembles, respectively (the design-level demands are as much as 1.4 times theµ demandsand 1.6 times theµp demands for the survival-level ensemble).
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
= 1, 2, 4, 6, 8 (thin → thick lines)RSD type, α = 0.10, S soilSD type, α = 0.10, S soil
D
E
= 1, 2, 4, 6, 8 (thin → thick lines)RSD type, α = 0.10, S soilSD type, α = 0.10, S soil
D
E
= 1, 2, 4, 6, 8 (thin → thick lines)RSD type, α = 0.10, S soilSD type, α = 0.10, S soil
D
E
= 1, 2, 4, 6, 8 (thin → thick lines)RSD type, α = 0.10, S soilSD type, α = 0.10, S soil
D
E
= 1, 2, 4, 6, 8 (thin → thick lines)RBE type, α = 0.10, S soilBE type, α = 0.10, S soil
D
E
= 1, 2, 4, 6, 8 (thin → thick lines)RBE type, α = 0.10, S soilBE type, α = 0.10, S soil
D
E
= 1, 2, 4, 6, 8 (thin → thick lines)RBE type, α = 0.10, S soilBE type, α = 0.10, S soil
D
E
= 1, 2, 4, 6, 8 (thin → thick lines)RBE type, α = 0.10, S soilBE type, α = 0.10, S soil
D
E
= 1, 2, 4, 6, 8 (thin → thick lines)R
BP type, α = 0.10, β = β = 1/3, S soils r E
BP type, α = 0.10, β = β = 1/3, S soilDs r
= 1, 2, 4, 6, 8 (thin → thick lines)R
BP type, α = 0.10, β = β = 1/3, S soils r E
BP type, α = 0.10, β = β = 1/3, S soilDs r
= 1, 2, 4, 6, 8 (thin → thick lines)R
BP type, α = 0.10, β = β = 1/3, S soils r E
BP type, α = 0.10, β = β = 1/3, S soilDs r
= 1, 2, 4, 6, 8 (thin → thick lines)R
BP type, α = 0.10, β = β = 1/3, S soils r E
BP type, α = 0.10, β = β = 1/3, S soilDs r
Figure 6.2: Effect of site soil characteristics (SD, BE, and BP hysteresis types,α = 0.10,SAC Los Angeles, design-level): (a)µ; (b) µp; (c) µr; (d) ny.
(a)
(b)
(c)
(d)
77
For theSD soil profile, on average, the design-level ensemble tends to result in largerµrdemands forT < 0.75-1.0 sec. andT > 1.75-3.0 sec., depending onR (Fig. 6.4c). For 0.75-1.0 <T< 1.75-3.0 sec.,µr for theSD soil design-level ensemble ranges from 1 to 0.5 times theµr for theSD soil survival-level ensemble, depending onR. For theSE soil profile, the design-levelµrdemands are on average slightly larger than the survival-levelµr demands for almost the entireperiod range, especially for largerR. The differences in theny demands for theSD soil design-level and survival-level ensembles are small forT < 1 sec. (Fig. 6.4d). ForT > 1 sec., thenydemands for the design-level ensemble can be as much as 1.5 times theny demands for the sur-vival-level ensemble. For theSE soil profile, theny demands for the design-level ensemble arelarger than theny demands for the survival-level ensemble for almost the entire period range (asmuch as 50% larger).
Similar observations can be made for the mean demand spectra of the SD, BE, and BPhysteresis types in Figure 6.5.
The increase in theµ, µp, µr, andny demands of long period structures from survival-levelto design-level ground motions is explained as follows. Comparing the average linear-elasticresponse spectra for the survival-level and the design-level ground motion ensembles (solid anddashed lines, respectively, Fig. 6.6a), it can be seen that the intensity of the survival-level groundmotion ensemble in the longer period range decreases at a faster rate as compared to the intensityof the design-level ensemble. Since the lateral force capacities,Fy, of the structures are based onthe IND response spectra, the survival-level ground motion ensemble results in largerR coeffi-cients to obtainµ, µp, µr, andny demands similar to the demands for the design-level ensemble.Thus, the reduced survival-level demands in Figures 6.4 and 6.5 for longer periods may be uncon-servative for structures where smooth design (DES) response spectra are used to determineFy asdescribed in more detail in Chapter 7. This finding is also evident when comparing the averageresponse spectra for different site seismicities (Fig. 6.6b) and different epicentral distances (Fig.6.6c), which is discussed in further detail in Sections 6.3 and 6.4, respectively.
0 30
2.0
T (sec)
LA02 response spectrumS soil, ξ = 5%
LOS ANGELES DESIGN-LEVEL S
(g
)a
0 30
2.0
T (sec)
R = 2
Design response spectrumUBC 1997, S soil, ξ = 5%
unconservative inconsistency in determining F.
LOS ANGELES DESIGN-LEVEL SOFT SOIL
S
(g)
a
R = 1 (linear-elastic)
y
D
shift of "effective" structure period
To,s Teff,s
LS02 response spectrumS soil, ξ = 5%
E
LS02 response spectrumS soil, ξ = 5%
E
E
To,l Teff,l
Figure 6.3: Response spectra: (a)SD soil ground motion spectrum versusSE soil groundmotion spectrum; (b) ground motion spectra versus smooth design spectra.
(a) (b)
78
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ = 1, 2, 4, 6, 8 (thin → thick lines)R
EP type, α = 0.10, survival-levelEP type, α = 0.10, design-level
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, survival-levelEP type, α = 0.10, design-level
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, survival-levelEP type, α = 0.10, design-level
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, survival-levelEP type, α = 0.10, design-level
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, survival-levelEP type, α = 0.10, design-level
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, survival-levelEP type, α = 0.10, design-level
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, survival-levelEP type, α = 0.10, design-level
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, survival-levelEP type, α = 0.10, design-level
Figure 6.4: Effect of seismic demand level (EP hysteresis type,α = 0.10, SAC Los Ange-les,SD andSE soil): (a)µ; (b) µp; (c) µr; (d) ny.
(a)
(b)
(c)
(d)SD soil SE soil
79
6.3 Site Seismicity
Figure 6.7 compares the mean demand spectra using the EP hysteresis type withα = 0.10under the Los Angeles and Boston survival-level ground motion ensembles for sites withSD andSE soil profiles. Theµ demands are smaller for the Boston ensemble as compared to the LosAngeles ensemble for almost the entire period range and, on average, the differences decrease asT increases (Fig. 6.7a). The same effect that was observed for the seismic demand level in the pre-vious section begin to appear forT > ~2.2 sec. for theSD soil profile and forT > ~1.75 sec. for theSE soil profile. This critical structure period increases asR increases. At these longer periods, the
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
= 1, 2, 4, 6, 8 (thin → thick lines)RSD type, α = 0.10, survival-levelSD type, α = 0.10, design-level
= 1, 2, 4, 6, 8 (thin → thick lines)R
BP type, α = 0.10, β = β = 1/3, design-levels r
BP type, α = 0.10, β = β = 1/3, survival-levels r
= 1, 2, 4, 6, 8 (thin → thick lines)RSD type, α = 0.10, survival-levelSD type, α = 0.10, design-level
= 1, 2, 4, 6, 8 (thin → thick lines)RSD type, α = 0.10, survival-levelSD type, α = 0.10, design-level
= 1, 2, 4, 6, 8 (thin → thick lines)RSD type, α = 0.10, survival-levelSD type, α = 0.10, design-level
= 1, 2, 4, 6, 8 (thin → thick lines)RBE type, α = 0.10, survival-levelBE type, α = 0.10, design-level
= 1, 2, 4, 6, 8 (thin → thick lines)RBE type, α = 0.10, survival-levelBE type, α = 0.10, design-level
= 1, 2, 4, 6, 8 (thin → thick lines)RBE type, α = 0.10, survival-levelBE type, α = 0.10, design-level
= 1, 2, 4, 6, 8 (thin → thick lines)RBE type, α = 0.10, survival-levelBE type, α = 0.10, design-level
= 1, 2, 4, 6, 8 (thin → thick lines)R
BP type, α = 0.10, β = β = 1/3, design-levels r
BP type, α = 0.10, β = β = 1/3, survival-levels r
= 1, 2, 4, 6, 8 (thin → thick lines)R
BP type, α = 0.10, β = β = 1/3, design-levels r
BP type, α = 0.10, β = β = 1/3, survival-levels r
= 1, 2, 4, 6, 8 (thin → thick lines)R
BP type, α = 0.10, β = β = 1/3, design-levels r
BP type, α = 0.10, β = β = 1/3, survival-levels r
Figure 6.5: Effect of seismic demand level (SD, BE, and BP hysteresis types,α = 0.10,SAC Los Angeles,SD soil): (a)µ; (b) µp; (c) µr; (d) ny.
(a)
(b)
(c)
(d)
80
Bostonµ demands are larger than the Los Angelesµ demands in the same fashion that the design-level µ demands are larger than the survival-levelµ demands in the previous section.
The increase in seismic demands at longer periods is more evident in theµp and nydemands. For theSD soil profile, the Bostonµp andny demands are larger than the Los Angelesdemands forT > ~1.75 sec. andT > ~0.5-0.75 sec., respectively (Figs. 6.7b and d). Theµpdemands for theSE soil profile are larger for the Boston ensemble as compared to the Los Angelesensemble forT > ~1.75 sec. Theny demands for theSE soil profile are always larger for the Bos-ton ensemble as compared to the Los Angeles ensemble, except forT ≅ 0.3 and 0.6 sec. andR= 2.
As shown in Figure 6.6b, the intensity of the Los Angeles ground motion ensemble in thelonger period range decreases at a faster rate as compared to the intensity of the Boston ensemble,in the same fashion that the intensity of the survival-level ensemble in the longer period rangedecreases at a faster rate as compared to the intensity of the design-level ensemble (Fig. 6.6a).This results in smaller long-period seismic demands for Los Angeles, which may be unconserva-
0 3.50
2.0
T (sec)
design-level averageresponse spectrum, ξ = 5%
SAC LOS ANGELES S
(g
)a
shift of "effective" structure period
To Teff
survival-level averageresponse spectrum, ξ = 5%
S SOILD
0 3.50
2.5
T (sec)
Boston averageresponse spectrumξ = 5%
S
(g)
a
shift of "effective" structure period
To Teff
Los Angeles averageresponse spectrumξ = 5%
SURVIVAL-LEVEL S SOILE
0 3.50
2.0
T (sec)
far-field averageresponse spectrum, ξ = 5%
SAC LOS ANGELES DESIGN-LEVEL
S
(g)
a
shift of "effective" structure period
To Teff
near-field averageresponse spectrum, ξ = 5%
S SOILD
Figure 6.6: Average response spectra: (a) design-level versus survival-level; (b) Bostonversus Los Angeles; (c) far-field versus near-field (NF).
(a) (b)
(c)
81
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ = 1, 2, 4, 6, 8 (thin → thick lines)R
EP type, α = 0.10, Los AngelesEP type, α = 0.10, Boston
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, SEP type, α = 0.10, S
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, Los AngelesEP type, α = 0.10, Boston
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/3r s
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, Los AngelesEP type, α = 0.10, Boston
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/3r s
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, Los AngelesEP type, α = 0.10, Boston
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, Los AngelesEP type, α = 0.10, Boston
20
40
60
80
100
µ p
00 0.5 1 1.5 2 2.5 3 3.5
T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, SEP type, α = 0.10, S
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, Los AngelesEP type, α = 0.10, Boston
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
1
2
3
4
µ r
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/3r s
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, Los AngelesEP type, α = 0.10, Boston
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
0
10
20
30
40
50
n y
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/3r s
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, Los AngelesEP type, α = 0.10, Boston
Figure 6.7: Effect of site seismicity (EP hysteresis type,α = 0.10, SAC survival-level,SDandSE soil): (a)µ; (b) µp; (c) µr; (d) ny.
(a)
(b)
(c)
(d)SD soil SE soil
82
tive for structures where smooth design (DES) response spectra are used to determineFy asdescribed in more detail in Chapter 7.
Figure 6.7c shows that the differences between theµr demands for the two seismicity lev-els are erratic but relatively small.
Similar observations can be made for the mean demand spectra under the design-levelground motion ensemble in Figure 6.8 and for the mean demand spectra of the SD, BE, and BPhysteresis types in Figure 6.9, except that the differences in theµp demands between the Bostonand Los Angeles ensembles are relatively small when using the BE and BP hysteresis types.
6.4 Epicentral Distance
Figure 6.10 compares the mean demand spectra using the EP hysteresis type withα = 0.10for the Los Angeles design-levelSD soil far-field and near-field ground motion ensembles. ForT <~1.0-1.5 sec., theµ andµp demands are larger for the near-field ensemble as compared to the far-field ensemble (up to 3.25 times as large forµ and 4 times as large forµp, Figs. 6.10a and b). Onaverage, the near-field ensemble has smaller or similarµr demands forT < ~0.5-1.10 sec. andT >~1.75-2.75 sec. (Fig. 6.10c), depending onR. For ~0.5-1.10 <T < ~1.75-2.75 sec., theµr demandfor the near-field ensemble can be as much as 2.5 times theµr demand for the far-field ensemble.Theny demands for the near-field ensemble are similar to theny demands for the far-field ensem-ble for T < ~0.5 sec. and can be as small as 0.5 times the demands for the far-field ensemble forT> ~0.5 sec. (Fig. 6.10d).
The differences between the near-field and far-field demands forµ andµp increase asRincreases forT < ~1.0-1.5 sec. and the differences remain relatively constant forT > ~1.0-1.5 sec.(Figs. 6.10a and b). The differences inµr are highly dependent on theR coefficient (Fig. 6.10c).The differences inny are somewhat independent of theR coefficient for the entire period range(Fig. 6.10d).
Similar to what was observed for the design-level ensemble versus survival-level ensem-ble discussed previously, there is an increase in theµ, µp, andny demands from near-field to far-field ground motions for longer period structures (T > ~0.5-1.5 sec.) since these results are basedon the IND spectra. Figure 6.6c shows that, in the longer period range, the intensity of the near-field ensemble decreases at a faster rate as compared to the intensity of the far-field ensemble. Asnoted earlier and discussed in detail in Chapter 7, the smaller long-period seismic demands fornear-field ground motion records in Figure 6.10 may be unconservative for structures wheresmooth design (DES) response spectra are used to determineFy.
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= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/3r s
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= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10BP type, α = 0.10, β = β = 1/3r s
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= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, Los AngelesEP type, α = 0.10, Boston
(a)
(b)
(c)
Figure 6.8: Effect of site seismicity (EP hysteresis type,α = 0.10, SAC design-level,SDandSE soil): (a)µ; (b) µp; (c) µr; (d) ny.
(d)SD soil SE soil
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= 1, 2, 4, 6, 8 (thin → thick lines)R
BP type, α = 0.10, β = β = 1/3, Bostons r
BP type, α = 0.10, β = β = 1/3, Los Angeless r
= 1, 2, 4, 6, 8 (thin → thick lines)RBE type, α = 0.10, Los AngelesBE type, α = 0.10, Boston
= 1, 2, 4, 6, 8 (thin → thick lines)RSD type, α = 0.10, Los AngelesSD type, α = 0.10, Boston
= 1, 2, 4, 6, 8 (thin → thick lines)RSD type, α = 0.10, Los AngelesSD type, α = 0.10, Boston
= 1, 2, 4, 6, 8 (thin → thick lines)RSD type, α = 0.10, Los AngelesSD type, α = 0.10, Boston
= 1, 2, 4, 6, 8 (thin → thick lines)RBE type, α = 0.10, Los AngelesBE type, α = 0.10, Boston
= 1, 2, 4, 6, 8 (thin → thick lines)RBE type, α = 0.10, Los AngelesBE type, α = 0.10, Boston
= 1, 2, 4, 6, 8 (thin → thick lines)RBE type, α = 0.10, Los AngelesBE type, α = 0.10, Boston
= 1, 2, 4, 6, 8 (thin → thick lines)R
BP type, α = 0.10, β = β = 1/3, Bostons r
BP type, α = 0.10, β = β = 1/3, Los Angeless r
= 1, 2, 4, 6, 8 (thin → thick lines)R
BP type, α = 0.10, β = β = 1/3, Bostons r
BP type, α = 0.10, β = β = 1/3, Los Angeless r
= 1, 2, 4, 6, 8 (thin → thick lines)R
BP type, α = 0.10, β = β = 1/3, Bostons r
BP type, α = 0.10, β = β = 1/3, Los Angeless r
Figure 6.9: Effect of site seismicity (SD, BE, and BP hysteresis types,α = 0.10, SAC sur-vival-level,SD soil): (a)µ; (b) µp; (c) µr; (d) ny.
(a)
(b)
(c)
(d)
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= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, far-fieldEP type, α = 0.10, near-field
= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, far-fieldEP type, α = 0.10, near-field
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= 1, 2, 4, 6, 8 (thin → thick lines)REP type, α = 0.10, far-fieldEP type, α = 0.10, near-field
Figure 6.10: Effect of epicentral distance (EP hysteresis type,α = 0.10, SAC Los Angeles,design-level,SD soil): (a)µ; (b) µp; (c) µr; (d) ny.
(a) (b)
(c) (d)
86
CHAPTER 7
EFFECT OF REFERENCE RESPONSE SPECTRA
As described earlier, the results shown in the previous chapters are based onFy = Felas/Rvalues determined using the linear-elastic acceleration response spectrum for each ground motion(i.e., IND spectra). In this chapter, the effect of the reference response spectrum (i.e., smooth ver-sus IND spectra) on theR-µ-T relationships is investigated. The results are presented as meanR-µspectra for different hysteresis types, site seismicities, seismic demand levels, site soil conditions,and epicentral distances using the SAC ground motion ensemble.
7.1 Low Seismicity (Boston), Stiff Soil Profile (SD)
Figures 7.1a-b show the meanR-µ spectra using the EP hysteresis type withα = 0.10 forsites with a stiff soil profile in a region with low seismicity under design-level and survival-levelground motions, respectively. The solid and dashed lines represent results obtained using the INDand AVG spectra, respectively. The differences between the results obtained using the two spectraare moderate. For long period structures (i.e.,T > 1 sec.), theµ demands estimated using the AVGspectrum are consistently larger than the demands estimated using the IND spectra.
Similarly, Figures 7.1c-d show comparisons between theR-µ spectra obtained using theIND and DES spectra. Under design-level ground motions, the differences between the DES andIND spectra (Fig. 7.1c) are similar to the differences between the AVG and IND spectra (Fig.7.1a). For survival-level ground motions, the increase in theµ demands when using the DES spec-trum is significantly larger forT < 2 sec. From these observations, it can be concluded thatR-µ-Trelationships developed using IND spectra can lead to underestimatedµ demands, and thus,unconservative designs.
7.2 Low Seismicity (Boston), Soft Soil Profile (SE)
Figures 7.1e-h show the meanR-µ spectra using the EP hysteresis type withα = 0.10 forsites with a soft soil profile in a region with low seismicity under design-level and survival-levelground motions. The solid lines represent results obtained using the IND spectra and the dashedlines represent results obtained using the AVG and DES spectra. Similar to the stiff soil profile,the increase in theµ demands is moderate when using the AVG spectrum and is significant for thesurvival-level ground motions when using the DES spectrum.
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= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraDES spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraDES spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraDES spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraDES spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraAVG spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraAVG spectrum
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= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraAVG spectrum
Figure 7.1:R-µ spectra for low seismicity (EP hysteresis type,α = 0.10): (a-d) stiff soilprofile,SD; (e-h) soft soil profile,SE.
(a) design-level (b) survival-level
(c) design-level (d) survival-level
(e) design-level (f) survival-level
(g) design-level (h) survival-level
88
It is noted that, for the survival-level ground motions, the increase in theµ demand due tothe use of the DES spectrum is much larger for sites with a soft soil profile (Fig. 7.1h) than forsites with a stiff soil profile (Fig. 7.1d: e.g., theµ demand from the DES spectrum can be about 2times theµ demand from the IND spectra as shown in Fig. 7.1h). These large increases occur foralmost the entire period range (i.e.,T > 0.25 sec.) and are especially severe for 1.0 <T < 1.5 sec.where the predominant period of the soil resides (see Fig. A.21). This period range corresponds toa “dip” in the R-µ spectra (especially for largeR coefficients) when using the IND spectra sincethe lateral strength,Fy, is increased, as mentioned earlier in Section 6.1. TheR-µ spectra for theAVG spectrum follow this dip (Fig. 7.1f). However, theR-µ spectra for the DES spectrum do notshow a dip since, in this case,Fy is not based on the linear-elastic acceleration response spectrumof each ground motion. It is obvious from these results that using IND spectra instead of DESspectra to developR-µ-T relationships can lead to significantly unconservative designs, especiallyfor medium- to long-period structures under survival-level, soft soil ground motions.
7.3 High Seismicity (Los Angeles), Stiff Soil Profile (SD)
Figures 7.2a-b show the meanR-µ spectra using the EP hysteresis type withα = 0.10 forsites with a stiff soil profile in a region with high seismicity under design-level and survival-levelground motions. The solid and dashed lines represent results obtained using the IND and AVGspectra, respectively. On average, the differences between the results obtained using the IND andAVG spectra are small, particularly for the design-level ground motions. For long period struc-tures (i.e.,T > 1 sec.) under survival-level ground motions, theµ demands estimated using theAVG spectrum are consistently larger than the demands estimated using the IND spectra.
Similarly, Figures 7.2c-d show comparisons between theR-µ spectra obtained using theIND and DES spectra. Contrary to the AVG spectrum, theµ demands under design-level groundmotions decrease when using the DES spectrum. Thus,R-µ-T relationships developed using theIND spectra would result in more conservative designs for design-level ground motions than rela-tionships developed using the DES spectrum.
Under survival-level ground motions, theµ demands obtained using the DES spectrumcan be significantly larger than the demands obtained using the IND spectra, particularly for largeRcoefficients and T > 1 sec. From this important observation, it can be concluded thatR-µ-T rela-tionships developed using IND spectra can lead to underestimated seismic demands and, thus,unconservative designs, especially under survival-level ground motions.
7.4 High Seismicity (Los Angeles), Soft Soil Profile (SE)
Figures 7.2e-h show the meanR-µ spectra using the EP hysteresis type withα = 0.10 forsites with a soft soil profile in a region with high seismicity under design-level and survival-levelground motions, respectively. Similar to the stiff soil profile, the differences between the resultsobtained using the IND and AVG spectra are small, particularly for the design-level groundmotions. However, there is a significant increase in the demands when using the DES spectrum,especially for the survival-level ground motions (e.g., theµ demand from the DES spectrum can
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= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraAVG spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraDES spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraDES spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraDES spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraDES spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraAVG spectrum
3.5
Figure 7.2:R-µ spectra for high seismicity (EP hysteresis type,α = 0.10): (a-d) stiff soil pro-file, SD; (e-h) soft soil profile,SE.
(a) design-level (b) survival-level
(c) design-level (d) survival-level
(e) design-level (f) survival-level
(g) design-level (h) survival-level
90
be about 2.5 times theµ demand from the IND spectra as shown in Fig. 7.2h). These largeincreases occur for almost the entire period range (i.e.,T > 0.25 sec.) and are especially severe for1.0 <T < 2.5 sec. where the predominant period of the soil resides (see Fig. A.25). It is obviousfrom these results that using IND spectra instead of DES spectra to developR-µ-T relationshipscan lead to significantly unconservative designs, especially for medium- to long-period structuresunder survival-level, soft soil ground motions in regions with high seismicity. Furthermore, forthe DES spectrum, theµ demands are extreme and possibly uncontrollable at higher values ofR,especially for survival-level, soft soil ground motions in regions with high seismicity. Thus, theRcoefficients specified in current seismic design provisions may be very unconservative for theseconditions. Either the smooth design response spectra in the current provisions need to be modi-fied (e.g., using AVG spectra instead) or theR coefficients recommended in the provisions shouldbe reduced under these conditions.
7.5 High Seismicity (Los Angeles), Near-Field (NF), Stiff Soil Profile (SD)
Figure 7.3 shows the meanR-µ spectra using the EP hysteresis type withα = 0.10 for siteswith a stiff soil profile in a region with high seismicity under near-field ground motions. The dif-ferences between the results obtained using the IND and AVG spectra are small (Fig. 7.3a). How-ever, the increase in theµ demands when using the DES spectrum is dramatic (theµ demand fromthe DES spectrum can be almost 3 times the demand from the IND spectra as shown in Fig. 7.3b).
The large amplification factors using the DES spectrum occur for almost the entire periodrange (i.e.,T > 0.25 sec.) and are especially severe forT > 0.5 sec. The pulse-like (large ampli-tude, long period) characteristics (see Fig. A.26) of theNF ground motions result in largeµdemands for both medium-period and long-period structures. From these observations, it is con-cluded thatR-µ-T relationships developed using IND spectra can lead to significantly unconserva-tive designs for structures within close proximity of an active fault. Furthermore, for the DESspectrum, the near-fieldµ demands are extreme and possibly uncontrollable at higher values ofR.Thus, theRcoefficients specified in current seismic design provisions may be very unconservativefor these conditions. Either the smooth design response spectra in the current provisions need to
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Figure 7.3:R-µ spectra for near-field (Los Angeles design-levelSD soil, EP hysteresistype,α = 0.10: (a) AVG versus IND spectra; (b) DES versus IND spectra.
(a) (b)
91
be modified (e.g., using AVG spectra instead) or theRcoefficients recommended in the provisionsshould be reduced under these conditions.
7.6 SD, BE, and BP Hysteresis Types
This section investigates the effect of hysteresis type on theR-µ-T relationships developedusing different reference response spectra. The results are presented as meanR-µ spectra for theSAC Los Angeles survival-levelSD soil ground motion ensemble.
Figure 7.4 shows the meanR-µ spectra for the SD, BE, and BP (βs = βr = 1/3) hysteresistypes (α = 0.10). The solid lines represent the results obtained using the IND spectra and thedashed lines represent the results obtained using the AVG and DES spectra. Figures 7.2 b and dprovide results for the EP hysteresis type (α = 0.10) for similar site conditions. The differencesbetween theµ demands for the IND spectra, and the AVG and DES spectra are not significantlyaffected by the hysteresis type. Thus, it is concluded that the effect of the reference response spec-tra on the meanµ demand is not significantly dependent on the hysteresis type.
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= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraAVG spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraDES spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraAVG spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraDES spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraAVG spectrum
= 1, 2, 4, 6, 8 (thin → thick lines)RIND spectraDES spectrum
Figure 7.4: Effect of hysteresis type and reference response spectra on theµ demand(Los Angeles survival-levelSD soil, α = 0.10): (a-b) SD hysteresis type; (c-d) BE hyster-esis type; (e-f) BP hysteresis type (βs = βr = 1/3).
(a) (b)
(c) (d)
(e) (f)
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CHAPTER 8
REGRESSION ANALYSES FORµ
In the previous chapters, the relationship betweenR, µ, andT is investigated in terms ofmeanR-µ spectra from the dynamic analyses. Based on these spectra, this chapter presentsclosed-formR-µ-T relationships in order to achieve two objectives: (1) to quantify the effect of thedifferent parameters evaluated in the research on theR-µ-T relationships; and (2) to provide rela-tionships that can be integrated into current and future seismic design provisions.
To achieve these objectives, theR-µ-T relationships are developed using the two-step non-linear regression analysis scheme and the form of the regression equations developed by Nassarand Krawinkler (Eqs. 2.1-2.3) as described previously in Section 3.7.1.
8.1 Comparison ofR-µ-T Relationships with Previous Results
Figure 8.1 compares theR-µ-T relationships developed in this study with relationshipsdeveloped by Nassar and Krawinkler (1991) in terms ofR-µ spectra. The results are presented forthe EP hysteresis type withα = 0.10 based on the N&K ground motion ensemble.
0 0.5 1 1.5 2 2.5 3 3.50
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R = 1, 2, 4, 6, 8 (thin → thick lines)IND spectra, constant−R: a = 1.35, b = 0.44 AVG spectrum, constant−R: a = 2.74, b = 0.57 DES spectrum, constant−R: a = 1.74, b = 0.55
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R = 1, 2, 4, 6, 8 (thin → thick lines)constant−µ, IND spectra: a = 0.80, b = 0.29 constant−R, IND spectra: a = 1.35, b = 0.44
(a) (b)Figure 8.1: Regression curves (EP hysteresis type,α = 0.10): (a) constant-µ versus constant-R approaches; (b) IND spectra versus AVG and DES spectra.
94
The regression coefficients,a andb, corresponding to theR-µ spectra in Figure 8.1 aregiven in Table 8.1. As explained in Section 4.3, there are two differences between the new coeffi-cients and previous research: (1) the use of the constant-R approach (this research) instead of theconstant-µ approach (previous research); and (2) the use of AVG and DES spectra (this research)instead of IND spectra (previous research). These differences are illustrated in Figures 8.1a and b,respectively, and quantified in Table 8.1. The results indicate that the previousR-µ-T relationshipscan result in underestimated seismic demands for design.
8.2 R-µ-T Relationships Developed in this Study
Table 8.2 shows thea andb regression coefficients developed in this study for the SACground motion ensemble using the analyses presented earlier. Regression coefficients are givenfor all three reference response spectra, where applicable. Figure 8.2 provides a sampling of com-parisons between the meanR-µ spectra and the regression curves developed through this research.A reasonably good fit is observed, except for the SAC Los Angeles, design-level,SD soil, NFground motion ensemble (Fig. 8.2h). Thus, care should be taken when using the regression rela-tionship developed for this case. The regression relationships developed for the other cases inves-tigated by the research (as shown in Table 8.2) provide a reasonably good representation of themean demands from the dynamic analyses.
The R-µ spectra based on the regression coefficients listed in Table 8.2 are provided inFigures 8.3 through 8.11 to reinforce the previous findings from the meanR-µ spectra. The figuresare organized in the same order that the results are presented in Chapters 5-7: (1) effect of hyster-esis type (Fig. 8.3); (2) effect of site conditions (Figs. 8.4-8.7); and (3) effect of referenceresponse spectra (Figs. 8.8-8.11).
The effect of the post-yield stiffness ratio,α, can be clearly seen in Figure 8.3a, especiallybetweenα = 0.00 andα = 0.10. For hysteresis type (Fig. 8.3b), the significant difference between
Table 8.1: Regression coefficientsa andb for the N&K ground motionensemble, EP hysteresis type,α = 0.10
Regression Coefficient
Approach a b
IND spectra, constant-µ(Nassar and Krawinkler, 1991)
0.80 0.29
IND spectra, constant-R(Farrow and Kurama)
1.35 0.44
AVG spectrum, constant-R(Farrow and Kurama)
2.74 0.57
DES spectrum, constant-R(Farrow and Kurama)
1.74 0.55
95
theµ demands for the EP and BE types is demonstrated. For the BP hysteresis type with varyingβs = βr (Fig. 8.3c), theµ demands decrease asβr increases. The dependence of theµ demands onsite conditions (i.e., site soil characteristics, seismic demand level, site seismicity, and epicentraldistance) is demonstrated in Figures 8.4-8.7.
The dramatic effect of using smooth design response spectra to determine the design lat-eral force capacity,Fy, can be clearly seen, especially for structures subjected to survival-levelground motions in Figures 8.8d, 8.9b and d. For the DES spectrum, theµ demands are extremeand possibly uncontrollable at higher values ofR, especially for soft soil and near-field groundmotions in regions with high seismicity (Figs. 8.9d and 8.10). Thus, theRcoefficients specified incurrent seismic design provisions may be very unconservative for these conditions. Either thesmooth design response spectra in the current provisions need to be modified (e.g., using AVGspectra instead) or theR coefficients recommended in the provisions should be reduced underthese conditions.
The lack of dependence of the effect of reference response spectra on hysteresis type canbe seen in Figure 8.11.
Table 8.2: Regression coefficientsa andb for the SAC ground motion ensemble
HysteresisType
α, βs, βr Site SeismicityDemand
LevelSite Soil/Distance
IND AVG DES
a b a b a b
EP
α = 0.00 Los Angeles Survival SD 0.53 0.82 -- -- -- --
α = 0.05 Los Angeles Survival SD 0.37 0.74 -- -- -- --
α = 0.10
Boston
SurvivalSD 0.48 0.45 1.08 0.49 0.65 0.62SE -0.63 0.52 0.11 0.57 0.41 0.84
DesignSD 0.56 0.45 1.21 0.49 1.08 0.42SE 0.12 0.41 0.82 0.48 0.87 0.43
Los Angeles
SurvivalSD 0.40 0.72 1.01 0.73 2.04 0.79SE -0.71 0.94 -0.07 1.00 2.32 1.31
Design
SD 1.89 0.68 1.92 0.66 1.71 0.59SE 0.15 0.85 0.41 0.86 0.80 0.88NF 0.35 0.89 0.67 0.92 6.23 1.34
SD α = 0.10
Boston Survival SD 0.54 0.48 -- -- -- --
Los Angeles
Survival SD 0.52 0.75 1.14 0.76 2.20 0.82
DesignSD 2.09 0.70 -- -- -- --SE 0.22 0.86 -- -- -- --
BE α = 0.10
Boston Survival SD 1.30 0.72 -- -- -- --
Los Angeles
Survival SD 1.53 0.97 2.20 1.00 3.68 1.06
DesignSD 4.74 0.94 -- -- -- --SE 0.94 1.09 -- -- -- --
BP
α = 0.10βs = βr = 1/6 Los Angeles Survival SD 1.25 0.93 -- -- -- --
α = 0.10βs = βr = 1/3
Boston Survival SD 0.93 0.62 -- -- -- --
Los Angeles
Survival SD 1.08 0.89 1.65 0.91 3.00 0.97
DesignSD 3.83 0.87 -- -- -- --SE 0.65 1.02 -- -- -- --
α = 0.10βs = βr = 1/2 Los Angeles Survival SD 0.96 0.87 -- -- -- --
96
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
= 1, 2, 4, 6, 8 (thin → thick lines)Rmeanregression: a = 0.40, b = 0.72
= 1, 2, 4, 6, 8 (thin → thick lines)Rmeanregression: a = 1.89, b = 0.68
= 1, 2, 4, 6, 8 (thin → thick lines)Rmeanregression: a = 0.15, b = 0.85
= 1, 2, 4, 6, 8 (thin → thick lines)Rmeanregression: a = -0.71, b = 0.94
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5T (sec)
µ
= 1, 2, 4, 6, 8 (thin → thick lines)Rmeanregression: a = -0.63, b = 0.52
= 1, 2, 4, 6, 8 (thin → thick lines)Rmeanregression: a = 6.23, b = 1.34
= 1, 2, 4, 6, 8 (thin → thick lines)Rmeanregression: a = 0.56, b =0.45
= 1, 2, 4, 6, 8 (thin → thick lines)Rmeanregression: a = 0.12, b = 0.41
Figure 8.2: Comparison between meanR-µ spectra and regression curves (EP hysteresistype,α = 0.10): (a-b) SAC Boston design-level; (c-d) SAC Los AngelesSD soil; (e-f)SAC Los AngelesSE soil; (g) SAC Boston survival-levelSE soil; (h) SAC Los Angelesdesign-levelSD soil, NF.
(a) IND spectra,SD soil (b) IND spectra,SE soil
(e) IND spectra, design-level (f) IND spectra, survival-level
(c) IND spectra, design-level (d) IND spectra, survival-level
(g) IND spectra (h) DES spectrum
97
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines)EP type, α = 0.00: a = 0.53, b = 0.82 EP type, α = 0.05: a = 0.37, b = 0.74 EP type, α = 0.10: a = 0.40, b = 0.72
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) BP type, α = 0.10, β
s = β
r = 1/6: a = 1.25, b = 0.93
BP type, α = 0.10, βs = β
r = 1/3: a = 1.08, b = 0.89
BP type, α = 0.10, βs = β
r = 1/2: a = 0.96, b = 0.87
Figure 8.3: Effect of hysteretic behavior on regression curves using IND spectra (SAC LosAngeles, survival-level,SD soil): (a) post-yield stiffness ratio,α; (b) hysteresis type; (c)βs= βr.
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines)EP type, α = 0.10: a = 0.40, b = 0.72 SD type, α = 0.10: a = 0.52, b = 0.75 BE type, α = 0.10: a = 1.53, b = 0.97 (a)
(c)
(b)
98
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) BE type, α = 0.10, S
D soil: a = 4.74, b = 0.94
BE type, α = 0.10, SE soil: a = 0.94, b = 1.09
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) BP type, α = 0.10, β
s = β
r = 1/3, S
D soil: a = 3.83, b = 0.87
BP type, α = 0.10, βs = β
r = 1/3, S
E soil: a = 0.65, b = 1.02
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) EP type, α = 0.10, S
D soil: a = 1.89, b = 0.68
EP type, α = 0.10, SE soil: a = 0.15, b = 0.85
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) SD type, α = 0.10, S
D soil: a = 2.09, b = 0.70
SD type, α = 0.10, SE soil: a = 0.22, b = 0.86
Figure 8.4: Effect of site soil characteristics on regression curves using IND spectra (SACLos Angeles, design-level): (a) EP hysteresis type; (b) SD hysteresis type; (c) BE hystere-sis type; (d) BP hysteresis type.
(a)
(c)
(b)
(d)
99
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) SD type, α = 0.10, survival−level: a = 0.52, b = 0.75SD type, α = 0.10, design−level: a = 2.09, b = 0.70
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) BP type, α = 0.10, β
s = β
r = 1/3, survivallevel: a = 1.08, b = 0.89
BP type, α = 0.10, βs = β
r = 1/3, design level: a = 3.83, b = 0.87
Figure 8.5: Effect of seismic demand level on regression curves using IND spectra (SACLos Angeles): (a-b) EP hysteresis type; (c) SD hysteresis type; (d) BE hysteresis type; (e)BP hysteresis type.
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) EP type, α = 0.10, survival−level: a = 0.40, b = 0.72EP type, α = 0.10, design−level: a = 1.89, b = 0.68
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) EP type, α = 0.10, survival−level: a = −0.71, b = 0.94EP type, α = 0.10, design−level: a = 0.15, b = 0.85
SAC Los Angeles
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) BE type, α = 0.10, survival−level: a = 1.53, b = 0.97BE type, α = 0.10, design−level: a = 4.74, b = 0.94
(a)
SD soil
(b)
SE soil
(c)
SD soil
(d)
SD soil
(e)
SD soil
100
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) EP type, α = 0.10, Los Angeles: a = 0.40, b = 0.72EP type, α = 0.10, Boston: a = 0.48, b = 0.45
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) EP type, α = 0.10, Los Angeles: a = −0.71, b = 0.94EP type, α = 0.10, Boston: a = −0.63, b = 0.52
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) EP type, α = 0.10, Los Angeles: a = 1.89, b = 0.68EP type, α = 0.10, Boston: a = 0.56, b = 0.45
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) EP type, α = 0.10, Los Angeles: a = 0.15, b = 0.85EP type, α = 0.10, Boston: a = 0.12, b = 0.41
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) SD type, α = 0.10, Los Angeles: a = 0.52, b = 0.75SD type, α = 0.10, Boston: a = 0.54, b = 0.48
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) BE type, α = 0.10, Los Angeles: a = 1.53, b = 0.97BE type, α = 0.10, Boston: a = 1.30, b = 0.72
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines) BP type, α = 0.10, β
s = β
r = 1/3, Los Angeles: a = 1.08, b = 0.89
BP type, α = 0.10, βs = β
r = 1/3, Boston: a = 0.93, b = 0.62
Figure 8.6: Effect of site seismicity on regression curves using IND spectra: (a-d) EP hys-teresis type; (e) SD hysteresis type; (f) BE hysteresis type; (g) BP hysteresis type.
(a)
survival-level
(b)
survival-level
(c)
design-level
(d)
design-level
(e)
survival-level
(f)
survival-level
(g)
survival-level
101
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines)EP type, α = 0.10, far−field: a = 1.89, b = 0.68 EP type, α = 0.10, near−field: a = 0.35, b = 0.89
Figure 8.7: Effect of epicentral distance on regression curves using IND spectra (SAC LosAngeles, design-level,SD soil).
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines)IND spectra: a = 0.56, b = 0.45 AVG spectrum: a = 1.21, b = 0.49 DES spectrum: a = 1.08, b = 0.42
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines)IND spectra: a = 0.12, b = 0.41 AVG spectrum: a = 0.82, b = 0.48 DES spectrum: a = 0.87, b = 0.43
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines)IND spectra: a = 0.48, b = 0.45 AVG spectrum: a = 1.08, b = 0.49 DES spectrum: a = 0.65, b = 0.62
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines)IND spectra: a = −0.63, b = 0.52 AVG spectrum: a = 0.11, b = 0.57 DES spectrum: a = 0.41, b = 0.84
Figure 8.8: Effect of reference response spectra on regression curves (SAC Boston, EP hys-teresis type,α = 0.10): (a-b)SD soil; (c-d)SE soil.
(a)
design-level
(b)
survival-level
(c)
design-level
(d)
survival-level
102
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines)IND spectra: a = 1.89, b = 0.68 AVG spectrum: a = 1.92, b = 0.66 DES spectrum: a = 1.71, b = 0.59
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines)IND spectra: a = 0.15, b = 0.85 AVG spectrum: a = 0.41, b = 0.86 DES spectrum: a = 0.80, b = 0.88
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines)IND spectra: a = 0.40, b = 0.72 AVG spectrum: a = 1.01, b = 0.73 DES spectrum: a = 2.04, b = 0.79
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)µ
R = 1, 2, 4, 6, 8 (thin → thick lines)IND spectra: a = −0.71, b = 0.94 AVG spectrum: a = −0.07, b = 1.00 DES spectrum: a = 2.32, b = 1.31
Figure 8.9: Effect of reference response spectra on regression curves (SAC Los Angeles, EPhysteresis type,α = 0.10): (a-b)SD soil; (c-d)SE soil.
(a)
design-level
(b)
survival-level
(c)
design-level
(d)
survival-level
Figure 8.10: Effect of reference spectra on regression curves (SAC Los Angeles, design-level,SD soil,NF, EP hysteresis type,α = 0.10).
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines)IND spectra: a = 0.35, b = 0.89 AVG spectrum: a = 0.67, b = 0.92 DES spectrum: a = 6.23, b = 1.34
103
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines)IND spectra: a = 0.52, b = 0.75 AVG spectrum: a = 1.14, b = 0.76 DES spectrum: a = 2.20, b = 0.82
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines)IND spectra: a = 1.53, b = 0.97 AVG spectrum: a = 2.20, b = 1.00 DES spectrum: a = 3.68, b = 1.06
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
µ
R = 1, 2, 4, 6, 8 (thin → thick lines)IND spectra: a = 1.08, b = 0.89 AVG spectrum: a = 1.65, b = 0.91 DES spectrum: a = 3.00, b = 0.97
Figure 8.11: Effect of reference response spectra on regression curves (SAC Los Angeles,survival-level,SD soil, α = 0.10): (a) SD hysteresis type; (b) BE hysteresis type; (c) BPhysteresis type (βs = βr = 1/3).
(a) (b)
(c)
104
CHAPTER 9
REGRESSION ANALYSES BETWEEN THE DEMAND INDICES
The previous chapter provided regression relationships betweenR, µ, andT. Similarly, thischapter provides regression relationships between the demand indices,µ, µp, µr, andny, in theform of Equations 3.43 and 3.44. A simple one-step nonlinear regression analysis is performed foreach relationship. Regression relationships developed using IND spectra are given in Section 9.1.Similarly, Section 9.2 presents regression relationships developed using AVG and DES spectra,including comparisons with relationships developed using IND spectra. All regression relation-ships presented in this chapter are based on the SAC ground motion ensembles.
9.1 Regression Relationships Developed Based on IND Spectra
Tables 9.1 and 9.2 show thed andf regression coefficients developed for the relationshipbetweenµ and the other demand indices,Λ = µp, µr, andny, based on Equation 3.43. Similarly,Tables 9.3 and 9.4 show theg andh regression coefficients developed for the relationship betweenthe demand indices other thanµ (i.e., Λ = µp, µr, andny) based on Equation 3.44. In Tables 9.3-9.4, the columns represent the independent demand indices,Λj, and the rows represent the depen-dent demand indices,Λi, in Equation 3.44. The lateral strength,Fy = Felas/R, of the systems usedin this section are based on the IND reference response spectra.
The correlation coefficients,ρ, between the demand indices as defined in Section 3.7.1, arelisted in Tables 9.5 and 9.6. Figures 9.1-9.20 provide matrix plots for the cross-correlationsbetween the demand indices. The figures are organized in the same order that the results are pre-sented in Chapters 5 and 6: (1) effect of hysteresis type (Figs. 9.1-9.3); and (2) effect of site con-ditions (Figs. 9.4-9.20). The dots represent the data from the dynamic analyses and the linesrepresent the regression relationships using Equations 3.43 and 3.44.
In Figures 9.1-9.20, the correlation betweenµ and the other demand indices is relativelystrong, especially forµp (correlation coefficient,ρ, is as high as 0.99, close to full correlation). Insome cases, the cross-correlations between the demand indices show weak correlation, indicatingthat these demand indices can carry independent measures of seismic demand (e.g.,µ versusnyandµp versusny in Fig. 9.3). Designers should be careful in using the regression relationshipsdeveloped for these cases. Note thatρ values that are very close to zero indicate that one of thedemand indices shows close to no variation as another demand index is varied (e.g.,µ versusµr,µp versusµr, andny versusµr for the BE hysteresis type sinceµr for the BE type is always zero,Fig. 9.2b).
105
Using the relationships betweenµ and the other demand indices, a more extensive perfor-mance-based seismic design procedure can be realized. For example, for a given “target-µ” as partof a displacement-based design approach, it is possible to design (or redesign) structures withenhanced objectives in mind (e.g., limiting residual displacement and/or cumulative damage).This is outlined later in Chapter 10.
Table 9.1: Regression coefficientsd andf: EP hysteresis type, IND spectra
HysteresisType
α Site SeismicityDemand
LevelSite Soil/Distance
Regression Coefficientsa
a (Eq. 3.43 repeated)
Demand Index,Λ d f
EP
α = 0.00 Los Angeles Survival SD
µp 1.41 0.72µr 0.85 1.17ny 2.59 1.23
α = 0.05 Los Angeles Survival SD
µp 1.69 0.71µr 0.71 2.16ny 2.13 0.97
α = 0.10
Boston
Survival
SD
µp 2.78 0.69µr 0.46 2.62ny 4.62 1.12
SE
µp 3.07 0.68µr 0.37 2.63ny 6.69 1.91
Design
SD
µp 2.76 0.68µr 0.51 2.91ny 5.79 1.30
SE
µp 2.87 0.70µr 0.53 3.61ny 5.50 1.42
Los Angeles
Survival
SD
µp 1.91 0.69µr 0.47 2.84ny 1.77 0.81
SE
µp 2.46 0.77µr 0.40 3.35ny 4.77 2.83
Design
SD
µp 2.30 0.71µr 0.46 2.10ny 2.44 0.88
SE
µp 2.13 0.70µr 0.45 2.98ny 4.71 2.04
NF
µp 1.70 0.71µr 0.56 3.80ny 1.39 0.85
Λ d µ 1–( )1 f⁄=
106
Table 9.2: Regression coefficientsd andf: SD, BE, and BP hysteresis types, IND spectra
HysteresisType
α, βs, βrSite
SeismicityDemand
LevelSite Soil/Distance
Regression Coefficientsa
a (Eq. 3.43 repeated)
Demand Index,Λ d f
SD α = 0.10
Boston Survival SD
µp 7.88 0.76µr 0.36 2.08ny 8.32 3.89
Los Angeles
Survival SD
µp 1.80 0.48µr 0.37 1.82ny 3.86 3.02
Design
SD
µp 3.61 0.58µr 0.28 1.30ny 5.21 4.69
SE
µp 7.16 0.88µr 0.32 1.80ny 3.20 3.44
BE α = 0.10
Boston Survival SD
µp 6.70 0.76µr 0 0ny 1.41 0.63
Los Angeles
Survival SD
µp 4.31 0.68µr 0 0ny 0.19 0.36
Design
SD
µp 5.41 0.73µr 0 0ny 0.61 0.45
SE
µp 6.18 0.84µr 0 0ny 2.59 1.00
BP
α = 0.10βs = βr = 1/6 Los Angeles Survival SD
µp 5.04 0.83µr 0.02 1.07ny 0.13 0.32
α = 0.10βs = βr = 1/3
Boston Survival SD
µp 4.49 0.72µr 0.05 2.80ny 4.56 1.09
Los Angeles
Survival SD
µp 4.71 0.86µr 0.03 1.87ny 0.20 0.35
Design
SD
µp 3.46 0.70µr 0.06 1.49ny 1.91 0.74
SE
µp 3.50 0.74µr 0.05 1.71ny 6.50 2.78
α = 0.10βs = βr = 1/2 Los Angeles Survival SD
µp 3.99 0.82µr 0.05 2.69ny 0.32 0.40
Λ d µ 1–( )1 f⁄=
107
Table 9.3: Regression coefficientsg andh: EP hysteresis type, IND spectra
HysteresisType
α Site SeismicityDemand
LevelSite Soil/Distance
Regression Coefficientsa
a (Eq. 3.44 repeated)
Λj
Λi
µp µr ny
g h g h g h
EP
α = 0.00 Los Angeles Survival SD
µp -- -- 1.56 0.57 0.26 0.34µr 0.64 1.75 -- -- 0.24 0.70ny 3.92 2.93 4.10 1.42 -- --
α = 0.05 Los Angeles Survival SD
µp -- -- 4.63 0.31 0.27 0.38µr 0.22 3.23 -- -- 0.18 1.75ny 3.73 2.65 5.63 0.57 -- --
α = 0.10
Boston
Survival
SD
µp -- -- 60.7 0.24 0.23 0.43µr 0.02 4.17 -- -- 0.04 1.89ny 4.42 2.31 26.7 0.53 -- --
SE
µp -- -- 364 0.18 0.20 0.32µr 0.003 5.56 -- -- 0.01 3.13ny 5.00 3.13 73.7 0.32 -- --
Design
SD
µp -- -- 58.2 0.21 0.22 0.43µr 0.02 4.76 -- -- 0.03 2.50ny 4.59 2.34 28.9 0.40 -- --
SE
µp -- -- 91.0 0.17 0.25 0.42µr 0.01 5.88 -- -- 0.04 2.38ny 4.05 2.36 26.4 0.42 -- --
Los Angeles
Survival
SD
µp -- -- 45.7 0.23 0.28 0.40µr 0.02 4.35 -- -- 0.05 2.63ny 3.59 2.49 21.0 0.38 -- --
SE
µp -- -- 189 0.20 0.27 0.28µr 0.005 5.00 -- -- 0.06 1.49ny 3.68 3.63 17.0 0.67 -- --
Design
SD
µp -- -- 24.1 0.30 0.25 0.40µr 0.04 3.33 -- -- 0.06 2.38ny 4.06 2.51 16.9 0.42 -- --
SE
µp -- -- 90.7 0.19 0.26 0.33µr 0.01 5.26 -- -- 0.06 1.85ny 3.79 3.04 17.5 0.54 -- --
NF
µp -- -- 38.6 0.18 0.40 0.45µr 0.03 5.56 -- -- 21.9 0.21ny 2.53 2.23 0.05 4.76 -- --
Λ j gΛi1 h⁄
=
108
Table 9.4: Regression coefficientsg andh: SD, BE, and BP hysteresis types, IND spectra
HysteresisType
α, βs, βrSite
SeismicityDemand
LevelSite Soil/Distance
Regression Coefficientsa
a (Eq. 3.44 repeated)
Λj
Λi
µp µr ny
g h g h g h
SD α = 0.10
Boston Survival SD
µp -- -- 170 0.32 0.29 0.31µr 0.006 3.13 -- -- 0.06 0.96ny 3.44 3.19 17.1 1.04 -- --
Los Angeles
Survival SD
µp -- -- 67.8 0.38 1.85 0.66µr 0.01 2.63 -- -- 0.11 1.82ny 0.54 1.51 9.17 0.55 -- --
Design
SD
µp -- -- 79.8 0.33 1.23 0.57µr 0.01 3.03 -- -- 0.09 2.38ny 0.81 1.75 11.3 0.42 -- --
SE
µp -- -- 74.6 0.53 1.39 0.53µr 0.01 1.89 -- -- 0.12 1.45ny 0.72 1.89 8.43 0.69 -- --
BE α = 0.10
Boston Survival SD
µp -- -- 0 0 0.23 0.34µr 0 0 -- -- 0 0ny 4.33 2.95 0 0 -- --
Los Angeles
Survival SD
µp -- -- 0 0 1.18 0.74µr 0 0 -- -- 0 0ny 0.85 1.35 0 0 -- --
Design
SD
µp -- -- 0 0 0.37 0.46µr 0 0 -- -- 0 0ny 2.73 2.19 0 0 -- --
SE
µp -- -- 0 0 0.30 0.34µr 0 0 -- -- 0 0ny 3.35 2.91 0 0 -- --
BP
α = 0.10βs = βr = 1/6 Los Angeles Survival SD
µp -- -- 307 1.25 0.39 0.49µr 0.001 0.80 -- -- 0.004 1.04ny 2.54 2.03 272 0.96 -- --
α = 0.10βs = βr = 1/3
Boston Survival SD
µp -- -- 446 1.12 0.18 0.33µr 0.002 0.89 -- -- 0.004 0.99ny 5.43 3.01 227 1.01 -- --
Los Angeles
Survival SD
µp -- -- 297 1.19 0.32 0.43µr 0.003 0.84 -- -- 0.004 1.06ny 3.17 2.35 250 0.94 -- --
Design
SD
µp -- -- 578 0.67 0.23 0.37µr 0.002 1.49 -- -- 0.003 1.64ny 4.27 2.73 376 0.61 -- --
SE
µp -- -- 433 0.79 0.23 0.31µr 0.002 1.27 -- -- 0.006 1.27ny 4.27 3.26 161 0.79 -- --
α = 0.10βs = βr = 1/2 Los Angeles Survival SD
µp -- -- 332 1.05 0.29 0.40µr 0.003 0.95 -- -- 0.004 1.15ny 3.49 2.53 263 0.87 -- --
Λ j gΛi1 h⁄
=
109
Table 9.5: Correlation coefficient,ρ: EP hysteresis type, IND spectra
HysteresisType
α Site SeismicityDemand
LevelSite Soil/Distance
Correlation Coefficient,ρ
µ µp µr ny
EP
α = 0.00 Los Angeles Survival SD
µ 1.00 0.99 0.97 0.90µp 0.99 1.00 0.96 0.89µr 0.97 0.96 1.00 0.95ny 0.90 0.89 0.95 1.00
α = 0.05 Los Angeles Survival SD
µ 1.00 0.99 0.79 0.86µp 0.99 1.00 0.77 0.84µr 0.79 0.77 1.00 0.70ny 0.86 0.84 0.70 1.00
α = 0.10
Boston
Survival
SD
µ 1.00 0.99 0.75 0.92µp 0.99 1.00 0.70 0.95µr 0.75 0.70 1.00 0.67ny 0.92 0.95 0.67 1.00
SE
µ 1.00 0.97 0.48 0.90µp 0.97 1.00 0.36 0.95µr 0.48 0.36 1.00 0.25ny 0.90 0.95 0.25 1.00
Design
SD
µ 1.00 0.99 0.73 0.93µp 0.99 1.00 0.68 0.96µr 0.73 0.68 1.00 0.61ny 0.93 0.96 0.61 1.00
SE
µ 1.00 0.99 0.68 0.96µp 0.99 1.00 0.65 0.98µr 0.68 0.65 1.00 0.65ny 0.96 0.98 0.65 1.00
Los Angeles
Survival
SD
µ 1.00 0.99 0.67 0.78µp 0.99 1.00 0.78 0.81µr 0.67 0.64 1.00 0.48ny 0.78 0.81 0.48 1.00
SE
µ 1.00 0.99 0.56 0.94µp 0.99 1.00 0.51 0.94µr 0.56 0.51 1.00 0.47ny 0.94 0.94 0.47 1.00
Design
SD
µ 1.00 0.98 0.72 0.78µp 0.98 1.00 0.78 0.86µr 0.72 0.67 1.00 0.54ny 0.78 0.86 0.54 1.00
SE
µ 1.00 0.99 0.99 0.96µp 0.99 1.00 0.49 0.98µr 0.58 0.49 1.00 0.46ny 0.96 0.98 0.46 1.00
NF
µ 1.00 0.99 0.62 0.80µp 0.99 1.00 0.59 0.86µr 0.62 0.59 1.00 0.33ny 0.80 0.86 0.33 1.00
110
Table 9.6: Correlation coefficient,ρ: SD, BE, and BP hysteresis types, IND spectra
HysteresisType
α, βs, βrSite
SeismicityDemand
LevelSite Soil/Distance
Correlation Coefficient,ρ
µ µp µr ny
SD α = 0.10
Boston Survival SD
µ 1.00 0.85 0.86 0.84µp 0.85 1.00 0.64 0.93µr 0.86 0.64 1.00 0.72ny 0.84 0.93 0.72 1.00
Los Angeles
Survival SD
µ 1.00 0.78 0.94 0.62µp 0.78 1.00 0.80 0.93µr 0.94 0.80 1.00 0.71ny 0.62 0.93 0.71 1.00
Design
SD
µ 1.00 0.73 0.89 0.53µp 0.73 1.00 0.53 0.87µr 0.89 0.53 1.00 0.47ny 0.53 0.87 0.47 1.00
SE
µ 1.00 0.92 0.90 0.58µp 0.92 1.00 0.80 0.77µr 0.90 0.80 1.00 0.64ny 0.58 0.77 0.64 1.00
BE α = 0.10
Boston Survival SD
µ 1.00 0.96 0.00 0.60µp 0.96 1.00 0.00 0.79µr 0.00 0.00 1.00 0.00ny 0.60 0.79 0.00 1.00
Los Angeles
Survival SD
µ 1.00 0.92 0.00 0.50µp 0.92 1.00 0.00 0.78µr 0.00 0.00 1.00 0.00ny 0.50 0.78 0.00 1.00
Design
SD
µ 1.00 0.92 0.00 0.44µp 0.92 1.00 0.00 0.74µr 0.00 0.00 1.00 0.00ny 0.44 0.74 0.00 1.00
SE
µ 1.00 0.95 0.00 0.64µp 0.95 1.00 0.00 0.82µr 0.00 0.00 1.00 0.00ny 0.64 0.82 0.00 1.00
BP
α = 0.10βs = βr = 1/6 Los Angeles Survival SD
µ 1.00 0.94 0.69 0.45µp 0.94 1.00 0.54 0.69µr 0.69 0.54 1.00 0.01ny 0.45 0.69 0.01 1.00
α = 0.10βs = βr = 1/3
Boston Survival SD
µ 1.00 0.98 0.52 0.79µp 0.98 1.00 0.44 0.88µr 0.52 0.44 1.00 0.34ny 0.79 0.88 0.34 1.00
Los Angeles
Survival SD
µ 1.00 0.96 0.62 0.47µp 0.96 1.00 0.51 0.68µr 0.62 0.51 1.00 0.10ny 0.47 0.68 0.10 1.00
Design
SD
µ 1.00 0.97 0.49 0.61µp 0.97 1.00 0.34 0.78µr 0.49 0.34 1.00 0.04ny 0.61 0.78 0.04 1.00
SE
µ 1.00 0.98 0.54 0.90µp 0.98 1.00 0.45 0.91µr 0.54 0.45 1.00 0.26ny 0.90 0.91 0.26 1.00
α = 0.10βs = βr = 1/2 Los Angeles Survival SD
µ 1.00 0.97 0.53 0.51µp 0.97 1.00 0.42 0.69µr 0.53 0.42 1.00 0.03ny 0.51 0.69 0.03 1.00
111
ρ = 0.99
ρ = 0.67
ρ = 0.78
ρ = 0.64
ρ = 0.81 ρ = 0.48
ρ = 0.99 ρ = 0.67 ρ = 0.78
ρ = 0.81ρ = 0.64
ρ = 0.48
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 10 0 250
50
µp
0
50
0 40
50d = 1.91, f = 0.69
d =0.47, f = 2.84
d = 1.77, f = 0.81
d = 1.91, f = 0.69 d = 0.47, f = 2.84 d = 1.77, f = 0.81
g = 0.02, h = 4.35
g = 45.7, h = 0.23 g = 0.28, h = 0.40
g = 3.59, h = 2.49
g = 0.05, h = 2.63
g = 21.0, h = 0.38
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 100
50
0 40
50
µp
0 250
50
dataregression
d = 1.41, f = 0.72
d = 0.85, f = 1.17
d = 2.59, f = 1.23
d = 1.41, f = 0.72 d = 0.85, f = 1.17 d = 2.59, f = 1.23
g = 0.64, h = 1.75
g = 1.56, h = 0.57 g = 0.26, h = 0.34
g = 3.92, h = 2.93
g = 0.24, h = 0.70
g = 4.10, h = 1.42
ρ = 0.99
ρ = 0.97
ρ = 0.90
ρ = 0.96
ρ = 0.89 ρ = 0.95
ρ = 0.99 ρ = 0.97 ρ = 0.90
ρ = 0.89ρ = 0.96
ρ = 0.95
Figure 9.1: Matrix plots of cross-correlations: effect of post-yield stiffness ratio,α (EPhysteresis type, SAC Los Angeles, survival-level,SD soil): (a)α = 0.00; (b)α = 0.05; (c)α = 0.10.
(a)
(c)
ρ = 0.99
ρ = 0.79
ρ = 0.86
ρ = 0.77
ρ = 0.84 ρ = 0.70
ρ = 0.99 ρ = 0.79 ρ = 0.86
ρ = 0.84ρ = 0.77
ρ = 0.70
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 1.69, f = 0.71
d = 0.71, f = 2.16
d = 2.13, f = 0.97
d = 1.69, f = 0.71 d = 0.71, f = 2.16 d = 2.13, f = 0.97
g = 0.22, h = 3.23
g = 4.63, h = 0.31 g = 0.27, h = 0.38
g = 3.73, h = 2.65
g = 0.18, h = 1.75
g = 5.63, h = 0.57
(b)
112
ρ = 0.92
ρ = 0.00
ρ = 0.50
ρ = 0.00
ρ = 0.78 ρ = 0.00
ρ = 0.92 ρ = 0.00 ρ = 0.50
ρ = 0.78ρ = 0.00
ρ = 0.00
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 4.31, f = 0.68
d =0.00, f = 0.00
d = 0.19, f = 0.36
d = 4.31, f = 0.68 d = 0.00, f = 0.00 d = 0.19, f = 0.36
g = 0.00, h = 0.00
g = 0.00, h = 0.00 g = 1.18, h = 0.74
g = 0.85, h = 1.35
g = 0.00, h = 0.00
g = 0.00, h = 0.00
ρ = 0.78
ρ = 0.94
ρ = 0.62
ρ = 0.80
ρ = 0.93 ρ = 0.71
ρ = 0.78 ρ = 0.94 ρ = 0.62
ρ = 0.93ρ = 0.80
ρ = 0.71
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µp
0 250
4
0 500
4
0 500
25
0 100
50
µr
0 250
50
0 40
50d = 1.80, f = 0.48
d =0.37, f = 1.82
d = 3.86, f = 3.02
d = 1.80, f = 0.48 d = 0.37, f = 1.82 d = 3.86, f = 3.02
g = 0.01, h = 2.63
g = 67.8, h = 0.38 g = 1.85, h = 0.66
g = 0.54, h = 1.51
g = 0.11, h = 1.82
g = 9.17, h = 0.55
Figure 9.2: Matrix plots of cross-correlations: effect of hysteresis type (α = 0.10, SAC LosAngeles, survival-level,SD soil): (a) SD hysteresis type; (b) BE hysteresis type.
(a) (b)
113
ρ = 0.96
ρ = 0.62
ρ = 0.47
ρ = 0.51
ρ = 0.68 ρ = 0.10
ρ = 0.96 ρ = 0.62 ρ = 0.47
ρ = 0.68ρ = 0.51
ρ = 0.10
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 4.71, f = 0.86
d =0.03, f = 1.87
d = 0.20, f = 0.35
d = 4.71, f = 0.86 d = 0.03, f = 1.87 d = 0.20, f = 0.35
g = 0.003h = 0.84
g = 297, h = 1.19 g = 0.32, h = 0.43
g = 3.17, h = 2.35
g = 0.004h = 1.06
g = 250, h = 0.94
ρ = 0.97
ρ = 0.53
ρ = 0.51
ρ = 0.42
ρ = 0.69 ρ = 0.03
ρ = 0.97 ρ = 0.53 ρ = 0.51
ρ = 0.69ρ = 0.42
ρ = 0.03
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 3.99, f = 0.82
d = 0.05, f = 2.69
d = 0.32, f = 0.40
d = 3.99, f = 0.82 d = 0.05, f = 2.69 d = 0.32, f = 0.40
g = 0.003h = 0.95
g = 332, h = 1.05 g = 0.29, h = 0.40
g = 3.49, h = 2.53
g = 0.004h = 1.15
g = 263, h = 0.87
ρ = 0.94
ρ = 0.69
ρ = 0.45
ρ = 0.54
ρ = 0.69 ρ = 0.01
ρ = 0.94 ρ = 0.69 ρ = 0.45
ρ = 0.69ρ = 0.54
ρ = 0.01
0 500
4
0 40
50
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
25
0 100
50
µp
0 250
50d = 5.04, f = 0.83
d =0.02, f = 1.07
d = 0.13, f = 0.32
d = 5.04, f = 0.83 d = 0.02, f = 1.07 d = 0.13, f = 0.32
g = 0.01h = 0.80
g = 307, h = 1.25 g = 0.63, h = 0.49
g = 2.54, h = 2.03
g = 0.003h = 1.04
g = 272, h = 0.96
Figure 9.3: Matrix plots of cross-correlations: effect ofβs = βr (BP hysteresis type,α = 0.10,SAC Los Angeles, survival-level,SD soil): (a)βs = βr = 1/6; (b)βs = βr = 1/3; (c)βs = βr = 1/2.
(a) (b)
(c)
114
ρ = 0.99
ρ = 0.58
ρ = 0.96
ρ = 0.49
ρ = 0.98 ρ = 0.46
ρ = 0.99 ρ = 0.58 ρ = 0.96
ρ = 0.98ρ = 0.49
ρ = 0.46
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 2.13, f = 0.70
d =0.45, f = 2.98
d = 4.71, f = 2.04
d = 2.13, f = 0.70 d = 0.45, f = 2.98 d = 4.71, f = 2.04
g = 0.01, h = 5.26
g = 90.7, h = 0.19 g = 0.26, h = 0.33
g = 3.79, h = 3.04
g = 0.06, h = 1.85
g = 17.5, h = 0.54
ρ = 0.98
ρ = 0.72
ρ = 0.78
ρ = 0.67
ρ = 0.86 ρ = 0.54
ρ = 0.98 ρ = 0.72 ρ = 0.78
ρ = 0.86ρ = 0.67
ρ = 0.54
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 2.30, f = 0.71
d =0.46, f = 2.10
d = 2.44, f = 0.88
d = 2.30, f = 0.71 d = 0.46, f = 2.10 d = 2.44, f = 0.88
g = 0.04, h = 3.33
g = 24.1, h = 0.30 g = 0.25, h = 0.40
g = 4.06, h = 2.51
g = 0.06, h = 2.38
g = 16.9, h = 0.42
Figure 9.4: Matrix plots of cross-correlations: effect of site soil characteristics (EP hystere-sis type,α = 0.10, SAC Los Angeles, design-level): (a)SD soil; (b)SE soil.
(a) (b)
115
ρ = 0.73
ρ = 0.89
ρ = 0.53
ρ = 0.53
ρ = 0.87 ρ = 0.47
ρ = 0.73 ρ = 0.89 ρ = 0.53
ρ = 0.87ρ = 0.53
ρ = 0.47
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 3.61, f = 0.58
d =0.28, f = 1.30
d = 5.21, f = 4.69
d = 3.61, f = 0.58 d = 0.28, f = 1.30 d = 5.21, f = 4.69
g = 0.01, h = 3.03
g = 79.8, h = 0.33 g = 1.23, h = 0.57
g = 0.81, h = 1.75
g = 0.09, h = 2.38
g = 11.3, h = 0.42
ρ = 0.92
ρ = 0.90
ρ = 0.58
ρ = 0.80
ρ = 0.77 ρ = 0.64
ρ = 0.92 ρ = 0.90 ρ = 0.58
ρ = 0.77ρ = 0.80
ρ = 0.64
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 7.16, f = 0.88
d =0.32, f = 1.80
d = 3.20, f = 3.44
d = 7.16, f = 0.88 d = 0.32, f = 1.80 d = 3.20, f = 3.44
g = 0.01, h = 1.89
g = 74.6, h = 0.53 g = 1.39, h = 0.53
g = 0.72, h = 1.89
g = 0.12, h =1.45
g = 8.43, h = 0.69
Figure 9.5: Matrix plots of cross-correlations: effect of site soil characteristics (SD hystere-sis type,α = 0.10, SAC Los Angeles, design-level): (a)SD soil; (b)SE soil.
(a) (b)
ρ = 0.95
ρ = 0.00
ρ = 0.64
ρ = 0.00
ρ = 0.82 ρ = 0.00
ρ = 0.95 ρ = 0.00 ρ = 0.64
ρ = 0.82ρ = 0.00
ρ = 0.00
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 6.18, f = 0.84
d =0.00, f = 0.00
d = 2.59, f = 1.00
d = 6.18, f = 0.84 d = 0.00, f = 0.00 d = 2.59, f = 1.00
g = 0.00, h = 0.00
g = 0.00, h = 0.00 g = 0.30, h = 0.34
g = 3.35, h = 2.91
g = 0.00, h = 0.00
g = 0.00, h = 0.00
ρ = 0.92
ρ = 0.00
ρ = 0.44
ρ = 0.00
ρ = 0.74 ρ = 0.00
ρ = 0.92 ρ = 0.00 ρ = 0.44
ρ = 0.74ρ = 0.00
ρ = 0.00
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 5.41, f = 0.73
d =0.00, f = 0.00
d = 0.61, f = 0.45
d = 5.41, f = 0.73 d = 0.00, f = 0.00 d = 0.63, f = 0.45
g = 0.00, h = 0.00
g = 0.00, h = 0.00 g = 0.37, h = 0.46
g = 2.73, h = 2.19
g = 0.00, h = 0.00
g = 0.00, h = 0.00
Figure 9.6: Matrix plots of cross-correlations: effect of site soil characteristics (BE hystere-sis type,α = 0.10, SAC Los Angeles, design-level): (a)SD soil; (b)SE soil.
(a) (b)
116
ρ = 0.97
ρ = 0.49
ρ = 0.61
ρ = 0.34
ρ = 0.78 ρ = 0.04
ρ = 0.97 ρ = 0.49 ρ = 0.61
ρ = 0.78ρ = 0.34
ρ = 0.04
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d =3.46, f = 0.70
d =0.06, f = 1.49
d =1.91, f = 0.74
d = 3.46, f = 0.70 d = 0.06, f = 1.49 d = 1.91, f = 0.74
g = 0.002h = 1.49
g = 578, h = 0.67 g = 0.23, h = 0.37
g = 4.27, h = 2.73
g = 0.003h = 1.64
g = 376, h = 0.61
ρ = 0.98
ρ = 0.54
ρ = 0.90
ρ = 0.45
ρ = 0.91 ρ = 0.26
ρ = 0.98 ρ = 0.54 ρ = 0.90
ρ = 0.91ρ = 0.45
ρ = 0.26
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d =3.50, f = 0.74
d =0.05, f = 1.71
d =6.50, f = 2.78
d = 3.50, f = 0.74 d = 0.05, f = 1.71 d = 6.50, f = 2.78
g = 0.00h =1.27
g = 433, h = 0.79 g = 0.64, h = 0.31
g = 4.27, h = 3.26
g = 0.002h = 1.27
g = 161, h = 0.79
Figure 9.7: Matrix plots of cross-correlations: effect of site soil characteristics (BP hystere-sis type,α = 0.10,βs = βr = 1/3, SAC Los Angeles, design-level): (a)SD soil; (b)SE soil.
(a) (b)
ρ = 0.98
ρ = 0.72
ρ = 0.78
ρ = 0.67
ρ = 0.86 ρ = 0.54
ρ = 0.98 ρ = 0.72 ρ = 0.78
ρ = 0.86ρ = 0.67
ρ = 0.54
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 2.30, f = 0.71
d =0.46, f = 2.10
d = 2.44, f = 0.88
d = 2.30, f = 0.71 d = 0.46, f = 2.10 d = 2.44, f = 0.88
g = 0.04, h = 3.33
g = 24.1, h = 0.30 g = 0.25, h = 0.40
g = 4.06, h = 2.51
g = 0.06, h = 2.38
g = 16.9, h = 0.42
ρ = 0.99
ρ = 0.67
ρ = 0.78
ρ = 0.64
ρ = 0.81 ρ = 0.48
ρ = 0.99 ρ = 0.67 ρ = 0.78
ρ = 0.81ρ = 0.64
ρ = 0.48
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 10 0 250
50
µp
0
50
0 40
50d = 1.91, f = 0.69
d =0.47, f = 2.84
d = 1.77, f = 0.81
d = 1.91, f = 0.69 d = 0.47, f = 2.84 d = 1.77, f = 0.81
g = 0.02, h = 4.35
g = 45.7, h = 0.23 g = 0.28, h = 0.40
g = 3.59, h = 2.49
g = 0.05, h = 2.63
g = 21.0, h = 0.38
Figure 9.8: Matrix plots of cross-correlations: effect of seismic demand level (EP hysteresistype,α = 0.10, SAC Los Angeles,SD soil): (a) survival-level; (b) design-level.
(a) (b)
117
ρ = 0.99
ρ = 0.58
ρ = 0.96
ρ = 0.49
ρ = 0.98 ρ = 0.46
ρ = 0.99 ρ = 0.58 ρ = 0.96
ρ = 0.98ρ = 0.49
ρ = 0.46
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 2.13, f = 0.70
d =0.45, f = 2.98
d = 4.71, f = 2.04
d = 2.13, f = 0.70 d = 0.45, f = 2.98 d = 4.71, f = 2.04
g = 0.01, h = 5.26
g = 90.7, h = 0.19 g = 0.26, h = 0.33
g = 3.79, h = 3.04
g = 0.06, h = 1.85
g = 17.5, h = 0.54
ρ = 0.99
ρ = 0.56
ρ = 0.94
ρ = 0.51
ρ = 0.94 ρ = 0.47
ρ = 0.99 ρ = 0.56 ρ = 0.94
ρ = 0.94ρ = 0.51
ρ = 0.47
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 2.46, f = 0.77
d =0.40, f = 3.35
d = 4.77, f = 2.83
d = 2.46, f = 0.77 d = 0.40, f = 3.35 d = 4.77, f = 2.83
g = 0.005h = 5.00
g = 189, h = 0.20 g = 0.27, h = 0.28
g = 3.68, h = 3.63
g = 0.06, h = 1.49
g = 17.0, h = 0.67
Figure 9.9: Matrix plots of cross-correlations: effect of seismic demand level (EP hysteresistype,α = 0.10, SAC Los Angeles,SE soil): (a) survival-level; (b) design-level.
(a) (b)
ρ = 0.73
ρ = 0.89
ρ = 0.53
ρ = 0.53
ρ = 0.87 ρ = 0.47
ρ = 0.73 ρ = 0.89 ρ = 0.53
ρ = 0.87ρ = 0.53
ρ = 0.47
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 3.61, f = 0.58
d =0.28, f = 1.30
d = 5.21, f = 4.69
d = 3.61, f = 0.58 d = 0.28, f = 1.30 d = 5.21, f = 4.69
g = 0.01, h = 3.03
g = 79.8, h = 0.33 g = 1.23, h = 0.57
g = 0.81, h = 1.75
g = 0.09, h = 2.38
g = 11.3, h = 0.42
ρ = 0.78
ρ = 0.94
ρ = 0.62
ρ = 0.80
ρ = 0.93 ρ = 0.71
ρ = 0.78 ρ = 0.94 ρ = 0.62
ρ = 0.93ρ = 0.80
ρ = 0.71
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µp
0 250
4
0 500
4
0 500
25
0 100
50
µr
0 250
50
0 40
50d = 1.80, f = 0.48
d =0.37, f = 1.82
d = 3.86, f = 3.02
d = 1.80, f = 0.48 d = 0.37, f = 1.82 d = 3.86, f = 3.02
g = 0.01, h = 2.63
g = 67.8, h = 0.38 g = 1.85, h = 0.66
g = 0.54, h = 1.51
g = 0.11, h = 1.82
g = 9.17, h = 0.55
Figure 9.10: Matrix plots of cross-correlations: effect of seismic demand level (SD hystere-sis type,α = 0.10, SAC Los Angeles,SD soil): (a) survival-level; (b) design-level.
(a) (b)
118
ρ = 0.92
ρ = 0.00
ρ = 0.44
ρ = 0.00
ρ = 0.74 ρ = 0.00
ρ = 0.92 ρ = 0.00 ρ = 0.44
ρ = 0.74ρ = 0.00
ρ = 0.00
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 5.41, f = 0.73
d =0.00, f = 0.00
d = 0.61, f = 0.45
d = 5.41, f = 0.73 d = 0.00, f = 0.00 d = 0.63, f = 0.45
g = 0.00, h = 0.00
g = 0.00, h = 0.00 g = 0.37, h = 0.46
g = 2.73, h = 2.19
g = 0.00, h = 0.00
g = 0.00, h = 0.00
ρ = 0.92
ρ = 0.00
ρ = 0.50
ρ = 0.00
ρ = 0.78 ρ = 0.00
ρ = 0.92 ρ = 0.00 ρ = 0.50
ρ = 0.78ρ = 0.00
ρ = 0.00
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 4.31, f = 0.68
d =0.00, f = 0.00
d = 0.19, f = 0.36
d = 4.31, f = 0.68 d = 0.00, f = 0.00 d = 0.19, f = 0.36
g = 0.00, h = 0.00
g = 0.00, h = 0.00 g = 1.18, h = 0.74
g = 0.85, h = 1.35
g = 0.00, h = 0.00
g = 0.00, h = 0.00
Figure 9.11: Matrix plots of cross-correlations: effect of seismic demand level (BE hystere-sis type,α = 0.10, SAC Los Angeles,SD soil): (a) survival-level; (b) design-level.
(a) (b)
119
ρ = 0.97
ρ = 0.49
ρ = 0.61
ρ = 0.34
ρ = 0.78 ρ = 0.04
ρ = 0.97 ρ = 0.49 ρ = 0.61
ρ = 0.78ρ = 0.34
ρ = 0.04
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d =3.46, f = 0.70
d =0.06, f = 1.49
d =1.91, f = 0.74
d = 3.46, f = 0.70 d = 0.06, f = 1.49 d = 1.91, f = 0.74
g = 0.002h = 1.49
g = 578, h = 0.67 g = 0.23, h = 0.37
g = 4.27, h = 2.73
g = 0.003h = 1.64
g = 376, h = 0.61
ρ = 0.96
ρ = 0.62
ρ = 0.47
ρ = 0.51
ρ = 0.68 ρ = 0.10
ρ = 0.96 ρ = 0.62 ρ = 0.47
ρ = 0.68ρ = 0.51
ρ = 0.10
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 4.71, f = 0.86
d =0.03, f = 1.87
d = 0.20, f = 0.35
d = 4.71, f = 0.86 d = 0.03, f = 1.87 d = 0.20, f = 0.35
g = 0.003h = 0.84
g = 297, h = 1.19 g = 0.32, h = 0.43
g = 3.17, h = 2.35
g = 0.004h = 1.06
g = 250, h = 0.94
Figure 9.12: Matrix plots of cross-correlations: effect of seismic demand level (BP hystere-sis type,α = 0.10,βs = βr = 1/3, SAC Los Angeles,SD soil): (a) survival-level; (b) design-level.
(a) (b)
ρ = 0.99
ρ = 0.75
ρ = 0.92
ρ = 0.70
ρ = 0.95 ρ = 0.67
ρ = 0.99 ρ = 0.75 ρ = 0.92
ρ = 0.95ρ = 0.70
ρ = 0.67
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 2.78, f = 0.69
d = 0.46, f = 2.62
d = 4.62, f = 1.12
d = 2.78, f = 0.69 d = 0.46, f = 2.62 d = 4.62, f = 1.12
g = 0.02, h = 4.17
g = 60.7, h = 0.42 g = 0.23, h = 0.43
g = 4.42, h = 2.31
g = 0.04, h = 1.89
g = 26.7, h = 0.53
ρ = 0.99
ρ = 0.67
ρ = 0.78
ρ = 0.64
ρ = 0.81 ρ = 0.48
ρ = 0.99 ρ = 0.67 ρ = 0.78
ρ = 0.81ρ = 0.64
ρ = 0.48
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 10 0 250
50
µp
0
50
0 40
50d = 1.91, f = 0.69
d =0.47, f = 2.84
d = 1.77, f = 0.81
d = 1.91, f = 0.69 d = 0.47, f = 2.84 d = 1.77, f = 0.81
g = 0.02, h = 4.35
g = 45.7, h = 0.23 g = 0.28, h = 0.40
g = 3.59, h = 2.49
g = 0.05, h = 2.63
g = 21.0, h = 0.38
Figure 9.13: Matrix plots of cross-correlations: effect of site seismicity (EP hysteresis type,α = 0.10, SAC survival-level,SD soil): (a) Los Angeles; (b) Boston.
(a) (b)
120
ρ = 0.99
ρ = 0.56
ρ = 0.94
ρ = 0.51
ρ = 0.94 ρ = 0.47
ρ = 0.99 ρ = 0.56 ρ = 0.94
ρ = 0.94ρ = 0.51
ρ = 0.47
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 2.46, f = 0.77
d =0.40, f = 3.35
d = 4.77, f = 2.83
d = 2.46, f = 0.77 d = 0.40, f = 3.35 d = 4.77, f = 2.83
g = 0.005h = 5.00
g = 189, h = 0.20 g = 0.27, h = 0.28
g = 3.68, h = 3.63
g = 0.06, h = 1.49
g = 17.0, h = 0.67
ρ = 0.97
ρ = 0.48
ρ = 0.90
ρ = 0.36
ρ = 0.95 ρ = 0.25
ρ = 0.97 ρ = 0.48 ρ = 0.90
ρ = 0.95ρ = 0.36
ρ = 0.25
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 3.07, f = 0.68
d =0.37, f = 2.63
d = 6.69, f = 1.91
d = 3.07, f = 0.68 d = 0.37, f = 2.63 d = 6.69, f = 1.91
g = 0.003h = 5.56
g = 3.64, h = 0.18 g = 0.20, h = 0.32
g = 5.00, h = 3.13
g = 0.01, h = 3.13
g = 73.7, h = 0.32
Figure 9.14: Matrix plots of cross-correlations: effect of site seismicity (EP hysteresis type,α = 0.10, SAC survival-level,SE soil): (a) Los Angeles; (b) Boston.
(a) (b)
ρ = 0.98
ρ = 0.72
ρ = 0.78
ρ = 0.67
ρ = 0.86 ρ = 0.54
ρ = 0.98 ρ = 0.72 ρ = 0.78
ρ = 0.86ρ = 0.67
ρ = 0.54
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 2.30, f = 0.71
d =0.46, f = 2.10
d = 2.44, f = 0.88
d = 2.30, f = 0.71 d = 0.46, f = 2.10 d = 2.44, f = 0.88
g = 0.04, h = 3.33
g = 24.1, h = 0.30 g = 0.25, h = 0.40
g = 4.06, h = 2.51
g = 0.06, h = 2.38
g = 16.9, h = 0.42
ρ = 0.99
ρ = 0.73
ρ = 0.93
ρ = 0.68
ρ = 0.96 ρ = 0.61
ρ = 0.99 ρ = 0.73 ρ = 0.93
ρ = 0.96ρ = 0.68
ρ = 0.61
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 2.76, f = 0.68
d =0.51, f = 2.91
d = 5.79, f = 1.301
d = 2.76, f = 0.68 d = 0.51, f = 2.91 d = 5.79, f = 1.30
g = 0.02, h = 4.76
g = 58.2, h = 0.21 g = 0.22, h = 0.43
g = 4.59, h = 2.34
g = 0.03, h = 2.50
g = 28.9, h = 0.40
Figure 9.15: Matrix plots of cross-correlations: effect of site seismicity (EP hysteresis type,α = 0.10, SAC design-level,SD soil): (a) Los Angeles; (b) Boston.
(a) (b)
121
ρ = 0.99
ρ = 0.58
ρ = 0.96
ρ = 0.49
ρ = 0.98 ρ = 0.46
ρ = 0.99 ρ = 0.58 ρ = 0.96
ρ = 0.98ρ = 0.49
ρ = 0.46
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 2.13, f = 0.70
d =0.45, f = 2.98
d = 4.71, f = 2.04
d = 2.13, f = 0.70 d = 0.45, f = 2.98 d = 4.71, f = 2.04
g = 0.01, h = 5.26
g = 90.7, h = 0.19 g = 0.26, h = 0.33
g = 3.79, h = 3.04
g = 0.06, h = 1.85
g = 17.5, h = 0.54
ρ = 0.99
ρ = 0.68
ρ = 0.96
ρ = 0.65
ρ = 0.98 ρ = 0.65
ρ = 0.99 ρ = 0.68 ρ = 0.96
ρ = 0.98ρ = 0.65
ρ = 0.65
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 2.87, f = 0.70
d =0.53, f = 3.61
d = 5.50, f = 1.42
d = 2.87, f = 0.70 d = 0.53, f = 3.61 d = 5.50, f = 1.42
g = 0.01, h = 5.88
g = 91.0, h = 0.17 g = 0.25, h = 0.42
g = 4.05, h = 2.36
g = 0.04, h =2.38
g = 26.4, h = 0.42
Figure 9.16: Matrix plots of cross-correlations: effect of site seismicity (EP hysteresis type,α = 0.10, SAC design-level,SE soil): (a) Los Angeles; (b) Boston.
(a) (b)
ρ = 0.78
ρ = 0.94
ρ = 0.62
ρ = 0.80
ρ = 0.93 ρ = 0.71
ρ = 0.78 ρ = 0.94 ρ = 0.62
ρ = 0.93ρ = 0.80
ρ = 0.71
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µp
0 250
4
0 500
4
0 500
25
0 100
50
µr
0 250
50
0 40
50d = 1.80, f = 0.48
d =0.37, f = 1.82
d = 3.86, f = 3.02
d = 1.80, f = 0.48 d = 0.37, f = 1.82 d = 3.86, f = 3.02
g = 0.01, h = 2.63
g = 67.8, h = 0.38 g = 1.85, h = 0.66
g = 0.54, h = 1.51
g = 0.11, h = 1.82
g = 9.17, h = 0.55
ρ = 0.85
ρ = 0.86
ρ = 0.84
ρ = 0.64
ρ = 0.93 ρ = 0.72
ρ = 0.85 ρ = 0.86 ρ = 0.84
ρ = 0.93ρ = 0.64
ρ = 0.72
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 7.88, f = 0.76
d =0.36, f = 2.08
d = 8.32, f = 3.89
d = 7.88, f = 0.76 d = 0.36, f = 2.08 d = 8.32, f = 3.89
g = 0.006h = 3.13
g = 170, h = 0.32 g = 0.29, h = 0.31
g = 3.44, h = 3.19
g = 0.06, h = 0.96
g = 17.1, h = 1.04
Figure 9.17: Matrix plots of cross-correlations: effect of site seismicity (SD hysteresis type,α = 0.10, SAC survival-level,SD soil): (a) Los Angeles; (b) Boston.
(a) (b)
122
ρ = 0.92
ρ = 0.00
ρ = 0.50
ρ = 0.00
ρ = 0.78 ρ = 0.00
ρ = 0.92 ρ = 0.00 ρ = 0.50
ρ = 0.78ρ = 0.00
ρ = 0.00
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 4.31, f = 0.68
d =0.00, f = 0.00
d = 0.19, f = 0.36
d = 4.31, f = 0.68 d = 0.00, f = 0.00 d = 0.19, f = 0.36
g = 0.00, h = 0.00
g = 0.00, h = 0.00 g = 1.18, h = 0.74
g = 0.85, h = 1.35
g = 0.00, h = 0.00
g = 0.00, h = 0.00
ρ = 0.96
ρ = 0.00
ρ = 0.60
ρ = 0.00
ρ = 0.79 ρ = 0.00
ρ = 0.96 ρ = 0.00 ρ = 0.60
ρ = 0.79ρ = 0.00
ρ = 0.00
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 6.70, f = 0.76
d =0.00, f = 0.00
d = 1.41, f = 0.63
d = 6.70, f = 0.76 d = 0.00, f = 0.00 d = 1.41, f = 0.63
g = 0.00, h = 0.00
g = 0.00, h = 0.00 g = 0.23, h = 0.34
g = 4.33, h = 2.95
g = 0.00, h = 0.00
g = 0.00, h = 0.00
Figure 9.18: Matrix plots of cross-correlations: effect of site seismicity (BE hysteresis type,α = 0.10, SAC survival-level,SD soil): (a) Los Angeles; (b) Boston.
(a) (b)
ρ = 0.96
ρ = 0.62
ρ = 0.47
ρ = 0.51
ρ = 0.68 ρ = 0.10
ρ = 0.96 ρ = 0.62 ρ = 0.47
ρ = 0.68ρ = 0.51
ρ = 0.10
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 4.71, f = 0.86
d =0.03, f = 1.87
d = 0.20, f = 0.35
d = 4.71, f = 0.86 d = 0.03, f = 1.87 d = 0.20, f = 0.35
g = 0.003h = 0.84
g = 297, h = 1.19 g = 0.32, h = 0.43
g = 3.17, h = 2.35
g = 0.004h = 1.06
g = 250, h = 0.94
ρ = 0.98
ρ = 0.52
ρ = 0.79
ρ = 0.44
ρ = 0.88 ρ = 0.34
ρ = 0.98 ρ = 0.52 ρ = 0.79
ρ = 0.88ρ = 0.44
ρ = 0.34
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 4.49, f = 0.72
d =0.05, f = 2.80
d = 4.56, f = 1.09
d = 4.49, f = 0.72 d = 0.05, f = 2.80 d = 4.56, f = 1.09
g = 0.002h = 0.89
g = 446, h = 1.12 g = 0.18, h = 0.33
g = 5.43, h = 3.01
g = 0.004h = 0.99
g = 227, h = 1.01
Figure 9.19: Matrix plots of cross-correlations: effect of site seismicity (BP hysteresis type,α = 0.10,βs = βr = 1/3, SAC survival-level,SD soil): (a) Los Angeles; (b) Boston.
(a) (b)
123
9.2 Effect of Reference Response Spectra on the Relationships Between the Demand Indices
As discussed earlier in Section 3.5, several different reference response spectra can beused to develop capacity-demand index relationships (i.e., IND, AVG, and DES response spectra).The effect of using these different reference spectra on theR-µ-T relationships is discussed inChapter 7. This section investigates the effect of reference response spectra on the relationshipsbetween the demand indicesµ, µp, µr, andny. For this purpose, Tables 9.7 through 9.10 providethed, f, g, andh regression coefficients based on the AVG and DES reference spectra. The corre-lation coefficients,ρ, are listed in Tables 9.11 and 9.12 and the corresponding matrix plots are pro-vided in Figures 9.21-9.32. The figures are organized in the same order that the results arepresented in Chapter 7: (1) regression relationships for regions with low-seismicity, Boston (Fig.9.21-9.24); (2) regression relationships for regions with high seismicity, Los Angeles (Fig. 9.25-9.28); (3) regression relationships for near-field conditions,NF (Fig. 9.29); and (4) regressionrelationships for different hysteresis types, SD, BE, and BP (Fig. 9.30-9.32). In similar order, Fig-ures 9.33-9.36 compare theΛ-µ regression relationships developed based on the AVG and DESreference spectra with the relationships presented in Section 9.1 based on the IND spectra todetermineFy = Felas/R.
It was demonstrated earlier in Chapter 7 that reference response spectra significantly affectthe R-µ-T relationships. Figures 9.33-9.36 show that the relationships betweenµ and the otherdemand indicesΛ = µp, µr, andny are similarly dependent on the reference response spectra. The
ρ = 0.99
ρ = 0.62
ρ = 0.80
ρ = 0.59
ρ = 0.86 ρ = 0.33
ρ = 0.99 ρ = 0.62 ρ = 0.80
ρ = 0.86ρ = 0.59
ρ = 0.33
dataregression
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
0 500
4
0 250
4
0 500
25
ny
0 100
50
0 40
50
µp
0 250
50d = 1.70, f = 0.71
d =0.56, f = 3.80
d = 1.39, f = 0.85
d = 1.70, f = 0.71 d = 0.56, f = 3.80 d = 1.39, f = 0.85
g = 0.03, h = 5.56
g = 38.6, h = 0.18 g = 0.40, h = 0.45
g = 2.53, h = 2.23
g = 21.9, h = 0.21
g = 0.05, h =4.76
ρ = 0.98
ρ = 0.72
ρ = 0.78
ρ = 0.67
ρ = 0.86 ρ = 0.54
ρ = 0.98 ρ = 0.72 ρ = 0.78
ρ = 0.86ρ = 0.67
ρ = 0.54
dataregression
µ
0 100
25
0 250
10
ny
0 40
10
0 500
10
0 40
25
0 100
4
µr
0 250
4
0 500
4
0 500
25
0 100
50
µp
0 250
50
0 40
50d = 2.30, f = 0.71
d =0.46, f = 2.10
d = 2.44, f = 0.88
d = 2.30, f = 0.71 d = 0.46, f = 2.10 d = 2.44, f = 0.88
g = 0.04, h = 3.33
g = 24.1, h = 0.30 g = 0.25, h = 0.40
g = 4.06, h = 2.51
g = 0.06, h = 2.38
g = 16.9, h = 0.42
Figure 9.20: Matrix plots of cross-correlations: effect of epicentral distance (EP hysteresistype,α = 0.10, SAC Los Angeles, design-level,SD soil): (a) far-field; (b) near-field.
(a) (b)
124
Λ-µ relationships are significantly affected by reference response spectra for: (1)µ versusµp andµ versusny relationships for the Boston design-levelSD andSE soil ground motions using the EPhysteresis type (Figs. 9.33a,c); (2)µ versusny relationship for the Los Angeles design-levelSDsoil NF ground motions using the EP hysteresis type (Fig. 9.35); and (3)µ versusny relationshipfor the Los Angeles survival-levelSD soil ground motions using the BE and BP hysteresis types(Figs. 9.36b,c). In general, the differences between the regression relationships developed basedon the IND and DES spectra are larger than the differences between the relationships developedbased on the IND and AVG spectra.
It is concluded that theµp, µr, andny demands should be estimated using regression rela-tionships developed based on the reference response spectrum used in the estimation ofµ.
Table 9.7: Regression coefficientsd andf: AVG spectra
HysteresisType
α, βs, βrSite
SeismicityDemand
LevelSite Soil/Distance
Regression Coefficientsa
a (Eq. 3.43 repeated)
Demand Index,Λ d f
EP α = 0.10
Boston
Survival
SD
µp 1.75 0.63
µr 0.32 1.45
ny 2.30 0.85
SE
µp 2.42 0.68
µr 0.30 1.82
ny 4.24 1.39
Design
SD
µp 1.53 0.59
µr 0.42 1.94
ny 2.34 0.84
SE
µp 1.90 0.65
µr 0.34 1.49
ny 3.08 1.08
Los Angeles
Survival
SD
µp 1.64 0.67
µr 0.43 2.10
ny 0.98 0.65
SE
µp 1.73 0.70
µr 0.32 2.00
ny 3.38 2.13
Design
SD
µp 1.69 0.65
µr 0.45 2.30
ny 1.87 0.84
SE
µp 1.69 0.66
µr 0.38 2.25
ny 3.62 1.64
NF
µp 1.33 0.66
µr 0.32 1.32
ny 0.57 0.60
SD α = 0.10 Los Angeles Survival SD
µp 1.85 0.50
µr 0.27 1.33
ny 3.48 3.04
BE α = 0.10 Los Angeles Survival SD
µp 2.23 0.54
µr 0 0
ny 0.44 0.47
BPα = 0.10
βs = βr = 1/3 Los Angeles Survival SD
µp 2.48 0.65
µr 0.03 1.44
ny 0.17 0.35
Λ d µ 1–( )1 f⁄=
125
Table 9.8: Regression coefficientsd andf: DES spectra
HysteresisType
α, βs, βrSite
SeismicityDemand
LevelSite Soil/Distance
Regression Coefficientsa
a (Eq. 3.43 repeated)
Demand Index,Λ d f
EP α = 0.10
Boston
Survival
SD
µp 1.09 0.52
µr 0.30 1.48
ny 1.32 0.65
SE
µp 2.57 0.68
µr 0.44 4.48
ny 4.32 1.37
Design
SD
µp 0.38 0.41
µr 0.33 1.46
ny 0.46 0.47
SE
µp 0.63 0.49
µr 0.30 1.32
ny 1.20 0.71
Los Angeles
Survival
SD
µp 1.73 0.68
µr 0.43 2.37
ny 1.80 0.88
SE
µp 2.72 0.82
µr 0.28 1.78
ny 4.45 2.93
Design
SD
µp 1.49 0.63
µr 0.40 1.96
ny 1.56 0.78
SE
µp 1.45 0.63
µr 0.45 2.66
ny 2.74 1.30
NF
µp 2.42 0.81
µr 0.33 1.41
ny 0.02 0.25
SD α = 0.10 Los Angeles Survival SD
µp 3.93 0.63
µr 0.27 1.32
ny 4.16 5.41
BE α = 0.10 Los Angeles Survival SD
µp 6.06 0.84
µr 0 0
ny 1.17 0.66
BPα = 0.10
βs = βr = 1/3 Los Angeles Survival SD
µp 3.46 0.76
µr 0.03 1.67
ny 1.09 0.62
Λ d µ 1–( )1 f⁄=
126
Table 9.9: Regression coefficientsg andh: AVG spectra
HysteresisType
α, βs, βrSite
SeismicityDemand
LevelSite Soil/Distance
Regression Coefficientsa
a (Eq. 3.44 repeated)
Λj
Λi
µp µr ny
g h g h g h
EP α = 0.10
Boston
Survival
SD
µp -- -- 25.4 0.37 0.39 0.58
µr 0.04 2.70 -- -- 0.06 1.56
ny 2.55 1.73 16.5 0.64 -- --
SE
µp -- -- 86.7 0.29 0.29 0.41
µr 0.01 3.45 -- -- 0.04 1.79
ny 3.47 2.41 26.4 0.56 -- --
Design
SD
µp -- -- 25.0 0.29 0.41 0.60
µr 0.04 3.45 -- -- 0.06 2.04
ny 2.43 1.68 17.8 0.49 -- --
SE
µp -- -- 25.0 0.39 0.37 0.51
µr 0.04 2.56 -- -- 0.07 1.43
ny 2.73 1.95 14.0 0.70 -- --
Los Angeles
Survival
SD
µp -- -- 23.7 0.34 0.36 0.48
µr 0.04 2.94 -- -- 0.07 2.33
ny 2.74 2.09 14.6 0.43 -- --
SE
µp -- -- 43.3 0.36 0.36 0.33
µr 0.02 2.78 -- -- 0.09 1.23
ny 2.78 3.00 11.2 0.81 -- --
Design
SD
µp -- -- 34.3 0.24 0.35 0.49
µr 0.03 4.17 -- -- 0.04 3.45
ny 2.87 2.05 22.4 0.29 -- --
SE
µp -- -- 52.2 0.26 0.30 0.36
µr 0.02 3.85 -- -- 0.07 1.43
ny 3.34 2.80 14.0 0.70 -- --
NFµp -- -- 13.4 0.49 0.60 0.58
µr 0.07 2.04 -- -- 0.13 2.22
ny 1.66 1.73 7.70 0.45 -- --
SD α = 0.10 Los Angeles Survival SD
µp -- -- 62.5 0.41 1.64 0.60
µr 0.02 2.44 -- -- 0.12 2.00
ny 0.61 1.67 8.04 0.50 -- --
BE α = 0.10 Los Angeles Survival SD
µp -- -- 0 0 0.89 0.65
µr 0 0 -- -- 0 0
ny 1.12 1.55 0 0 -- --
BP α = 0.10βs = βr = 1/3
Los Angeles Survival SD
µp -- -- 327 1.13 0.49 0.52
µr 0.003 0.88 -- -- 0.004 1.10
ny 2.03 1.91 226 0.91 -- --
Λ j gΛi1 h⁄
=
127
Table 9.10: Regression coefficientsg andh: DES spectra
HysteresisType
α, βs, βrSite
SeismicityDemand
LevelSite Soil/Distance
Regression Coefficientsa
a (Eq. 3.44 repeated)
Λj
Λi
µp µr ny
g h g h g h
EP α = 0.10
Boston
Survival
SD
µp -- -- 33.3 0.34 0.55 0.68
µr 0.03 2.94 -- -- 0.05 2.17
ny 1.82 1.47 20.7 0.46 -- --
SE
µp -- -- 470 0.16 0.30 0.43
µr 0.002 6.25 -- -- 0.003 6.25
ny 3.38 2.35 289 0.16 -- --
Design
SD
µp -- -- 22.2 0.22 0.51 0.66
µr 0.05 4.55 -- -- 0.05 4.17
ny 1.95 1.51 18.2 0.24 -- --
SE
µp -- -- 16.3 0.33 0.45 0.56
µr 0.06 3.03 -- -- 0.09 2.17
ny 2.23 1.77 11.3 0.46 -- --
Los Angeles
Survival
SD
µp -- -- 33.3 0.30 0.33 0.43
µr 0.03 3.33 -- -- 0.05 2.50
ny 3.07 2.31 19.1 0.40 -- --
SE
µp -- -- 58.6 0.34 0.31 0.29
µr 0.02 2.94 -- -- 0.13 0.003
ny 3.21 3.40 7.74 357 -- --
Design
SD
µp -- -- 27.7 0.31 0.40 0.53
µr 0.04 3.23 -- -- 0.06 2.33
ny 2.49 1.87 16.8 0.43 -- --
SE
µp -- -- 71.4 0.17 0.35 0.40
µr 0.01 5.88 -- -- 0.008 8.33
ny 2.89 2.53 119 0.12 -- --
NFµp -- -- 17.0 0.71 0.60 0.62
µr 0.06 1.41 -- -- 0.11 1.64
ny 1.67 1.62 9.36 0.61 -- --
SD α = 0.10 Los Angeles Survival SD
µp -- -- 65.0 0.48 0.87 0.43
µr 0.02 2.08 -- -- 0.11 1.75
ny 1.15 2.32 8.95 0.57 -- --
BE α = 0.10 Los Angeles Survival SD
µp -- -- 0 0 0.33 0.36
µr 0 0 -- -- 0 0
ny 3.07 2.79 0 0 -- --
BP α = 0.10βs = βr = 1/3
Los Angeles Survival SD
µp -- -- 314 1.15 0.26 0.32
µr 0.003 0.87 -- -- 0.006 1.01
ny 3.80 3.14 176 0.99 -- --
Λ j gΛi1 h⁄
=
128
Table 9.11: Correlation coefficient,ρ: AVG spectra
HysteresisType
α, βs, βrSite
SeismicityDemand
LevelSite Soil/Distance
Correlation Coefficient,ρ
µ µp µr ny
EP α = 0.10
Boston
Survival
SD
µ 1.00 0.97 0.66 0.92µp 0.97 1.00 0.58 0.97µr 0.66 0.58 1.00 0.54ny 0.92 0.97 0.54 1.00
SE
µ 1.00 0.98 0.60 0.94µp 0.98 1.00 0.53 0.97µr 0.60 0.53 1.00 0.45ny 0.94 0.97 0.45 1.00
Design
SD
µ 1.00 0.97 0.66 0.91µp 0.97 1.00 0.55 0.97µr 0.66 0.55 1.00 0.45ny 0.91 0.97 0.45 1.00
SE
µ 1.00 0.98 0.76 0.93µp 0.98 1.00 0.70 0.97µr 0.76 0.70 1.00 0.64ny 0.93 0.97 0.64 1.00
Los Angeles
Survival
SD
µ 1.00 0.97 0.82 0.70µp 0.97 1.00 0.79 0.84µr 0.82 0.79 1.00 0.55ny 0.70 0.84 0.55 1.00
SE
µ 1.00 0.99 0.76 0.94µp 0.99 1.00 0.78 0.95µr 0.76 0.78 1.00 0.70ny 0.94 0.95 0.70 1.00
Design
SD
µ 1.00 0.98 0.73 0.84µp 0.98 1.00 0.68 0.91µr 0.73 0.68 1.00 0.52ny 0.84 0.91 0.52 1.00
SE
µ 1.00 0.98 0.75 0.93µp 0.98 1.00 0.68 0.96µr 0.75 0.68 1.00 0.67ny 0.93 0.96 0.67 1.00
NF
µ 1.00 0.96 0.88 0.70µp 0.96 1.00 0.78 0.85µr 0.88 0.78 1.00 0.46ny 0.70 0.85 0.46 1.00
SD α = 0.10 Los Angeles Survival SD
µ 1.00 0.75 0.97 0.66µp 0.75 1.00 0.71 0.94µr 0.97 0.71 1.00 0.63ny 0.66 0.94 0.63 1.00
BE α = 0.10 Los Angeles Survival SD
µ 1.00 0.90 0.00 0.57µp 0.90 1.00 0.00 0.86µr 0.00 0.00 1.00 0.00ny 0.57 0.86 0.00 1.00
BPα = 0.10
βs = βr = 1/3 Los Angeles Survival SD
µ 1.00 0.91 0.70 0.46µp 0.91 1.00 0.57 0.77µr 0.70 0.57 1.00 0.17ny 0.46 0.77 0.17 1.00
129
Table 9.12: Correlation coefficient,ρ: DES spectra
HysteresisType
α, βs, βrSite
SeismicityDemand
LevelSite Soil/Distance
Correlation Coefficient,ρ
µ µp µr ny
EP α = 0.10
Boston
Survival
SD
µ 1.00 0.95 0.70 0.86µp 0.95 1.00 0.62 0.96µr 0.70 0.62 1.00 0.53ny 0.86 0.96 0.53 1.00
SE
µ 1.00 0.96 0.26 0.85µp 0.96 1.00 0.15 0.95µr 0.26 0.15 1.00 0.02ny 0.85 0.95 0.02 1.00
Design
SD
µ 1.00 0.82 0.78 0.72µp 0.82 1.00 0.75 0.98µr 0.78 0.75 1.00 0.65ny 0.72 0.98 0.65 1.00
SE
µ 1.00 0.81 0.82 0.68µp 0.81 1.00 0.89 0.96µr 0.82 0.89 1.00 0.79ny 0.68 0.96 0.79 1.00
Los Angeles
Survival
SD
µ 1.00 0.98 0.82 0.72µp 0.98 1.00 0.78 0.81µr 0.82 0.78 1.00 0.46ny 0.72 0.81 0.46 1.00
SE
µ 1.00 0.98 0.41 0.75µp 0.98 1.00 0.36 0.80µr 0.41 0.36 1.00 0.07ny 0.75 0.80 0.07 1.00
Design
SD
µ 1.00 0.97 0.85 0.84µp 0.97 1.00 0.84 0.93µr 0.85 0.84 1.00 0.72ny 0.84 0.93 0.72 1.00
SE
µ 1.00 0.96 0.65 0.74µp 0.96 1.00 0.47 0.87µr 0.65 0.47 1.00 0.17ny 0.74 0.87 0.17 1.00
NF
µ 1.00 0.92 0.62 0.62µp 0.92 1.00 0.59 0.66µr 0.62 0.59 1.00 0.37ny 0.36 0.66 0.37 1.00
SD α = 0.10 Los Angeles Survival SD
µ 1.00 0.63 0.95 0.48µp 0.63 1.00 0.49 0.88µr 0.95 0.49 1.00 0.37ny 0.48 0.88 0.37 1.00
BE α = 0.10 Los Angeles Survival SD
µ 1.00 0.91 0.00 0.40µp 0.91 1.00 0.00 0.72µr 0.00 0.00 1.00 0.00ny 0.40 0.72 0.00 1.00
BPα = 0.10
βs = βr = 1/3 Los Angeles Survival SD
µ 1.00 0.95 0.68 0.38µp 0.95 1.00 0.54 0.64µr 0.68 0.54 1.00 0.02ny 0.38 0.64 0.02 1.00
130
ρ = 0.97
ρ = 0.66
ρ = 0.91
ρ = 0.55
ρ = 0.97 ρ = 0.45
ρ = 0.97 ρ = 0.66 ρ = 0.91
ρ = 0.97ρ = 0.55
ρ = 0.45
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 1.53, f = 0.59
d = 0.42, f = 1.94
d = 2.34, f = 0.84
d = 1.53, f = 0.59 d = 0.42, f = 1.94 d = 2.34, f = 0.84
g = 0.04, h = 3.45
g = 25.0, h = 0.29 g = 0.41, h = 0.60
g = 2.43, h = 1.68
g = 0.06, h = 2.04
g = 17.8, h = 0.49
ρ = 0.82
ρ = 0.78
ρ = 0.72
ρ = 0.75
ρ = 0.98 ρ = 0.65
ρ = 0.82 ρ = 0.78 ρ = 0.72
ρ = 0.98ρ = 0.75
ρ = 0.65
µ
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0 0 0
d = 0.38, f = 0.41
d = 0.33, f = 1.46
d = 0.46, f = 0.47
d = 0.38, f = 0.41 d = 0.33, f = 1.46 d = 0.46, f = 0.47
g = 0.05, h = 4.55
g = 22.2, h = 0.22 g = 0.51, h = 0.66
g = 1.95, h = 1.51
g = 0.05, h = 4.17
g = 18.2, h = 0.24
Figure 9.21: Matrix plots of cross-correlations: effect of reference response spectrum(EP hysteresis type,α = 0.10, SAC Boston, design-level,SD soil): (a) AVG spectrum; (b)DES spectrum.
(a) (b)
ρ = 0.97
ρ = 0.66
ρ = 0.92
ρ = 0.58
ρ = 0.97 ρ = 0.54
ρ = 0.97 ρ = 0.66 ρ = 0.92
ρ = 0.97ρ = 0.58
ρ = 0.54
µ
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0 0 0
d = 1.75, f = 0.63
d = 0.32, f = 1.45
d = 2.30, f = 0.85
d = 1.75, f = 0.63 d = 0.32, f = 1.45 d = 2.30, f = 0.85
g = 0.04, h = 2.70
g = 25.4, h = 0.37 g = 0.39, h = 0.58
g = 2.55, h = 1.73
g = 0.06, h = 1.56
g = 16.5, h = 0.64
ρ = 0.95
ρ = 0.70
ρ = 0.86
ρ = 0.62
ρ = 0.96 ρ = 0.53
ρ = 0.95 ρ = 0.70 ρ = 0.86
ρ = 0.96ρ = 0.62
ρ = 0.53
µ
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0 0 0
d = 1.09, f = 0.52
d = 0.30, f = 1.48
d = 1.32, f = 0.65
d = 1.09, f = 0.52 d = 0.30, f = 1.48 d = 1.32, f = 0.65
g = 0.03, h = 2.94
g = 33.3, h = 0.34 g = 0.55, h = 0.68
g = 1.82, h = 1.47
g = 0.05, h = 2.17
g = 20.7, h = 0.46
Figure 9.22: Matrix plots of cross-correlations: effect of reference response spectrum(EP hysteresis type,α = 0.10, SAC Boston, survival-level,SD soil): (a) AVG spectrum;(b) DES spectrum.
(a) (b)
131
ρ = 0.98
ρ = 0.76
ρ = 0.93
ρ = 0.70
ρ = 0.97 ρ = 0.64
ρ = 0.98 ρ = 0.76 ρ = 0.93
ρ = 0.97ρ = 0.70
ρ = 0.64
µ
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0 0 0
d = 1.90, f = 0.65
d = 0.34, f = 1.49
d = 3.08, f = 1.08
d = 1.90, f = 0.65 d = 0.34, f = 1.49 d = 3.08, f = 1.08
g = 0.04, h = 2.56
g = 25.0, h = 0.39 g = 0.37, h = 0.51
g = 2.73, h = 1.95
g = 0.07, h = 1.43
g = 14.0, h = 0.70
ρ = 0.81
ρ = 0.82
ρ = 0.68
ρ = 0.89
ρ = 0.96 ρ = 0.79
ρ = 0.81 ρ = 0.82 ρ = 0.68
ρ = 0.96ρ = 0.89
ρ = 0.79
µ
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0 0 0
d = 0.63, f = 0.49
d = 0.30, f = 1.32
d = 1.20, f = 0.71
d = 0.63, f = 0.49 d = 0.30, f = 1.32 d = 1.20, f = 0.71
g = 0.06, h = 3.03
g = 16.3, h = 0.33 g = 0.45, h = 0.56
g = 2.23, h = 1.77
g = 0.09, h = 2.17
g = 11.3, h = 0.46
Figure 9.23: Matrix plots of cross-correlations: effect of reference response spectrum(EP hysteresis type,α = 0.10, SAC Boston, design-level,SE soil): (a) AVG spectrum; (b)DES spectrum.
(a) (b)
ρ = 0.98
ρ = 0.60
ρ = 0.94
ρ = 0.53
ρ = 0.97 ρ = 0.45
ρ = 0.98 ρ = 0.60 ρ = 0.94
ρ = 0.97ρ = 0.53
ρ = 0.45
µ
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dataregression
0 0 0
d = 2.42, f = 0.68
d = 0.30, f = 1.82
d = 4.24, f = 1.39
d = 2.42, f = 0.68 d = 0.30, f = 1.82 d = 4.24, f = 1.39
g = 0.01, h = 3.45
g = 86.7, h = 0.29 g = 0.29, h = 0.41
g = 3.47, h = 2.41
g = 0.04, h = 1.79
g = 16.5, h = 0.64
ρ = 0.96
ρ = 0.26
ρ = 0.85
ρ = 0.15
ρ = 0.95 ρ = 0.02
ρ = 0.96 ρ = 0.26 ρ = 0.85
ρ = 0.95ρ = 0.15
ρ = 0.02
µ
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0 0 0
d = 2.57, f = 0.68
d = 0.44, f = 4.48
d = 4.32, f = 1.37
d = 2.57, f = 0.68 d = 0.44, f = 4.48 d = 4.32, f = 1.37
g = 0.002h = 6.25
g = 470, h = 0.16 g = 0.30, h = 0.43
g = 3.38, h = 2.35
g = 0.003h = 6.25
g = 289, h = 0.16
Figure 9.24: Matrix plots of cross-correlations: effect of reference response spectrum(EP hysteresis type,α = 0.10, SAC Boston, survival-level,SE soil): (a) AVG spectrum;(b) DES spectrum.
(a) (b)
132
ρ = 0.98
ρ = 0.73
ρ = 0.84
ρ = 0.68
ρ = 0.91 ρ = 0.52
ρ = 0.98 ρ = 0.73 ρ = 0.84
ρ = 0.91ρ = 0.68
ρ = 0.52
µ
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0 0 0
d = 1.69, f = 0.65
d = 0.45, f = 2.30
d = 1.87, f = 0.84
d = 1.69, f = 0.65 d = 0.45, f = 2.30 d = 1.87, f = 0.84
g = 0.03, h = 4.17
g = 34.3, h = 0.24 g = 0.35, h = 0.49
g = 2.87, h = 2.05
g = 0.04, h = 3.45
g = 22.4, h = 0.29
ρ = 0.97
ρ = 0.85
ρ = 0.84
ρ = 0.84
ρ = 0.93 ρ = 0.72
ρ = 0.97 ρ = 0.85 ρ = 0.84
ρ = 0.93ρ = 0.84
ρ = 0.72
µ
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0 0 0
d = 1.49, f = 0.63
d = 0.40, f = 1.96
d = 1.56, f = 0.78
d = 1.49, f = 0.63 d = 0.40, f = 1.96 d = 1.56, f = 0.78
g = 0.04, h = 3.23
g = 27.7, h = 0.31 g = 0.40, h = 0.53
g = 2.49, h = 1.87
g = 0.06, h = 2.33
g = 16.8, h = 0.43
Figure 9.25: Matrix plots of cross-correlations: effect of reference response spectrum(EP hysteresis type,α = 0.10, SAC Los Angeles, design-level,SD soil): (a) AVG spec-trum; (b) DES spectrum.
(a) (b)
ρ = 0.97
ρ = 0.82
ρ = 0.70
ρ = 0.79
ρ = 0.84 ρ = 0.55
ρ = 0.97 ρ = 0.82 ρ = 0.70
ρ = 0.84ρ = 0.79
ρ = 0.55
µ
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0 0 0
d = 1.64, f = 0.67
d = 0.43, f = 2.10
d = 0.98, f = 0.65
d = 1.64, f = 0.67 d = 0.43, f = 2.10 d = 0.98, f = 0.65
g = 0.04, h = 2.94
g = 23.7, h = 0.34 g = 0.36, h = 0.48
g = 2.74, h = 2.09
g = 0.07, h = 2.33
g = 14.6, h = 0.43
ρ = 0.98
ρ = 0.82
ρ = 0.72
ρ = 0.78
ρ = 0.81 ρ = 0.46
ρ = 0.98 ρ = 0.82 ρ = 0.72
ρ = 0.81ρ = 0.78
ρ = 0.46
µ
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0 0 0
d = 1.73, f = 0.68
d = 0.43, f = 2.37
d = 1.80, f = 0.88
d = 1.73, f = 0.68 d = 0.43, f = 2.37 d = 1.80, f = 0.88
g = 0.03, h = 3.33
g = 33.3, h = 0.30 g = 0.33, h = 0.43
g = 3.07, h = 2.31
g = 0.05, h = 2.50
g = 19.1, h = 0.40
Figure 9.26: Matrix plots of cross-correlations: effect of reference response spectrum(EP hysteresis type,α = 0.10, SAC Los Angeles, survival-level,SD soil): (a) AVG spec-trum; (b) DES spectrum.
(a) (b)
133
ρ = 0.96
ρ = 0.65
ρ = 0.74
ρ = 0.47
ρ = 0.87 ρ = 0.17
ρ = 0.96 ρ = 0.65 ρ = 0.74
ρ = 0.87ρ = 0.47
ρ = 0.17
µ
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0 0 0
d = 1.45, f = 0.63
d = 0.45, f = 2.66
d = 2.74, f = 1.30
d = 1.45, f = 0.63 d = 0.45, f = 2.66 d = 2.74, f = 1.30
g = 0.01, h = 5.88
g = 71.4, h = 0.17 g = 0.35, h = 0.40
g = 2.89, h = 2.53
g = 0.008h = 8.33
g = 119, h = 0.12
ρ = 0.98
ρ = 0.75
ρ = 0.93
ρ = 0.68
ρ = 0.96 ρ = 0.67
ρ = 0.98 ρ = 0.75 ρ = 0.93
ρ = 0.96ρ = 0.68
ρ = 0.67
µ
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0 0 0
d = 1.69, f = 0.66
d = 0.38, f = 2.25
d = 3.62, f = 1.64
d = 1.69, f = 0.66 d = 0.38, f = 2.25 d = 3.62, f = 1.64
g = 0.02, h = 3.85
g = 52.2, h = 0.26 g = 0.30, h = 0.36
g = 3.34, h = 2.80
g = 0.07, h = 1.43
g = 14.0, h = 0.70
Figure 9.27: Matrix plots of cross-correlations: effect of reference response spectrum(EP hysteresis type,α = 0.10, SAC Los Angeles, design-level,SE soil): (a) AVG spec-trum; (b) DES spectrum.
(a) (b)
ρ = 0.99
ρ = 0.76
ρ = 0.94
ρ = 0.78
ρ = 0.95 ρ = 0.70
ρ = 0.99 ρ = 0.76 ρ = 0.94
ρ = 0.95ρ = 0.78
ρ = 0.70
µ
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0 0 0
d = 1.73, f = 0.70
d = 0.32, f = 2.00
d = 3.38, f = 2.13
d = 1.73, f = 0.70 d = 0.32, f = 2.00 d = 3.38, f = 2.13
g = 0.02, h = 2.78
g = 43.3, h = 0.36 g = 0.36, h = 0.33
g = 2.78, h = 3.00
g = 0.09, h = 1.23
g = 11.2, h = 0.81
ρ = 0.98
ρ = 0.41
ρ = 0.75
ρ = 0.36
ρ = 0.80 ρ = 0.07
ρ = 0.98 ρ = 0.41 ρ = 0.75
ρ = 0.80ρ = 0.36
ρ = 0.07
µ
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0 0 0
d = 2.72, f = 0.82
d = 0.28, f = 1.78
d = 4.45, f = 2.93
d = 2.72, f = 0.82 d = 0.28, f = 1.78 d = 4.45, f = 2.93
g = 0.02, h = 2.94
g = 58.6, h = 0.34 g = 0.31, h = 0.29
g = 3.21, h = 3.40
g = 0.13h = 0.003
g = 7.74, h =357
Figure 9.28: Matrix plots of cross-correlations: effect of reference response spectrum(EP hysteresis type,α = 0.10, SAC Los Angeles, survival-level,SE soil): (a) AVG spec-trum; (b) DES spectrum.
(a) (b)
134
ρ = 0.96
ρ = 0.88
ρ = 0.70
ρ = 0.78
ρ = 0.85 ρ = 0.46
ρ = 0.96 ρ = 0.88 ρ = 0.70
ρ = 0.85ρ = 0.78
ρ = 0.46
µ
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0 0 0
d = 1.33, f = 0.66
d = 0.32, f = 1.32
d = 0.57, f = 0.60
d = 1.33, f = 0.66 d = 0.32, f = 1.32 d = 0.57, f = 0.60
g = 0.07, h = 2.04
g = 13.4, h = 0.49 g = 0.60, h = 0.58
g = 1.66, h = 1.73
g = 0.13, h = 2.22
g = 7.70, h = 0.45
ρ = 0.92
ρ = 0.62
ρ = 0.36
ρ = 0.59
ρ = 0.66 ρ = 0.37
ρ = 0.92 ρ = 0.62 ρ = 0.36
ρ = 0.66ρ = 0.59
ρ = 0.37
µ
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0 0 0
d = 2.42, f = 0.81
d = 0.33, f = 1.41
d = 0.02, f = 0.25
d = 2.42, f = 0.81 d = 0.33, f = 1.41 d = 0.02, f = 0.25
g = 0.06, h = 1.41
g = 17.0, h = 0.71 g = 0.60, h = 0.62
g = 1.67, h = 1.62
g = 0.11, h = 1.64
g = 9.36, h = 0.61
Figure 9.29: Matrix plots of cross-correlations: effect of reference response spectrum(EP hysteresis type,α = 0.10, SAC Los Angeles, design-level,SD soil,NF): (a) AVGspectrum; (b) DES spectrum.
(a) (b)
ρ = 0.75
ρ = 0.97
ρ = 0.66
ρ = 0.71
ρ = 0.94 ρ = 0.63
ρ = 0.75 ρ = 0.97 ρ = 0.66
ρ = 0.94ρ = 0.71
ρ = 0.63
µ
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0 0 0
d = 1.85, f = 0.50
d = 0.27, f = 1.33
d = 3.48, f = 3.04
d = 1.85, f = 0.50 d = 0.27, f = 1.33 d = 3.48, f = 3.04
g = 0.02, h = 2.44
g = 62.5, h = 0.41 g = 1.64, h = 0.60
g = 0.61, h = 1.67
g = 0.12, h = 2.00
g = 8.04, h = 0.50
ρ = 0.63
ρ = 0.95
ρ = 0.48
ρ = 0.49
ρ = 0.88 ρ = 0.37
ρ = 0.63 ρ = 0.95 ρ = 0.48
ρ = 0.88ρ = 0.49
ρ = 0.37
µ
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0 0 0
d = 3.93, f = 0.63
d = 0.27, f = 1.32
d = 4.16, f = 5.41
d = 3.93, f = 0.63 d = 0.27, f = 1.32 d = 4.16, f = 5.41
g = 0.02, h = 2.08
g = 65.0, h = 0.48 g = 0.87, h = 0.43
g = 1.15, h = 2.32
g = 0.11, h = 1.75
g = 8.95, h = 0.57
Figure 9.30: Matrix plots of cross-correlations: effect of reference response spectrum(SD hysteresis type,α = 0.10, SAC Los Angeles, survival-level,SD soil): (a) AVG spec-trum; (b) DES spectrum.
(a) (b)
135
ρ = 0.90
ρ = 0.00
ρ = 0.57
ρ = 0.00
ρ = 0.86 ρ = 0.00
ρ = 0.90 ρ = 0.00 ρ = 0.57
ρ = 0.86ρ = 0.00
ρ = 0.00
µ
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0 0 0
d = 2.23, f = 0.54
d = 0.00, f = 0.00
d = 0.44, f = 0.47
d = 2.23, f = 0.54 d = 0.00, f = 0.00 d = 0.44, f = 0.47
g = 0.00, h = 0.00
g = 0.00, h = 0.00 g = 0.89, h = 0.65
g = 1.12, h = 1.55
g = 0.00, h = 0.00
g = 0.00, h = 0.00
ρ = 0.91
ρ = 0.00
ρ = 0.40
ρ = 0.00
ρ = 0.72 ρ = 0.00
ρ = 0.91 ρ = 0.00 ρ = 0.40
ρ = 0.72ρ = 0.00
ρ = 0.00
µ
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0 0 0
d = 6.06, f = 0.84
d = 0.00, f = 0.00
d = 1.17, f = 0.66
d = 6.06, f = 0.84 d = 0.00, f = 0.00 d = 1.17, f = 0.66
g = 0.00, h = 0.00
g = 0.00, h = 0.00 g = 0.33, h = 0.36
g = 3.07, h = 2.79
g = 0.00, h = 0.00
g = 0.00, h = 0.00
Figure 9.31: Matrix plots of cross-correlations: effect of reference response spectrum(BE hysteresis type,α = 0.10, SAC Los Angeles, survival-level,SD soil): (a) AVG spec-trum; (b) DES spectrum.
(a) (b)
ρ = 0.91
ρ = 0.70
ρ = 0.46
ρ = 0.57
ρ = 0.77 ρ = 0.17
ρ = 0.91 ρ = 0.70 ρ = 0.46
ρ = 0.77ρ = 0.57
ρ = 0.17
µ
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0 0 0
1
2
3
4
d = 2.48, f = 0.65
d = 0.03, f = 1.44
d = 0.17, f = 0.35
d = 2.48, f = 0.65 d = 0.03, f = 1.44 d = 0.17, f = 0.35
g = 0.003h = 0.88
g = 327, h = 1.13 g = 0.49, h = 0.52
g = 2.03, h = 1.91
g = 0.004h = 1.10
g = 226, h = 0.91
ρ = 0.95
ρ = 0.68
ρ = 0.38
ρ = 0.54
ρ = 0.64 ρ = 0.02
ρ = 0.95 ρ = 0.68 ρ = 0.38
ρ = 0.64ρ = 0.54
ρ = 0.02
µ
0 40
10
0 500
10
0 100
25
0 250
10
0 40
25
0 100
4
µr
5000
4
0 250
4
0 500
25
ny
0 10
50
0 4
50
µp
0 25
50
dataregression
0 0 0
d = 3.46, f = 0.76
d = 0.03, f = 1.67
d = 1.09, f = 0.62
d = 3.46, f = 0.76 d = 0.03, f = 1.67 d = 1.09, f = 0.62
g = 0.003h = 0.87
g = 314, h = 1.15 g = 0.26, h = 0.32
g = 3.80, h = 3.14
g = 0.006h = 1.01
g = 176, h = 0.99
Figure 9.32: Matrix plots of cross-correlations: effect of reference response spectrum(BP hysteresis type,α = 0.10,βs = βr = 1/3, SAC Los Angeles, survival-level,SD soil):(a) AVG spectrum; (b) DES spectrum.
(a) (b)
136
0 2 4 6 80
4
µ
µ r
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
50
µ
µ p
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
25
µn y
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
4
µ
µ r
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
50
µ
µ p
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
25
µ
n y
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
4
µ
µ r
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
50
µ
µ p
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
25
µ
n y
IND spectraAVG spectrum DES spectrum
Figure 9.33:Λ-µ regression curves using IND, AVG, and DES spectra (EP hysteresis type,α = 0.10, SAC Boston): (a-b)SD soil; (c-d)SE soil.
0 2 4 6 80
4
µ
µ r
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
50
µ
µ pIND spectraAVG spectrum DES spectrum
0 2 4 6 80
25
µ
n y
IND spectraAVG spectrum DES spectrum
(a) design-level (b) survival-level
(c) design-level (d) survival-level
IND spectra: d = 0.51, f = 2.91 AVG spectrum: d = 0.42, f = 1.94 DES spectrum: d = 0.33, f = 1.46
IND spectra: d = 2.76, f = 0.68 AVG spectrum: d = 1.53, f = 0.59 DES spectrum: d = 0.38, f = 0.41
IND spectra: d = 5.79, f = 1.30 AVG spectrum: d = 2.34, f = 0.84 DES spectrum: d = 0.46, f = 0.47
IND spectra: d = 0.53, f = 3.61 AVG spectrum: d = 0.34, f = 1.49 DES spectrum: d = 0.30, f = 1.32
IND spectra: d = 2.87, f = 0.70 AVG spectrum: d = 1.90, f = 0.65 DES spectrum: d = 0.63, f = 0.49
IND spectra: d = 5.50, f = 1.42 AVG spectrum: d = 3.08, f = 1.08 DES spectrum: d = 1.20, f = 0.71
IND spectra: d = 0.46, f = 2.62 AVG spectrum: d = 0.32, f = 1.45 DES spectrum: d = 0.30, f = 1.48
IND spectra: d = 2.78, f = 0.69 AVG spectrum: d = 1.75, f = 0.63 DES spectrum: d = 1.09, f = 0.52
IND spectra: d = 4.62, f = 1.12 AVG spectrum: d = 2.30, f = 0.85 DES spectrum: d = 1.32, f = 0.65
IND spectra: d = 0.37, f = 2.63 AVG spectrum: d = 0.30, f = 1.82 DES spectrum: d = 0.44, f = 4.48
IND spectra: d = 3.07, f = 0.68 AVG spectrum: d = 2.42, f = 0.68 DES spectrum: d = 2.57, f = 0.68
IND spectra: d = 6.69, f = 1.91 AVG spectrum: d = 4.24, f = 1.39 DES spectrum: d = 4.32, f = 1.37
137
0 2 4 6 80
4
µ
µ r
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
50
µ
µ p
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
25
µn y
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
4
µ
µ r
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
50
µ
µ p
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
25
µ
n y
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
4
µ
µ r
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
50
µ
µ p
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
25
µ
n y
IND spectraAVG spectrum DES spectrum
Figure 9.34:Λ-µ regression curves using IND, AVG, and DES spectra (EP hysteresis type,α = 0.10, SAC Los Angeles): (a-b)SD soil; (c-d)SE soil.
0 2 4 6 80
4
µ
µ r
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
50
µ
µ pIND spectraAVG spectrum DES spectrum
0 2 4 6 80
25
µ
n y
IND spectraAVG spectrum DES spectrum
(a) design-level (b) survival-level
(c) design-level (d) survival-level
IND spectra: d = 0.46, f = 2.10 AVG spectrum: d = 0.45, f = 2.30 DES spectrum: d = 0.40, f = 1.96
IND spectra: d = 2.30, f = 0.71 AVG spectrum: d = 1.69, f = 0.65 DES spectrum: d = 1.49, f = 0.63
IND spectra: d = 2.44, f = 0.88 AVG spectrum: d = 1.87, f = 0.84 DES spectrum: d = 1.56, f = 0.78
IND spectra: d = 0.47, f = 2.84 AVG spectrum: d = 0.43, f = 2.10 DES spectrum: d = 0.43, f = 2.37
IND spectra: d = 1.91, f = 0.69 AVG spectrum: d = 1.64, f = 0.67 DES spectrum: d = 1.73, f = 0.68
IND spectra: d = 1.77, f = 0.81 AVG spectrum: d = 0.98, f = 0.65 DES spectrum: d = 1.80, f = 0.88
IND spectra: d = 0.53, f = 3.61
IND spectra: d = 2.87, f = 0.70
IND spectra: d = 5.50, f = 1.42
IND spectra: d = 0.45, f = 2.98 AVG spectrum: d = 0.38, f = 2.25 DES spectrum: d = 0.45, f = 2.66
IND spectra: d = 2.13, f = 0.70 AVG spectrum: d = 1.69, f = 0.66 DES spectrum: d = 1.45, f = 0.63
IND spectra: d = 4.71, f = 2.04 AVG spectrum: d = 3.62, f = 1.64 DES spectrum: d = 2.74, f = 1.30
IND spectra: d = 0.40, f = 3.35 AVG spectrum: d = 0.32, f = 2.00 DES spectrum: d = 0.28, f = 1.78
IND spectra: d = 2.46, f = 0.77 AVG spectrum: d = 1.73, f = 0.70 DES spectrum: d = 2.72, f = 0.82
IND spectra: d = 4.77, f = 2.83 AVG spectrum: d = 3.38, f = 2.13 DES spectrum: d = 4.45, f = 2.93
138
Figure 9.35:Λ-µ regression curves using IND, AVG, and DES spectra (EP hysteresis type,α = 0.10, SAC Los Angeles, design-level,SD soil,NF).
0 2 4 6 80
4
µ
µ r
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
50
µ
µ p
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
25
µ
n y
IND spectraAVG spectrum DES spectrum
IND spectra: d = 0.56, f = 3.80 AVG spectrum: d = 0.32, f = 1.32 DES spectrum: d = 0.33, f = 1.41
IND spectra: d = 1.70, f = 0.71 AVG spectrum: d = 1.33, f = 0.66 DES spectrum: d = 2.42, f = 0.81
IND spectra: d = 1.39, f = 0.85 AVG spectrum: d = 0.57, f = 0.60 DES spectrum: d = 0.02, f = 0.25
139
0 2 4 6 80
4
µ
µ r
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
50
µ
µ p
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
25
µn y
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
4
µ
µ r
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
50
µ
µ p
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
25
µ
n y
IND spectraAVG spectrum DES spectrum
Figure 9.36:Λ-µ regression curves using IND, AVG, and DES spectra (α = 0.10, SAC LosAngeles, survival-level,SD soil): (a) SD hysteresis type; (b) BE hysteresis type; (c) BPhysteresis type,βs = βr = 1/3.
0 2 4 6 80
4
µ
µ r
IND spectraAVG spectrum DES spectrum
0 2 4 6 80
50
µ
µ pIND spectraAVG spectrum DES spectrum
0 2 4 6 80
25
µ
n y
IND spectraAVG spectrum DES spectrum
(a) (b)
(c)
IND spectra: d = 0.37, f = 1.82 AVG spectrum: d = 0.27, f = 1.33 DES spectrum: d = 0.27, f = 1.32
IND spectra: d = 1.80, f = 0.48 AVG spectrum: d = 1.85, f = 0.50 DES spectrum: d = 3.93, f = 0.63
IND spectra: d = 3.86, f = 3.02 AVG spectrum: d = 3.48, f = 3.04 DES spectrum: d = 4.16, f = 5.41
IND spectra: d = 0.00, f = 0.00 AVG spectrum: d = 0.00, f = 0.00 DES spectrum: d = 0.00, f = 0.00
IND spectra: d = 4.31, f = 0.68 AVG spectrum: d = 2.23, f = 0.54 DES spectrum: d = 6.06, f = 0.84
IND spectra: d = 0.19, f = 0.36 AVG spectrum: d = 0.44, f = 0.47 DES spectrum: d = 1.17, f = 0.66
IND spectra: d = 0.03, f = 1.87 AVG spectrum: d = 0.03, f = 1.44 DES spectrum: d = 0.03, f = 1.67
IND spectra: d = 4.71, f = 0.86 AVG spectrum: d = 2.48, f = 0.65 DES spectrum: d = 3.46, f = 0.76
IND spectra: d = 0.20, f = 0.35 AVG spectrum: d = 0.17, f = 0.35 DES spectrum: d = 1.09, f = 0.62
140
CHAPTER 10
RECOMMENDED DESIGN PROCEDURE USING
CAPACITY-DEMAND INDEX RELATIONSHIPS
There is a need for implementing performance-based seismic design approaches whichuse nonlinear static procedures with inelastic demand spectra in future seismic design codes(Reinhorn, 1997; Chopra and Goel, 1999). Capacity-demand index relationships consistent withsmooth design response spectra, such as the relationships developed in this research, provide thenecessary tools for this purpose.
10.1 Inelastic Demand Spectra
In Figures 10.1 through 10.4, inelastic demand spectra are developed using the DES spec-trum a andb regression coefficients in Table 8.2. The figures are organized in the same order thatthe results are presented in Chapter 7: (1) demand spectra for regions with low seismicity, Boston(Fig. 10.1); (2) demand spectra for regions with high seismicity, Los Angeles (Fig. 10.2); (3)demand spectra for near-field conditions,NF (Fig. 10.3); and (4) demand spectra for differenthysteresis types, SD, BE, and BP (Fig. 10.4). As described previously in Section 2.3.2, these spec-tra are constructed using a displacement-based approach by specifying target ductilities,µt, andusing Equations 2.1-2.3, 2.12, and 2.13. Inelastic demand spectra developed based on this proce-dure can be used within the framework of a capacity spectrum method as demonstrated in thedesign example below.
10.2 Design Example
This section presents a design example using the capacity-demand index relationshipsdeveloped in this research. The capacity-spectrum design procedure with inelastic demand spectrais implemented (Sect. 2.3.2). As indicated by Chopra and Goel (1999), a graphical implementa-tion of this procedure in design is not essential and a purely numerical treatment can be per-formed. Nevertheless, the procedure in this example is graphically implemented for illustrativepurposes.
141
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
Sd (cm)
Sa (
g)
2
4
68
SAC Boston, SD soil, design level
µ = 1
(linear-elastic)
a = 1.08b = 0.42
0 2 4 6 8 100
0.2
0.4
0.6
0.8
2
4
68
SAC Boston, SE soil, design level
Sd (cm)
Sa (
g)
µ = 1
(linear-elastic)
a = 0.87b = 0.43
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
2
46
8
SAC Boston, SD soil, survival level
Sd (cm)
Sa (
g)
µ = 1
(linear-elastic)
a = 0.65b = 0.62
0 2 4 6 8 100
0.2
0.4
0.6
0.8
2
4
6
8
SAC Boston, SE soil, survival level
Sd (cm)
Sa (
g)
µ = 1
(linear-elastic)
a = 0.41b = 0.84
Figure 10.1: Inelastic demand spectra based on DES response spectra (EP hysteresis type,α= 0.10): (a-b) SAC Boston,SD soil; (c-d) SAC Boston,SE soil.
(a) (b)
(c) (d)
ξ = 5% ξ = 5%
ξ = 5% ξ = 5%
0 10 20 30 40 500
0.5
1
1.5
2
2.5
2
4
68
SAC Los Angeles, SD soil, design level
Sd (cm)
Sa (
g)
µ = 1
(linear-elastic)
a = 1.71b = 0.59
0 10 20 30 40 500
0.5
1
1.5
2
46
8
SAC Los Angeles, SE soil, design level
Sd (cm)
Sa (
g)
µ = 1
(linear-elastic)
a = 0.80b = 0.88
0 10 20 30 40 500
0.5
1
1.5
2
2.5
2
46
8
SAC Los Angeles, SD soil, survival level
Sd (cm)
Sa (
g)
µ = 1
(linear-elastic)
a = 2.04b = 0.79
0 10 20 30 40 500
0.5
1
1.5
2
4
6
8
SAC Los Angeles, SE soil, survival level
Sd (cm)
Sa (
g)
µ = 1
(linear-elastic)
a = 2.32b = 1.31
(a) (b)
(c) (d)
Figure 10.2: Inelastic demand spectra based on DES response spectra (EP hysteresis type,α= 0.10): (a-b) SAC Los Angeles,SD soil; (c-d) SAC Los Angeles,SE soil.
ξ = 5% ξ = 5%
ξ = 5%
ξ = 5%
142
0 10 20 30 40 500
0.5
1
1.5
2
46
8
SAC Los Angeles, NF, design level
Sd (cm)
Sa (
g)
µ = 1
(linear-elastic)
a = 6.23b = 1.34
ξ = 5%
Figure 10.3: Inelastic demand spectra based on DES response spectra (EP hysteresis type,α= 0.10): SAC Los Angeles,SD soil,NF.
2
4
68
0 10 20 30 40 500
0.5
1
1.5
2
2.5
SAC Los Angeles, SD soil, survival level
Sd (cm)
Sa (
g)
µ = 1
(linear-elastic)
a = 2.20b = 0.82
2
4
68
0 10 20 30 40 500
0.5
1
1.5
2
2.5
SAC Los Angeles, SD soil, survival level
Sd (cm)
Sa (
g)
µ = 1
(linear-elastic)
a = 3.68b = 1.06
2
4
68
0 10 20 30 40 500
0.5
1
1.5
2
2.5
SAC Los Angeles, SD soil, survival level
Sd (cm)
Sa (
g)
µ = 1
(linear-elastic)
a = 3.00b = 0.97
Figure 10.4: Inelastic demand spectra based on DES response spectra (SAC Los Angeles,survival-level,SD soil, α = 0.10): (a) SD hysteresis type; (b) BE hysteresis type; (c) BP hys-teresis type,βs = βr = 1/3.
(a) (b)
(c)
ξ = 5% ξ = 5%
ξ = 5%
143
The design process requires the estimation of the seismic demand and the structure capac-ity as described below. The seismic demand is determined using theIBC 2000smooth designresponse spectra and the regression coefficients presented in Table 8.2. A displacement-baseddesign procedure prescribing target ductility values,µt, is used in the example.
It is assumed that the global lateral force-displacement relationship of the structures in thedesign example can be represented using the EP hysteresis type. The design procedure is illus-trated for six systems withα = 0.10, defined by two structure periods,T = 0.5 sec. and 1.5 sec.,and three target ductilities,µt = 2, 4, and 8. The results are presented for capacity-demand indexrelationships developed using IND and DES spectra to demonstrate the effect of referenceresponse spectrum on design. The seismic parameters used in the example are as follows:
• Site seismicity: high, Los Angeles
• Soil profile: soft (SE) soil
• Seismic demand level: survival-level
STEP 1: Determine linear-elastic demand spectrum
The linear-elastic demand spectrum (corresponding toR = µ =1) is determinedaccording to Figures 2.4 and 3.15a using values of 1.35 and 1.44 (Table 3.7) for theseismic coefficientsSdsandSd1, respectively (corresponding to the seismic param-eters for the example, i.e., Los Angeles,SE soil, survival-level), as shown in Figure10.5.
STEP 2: Determine regression coefficients,a andb
Thea andb regression coefficients corresponding to the design parameters for theexample (i.e., Los Angeles,SE soil, survival-level) are taken from Table 8.2. Theregression coefficients based on the DES reference spectrum are:a = 2.32,b =1.31. Similarly, the regression coefficients based on the IND reference spectra are:a = -0.71,b = 0.94.
STEP 3: DetermineR-µ-T relationships
The R-µ-T relationships are determined using Equations 2.1-2.3 and thea andbcoefficients fromStep 2. These relationships are used in the next step to determinethe inelastic demand spectra.
STEP 4: Determine inelastic demand spectra
The inelastic demand spectra are constructed based on inelastic acceleration,Sai,and displacement,Sdi, demands as shown in Figure 2.4b using Equations 2.12 and2.13, and theR-µ-T relationships fromStep 3. The inelastic demand spectra basedon the DES and INDa and b coefficients used in the example (i.e.,Step 2) are
144
shown in Figures 10.5a and b, respectively. Note that the demand spectra in Figure10.5a are the same as the spectra in Figure 10.2d.
STEP 5: Superimpose capacity curve
The nonlinear-static pushover curve of the structure, developed using nonlinear-static procedures as described in current seismic design provisions (ATC, 1996;ASCE, 2000), is then converted into an equivalent SDOF capacity curve usingEquation 2.11. If the acceleration demand,Sai, is less than or equal to the accelera-tion capacity,a*
y, and the displacement demand,Sdi, is less than or equal to thedisplacement capacity,∆*
ult, then the current design of the structure is adequate.Otherwise, redesign is required.
A
0 20 40 60 80 1000
0.5
1
1.5
S (
g)a
S (cm)d
B
C
T = 0.5 sec.a = 2.32b = 1.31
µ = 1
2
4
8
0 20 40 60 80 1000
0.5
1
1.5
S (
g)a
T = 1.5 sec.a = 2.32b = 1.31
D
E
F
µ = 1
2
4
8
(linear-elastic)
(linear-elastic)
S (cm)d
0 20 40 60 80 1000
0.5
1
1.5
S (
g)a
S (cm)d
T = 0.5 sec.a = -0.71b = 0.94
µ = 1
(linear-elastic)
A
B
C
2
4
8
0 20 40 60 80 1000
0.5
1
1.5
S (
g)a
S (cm)d
D
E
F
2
4
8
T = 1.5 sec.a = -0.71b = 0.94
µ = 1
(linear-elastic)
Figure 10.5: Capacity curve-demand spectra for design example: (a) using DES spectrum;(b) using IND spectra.
(a)
(b)
145
For this example, the required idealized capacity curves that satisfy the target duc-tilities, µt, for the structures are plotted in Figures 10.5a and b for the DES andIND spectra, respectively. These capacity curves are constructed as illustrated inFigure 2.4b. The resultingSai andSdi values, which are graphically represented bythe intersection of the capacity curves and the inelastic demand spectra, are sum-marized in Table 10.1. In comparing Figures 10.5a and b and the values in Table10.1, for a given target displacement ductility,µt, the IND reference spectra resultin largerR coefficients (i.e., smaller lateral strengths,Fy) to be used in design thanthe DES reference spectrum, especially for long-period structures and for largerµtvalues (see Table 10.1, structure F). Thus, it is clear that using capacity-demandindex relationships developed based on IND spectra in design could lead toseverely underestimated seismic demands.
STEP 6: Estimate other demand indices
With the prescribed “target-µ”, µt, it is possible to estimate the residual displace-ment (µr) and cumulative (µp andny) damage demands as part of a performance-based seismic design approach. Theµp, µr, andny demands can be estimated usingEquation 3.43 (with regression coefficients,d andf, from Tables 9.1 and 9.8) andcan be compared with: (1) allowable values to achieve specified performancegoals; and/or (2) corresponding structural capacities.
For this example, the estimatedµp, µr, andny demands using Equation 3.43 andthed andf coefficients from Tables 9.1 and 9.8 are listed in Table 10.2. Notice thatthe structure period,T, and theR coefficient are not required to determineµp, µr,andny onceµ = µt is known. Additionally, note that the demands for this designexample are relatively independent of the reference response spectrum used.
Table 10.1: Structure properties and results for the design example
DES spectrum, constant-R(Farrow and Kurama)
IND spectra, constant-R(Farrow and Kurama)
StructureT
(sec)Felas/W
(g)µt R Sai
(g)
Sde
(cm)
Sdi
(cm)R Sai
(g)
Sde
(cm)
Sdi
(cm)
A
0.5 1.35
2 1.6 0.84 5.20 10.4 1.7 0.82 5.08 10.2
B 4 2.2 0.61 3.76 15.0 2.4 0.57 3.56 14.3
C 8 3.0 0.46 2.84 22.7 3.2 0.42 2.61 20.9
D
1.5 0.96
2 1.8 0.53 29.5 59.0 2.0 0.49 27.1 54.2
E 4 3.0 0.32 17.8 71.4 3.9 0.25 13.9 55.5
F 8 4.8 0.20 11.2 89.5 7.5 0.13 7.15 57.2
146
Table 10.2:µp, µr, andny demands for the design example
DES spectrum, constant-R(Farrow and Kurama)
IND spectra, constant-R(Farrow and Kurama)
StructureT
(sec)Felas/W
(g)µt R µp µr ny R µp µr ny
A
0.5 1.35
2 1.6 3.4 0.5 1.7 1.7 3.2 0.8 1.7
B 4 2.2 13.0 0.9 2.4 2.4 13.0 1.1 2.6
C 8 3.0 40.0 1.5 3.2 3.2 40.0 1.4 3.4
D
1.5 0.96
2 1.8 3.4 0.5 1.7 2.0 3.2 0.8 1.7
E 4 3.0 13.0 0.9 2.4 3.9 13.0 1.1 2.6
F 8 4.8 40.0 1.5 3.2 7.5 40.0 1.4 3.4
147
CHAPTER 11
EFFECT OF GROUND MOTION SCALING METHOD
This chapter compares the seven different ground motion scaling methods described inSection 3.4 based on the scatter in the dimensional (unit dependent) maximum displacementdemand,∆nlin, and the non-dimensional maximum displacement ductility demand,µ = ∆nlin/∆y,from the SDOF nonlinear dynamic time-history analyses. The evaluation of scatter for the MDOFanalyses is presented as normalized floor and roof displacement,∆, and normalized interstorydrift, θ, response profiles.
The SDOF analysis results are discussed first. Then, the findings from the SDOF analysesare reinforced using the results from the MDOF analyses. The evaluation of scatter is presentedas: (1) COV-spectra; and (2)γ-spectra, defined as the ratio of the COV-spectra for two differentsets of parameters (i.e., scaling method, hysteresis type, and site conditions). The lateral forcecapacities,Fy = Felas/R, of the SDOF systems described in this chapter are determined based onthe linear-elastic acceleration response spectrum for each ground motion (i.e., IND spectra). Thelateral force capacities of the MDOF structures are determined based on smooth design (DES)response spectra as described in Section 3.1.2. The results are presented for the UND groundmotion ensemble and the SAC Los Angeles design-level stiff (SD) soil near-field (NF) groundmotion ensemble.
Note that, as stated earlier in Section 3.3, thea andb regression coefficients used in theestimation of the effective structure period,Tµ, for the scaling method take intoaccount the site conditions and hysteresis types used in the dynamic analyses. Thesea andb coef-ficients are listed in Table 3.6.
11.1 EP Hysteresis Type
Figure 11.1 shows the scatter in the maximum displacement demand,∆nlin, using theresults from the UNDSD soil ground motion ensemble for the EP hysteresis type withα = 0.10.The COV-spectra are presented on the left side of Figure 11.1. The solid lines in the COV-spectrarepresent results from the ground motions scaled to the average peak ground acceleration (i.e.,PGA scaling method) while the dotted lines represent results from the same ensemble for theother scaling methods. Theγ-spectra are presented on the right side of Figure 11.1.
For the PGA, EPA, andA95 scaling methods, the results indicate that the scatter tends toincrease as the structure period increases, similar to previous investigations by Nau and Hall
Saˆ To Tµ→( )
148
0 2.00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 2.0T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAEPA PGA
EPA = 1, 2, 4, 6, 8 (thin → thick lines)R
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAEPV PGA
EPV = 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)2.0 2.0
2.0
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAMIV PGA
MIV = 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec) 2.0
2.0
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAA95 PGA
A95 = 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec) 2.0
2.0
= 1, 2, 4, 6, 8 (thin → thick lines)RPGA
PGA
= 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
S (T )a o
ˆS (T )
a oˆ
2.0
2.0
= 1, 2, 4, 6, 8 (thin → thick lines)RPGA
PGA
= 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
µS (T →T )
a oˆ
µS (T →T )
a oˆ
2.0
Figure 11.1: Scatter in∆nlin for the UNDSD soil ensemble (EP hysteresis type,α = 0.10)- PGA method compared to: (a) EPA method; (b) EPV method; (c) MIV method; (d)A95method; (e) method; (f) method.Sa
ˆ To( ) Saˆ To Tµ→( )
(a)
(b)
(c)
(d)
(e)
(f)
CO
V(∆
nlin
)C
OV
(∆n
lin)
CO
V(∆
nlin
)C
OV
(∆n
lin)
CO
V(∆
nlin
)C
OV
(∆n
lin)
γ CO
V(∆
nlin
)γ C
OV
(∆n
lin)
γ CO
V(∆
nlin
)γ C
OV
(∆n
lin)
γ CO
V(∆
nlin
)γ C
OV
(∆n
lin)
149
(1982) and Miranda (1993). This is expected for the PGA and EPA methods since these scalingmethods control the short-period range of the ground motion acceleration spectra. Overall, thedifferences between the three scaling methods are negligible, except for short period structureswith smallR coefficients.
As compared to the PGA method, the EPV method is, on average, considerably moreeffective in reducing the scatter forT > 0.5 sec. (Fig. 11.1b). AtT ≅ 1 sec., the EPV method ismost effective in reducing scatter since this method scales the ground motions to the average spec-tral pseudo-velocity at around a structure period of 1 sec. (see definition of EPV in Section 3.3).
The PGA method is compared to scaling the ground motion ensemble to the average max-imum incremental velocity (i.e., MIV scaling method) in Figure 11.1c. The results indicate that,regardless of theRcoefficient, the MIV method is modestly more effective in reducing the scatterfor T > 0.5 sec. ForT < 0.5 sec., the effectiveness of the MIV method increases withR. The scatterfor the MIV method approaches the scatter for the PGA method asT increases.
The PGA method is compared to scaling the ground motion ensemble to the average spec-tral intensity at each linear-elastic structure period (i.e., method) in Figure 11.1e and toscaling the ground motion ensemble to the average spectral intensity over a range of structureperiods (i.e., method) in Figure 11.1f. It can be expected that all three methods pro-duce similar scatter for extremely short-period structures since, atT = 0, the and
methods are equivalent to the PGA method. ForR = 1 (linear-elastic behavior), thescatter for the and methods is zero for the entire period range since theground motion records are scaled to the average linear-elastic spectral acceleration at the linear-elastic structure period.
Unlike the PGA method which is relatively independent ofR, the COV-spectra for themethod increase asR increases. With an increase inR, µ increases and, in turn, the effec-
tive structure period,Tµ, increases (see Eq. 3.38). Therefore, the effectiveness of themethod, based on the linear-elastic structure period,To, decreases for largerR. Since the
method accounts for the period elongation due to the nonlinear behavior expectedin the structure, the scatter is markedly reduced, regardless of theR coefficient forT > 0.75 sec.For T < 0.75 sec., the effectiveness of the method tends to decrease asR increases.
Comparing the results for the method and the MIV method (solid lines, Fig.11.2a), it can be seen that the MIV method is more effective for 0.2 <T < 0.3 sec. withR = 6 andfor 0.1 <T < 0.45 sec. withR = 8. The MIV method is even more effective when compared to the
method (dashed lines) for 0.1 <T < 0.5 sec. withR = 6 and for 0.1 <T < 0.65 sec. withR= 8. Thus, it is evident that scaling ground motions based on spectral intensity measures (i.e.,
and ) is not necessarily effective for all period ranges and strength levels.This finding is more prominent for the soft (SE) soil profile (Fig. 11.2c) which will be discussed infurther detail later in Section 11.3. Figures 11.2b and d will also be described later in the chapter.
It was previously reported by Nassar and Krawinkler (1991) that the scatter in the maxi-mum displacement ductility demand,µ, tends to increase with theR coefficient. Inspecting thestandard deviation,σ, for theµ spectra (Fig. 11.3a), it is evident that the scatter inµ does increase
Saˆ To( )
Saˆ To Tµ→( )
Saˆ To( )
Saˆ To Tµ→( )
Saˆ To( ) Sa
ˆ To Tµ→( )
Saˆ To( )
Saˆ To( )
Saˆ To Tµ→( )
Saˆ To Tµ→( )
Saˆ To Tµ→( )
Saˆ To( )
Saˆ To( ) Sa
ˆ To Tµ→( )
150
asR increases. However, using the standard deviation alone as a measure of scatter can be mis-leading. The coefficient of variation, COV, which normalizes the sample standard deviation by thesample mean, is a better means to evaluate scatter. As illustrated in Figure 11.3b, scatter relativeto the sample mean (COV) does not increase withR as much as the sample standard deviation,σ(Fig. 11.3a).
It should be noted that dimensionless demand estimates, such as the maximum displace-ment ductility demand,µ = ∆nlin/∆y, are independent of scaling when the individual (IND) linear-elastic ground motion response spectra are used to determine the lateral force capacity,Fy = Felas/R (Nassar and Krawinkler, 1991). For example, scaling a ground motion record by a constant fac-tor will equally affect both the linear-elastic lateral force demand,Felas, and the lateral forcecapacity,Fy, (for a givenR coefficient) resulting in the sameµ. Thus, the scatter inµ shown inFigure 11.3 is not affected by the method used in the scaling of the ground motion records.
11.2 SD, BE, and BP Hysteresis Types
The effect of the structure hysteresis type on the scatter from the different ground motionscaling methods is examined in this section. The COV-spectra for the scatter in∆nlin using theseven scaling methods are presented as is done in the previous section for the EP hysteresis type.The SD, BE, and BP (βs = βr = 1/3) hysteresis type (α = 0.10) trends, provided in Figure 11.4, arealmost identical to the trends observed for the EP type in Figure 11.1. Theγ-spectra for the scatterin ∆nlin between the different hysteresis types range from 1.0 to 1.25. Theγ-spectra for the scatter
2
00 T (sec) 2.0
= 1, 2, 4, 6, 8 (thin → thick lines)R2
00 T (sec) 2.0
= 1, 2, 4, 6, 8 (thin → thick lines)R2
γ CO
V(∆
nlin
)
00 T (sec) 2.0
= 1, 2, 4, 6, 8 (thin → thick lines)R2
00 T (sec) 2.0
MIV)S (T
a oˆ
MIVT )
µS (T →
a oˆ
MIV)S (T
a oˆ
MIVT )
µS (T →
a oˆ
MIV)S (T
a oˆ
MIVT )
µS (T →
a oˆ
MIV)S (T
a oˆ
MIVT )
µS (T →
a oˆ
= 1, 2, 4, 6, 8 (thin → thick lines)R
γ CO
V(∆
nlin
)
γ CO
V(∆
nlin
)
γ CO
V(∆
nlin
)
Figure 11.2: Scatter in∆nlin using the and methods compared tothe MIV method (EP hysteresis type,α = 0.10): (a) UNDSD soil; (b) UND SC soil; (c)UND SE soil; (d) SACSD soil,NF.
Saˆ To( ) Sa
ˆ To Tµ→( )(c)
(a) (b)
(d)
151
in µ between the different hysteresis types have similar values as theγ-spectra for the scatter in∆nlin between the different hysteresis types. Thus, it is concluded that the scatter in∆nlin and thescatter inµ are not significantly affected by the hysteresis type or by the amount of inelasticenergy dissipated by the system.
Note that theγ-spectra for the scatter inµ = ∆nlin/∆y between the different hysteresis typesare slightly different from theγ-spectra for the scatter in∆nlin, because, the use of individual(IND) linear-elastic ground motion response spectra to determine the lateral force capacities,Fy =Felas/R, as described before results in a scatter in the yield displacement,∆y, of the structures. Forcases where the ground motions are scaled using the method or where a smooth designreference response spectrum is used to determine the lateral force capacities,Fy, the yield dis-placements,∆y, remain the same regardless of which ground motion is used (i.e., there is no scat-ter in∆y). Under these conditions, theγ-spectra for the scatter inµ would be equal to theγ-spectrafor the scatter in∆nlin.
11.3 Site Soil Characteristics
The effect of site soil characteristics on the scatter from the different ground motion scal-ing methods is examined in this section. Figures 11.5 and 11.6 show the COV-spectra for the scat-ter in ∆nlin using the UNDSC and SE soil ground motion ensembles, respectively. ComparingFigures 11.1 and 11.5, the trends observed for theSC soil profile are similar to the trends for theSD soil profile. TheSC soil ensemble exhibits slightly more scatter than theSD soil ensemble foralmost the entire period range, except for the EPV and MIV methods. For the EPV method, thescatter from theSC soil ensemble is less than the scatter from theSD soil ensemble forT < 0.5 sec.and is more or less similar to that from theSD soil ensemble forT > 0.5 sec. For the MIV method,the scatter from theSC soil ensemble is similar to that from theSD soil ensemble for the entireperiod range.
thick lines) = 1, 2, 4, 6, 8 (thin →R
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6
8
10
12
14
T (sec)
σ(µ)
0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
T (sec)
CO
V(µ)
thick lines) = 1, 2, 4, 6, 8 (thin →R
Figure 11.3: Scatter inµ (UND SD soil ensemble, EP hysteresis type,α = 0.10): (a) stan-dard deviation,σ; (b) coefficient of variation, COV.
(a) (b)
Saˆ To( )
152
Comparing Figures 11.1, 11.5, and 11.6, the trends observed for theSE soil profile are sig-nificantly different than the trends for theSC andSD soil profiles. TheSE soil ensemble exhibitslarger scatter for almost the entire period range, except for the PGA, MIV, andA95 methods. It isinteresting to observe that forT < 1.1-1.5 sec., the dependency of scatter on theR coefficient issignificantly larger for theSE soil profile as compared to theSC andSD soil profiles. The scattertends to increase asR increases, except for the EPV (withT < 0.7 sec.), MIV, andA95 methods. Asthe amount of nonlinear behavior increases (due to higherR), the effective structure period
2.0
thick lines) = 1, 2, 4, 6, 8 (thin →RPGAMIV PGA
MIV = 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2γ C
OV
(Sd)
00 T (sec) 2.0
2.000
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec) 2.0
thick lines) = 1, 2, 4, 6, 8 (thin →RPGAEPV PGA
EPV = 1, 2, 4, 6, 8 (thin → thick lines)R
2.0
thick lines) = 1, 2, 4, 6, 8 (thin →RPGAA95 PGA
A95 = 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec) 2.0
0 2.00
1.5
T (sec)
CO
V(S
d)2
γ CO
V(S
d)
00 2.0T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAEPA PGA
EPA = 1, 2, 4, 6, 8 (thin → thick lines)R
2.0 2.000
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAS (T )
a oˆ
PGA
= 1, 2, 4, 6, 8 (thin → thick lines)RS (T )
a oˆ
2.0 2.000
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
T )
= 1, 2, 4, 6, 8 (thin → thick lines)RPGA
µS (T →
a oˆ
PGA
= 1, 2, 4, 6, 8 (thin → thick lines)RT )
µS (T →
a oˆ
2.0
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAEPA
thick lines)
PGAEPA
= 1, 2, 4, 6, 8 (thin →R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec) 2.0
2.0
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAEPV
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec) 2.0
PGAEPV
= 1, 2, 4, 6, 8 (thin → thick lines)R
2.0
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAMIV PGA
MIV = 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)2
γ CO
V(S
d)
00 T (sec) 2.0
2.0 2.0
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAA95 PGA
A95 = 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
2.0 2.000
1.5
T (sec)
CO
V(S
d)
2γ C
OV
(Sd)
00 T (sec)
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAS (T )
a oˆ
PGA
= 1, 2, 4, 6, 8 (thin → thick lines)RS (T )
a oˆ
2.0 2.000
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
T )
= 1, 2, 4, 6, 8 (thin → thick lines)RPGA
µS (T →
a oˆ
PGA
= 1, 2, 4, 6, 8 (thin → thick lines)RT )
µS (T →
a oˆ
2.000
1.5
T (sec)
CO
V(S
d)
= 1, 2, 4, 6, 8 (thin → thick lines)R
T )
PGA
µS (T →
a oˆ
0 2.00
1.5
T (sec)
CO
V(S
d)
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAMIV
0 2.00
1.5
T (sec)C
OV
(Sd)
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAA95
0 2.00
1.5
T (sec)
CO
V(S
d)
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAEPV
0 2.00
1.5
T (sec)
CO
V(S
d)
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAEPA
2.000
1.5
T (sec)
CO
V(S
d)
= 1, 2, 4, 6, 8 (thin → thick lines)RPGAS (T )
a oˆ
Figure 11.4: Effect of hysteresis type on the scatter in∆nlin (UND SD soil ensemble,α =0.10): (a) SD type; (b) BE type; (c) BP type (βs = βr = 1/3).
(a) (b) (c)
CO
V(∆
nlin
)C
OV
(∆n
lin)
CO
V(∆
nlin
)C
OV
(∆n
lin)
CO
V(∆
nlin
)C
OV
(∆n
lin)
CO
V(∆
nlin
)C
OV
(∆n
lin)
CO
V(∆
nlin
)C
OV
(∆n
lin)
CO
V(∆
nlin
)C
OV
(∆n
lin)
CO
V(∆
nlin
)C
OV
(∆n
lin)
CO
V(∆
nlin
)C
OV
(∆n
lin)
CO
V(∆
nlin
)C
OV
(∆n
lin)
153
thick lines) = 1, 2, 4, 6, 8 (thin →R
PGAEPA PGA
EPA
= 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)2.0 2.0
2.0 2.0
thick lines) = 1, 2, 4, 6, 8 (thin →RPGAEPV PGA
EPV = 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
2.0 2.0
thick lines) = 1, 2, 4, 6, 8 (thin →RPGAMIV PGA
MIV = 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
thick lines) = 1, 2, 4, 6, 8 (thin →RPGAA95 PGA
A95 = 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)2.0 2.0
2.0
thick lines) = 1, 2, 4, 6, 8 (thin →R = 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
PGAS (T )
a oˆ
PGA
S (T )a o
ˆ
2.0
thick lines) = 1, 2, 4, 6, 8 (thin →RPGA
PGA
= 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)2.0 2.0
T )µ
S (T →a o
ˆT )
µS (T →
a oˆ
(a)
(b)
(c)
(d)
(e)
(f)
Figure 11.5: Scatter in∆nlin for the UNDSC soil ensemble (EP hysteresis type,α = 0.10) -PGA method compared to: (a) EPA method; (b) EPV method; (c) MIV method; (d)A95method; (e) method; (f) method.Sa
ˆ To( ) Saˆ To Tµ→( )
CO
V(∆
nlin
)C
OV
(∆n
lin)
CO
V(∆
nlin
)C
OV
(∆n
lin)
CO
V(∆
nlin
)C
OV
(∆n
lin)
γ CO
V(∆
nlin
)γ C
OV
(∆n
lin)
γ CO
V(∆
nlin
)γ C
OV
(∆n
lin)
γ CO
V(∆
nlin
)γ C
OV
(∆n
lin)
154
2.0 2.0
thick lines) = 1, 2, 4, 6, 8 (thin →R
PGA
PGA
= 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
EPAEPA
2.0 2.0
thick lines) = 1, 2, 4, 6, 8 (thin →R
PGAPGA
= 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
EPVEPV
thick lines) = 1, 2, 4, 6, 8 (thin →RPGA
PGA
= 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
MIVMIV
2.0 2.0
thick lines) = 1, 2, 4, 6, 8 (thin →R
PGAPGA
= 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
A95A95
2.0 2.0
2.0 2.0
thick lines) = 1, 2, 4, 6, 8 (thin →R
PGA
PGA
= 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
S (T )a o
ˆS (T )
a oˆ
2.0 2.0
thick lines) = 1, 2, 4, 6, 8 (thin →R = 1, 2, 4, 6, 8 (thin → thick lines)R
00
1.5
T (sec)
CO
V(S
d)
2
γ CO
V(S
d)
00 T (sec)
PGAT )
µS (T →
a oˆ
PGA
T )µ
S (T →a o
ˆ
Figure 11.6: Scatter in∆nlin for the UNDSE soil ensemble (EP hysteresis type,α = 0.10) -PGA method compared to: (a) EPA method; (b) EPV method; (c) MIV method; (d)A95method; (e) method; (f) method.Sa
ˆ To( ) Saˆ To Tµ→( )
(a)
(b)
(c)
(d)
(e)
(f)
CO
V(∆
nlin
)C
OV
(∆n
lin)
CO
V(∆
nlin
)C
OV
(∆n
lin)
CO
V(∆
nlin
)C
OV
(∆n
lin)
γ CO
V(∆
nlin
)γ C
OV
(∆n
lin)
γ CO
V(∆
nlin
)γ C
OV
(∆n
lin)
γ CO
V(∆
nlin
)γ C
OV
(∆n
lin)
155
increases in a similar fashion. This period elongation runs the structure into the energy-rich regionof theSE soil ensemble, increasing the nonlinear response and the variability in the response forT< 1.1-1.5 sec.
Figures 11.6e and f demonstrate that the effectiveness of the and themethods significantly decreases as the period decreases within the ranges of 0.75 <
T < 1.5 sec. and 0.75 <T < 1.25 sec., respectively. Note that, as mentioned at the beginning of thischapter, thea andb regression coefficients used in the estimation of the effective structure period,Tµ, for the method take into account site soil characteristics. Even so, the scatter inthe demand estimates forT < 0.75 sec. using the method is larger than the scatterusing the PGA method forR = 4, 6, and 8.
In examining Figure A.16, it is evident that there is considerable scatter in the linear-elas-tic ground motion response spectra for theSE soil ensemble at longer periods. Thus, once a struc-ture runs into this energy-rich region following yield, the responses themselves tend to haveconsiderable scatter. It is for this reason that even the method cannot account forthe high variability in the nonlinear response.
Comparing results between the scatter using the method and the MIVmethod, it can be seen that the MIV method is more effective for 0.1 <T < 0.5 sec. withR = 6, 8for theSC soil profile (Fig. 11.2b) and for 0.2 <T < 1.15 sec. withR= 4, 6, 8 for theSE soil profile(Fig. 11.2c). The MIV method is even more effective when compared to the method for0.1 <T < 0.65 sec. withR= 6, 8 for theSC soil profile and for 0.16 <T < 1.25-1.5 sec. withR= 4,6, 8 for theSE soil profile. The reduction in the scatter when using the MIV method as comparedto the and methods is significant for theSE soil profile atT ≅ 0.5 sec. (γ ≅0.2 forR = 6, 8). Thus, it is evident that scaling ground motions based on spectral intensity mea-sures (i.e., and methods) is not necessarily effective for all strength levels,period ranges, and site soil characteristics. This finding will be reinforced using results from theMDOF analyses later in Section 11.5.
11.4 Epicentral Distance
Figure 11.7 shows the COV-spectra for the scatter in∆nlin using the SAC Los Angelesdesign-levelSD soil NF ground motion ensemble. Comparing Figures 11.1 and 11.7, theNFensemble tends to exhibit less or similar scatter than the far-field ensemble for the PGA, EPA, andA95 methods, especially for long periods. The dependency of the scatter for these scaling methodson the period is significantly decreased for the near-field ensemble. Figure 11.7a shows that theEPA method results in significantly larger scatter as compared to the PGA method, except forshort-period structures with smallR coefficients. In contrast, the effectiveness of the EPV methodas compared to the PGA method is highly dependent on bothT andR (Fig. 11.7b).
The PGA method is compared to the MIV method in Figure 11.7c and to theA95 methodin Figure 11.7d. The MIV method is modestly more effective in reducing scatter for largerRcoef-ficients in shorter period ranges and for smallerR coefficients in longer period ranges. The differ-ences between theA95 method and the PGA method are negligible, except forT < 0.5 sec.
Saˆ To( )
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Figure 11.7: Scatter in∆nlin for the SAC Los Angeles, design-level,SD soil, NF ensemble(EP hysteresis type,α = 0.10) - PGA method compared to: (a) EPA method; (b) EPVmethod; (c) MIV method; (d)A95 method; (e) method; (f) method.Sa
ˆ To( ) Saˆ To Tµ→( )
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The PGA method is compared to the method in Figure 11.7e and to themethod in Figure 11.7f. As compared to the far-field ground motions in Figures
11.1e and f, the effectiveness of both methods with respect to the PGA method is significantlydecreased for the entire range of periods forR > 1, especially for large values ofR. Except forlong periods and smallR coefficients, the method is less effective or only slightly moreeffective than the PGA method. As compared to the method, the scatter for the
method is significantly reduced, especially for largerR coefficients. The depen-dence of scatter for both methods onR is highly noticeable.
Figure 11.2d shows that the MIV method does not provide any significant improvement ascompared to the method. However, the MIV method is more effective than the
method in reducing scatter for 0.15 <T < 1.15 sec. withR = 6, 8. Thus, the benefit ofusing the MIV method over the method is still evident for near-field conditions.
11.5 Results for the MDOF Frame Structures
As discussed previously in this chapter, the MIV method provides better reduction in scat-ter than the and methods for a significant range of structure periods, espe-cially on soft (SE) soil profiles and for largeR coefficients. This is because the MIV method isable to capture the impulsive characteristics typical of ground motions on soft soil profiles. Toshow that the results obtained from the SDOF analyses are applicable to MDOF systems, theamount of scatter in the lateral displacements of the four-story and eight-story structuresdescribed in Section 3.1.2 is investigated below.
As described in Section 3.6.2, the UND soft (SE) soil ground motion ensemble is used toexcite the MDOF systems. After scaling each ground motion based on the MIV method or the
method, the entire ground motion ensemble was scaled so that the average linear-elastic acceleration response spectrum of the ensemble is not less than 1.4 times the 5%-dampedUBC 1997(ICBO, 1997) Los Angeles design-levelSE soil design response spectrum for periodsbetween 0.2To and 1.5To, whereTo is the structure fundamental period (see Fig. 3.22).
Figure 11.8 shows the normalized maximum floor and roof lateral displacement,∆, andinterstory drift, θ, scatter profiles for the four-story and eight-story structures subjected to theUND SE soil ground motion ensemble scaled as described above. The lateral displacement,∆, val-ues are normalized with the maximum mean floor or roof displacement over the height of thestructure,∆max(see Section 3.7.2). Similarly, the interstory drift,θ, values are normalized with themaximum mean interstory drift over the height of the structure,θmax. It is evident from the narrow± standard deviation,σ, bands that, in each case, the MIV method provides a significant reductionin scatter as compared to the method. This observation is especially apparent forthe four-story structure (Fig. 11.8a). Based on the SDOF results shown previously, the reductionin scatter as compared to the method is expected to be even more apparent.
The reduced scatter in the seismic demands using the MIV method, coupled with its sim-plicity, makes the MIV method advantageous over the other scaling methods described herein fora wide range of site and structure characteristics. The advantages of the MIV method over the
Saˆ To( )
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and methods come from: (1) the MIV method is based only on groundmotion parameters and not on structure parameters (e.g., linear-elastic structure period,To), whichmay not be known in advance; and (2) ground motions scaled using the MIV method can be usedto analyze structures with different properties (e.g., with differentTo) since the ground motions donot need to be rescaled for each structure. Thus, it is recommended that nonlinear dynamic analy-ses are conducted using ground motion records scaled based on the MIV method, particularly forlargerR coefficients and soft soil profiles for the period ranges shown in Figure 11.2.
Figure 11.9 shows the maximum floor and roof displacement,∆, and interstory drift,θ,COV profiles for the four-story and eight-story structures, which display similar trends as the cor-responding normalized displacement and drift values in Figure 11.8. Furthermore, the average(taken over the height of the structure) COV values of the maximum floor and roof displacement,
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Figure 11.8: Scatter in MDOF demands for the UNDSE soil ensemble using the MIVmethod and the method: (a) four-story structure; (b) eight-story structure.Sa
ˆ To Tµ→( )
(a)
(b)
∆i/∆maxθi/θmax
∆i/∆max θi/θmax
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∆, for the four-story and eight-story structures in Figure 11.9 (which are equal to 0.75 and 0.25 forthe four-story structure using the and MIV methods, respectively, and 0.52 and0.35 for the eight-story structure using the and MIV methods, respectively) com-pare well with the COV values for the SDOF models atT = 0.49 and 0.87 sec. (i.e., the fundamen-tal periods of the four-story and eight-story structures), respectively, in Figures 11.6c and f (forR= 8). Thus, the results from the MDOF analyses seem to satisfactorily verify the results from theSDOF analyses.
Since the four-story and eight-story structures were designed based on smooth designresponse spectra as specified by theUBC 1997provisions, the observations made above for thescatter in the maximum lateral floor and roof displacement,∆, and interstory drift,θ, demands arealso applicable to the scatter in the maximum displacement ductility demand,µ. This is becausethe structure lateral force capacity, and thus, the “yield displacement” remain the same regardlessof which ground motion is used. Thus, the scatter inµ is the same as the scatter in∆.
Note that as the scatter in the maximum displacement ductility demand,µ, increases, thescatter in the other demand indices (e.g.,µp, µr, ny) are expected to increase. Thus, reducing scat-ter in ∆, θ, andµ is very important for the use of performance-based seismic design procedureswith multiple demand indices, as outlined in Chapter 10.
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i
ii
Figure 11.9: Covariance in the MDOF demands for the UNDSE soil ensemble using theMIV method and the method: (a) four-story structure; (b) eight-story struc-ture.
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CHAPTER 12
SUMMARY, CONCLUSIONS, AND FUTURE RESEARCH
12.1 Summary
The broad objective of this research is to address some of the research needs for the imple-mentation of performance-based procedures in current U.S. seismic design provisions. Theresearch has four specific objectives: (1) to develop new nonlinear single-degree-of-freedom(SDOF) capacity-demand index relationships based on linear-elastic smooth design responsespectra consistent with current seismic code provisions; (2) to develop new relationships thatquantify cumulative damage, hysteretic energy, and residual displacement demands; (3) to investi-gate the effects of the structure fundamental period, strength level, hysteretic behavior, and siteconditions on the demand estimates; and (4) to investigate the effect of the ground motion scalingmethod on the scatter in the demand estimates.
To accomplish these objectives, SDOF and multi-degree-of-freedom (MDOF) nonlineardynamic time-history analyses are conducted to investigate the relationships between the structurelateral force capacity and the selected seismic demand indices using ground motions ensemblescategorized by: (1) site soil characteristics; (2) seismic demand level; (3) site seismicity; and (4)epicentral distance. Three major suites of ground motion records are used: (1) ground motionscompiled by the authors at the University of Notre Dame; (2) ground motions compiled by theSAC steel project (SAC, 1997; Somerville et al., 1997); and (3) ground motions compiled by Nas-sar and Krawinkler (1991).
The University of Notre Dame (UND) ground motions are used to investigate the effect ofthe ground motion scaling method on the scatter in the demand estimates. The SAC steel project(SAC) ground motions are primarily used to develop new capacity-demand index relationships,including relationships based on linear-elastic smooth design response spectra. Finally, the Nassarand Krawinkler (N&K) ground motions are used to validate the analytical procedure used in thisresearch by comparing the results with already existing results.
The effects of variations in structural parameters on the capacity-demand index relation-ships are examined in the SDOF analyses, such as fundamental structure period, post-yield stiff-ness ratio, hysteresis type, and strength level (i.e., response modification coefficient,R, andreference response spectra). Five hysteresis types are considered: (1) linear-elastic (LE); (2) bilin-ear elasto-plastic (EP); (3) stiffness degrading (SD); (4) bilinear-elastic (BE); and (5) combinedbilinear-elastic/elasto-plastic (BP).
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In order to investigate the effect of reference response spectra on the capacity-demandindex relationships, three types of linear-elastic acceleration response spectra are used to calculatethe structure lateral force capacity,Fy = Felas/R, namely: (1) response spectra based on the indi-vidual ground motion records (IND); (2) average ground motion response spectra based on theground motion ensembles (AVG); and (3) smooth design response spectra from current U.S. seis-mic design provisions (DES).
Capacity-demand index relationships are provided in terms of nonlinear regression curvesfor: (1) the maximum displacement ductility demand,µ; and (2) demand indices that quantifycumulative damage, hysteretic energy, and residual displacement (i.e.,Λ = µp, µr, andny). Theresults of the regression analyses are interpreted for the purpose of producing viable structuraldesigns using performance-based engineering concepts (e.g., inelastic capacity-demand spectramethods).
The effect of seven ground motion scaling methods on the scatter in the demand estimatesis examined, including the following parameters: (1) input parameters such as site soil character-istics and epicentral distance; and (2) structural parameters such as fundamental period, hysteresistype, and strength level (i.e., response modification coefficient,R).
In addition to the SDOF models, two MDOF models representative of four-story andeight-story cast-in-place reinforced-concrete special moment-resisting office building framestructures are considered. These structures are designed according to theUBC 1997equivalentlateral force procedure for a region with high seismicity (Los Angeles) and for the soft (SE) soilprofile. Nonlinear dynamic time-history analyses of the MDOF models are conducted to reinforcethe findings from the SDOF analyses on ground motion scaling method. In total, 300,000 SDOFanalyses and 80 MDOF analyses are conducted by the research.
In summary, this research shows that: (1) previous capacity-demand index relationshipsdeveloped using linear-elastic ground motion response spectra can be significantly different thanthose developed using smooth design response spectra and can lead to unconservative designs,particularly for survival-level, soft soil, and near-field conditions; (2) the correlation between themaximum displacement ductility demand,µ, and the other demand indices,Λ, is relatively strong;and (3) scaling methods that work well for ground motions recorded on stiff soil and far-field con-ditions lose their effectiveness for ground motions on soft soil and near-field conditions.
Successful completion of this research has resulted in the development of recommenda-tions and observations for: (1) new relationships to estimate the maximum displacement ductilitydemand,µ; (2) new relationships for other demand indices (Λ = µp, µr, andny); (3) the effect ofreference response spectra on these relationships; (4) the effect of ground motion scaling methodon the scatter in the demand estimates; and (5) the effects of ground motion characteristics (e.g.,site conditions, demand level) and structure characteristics (e.g., lateral strength, hysteresis type,structure period) on the demand estimates. Ultimately, these results have been incorporated intoperformance-based guidelines and recommendations for current and future seismic design provi-sions.
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12.2 Conclusions
The major conclusions based on the research are summarized below. The conclusions arepresented in the following order: (1) capacity-demand index relationships; (2) effect of referenceresponse spectra; (3) regression analyses; and (4) ground motion scaling methods.
Capacity-Demand Index Relationships
The following conclusions are based on mean demands from SDOF nonlinear dynamictime-history analyses where the structural lateral force capacities,Fy = Felas/R, are determinedusing the individual (IND) linear-elastic ground motion acceleration response spectra.
(1) The constant-Rapproach is more conservative than the constant-µ approach even whenan iteration procedure that selects the smallestR coefficient is used. Thus, the constant-Rapproach should be used to develop seismic capacity-demand index relationships.
(2) For the EP hysteresis type, increased post-yield stiffness ratio,α, decreases the meanµandµr demands. The decrease in theµr demand with an increase inα is especially large for smallvalues ofα and large values ofR. Theµp demands increase for increasedα. Theny demands arevirtually independent ofα.
(3) The meanµ demand is slightly larger for the SD hysteresis type than for the EP hyster-esis type. There are large increases inµp and decreases inny due to the shooting branch of the SDtype. At higher levels ofR and for largerα, the SD type behaves similar to the EP type.
(4) For the BE hysteresis type, all mean demand indices, exceptµr, are significantly largerthan for the EP hysteresis type due to the lack of inelastic energy dissipation. The BE type is supe-rior to the EP type in terms ofµr since the BE type always returns to the zero displacement posi-tion after an earthquake event.
(5) As compared to the BE hysteresis type, all mean demand indices, exceptµr, decreasesomewhat proportionally to the increase inβr for the BP hysteresis type. There is a slight, butmostly negligible, increase inµr for the BP type with respect to the BE type. Thus, the objectiveof reducing theµ, µp, andny demands while keeping theµr demand within allowable limits can bereasonably achieved with the BP hysteresis type.
(6) For structures with periodT < ~1.0-1.5 sec., the meanµ, µp, andµr demands are largerfor theSE soil profile than for theSD soil profile. For longer period structures (i.e.,T > ~1.0-1.5sec.), the meanµ, µp, andµr demands are smaller for theSE soil profile than for theSD soil pro-file. This is because individual linear-elastic ground motion acceleration response spectra wereused to determine the structure force capacities,Fy = Felas/R, resulting in increasedFy values forthe structures near the predominant ground motion period.
(7) For seismic demand level, site seismicity, and epicentral distance, the long-periodmeanµ, µp, andny demands for the survival-level, Los Angeles, and near-field ground motionensembles are smaller than the corresponding demands for the design-level, Boston, and far-field
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ensembles, respectively. In these cases, the average linear-elastic acceleration response spectra ofthe survival-level, Los Angeles, and near-field ground motion ensembles decrease at a faster rate(asT increases) than the acceleration response spectra of the design-level, Boston, and far-fieldensembles, respectively, resulting in a significant reduction in the demands as the structure yields.
Reference Response Spectra
The following conclusions are based on mean demands from SDOF nonlinear dynamictime-history analyses where the structure lateral force capacities,Fy = Felas/R, are determinedbased on different reference response spectra.
(1) Capacity-demand index relationships developed using individual (IND) ground motionacceleration response spectra for the basis ofFy can lead to unconservative designs, especially forsurvival-level, soft (SE) soil, and near-field conditions. Using smooth design (DES) response spec-tra to determineFy provides demand estimates that are more consistent with current design provi-sions.
(2) In general, the differences between the meanµ demands based on the IND spectra andthe meanµ demands based on the average (AVG) spectra are small.
(3) Modest increases in meanµ demands are usually observed when using smooth (AVGand DES) response spectra in the development of capacity-demand index relationships for siteswith a stiff (SD) soil profile, with low seismicity (Boston), and under design-level ground motions.In some cases, the IND spectra can result in slightly largerµ demands as compared with thesmooth reference response spectra.
(4) Using design (DES) response spectra from current U.S. seismic provisions (ICBO,1997; ICC, 2000) to determineFy can result in meanµ demands that are extreme and possiblyuncontrollable for sites with a soft (SE) soil profile, with high seismicity (e.g., Los Angeles),within close proximity to an active fault (NF), or under survival-level ground motions. Thus,either the DES spectra in current seismic provisions need to be modified (e.g., using AVG spectrainstead) or theR coefficients recommended in the provisions should be reduced under these con-ditions.
(5) The differences between the meanµ demands based on the IND spectra and thesmooth (AVG and DES) response spectra are not significantly affected by the hysteresis type.
(6) Cross-correlations betweenµ and the other demand indices,Λ = µp, µr, andny, aredependent on the reference response spectra used to determineFy = Felas/R. TheΛ-µ relationshipsare significantly affected by reference response spectra for: (1)µ versusµp andµ versusny rela-tionships for the Boston design-levelSD andSE soil ground motions using the EP hysteresis type;(2) µ versusny relationship for the Los Angeles design-levelSD soil NF ground motions using theEP hysteresis type; and (3)µ versusny relationship for the Los Angeles survival-levelSD soilground motions using the BE and BP hysteresis types.
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Regression Analyses
The following conclusions are based on regression analyses of mean demand estimatesfrom SDOF nonlinear dynamic time-history analysis results.
(1) R-µ-T relationships were developed based on the dynamic analysis results using thetwo-step nonlinear regression analysis scheme and the form of the regression equations by Nassarand Krawinkler (Eqs. 2.1-2.3). The resultingR-µ-T relationships provide a reasonable representa-tion of the mean demands from the dynamic analyses. TheR-µ spectra from these regression anal-yses reinforce the previous findings for the meanR-µ spectra from the dynamic analyses.
(2) Relationships between the demand indices,µ, µp, µr, andny, were developed in theform of Equations 3.43 and 3.44 using a simple one-step nonlinear regression analysis. The corre-lation betweenµ and the other demand indices is relatively strong, especially forµp. In somecases, the cross-correlations between the demand indices show weak to no correlation, indicatingthat these demand indices can carry independent measures of seismic demand. Designers shouldbe careful in using the regression relationships developed for these cases.
(3) The regression relationships developed by the research, which take into account a widevariety of seismic and structural parameters, can be integrated into current and future seismicdesign approaches. As an example, a “capacity spectrum procedure” can be easily implementedeither by using the inelastic demand spectra provided in Figures 10.1 through 10.4 or by directcalculation using Equations 2.1-2.3, 2.12, and 2.13 (with regression coefficientsa and b fromTable 8.2), and smooth design response spectra from current seismic design provisions. Further-more, using the relationships betweenµ and the other demand indices (Eqs. 3.43 and 3.44 andregression coefficientsd, f, g, andh from Tables 9.1-9.4 and 9.7-9.10), a more extensive perfor-mance-based seismic design procedure can be realized. For example, for a given “target-µ” as partof a displacement-based design procedure, it is possible to design (or redesign) structures withenhanced objectives in mind (e.g., limiting residual displacement and/or cumulative damage). Adesign example that demonstrates this procedure is provided in the research.
Ground Motion Scaling Method and Scatter in the Demand Estimates
The following conclusions are based on demand estimates from SDOF and MDOF nonlin-ear dynamic time-history analyses under ground motions scaled using different methods.
(1) For the very dense (SC) and stiff (SD) soil profiles atT ≅ 1 sec., the EPV method is sig-nificantly more effective in reducing scatter in the maximum displacement demand than the PGAmethod. This can be expected since the EPV method scales the ground motions to the averagespectral pseudo-velocity at around a period of 1 sec.
(2) For theSC, SD, and soft (SE) soil profiles at longer periods, the MIV method is moreeffective in reducing the scatter in the maximum displacement demand than the PGA method,regardless of theR coefficient. The effectiveness of the MIV method increases withR for shorterperiods. The scatter for the MIV method approaches the scatter for the PGA method asTincreases.
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(3) Except for short and very short-period structures, the method is significantlymore effective in reducing the scatter than the PGA method for theSC andSD soil profiles, espe-cially for smallR coefficients.
(4) Since the method accounts for the amount of nonlinear behaviorexpected in the structure, the scatter using this scaling method is smaller than the scatter using the
method, particularly for largerR values. However, the differences between the two meth-ods are usually not very large.
(5) The SD, BE, and BP hysteresis type trends are almost identical to the trends observedfor the EP type. The dependency of scatter (in terms of covariance of the maximum displacementdemand and covariance of the maximum displacement ductility demand) on the hysteretic behav-ior of the structure is very small. Thus, it is concluded that the amount of scatter in the maximumdisplacement demand and the amount of scatter in the maximum displacement ductility demandare not significantly affected by the hysteresis type or by the amount of inelastic energy dissipatedby the structure.
(6) For theSC, SD, andSE soil profiles, the MIV method is more effective in reducing thescatter in the maximum displacement demand than the and methods for awide range of periods, especially for largerRcoefficients. The reduction in the scatter when usingthe MIV method is particularly significant for theSE soil profile. Thus, it is evident that scalingground motions based on spectral intensity measures (i.e., and ) is not nec-essarily effective for all site soil characteristics, structure lateral strengths, and periods.
(7) For theSE and near-field (NF) ground motion ensembles, the effectiveness of theand methods with respect to the PGA method is significantly decreased.
Except for long periods, the and methods are less effective than the PGAmethod for theSE soil profile. For theNF ground motion ensemble, the effectiveness of the
and methods in the long period range decreases asR increases.
(8) In general, the dependency of the scatter in the maximum displacement demand onR islarger for theNF andSE soil ground motion ensembles than for theSC andSD soil ensembles.
(9) For the PGA, EPA, andA95 scaling methods, the dependency of the scatter in the max-imum displacement demand on the period,T, is significantly decreased for the near-field (NF)ground motion ensemble.
(10) The COV values (i.e., the sample standard deviation normalized by the sample mean)for the scatter in the maximum lateral displacements of the MDOF four-story and eight-storystructures using the MIV and methods compare well with the corresponding COVvalues from the SDOF models. Thus, the results from the MDOF analyses seem to satisfactorilyverify the results from the SDOF analyses.
(11) The ratios between the COV values based on the MIV method and themethod using the MDOF structures compare well with the ratios using the SDOF models. Thus,the MDOF analyses support the finding from the SDOF analyses that the MIV method provides a
Saˆ To( )
Saˆ To Tµ→( )
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ˆ To Tµ→( )
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ˆ To Tµ→( )
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166
significant reduction in scatter as compared to the method for theSE soil profilewithin a wide range of periods, especially for largerR coefficients.
(12) The increased effectiveness of the MIV method over the andmethods, coupled with its simplicity, make the MIV method advantageous over the other scalingmethods described herein for a wide range of site and structure characteristics. The advantages ofthe MIV method over the and methods come from: (1) the MIV method isbased only on ground motion parameters and not on structure parameters (e.g., linear-elasticstructure period,To), which may not be known in advance; and (2) ground motions scaled usingthe MIV method can be used to analyze structures with different properties (e.g., with differentTo) since the ground motions do not need to be rescaled for each structure. Thus, it is recom-mended that nonlinear dynamic analyses are conducted using ground motion records scaled basedon the MIV method, particularly for largerR coefficients and soft soil profiles for the periodranges given in the research.
(13) The scatter in the maximum displacement ductility demand,µ, in terms of the samplestandard deviation,σ, increases asR increases. The scatter forµ in terms of the coefficient of vari-ation, COV, which normalizes the sample standard deviation by the sample mean, does notincrease withR as much as the increase in terms of the sample standard deviation,σ.
(14) Dimensionless demand estimates, such asµ = ∆nlin/∆y, are independent of the scalingmethod when the individual (IND) linear-elastic ground motion response spectra are used todetermine the lateral force capacity,Fy = Felas/R.
12.3 Future Research
Future research is needed to develop capacity-demand index relationships for structuresthat have significant multi-mode effects since the research described in this research is limited tostructures that can be reasonably modeled with a SDOF representation. Furthermore, research isneeded to develop reliable methods to estimate local force and deformation demands (i.e.,demands in the structural members and joints).
Saˆ To Tµ→( )
Saˆ To( ) Sa
ˆ To Tµ→( )
Saˆ To( ) Sa
ˆ To Tµ→( )
167
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174
APPENDIX A
CHARACTERISTICS, TIME HISTORIES, AND RESPONSE SPECTRA
OF GROUND MOTION RECORDS
Table A.1: University of Notre Dame (UND) very dense (SC) soil ensemble
Ground motionrecord
Station AbbreviationSite soil
description
Epicentraldistance
(km)
PeakGroundAccel.,PGA(g)
Max.IncrementalVel., MIV(cm/sec)
EffectivePeak Accel.,
EPA(g)
Loma Prieta, 1989 San Francisco-Presidio LPPR rock 102 0.20 45.4 0.16
Quebec, 1988 St. Andre Du Lac-St. Jean SASJ bedrock 64 0.09 1.83 0.04
Michoacan, 1985 Zihuatanero-Aeropuerto MIZI rock 166 0.17 26.4 0.21
San Francisco, 1957 San Francisco--Golden Gate Park SFSF bedrock, chert 16 0.08 5.50 0.09
Parkfield, 1966 Parkfield--Cholame Shandon #2 PACH alluvium oversstone
21 0.49 100.59 0.38
San Fernando, 1971 Castaic--Old Ridge Road SFCA bedrock, sand-stone
27 0.32 25.60 0.34
Morgan Hill, 1984 Gilroy #6 MHGI bedrock 37 0.42 24.65 0.33
Central Chile, 1985 Valparaiso CCVA bedrock, volcanic 26 0.18 22.41 0.26
Michoacan, 1985 La Union MILU bedrock, meta-andesite brec
80 0.15 16.90 0.25
Michoacan, 1985 La Villita MILV bedrock, tonalite 40 0.12 23.02 0.11
San Salvador, 1986 National Geographic Institute SSNG bedrock, fluviatilepumice
14 0.53 115.54 0.43
San Salvador, 1986 Institute of Urban Construction SSUC bedrock, fluviatilepumice
15 0.68 81.52 0.53
San Salvador, 1986 Geotechnical Research Center SSGR bedrock, fluviatilepumice
15 0.69 91.42 0.66
Whittier, 1987 Mt. Wilson WHMW quartz diorite 18 0.17 3.67 0.06
Loma Prieta, 1989 Corralitos LPCO alluvium, land-slide deposits
1 0.48 83.78 0.44
Loma Prieta, 1989 Santa Cruz LPSC bedrock, lime-stone
23 0.41 29.72 0.52
Loma Prieta, 1989 San Francisco--Cliff House LPCH bedrock, fran-ciscan melange
104 0.11 33.65 0.07
Loma Prieta, 1989 San Francisco--Pacific Heights LPPH bedrock, fran-ciscan melange
100 0.06 23.39 0.04
Loma Prieta, 1989 San Francisco--Rincon Hill LPRH bedrock, fran-ciscan melange
98 0.09 13.30 0.07
Loma Prieta, 1989 Yerba Buena Island LPYB bedrock, fran-ciscan melange
99 0.07 14.43 0.06
175
Table A.2: University of Notre Dame (UND) stiff (SD) soil ensemble
Ground motion record Station AbbreviationSite soil
description
Epicentraldistance
(km)
PeakGroundAccel.,PGA(g)
Max.IncrementalVel., MIV(cm/sec)
EffectivePeak Accel.,
EPA(g)
Northridge, 1994 Newhall-LA Co. Fire Station NONW alluvium 20 0.59 153.07 0.70
Loma Prieta, 1989 Hollister LPHO alluvium 50 0.18 40.33 0.21
Landers, 1992 Yermo LAYE deep alluvium 84 0.24 66.90 0.23
Northridge, 1994 Sylmar NOSY alluvium 16 0.84 148.70 0.81
San Fernando, 1971 Orion Blvd. SFOR deep cohesionless 21 0.25 45.81 0.29
Imperial Valley, 1940, El Centro ELCN alluvium 10 0.68 96.12 0.71
Kern County, 1952 Taft--Lincoln School Tunnel KCLS alluvium 56 0.15 20.32 0.19
San Fernando, 1971 Figueroa SFFI alluvium 41 0.15 23.57 0.14
San Fernando, 1971 Hollywood SFHO alluvium 35 0.21 38.02 0.29
San Fernando, 1971 Ave. of the Stars SFAS silt and sand layers 38 0.14 11.09 0.17
Whittier, 1987 Tarzana--Cedar Hill Nursery WHTA alluvium over silt-stone
110 0.54 21.77 0.48
Whittier, 1987 7215 Bright Ave. WHBT alluvium 21 0.61 54.88 0.61
Imperial Valley, 1979 Bonds Corner IVBC alluvium 3 0.58 75.96 0.68
Imperial Valley, 1979 James Road IVJR alluvium 22 0.52 52.04 0.52
Imperial Valley, 1979 Imperial Valley College IVIV alluvium 21 0.33 64.30 0.25
Central Chile, 1985 El Almendral CCEA compacted fill 84 0.29 47.18 0.33
Whittier, 1987 Long Beach--Ranchos LosCerritos
WHLC alluvium 47 0.24 17.56 0.23
Loma Prieta, 1989 Oakland--two story office bldg LPOA alluvium 49 0.26 54.59 0.21
Whittier, 1987 Alhambra WHAH alluvium 7 0.40 11.90 0.25
Whittier, 1987 Altadena WHAD alluvium 13 0.31 8.94 0.21
176
Table A.3: University of Notre Dame (UND) soft (SE) soil ensemble
Ground motion record Station AbbreviationSite soil
description
Epicentraldistance
(km)
PeakGroundAccel.,PGA(g)
Max.IncrementalVel., MIV(cm/sec)
EffectivePeak
Accel., EPA(g)
Loma Prieta, 1989 Foster City LPFO bay mud 65 0.28 75.10 0.24
Bucharest, Romania, 1977 Bucharest RUBU soft 174 0.21 120.46 0.13
Michoacan, Mexico City, 1985 Secretaria de Comunica-ciones Transportes
MISE soft clay 400 0.17 117.96 0.09
Loma Prieta, 1989 Treasure Island LPTR fill 98 0.16 57.15 0.14
Eureka, 1954 Basement, Eureka FederalBldg.
EUF1 partially consoli-dated sediment
1102 0.17 37.70 0.20
Eureka, 1954 Basement, Eureka FederalBldg.
EUF2 partly consoli-dated sediment
1102 0.26 36.33 0.32
Loma Prieta, 1989 Oakland, Outer HarborWharf
OHW1 alluvium, baymud/fill
98 0.27 72.11 0.21
Loma Prieta, 1989 Oakland, Outer HarborWharf
OHW2 alluvium, baymud/fill
98 0.29 57.58 0.24
Bucharest, Romania, 1977 Building Research Institute RUB1 soft 107 0.11 15.05 0.10
Bucharest, Romania, 1977 Building Research Institute RUB2 soft 107 0.18 53.46 0.13
Michoacan, Mexico City, 1985 Tlahuac BombasXochimilco
TLB1 alluvial, soft soil,clay
381 0.11 59.41 0.05
Michoacan, Mexico City, 1985 Tlahuac BombasXochimilco
TLB2 alluvial, soft soil,clay
381 0.14 65.49 0.06
Whittier, 1987 Alhambra--Fremont WHA1 unconsolidatedsediment (soft)
7.3 0.25 37.95 0.35
Whittier, 1987 Alhambra--Fremont WHA2 unconsolidatedsediment (soft)
7.3 0.29 20.00 0.28
Michoacan, Mexico City, 1985 Tlahuac DeportivoXochimilco
MIDX alluvial, soft, clay 97 0.11 44.67 0.06
Loma Prieta, 1989 Oakland--Outer HarborWharf
LPOH bay mud 95 0.27 72.11 0.21
Michoacan, Mexico City, 1985 Central de Abastos--Frig-orifico
MIFI soft clay 389 0.10 72.61 0.05
Michoacan, Mexico City, 1985 Central de Abastos--Oficina MIOF soft clay 389 0.08 74.31 0.04
Loma Prieta, 1989 San Francisco Int. Airport LPIA bay mud 79 0.33 48.90 0.29
Loma Prieta, 1989 San Francisco comm. build-ing
LPCB fill over mud 95 0.16 28.65 0.15
177
Table A.4: Nassar and Krawinkler (N&K) 15s very dense (SC) soil ensemble
Ground motion record Station AbbreviationSite soil
description
Epicentraldistance
(km)
PeakGroundAccel.,PGA(g)
Max.IncrementalVel., MIV(cm/sec)
EffectivePeakAccel.,
EPA(g)
Kern County, 1952 Taft Lincoln School Tunnel eq03 rock 43.0 0.16 20.32 0.14
Kern County, 1952 Taft Lincoln School Tunnel eq04 rock 43.0 0.18 27.05 0.15
Lower California, 1934 Imperial Valley eq05 rock 64.0 0.16 22.29 0.15
Lower California, 1934 Imperial Valley eq06 rock 64.0 0.18 21.56 0.17
Western Washington, 1949 Olympia Washington High-way
eq07 rock 16.0 0.16 23.53 0.18
Western Washington, 1949 Olympia Washington High-way
eq08 rock 16.0 0.28 32.97 0.22
Puget Sound, 1965 Olympia Washington High-way
eq09 rock 61.0 0.20 16.40 0.16
San Fernando, 1971 Castaic Old Ridge eq10 rock 29.0 0.27 37.30 0.22
Long Beach, 1933 Public Utilities Bldg. eq11 rock 27.0 0.20 34.01 0.18
Long Beach, 1933 Public Utilities Bldg. eq12 rock 27.0 0.16 23.61 0.16
Imperial Valley, 1979 Holtville P.O. eq14 rock 19.0 0.25 58.42 0.23
Imperial Valley, 1979 Calexico Fire Station eq15 rock 15.0 0.27 28.93 0.25
Coyote Lake, 1979 San Yasidro School eq16 rock 12.0 0.25 48.26 0.21
Coyote Lake, 1979 San Yasidro School eq17 rock 12.0 0.23 37.60 0.21
Coalinga, 1983 Parkfield Zone 16 eq20 rock 39.1 0.18 21.96 0.16
178
Table A.5: SAC Boston design-level stiff (SD) soil ensemble
Ground motion record Station AbbreviationSite soil
description
Epicentraldistance
(km)
PeakGroundAccel.,PGA(g)
Max.IncrementalVel., MIV(cm/sec)
EffectivePeak
Accel., EPA(g)
Hanging Wall (generated) -- bo01 firm 30 0.12 15.07 0.39
Hanging Wall (generated) -- bo02 firm 30 0.07 9.47 0.39
Foot Wall (generated) -- bo03 firm 30 0.14 15.04 0.54
Foot Wall (generated) -- bo04 firm 30 0.11 19.00 0.54
New Hampshire, 1982 Franklin Falls bo05 firm 8.4 0.58 22.84 10.75
New Hampshire, 1982 Franklin Falls bo06 firm 8.4 0.32 20.73 10.75
Nahanni, 1985 Station 1 bo07 firm 9.6 0.09 5.70 0.09
Nahanni, 1985 Station 1 bo08 firm 9.6 0.08 6.91 0.09
Nahanni, 1985 Station 2 bo09 firm 6.1 0.06 9.53 0.20
Nahanni, 1985 Station 2 bo10 firm 6.1 0.07 11.16 0.20
Nahanni, 1985 Station 3 bo11 firm 18 0.13 4.56 0.92
Nahanni, 1985 Station 3 bo12 firm 18 0.14 3.87 0.92
Saguenay, 1988 sm07 bo13 firm 96 0.20 11.02 1.57
Saguenay, 1988 sm07 bo14 firm 96 0.29 16.73 1.57
Saguenay, 1988 sm08 bo15 firm 98 0.52 27.71 3.21
Saguenay, 1988 sm08 bo16 firm 98 0.25 10.78 3.21
Saguenay, 1988 sm09 bo17 firm 118 0.18 14.95 3.25
Saguenay, 1988 sm09 bo18 firm 118 0.23 21.09 3.25
Saguenay, 1988 sm10 bo19 firm 132 0.18 17.80 3.34
Saguenay, 1988 sm10 bo20 firm 132 0.27 25.88 3.34
179
Table A.6: SAC Boston survival-level stiff (SD) soil ensemble
Ground motion record Station AbbreviationSite soil
description
Epicentraldistance
(km)
PeakGroundAccel.,PGA(g)
Max.IncrementalVel., MIV(cm/sec)
EffectivePeak Accel.,
EPA(g)
Foot Wall (generated) -- bo21 firm 30 0.32 34.40 0.99
Foot Wall (generated) -- bo22 firm 30 0.36 41.92 0.99
Foot Wall (generated) -- bo23 firm 30 0.34 25.18 0.84
Foot Wall (generated) -- bo24 firm 30 0.24 25.26 0.84
Foot Wall (generated) -- bo25 firm 30 0.29 39.78 0.63
Foot Wall (generated) -- bo26 firm 30 0.31 35.59 0.63
Nahanni, 1985 Station 1 bo27 firm 9.6 0.25 16.32 0.27
Nahanni, 1985 Station1 bo28 firm 9.6 0.24 19.78 0.27
Nahanni, 1985 Station2 bo29 firm 6.1 0.17 27.28 0.56
Nahanni, 1985 Station2 bo30 firm 6.1 0.21 31.94 0.56
Nahanni, 1985 Station3 bo31 firm 18 0.38 13.04 2.63
Nahanni, 1985 Station3 bo32 firm 18 0.39 11.06 2.63
Saguenay, 1988 sm07 bo33 firm 96 0.57 31.53 4.48
Saguenay, 1988 sm07 bo34 firm 96 0.78 47.89 4.48
Saguenay, 1988 sm08 bo35 firm 98 1.50 79.59 9.21
Saguenay, 1988 sm08 bo36 firm 98 0.71 30.95 9.21
Saguenay, 1988 sm09 bo37 firm 118 0.52 42.82 9.30
Saguenay, 1988 sm09 bo38 firm 118 0.65 60.41 9.30
Saguenay, 1988 sm10 bo39 firm 132 0.51 51.00 9.58
Saguenay, 1988 sm10 bo40 firm 132 0.78 74.14 9.58
180
Table A.7: SAC Boston design-level soft (SE) soil ensemble
Ground motionrecord
Station AbbreviationSite soil
description
Epicentraldistance
(km)
PeakGroundAccel.,PGA(g)
Max.IncrementalVel., MIV(cm/sec)
EffectivePeak
Accel.,EPA(g)
BN-ST06 N -- bs01 soft -- 0.14 24.76 0.10
BN-ST06 P -- bs02 soft -- 0.09 19.38 0.09
BN-ST12 N -- bs03 soft -- 0.14 42.02 0.13
BN-ST12 P -- bs04 soft -- 0.10 16.86 0.09
NA-STA1 N -- bs05 soft -- 0.08 12.29 0.05
NA-STA1 P -- bs06 soft -- 0.09 13.49 0.07
NA-STA2 N -- bs07 soft -- 0.08 12.13 0.04
NA-STA2 P -- bs08 soft -- 0.08 18.94 0.04
NA-STA3 N -- bs09 soft -- 0.08 10.70 0.07
NA-STA3 P -- bs10 soft -- 0.08 5.71 0.07
NH-FFAL N -- bs11 soft -- 0.34 29.17 0.26
NH-FFAL P -- bs12 soft -- 0.31 31.79 0.29
SG-SM07 N -- bs13 soft -- 0.18 15.04 0.20
SG-SM07 P -- bs14 soft -- 0.25 19.88 0.23
SG-SM08 N -- bs15 soft -- 0.26 33.28 0.20
SG-SM08 P -- bs16 soft -- 0.28 31.43 0.27
SG-SM09 N -- bs17 soft -- 0.18 13.66 0.18
SG-SM09 P -- bs18 soft -- 0.21 29.05 0.18
SG-SM10 N -- bs19 soft -- 0.21 28.00 0.18
SG-SM10 P -- bs20 soft -- 0.24 36.46 0.17
181
Table A.8: SAC Boston survival-level soft (SE) soil ensemblea
aGround motions generated using EERA site response analysis program (Bardet et al., 2000), asdescribed in Section 3.3
Ground motion record Station AbbreviationSite soil
description
Epicentraldistance
(km)
PeakGroundAccel.,PGA(g)
Max.IncrementalVel., MIV(cm/sec)
EffectivePeak Accel.,
EPA(g)
Foot Wall -- bs21 soft 30 0.34 96.8 0.25
Foot Wall -- bs22 soft 30 0.40 80.5 0.36
Foot Wall -- bs23 soft 30 0.34 53.8 0.25
Foot Wall -- bs24 soft 30 0.28 76.2 0.16
Foot Wall -- bs25 soft 30 0.33 86.9 0.24
Foot Wall -- bs26 soft 30 0.44 73.7 0.25
Nahanni, 1985 -- bs27 soft 9.6 0.24 30.6 0.20
Nahanni, 1985 -- bs28 soft 9.6 0.20 56.9 0.14
Nahanni, 1985 -- bs29 soft 6.1 0.21 67.3 0.11
Nahanni, 1985 -- bs30 soft 6.1 0.26 52.2 0.16
Nahanni, 1985 -- bs31 soft 18 0.26 74.0 0.17
Nahanni, 1985 -- bs32 soft 18 0.19 39.1 0.17
Saguenay, 1988 -- bs33 soft 96 0.47 51.0 0.53
Saguenay, 1988 -- bs34 soft 96 0.56 58.6 0.64
Saguenay, 1988 -- bs35 soft 98 0.67 119.3 0.56
Saguenay, 1988 -- bs36 soft 98 0.65 43.7 0.46
Saguenay, 1988 -- bs37 soft 118 0.50 110.0 0.34
Saguenay, 1988 -- bs38 soft 118 0.53 66.2 0.55
Saguenay, 1988 -- bs39 soft 132 0.52 80.9 0.37
Saguenay, 1988 -- bs40 soft 132 0.75 102.4 0.48
182
Table A.9: SAC Los Angeles design-level stiff (SD) soil ensemble
Ground motion record Station AbbreviationSite soil
description
Epicentraldistance
(km)
PeakGroundAccel.,PGA(g)
Max.IncrementalVel., MIV(cm/sec)
EffectivePeak
Accel., EPA(g)
Imperial Valley, 1940 El Centro la01 firm 10 0.46 89.18 2.01
Imperial Valley, 1940 El Centro la02 firm 10 0.68 96.12 2.01
Imperial Valley, 1979 Array #5 la03 firm 4.1 0.39 103.0 1.01
Imperial Valley, 1979 Array #5 la04 firm 4.1 0.49 74.75 1.01
Imperial Valley, 1979 Array #6 la05 firm 1.2 030 106.4 0.84
Imperial Valley, 1979 Array #6 la06 firm 1.2 0.23 81.73 0.84
Landers, 1992 Barstow la07 firm 36 0.42 59.27 3.20
Landers, 1992 Barstow la08 firm 36 0.43 71.85 3.20
Landers, 1992 Yermo la09 firm 25 0.52 135.4 2.17
Landers, 1992 Yermo la10 firm 25 0.36 76.82 2.17
Loma Prieta, 1989 Gilroy la11 firm 12 0.67 79.66 1.79
Loma Prieta, 1989 Gilroy la12 firm 12 0.97 88.94 1.79
Northridge, 1994 Newhall la13 firm 6.7 0.68 138.4 1.03
Northridge, 1994 Newhall la14 firm 6.7 0.66 132.1 1.03
Northridge, 1994 Rinaldi la15 firm 7.5 0.53 124.2 0.79
Northridge, 1994 Rinaldi la16 firm 7.5 0.58 165.1 0.79
Northridge, 1994 Sylmar la17 firm 6.4 0.57 102.5 0.99
Northridge, 1994 Sylmar la18 firm 6.4 0.82 139.3 0.99
North Palm Springs, 1986 dhsp la19 firm 6.7 1.02 99.50 2.97
North Palm Springs, 1986 dhsp la20 firm 6.7 0.99 150.3 2.97
183
Table A.10: SAC Los Angeles survival-level stiff (SD) soil ensemble
Ground motion record Station AbbreviationSite soil
description
Epicentraldistance
(km)
PeakGroundAccel.,PGA(g)
Max.IncrementalVel., MIV(cm/sec)
EffectivePeak
Accel., EPA(g)
Kobe, 1995 Kobe JMA la21 firm 3.4 1.28 275.1 1.15
Kobe, 1995 Kobe JMA la22 firm 3.4 0.92 241.8 1.15
Loma Prieta, 1989 Los Gatos la23 firm 3.5 0.42 87.07 0.82
Loma Prieta, 1989 Los Gatos la24 firm 3.5 0.47 210.8 0.82
Northridge, 1994 Rinaldi la25 firm 7.5 0.87 202.1 1.29
Northridge, 1994 Rinaldi la26 firm 7.5 0.94 268.7 1.29
Northridge, 1994 Sylmar la27 firm 6.4 0.93 166.9 1.61
Northridge, 1994 Sylmar la28 firm 6.4 1.33 226.7 1.61
Tabas, 1974 Tabas la29 firm 1.2 0.81 93.13 1.08
Tabas, 1974 Tabas la30 firm 1.2 0.99 129.7 1.08
Elysian Park (simulation) -- la31 firm 17.5 1.30 208.1 1.43
Elysian Park (simulation) -- la32 firm 17.5 1.19 260.2 1.43
Elysian Park (simulation) -- la33 firm 10.7 0.78 188.1 0.97
Elysian Park (simulation) -- la34 firm 10.7 0.68 161.5 0.97
Elysian Park (simulation) -- la35 firm 11.2 0.99 343.8 1.10
Elysian Park (simulation) -- la36 firm 11.2 1.10 329.5 1.10
Palos Verdes (simulation) -- la37 firm 1.5 0.71 263.1 0.90
Palos Verdes (simulation) -- la38 firm 1.5 0.78 302.1 0.90
Palos Verdes (simulation) -- la39 firm 1.5 0.50 117.0 0.88
Palos Verdes (simulation) -- la40 firm 1.5 0.63 279.0 0.88
184
Table A.11: SAC Los Angeles design-level soft (SE) soil ensemble
Ground motionrecord
Station AbbreviationSite soil
description
Epicentraldistance
(km)
PeakGroundAccel.,PGA(g)
Max.IncrementalVel., MIV(cm/sec)
EffectivePeak
Accel.,EPA(g)
40-IVIR N -- ls01 soft -- 0.36 99.53 0.25
40-IVIR P -- ls02 soft -- 0.39 154.34 0.24
IV-AR05 N -- ls03 soft -- 0.33 184.71 0.21
IV-AR05 P -- ls04 soft -- 0.36 78.58 0.27
IV-AR06 N -- ls05 soft -- 0.38 194.23 0.19
IV-AR06 P -- ls06 soft -- 0.28 115.80 0.15
LN-BARS N -- ls07 soft -- 0.42 133.83 0.21
LN-BARS P -- ls08 soft -- 0.24 79.07 0.20
LN-YERM N -- ls09 soft -- 0.44 178.09 0.21
LN-YERM P -- ls10 soft -- 0.40 118.27 0.26
LP-GIL3 N -- ls11 soft -- 0.67 132.93 0.39
LP-GIL3 P -- ls12 soft -- 0.49 176.72 0.31
NR-NEWH N -- ls13 soft -- 0.54 170.06 0.30
NR-NEWH P -- ls14 soft -- 0.40 84.09 0.36
NR-RRS N -- ls15 soft -- 0.60 203.00 0.28
NR-RRS P -- ls16 soft -- 0.30 106.61 0.23
NR-SYLM N -- ls17 soft -- 0.52 253.22 0.28
NR-SYLM P -- ls18 soft -- 0.47 137.27 0.32
PS-DHSP N -- ls19 soft -- 0.55 124.54 0.30
PS-DHSP P -- ls20 soft -- 0.48 113.68 0.36
185
Table A.12: SAC Los Angeles survival-level soft (SE) soil ensemblea
aGround motions generated using EERA site response analysis program (Bardet et al., 2000), asdescribed in Section 3.3
Ground motion record Station AbbreviationSite soil
description
Epicentraldistance
(km)
PeakGroundAccel.,PGA(g)
Max.IncrementalVel., MIV(cm/sec)
EffectivePeak
Accel., EPA(g)
Kobe, 1995 -- ls21 soft 3.4 1.24 376.0 0.60
Kobe, 1995 -- ls22 soft 3.4 1.18 316.7 0.58
Loma Prieta, 1989 -- ls23 soft 3.5 0.45 181.1 0.23
Loma Prieta, 1989 -- ls24 soft 3.5 0.81 420.4 0.37
Northridge, 1994 -- ls25 soft 7.5 0.73 234.6 0.40
Northridge, 1994 -- ls26 soft 7.5 1.33 438.6 0.66
Northridge, 1994 -- ls27 soft 6.4 1.02 315.0 0.51
Northridge, 1994 -- ls28 soft 6.4 1.07 386.2 0.69
Tabas, 1974 -- ls29 soft 1.2 0.46 152.2 0.28
Tabas, 1974 -- ls30 soft 1.2 0.88 191.5 0.81
Elysian Park -- ls31 soft 17.5 1.00 325.1 0.57
Elysian Park -- ls32 soft 17.5 0.73 230.0 0.43
Elysian Park -- ls33 soft 10.7 0.64 400.9 0.39
Elysian Park -- ls34 soft 10.7 0.83 402.4 0.45
Elysian Park -- ls35 soft 11.2 1.59 706.5 0.85
Elysian Park -- ls36 soft 11.2 1.61 765.7 0.75
Palos Verdes -- ls37 soft 1.5 0.73 410.8 0.36
Palos Verdes -- ls38 soft 1.5 1.15 560.8 0.49
Palos Verdes -- ls39 soft 1.5 0.54 197.5 0.25
Palos Verdes -- ls40 soft 1.5 1.05 574.7 0.45
186
Table A.13: SAC Los Angeles near-field design-level stiff (SD) soil ensemble
Ground motionrecord
Station AbbreviationSite soil
description
Epicentraldistance
(km)
PeakGroundAccel.,PGA(g)
Max.IncrementalVel., MIV(cm/sec)
EffectivePeak
Accel.,EPA(g)
Tabas, 1978 Tabas nf01 firm 1.2 0.90 90.70 1.22
Loma Prieta, 1989 Los Gatos nf03 firm 3.5 0.72 268.87 0.59
Loma Prieta, 1989 Lexington Dam nf05 firm 6.3 0.69 243.65 0.35
C. Mendocino, 1992 Petrolia nf07 firm 8.5 0.64 206.38 0.42
Erzincan, 1992 Erzincan nf09 firm 2.0 0.43 153.99 0.35
Landers, 1992 Landers nf11 firm 1.1 0.72 59.66 0.63
Northridge, 1994 Rinaldi nf13 firm 7.5 0.89 254.66 0.73
Northridge, 1994 Olive View nf15 firm 6.4 0.73 142.39 0.65
Kobe, 1995 Kobe nf17 firm 3.4 1.09 297.78 0.74
Kobe, 1995 Takatori nf19 firm 4.3 0.79 296.32 0.50
Elysian Park 1 Elysian Park 1 nf21 firm 17.5 0.86 186.04 0.97
Elysian Park 2 Elysian Park 2 nf23 firm 10.7 1.80 582.84 1.46
Elysian Park 3 Elysian Park 3 nf25 firm 11.2 1.01 251.98 1.08
Elysian Park 4 Elysian Park 4 nf27 firm 13.2 0.92 417.15 0.96
Elysian Park 5 Elysian Park 5 nf29 firm 13.7 1.16 497.39 1.00
Palos Verdes 1 Palos Verdes 1 nf31 firm 1.5 0.97 430.46 0.64
Palos Verdes 2 Palos Verdes 2 nf33 firm 1.5 0.97 438.77 0.57
Palos Verdes 3 Palos Verdes 3 nf35 firm 1.5 0.87 404.87 0.59
Palos Verdes 4 Palos Verdes 4 nf37 firm 1.5 0.79 295.83 0.54
Palos Verdes 5 Palos Verdes 5 nf39 firm 1.5 0.92 394.90 0.64
187
0 10 20 30 40−0.2
−0.1
0
0.1
0.2LPPR
acce
lera
tion
(g)
0 10 20 30−0.1
−0.05
0
0.05
0.1SASJ
0 20 40 60 80−0.2
−0.1
0
0.1
0.2MIZI
acce
lera
tion
(g)
0 10 20 30 40−0.1
−0.05
0
0.05
0.1SFSF
0 20 40 60−0.5
0
0.5PACH
acce
lera
tion
(g)
0 20 40 60 80−0.5
0
0.5SFCA
0 20 40 60−0.5
0
0.5MHGI
acce
lera
tion
(g)
0 20 40 60 80−0.2
−0.1
0
0.1
0.2CCVA
0 20 40 60 80−0.2
−0.1
0
0.1
0.2MILU
time (sec)
acce
lera
tion
(g)
0 20 40 60 80−0.2
−0.1
0
0.1
0.2MILV
time (sec)
Figure A.1: Ground motion records: University of Notre Dame (UND) verydense (SC) soil ensemble.
188
0 10 20 30−1
−0.5
0
0.5
1SSNG
acce
lera
tion
(g)
0 5 10 15−1
−0.5
0
0.5
1SSUC
0 2 4 6 8 10−1
−0.5
0
0.5
1SSGR
acce
lera
tion
(g)
0 5 10 15 20−0.2
−0.1
0
0.1
0.2WHMW
0 10 20 30 40−0.5
0
0.5LPCO
acce
lera
tion
(g)
0 10 20 30 40−0.5
0
0.5LPSC
0 10 20 30 40−0.2
−0.1
0
0.1
0.2LPCH
acce
lera
tion
(g)
0 10 20 30 40−0.1
−0.05
0
0.05
0.1LPPH
0 10 20 30 40−0.1
−0.05
0
0.05
0.1LPRH
time (sec)
acce
lera
tion
(g)
0 10 20 30 40−0.1
−0.05
0
0.05
0.1LPYB
time (sec)
Figure A.1 (continued): Ground motion records: University of Notre Dame(UND) very dense (SC) soil ensemble.
189
0 20 40 60−1
−0.5
0
0.5
1NONW
acce
lera
tion
(g)
0 20 40 60−0.2
−0.1
0
0.1
0.2LPHO
0 20 40 60 80−0.5
0
0.5LAYE
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1NOSY
0 20 40 60−0.5
0
0.5SFOR
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1la02
0 20 40 60−0.2
−0.1
0
0.1
0.2KCLS
acce
lera
tion
(g)
0 20 40 60−0.2
−0.1
0
0.1
0.2SFFI
0 20 40 60 80−0.5
0
0.5SFHO
time (sec)
acce
lera
tion
(g)
0 20 40 60−0.2
−0.1
0
0.1
0.2SFAS
time (sec)
Figure A.2: Ground motion records: University of Notre Dame (UND) stiff(SD) soil ensemble.
190
0 5 10 15 20−1
−0.5
0
0.5
1WHTA
acce
lera
tion
(g)
0 10 20 30 40−1
−0.5
0
0.5
1WHBT
0 10 20 30 40−1
−0.5
0
0.5
1IVBC
acce
lera
tion
(g)
0 10 20 30 40−1
−0.5
0
0.5
1IVJR
0 10 20 30 40−0.5
0
0.5IVIV
acce
lera
tion
(g)
0 20 40 60 80 100−0.5
0
0.5CCEA
0 5 10 15 20−0.5
0
0.5WHLC
acce
lera
tion
(g)
0 10 20 30 40−0.5
0
0.5LPOA
0 5 10 15 20−0.5
0
0.5WHAH
time (sec)
acce
lera
tion
(g)
0 5 10 15 20−0.5
0
0.5WHAD
time (sec)
Figure A.2 (continued): Ground motion records: University of Notre Dame(UND) stiff (SD) soil ensemble.
191
0 20 40 60−0.5
0
0.5LPFO
acce
lera
tion
(g)
0 5 10 15 20−0.5
0
0.5RUBU
0 50 100 150 200−2
−1
0
1
2x 10
−4 MISE
acce
lera
tion
(g)
0 20 40 60−0.2
−0.1
0
0.1
0.2LPTR
0 20 40 60 80−0.2
−0.1
0
0.1
0.2EUF1
acce
lera
tion
(g)
0 20 40 60 80−0.5
0
0.5EUF2
0 10 20 30 40−0.5
0
0.5OHW1
acce
lera
tion
(g)
0 10 20 30 40−0.5
0
0.5OHW2
0 5 10 15 20−0.2
−0.1
0
0.1
0.2RUB1
time (sec)
acce
lera
tion
(g)
0 5 10 15 20−0.2
−0.1
0
0.1
0.2RUB2
time (sec)
Figure A.3: Ground motion records: University of Notre Dame (UND) soft(SE) soil ensemble.
192
0 20 40 60 80 100−0.2
−0.1
0
0.1
0.2TLB1
acce
lera
tion
(g)
0 20 40 60 80 100−0.2
−0.1
0
0.1
0.2TLB2
0 10 20 30 40−0.5
0
0.5WHA1
acce
lera
tion
(g)
0 10 20 30 40−0.5
0
0.5WHA2
0 20 40 60 80 100−0.2
−0.1
0
0.1
0.2MIDX
acce
lera
tion
(g)
0 10 20 30 40−0.5
0
0.5LPOH
0 20 40 60−0.1
−0.05
0
0.05
0.1MIFI
acce
lera
tion
(g)
0 50 100 150 200−0.1
−0.05
0
0.05
0.1MIOF
0 10 20 30 40−0.5
0
0.5LPIA
time (sec)
acce
lera
tion
(g)
0 10 20 30 40−0.2
−0.1
0
0.1
0.2LPCB
time (sec)
Figure A.3 (continued): Ground motion records: University of Notre Dame(UND) soft (SE) soil ensemble.
193
0 20 40 60−0.2
−0.1
0
0.1
0.2eq03
acce
lera
tion
(g)
0 20 40 60−0.2
−0.1
0
0.1
0.2eq04
0 20 40 60 80 100−0.2
−0.1
0
0.1
0.2eq05
acce
lera
tion
(g)
0 20 40 60 80 100−0.2
−0.1
0
0.1
0.2eq06
0 20 40 60 80 100−0.2
−0.1
0
0.1
0.2eq07
acce
lera
tion
(g)
0 20 40 60 80 100−0.5
0
0.5eq08
0 20 40 60 80 100−0.2
−0.1
0
0.1
0.2eq09
acce
lera
tion
(g)
0 20 40 60 80−0.5
0
0.5eq10
0 20 40 60 80 100−0.2
−0.1
0
0.1
0.2eq11
time (sec)
acce
lera
tion
(g)
0 20 40 60 80 100−0.2
−0.1
0
0.1
0.2eq12
time (sec)
Figure A.4: Ground motion records: Nassar and Krawinkler 15s very dense(SC) soil ensemble.
194
0 10 20 30 40−0.5
0
0.5eq14
acce
lera
tion
(g)
0 10 20 30 40−0.5
0
0.5eq15
0 10 20 30−0.5
0
0.5eq16
acce
lera
tion
(g)
0 10 20 30−0.5
0
0.5eq17
0 20 40 60 80−0.2
−0.1
0
0.1
0.2eq20
acce
lera
tion
(g)
Figure A.4 (continued): Ground motion records: Nassar and Krawinkler 15svery dense (SC) soil ensemble.
195
0 10 20 30−0.2
−0.1
0
0.1
0.2bo01
acce
lera
tion
(g)
0 10 20 30−0.1
−0.05
0
0.05
0.1bo02
0 10 20 30−0.2
−0.1
0
0.1
0.2bo03
acce
lera
tion
(g)
0 10 20 30−0.2
−0.1
0
0.1
0.2bo04
0 5 10 15 20−1
−0.5
0
0.5
1bo05
acce
lera
tion
(g)
0 5 10 15 20−0.5
0
0.5bo06
0 10 20 30−0.1
−0.05
0
0.05
0.1bo07
acce
lera
tion
(g)
0 10 20 30−0.1
−0.05
0
0.05
0.1bo08
0 5 10 15 20−0.1
−0.05
0
0.05
0.1bo09
time (sec)
acce
lera
tion
(g)
0 5 10 15 20−0.1
−0.05
0
0.05
0.1bo10
time (sec)
Figure A.5: Ground motion records: SAC Boston design-level stiff (SD) soilensemble.
196
0 5 10 15 20−0.2
−0.1
0
0.1
0.2bo11
acce
lera
tion
(g)
0 5 10 15 20−0.2
−0.1
0
0.1
0.2bo12
0 5 10 15 20−0.5
0
0.5bo13
acce
lera
tion
(g)
0 5 10 15 20−0.5
0
0.5bo14
0 10 20 30−1
−0.5
0
0.5
1bo15
acce
lera
tion
(g)
0 10 20 30−0.5
0
0.5bo16
0 10 20 30 40−0.2
−0.1
0
0.1
0.2bo17
acce
lera
tion
(g)
0 10 20 30 40−0.5
0
0.5bo18
0 10 20 30 40−0.2
−0.1
0
0.1
0.2bo19
time (sec)
acce
lera
tion
(g)
0 10 20 30 40−0.5
0
0.5bo20
time (sec)
Figure A.5 (continued): Ground motion records: SAC Boston design-level stiff(SD) soil ensemble.
197
0 10 20 30−0.5
0
0.5bo21
acce
lera
tion
(g)
0 10 20 30−0.5
0
0.5bo22
0 10 20 30−0.5
0
0.5bo23
acce
lera
tion
(g)
0 10 20 30−0.5
0
0.5bo24
0 10 20 30−0.5
0
0.5bo25
acce
lera
tion
(g)
0 10 20 30−0.5
0
0.5bo26
0 10 20 30−0.5
0
0.5bo27
acce
lera
tion
(g)
0 10 20 30−0.5
0
0.5bo28
0 5 10 15 20−0.2
−0.1
0
0.1
0.2bo29
time (sec)
acce
lera
tion
(g)
0 5 10 15 20−0.5
0
0.5bo30
time (sec)
Figure A.6: Ground motion records: SAC Boston survival-level stiff (SD) soilensemble.
198
0 5 10 15 20−0.5
0
0.5bo31
acce
lera
tion
(g)
0 5 10 15 20−0.5
0
0.5bo32
0 5 10 15 20−1
−0.5
0
0.5
1bo33
acce
lera
tion
(g)
0 5 10 15 20−1
−0.5
0
0.5
1bo34
0 10 20 30−2
−1
0
1
2bo35
acce
lera
tion
(g)
0 10 20 30−1
−0.5
0
0.5
1bo36
0 10 20 30 40−1
−0.5
0
0.5
1bo37
acce
lera
tion
(g)
0 10 20 30 40−1
−0.5
0
0.5
1bo38
0 10 20 30 40−1
−0.5
0
0.5
1bo39
time (sec)
acce
lera
tion
(g)
0 10 20 30 40−1
−0.5
0
0.5
1bo40
time (sec)
Figure A.6 (continued): Ground motion records: SAC Boston survival-levelstiff (SD) soil ensemble.
199
0 10 20 30 40 50−0.2
−0.1
0
0.1
0.2bs01
acce
lera
tion
(g)
0 10 20 30 40 50−0.1
−0.05
0
0.05
0.1bs02
0 10 20 30 40 50−0.2
−0.1
0
0.1
0.2bs03
acce
lera
tion
(g)
0 10 20 30 40 50−0.1
−0.05
0
0.05
0.1bs04
0 10 20 30 40 50−0.1
−0.05
0
0.05
0.1bs05
acce
lera
tion
(g)
0 10 20 30 40 50−0.1
−0.05
0
0.05
0.1bs06
0 5 10 15 20 25−0.1
−0.05
0
0.05
0.1bs07
acce
lera
tion
(g)
0 5 10 15 20 25−0.1
−0.05
0
0.05
0.1bs08
0 5 10 15 20 25−0.1
−0.05
0
0.05
0.1bs09
time (sec)
acce
lera
tion
(g)
0 5 10 15 20 25−0.1
−0.05
0
0.05
0.1bs10
time (sec)
Figure A.7: Ground motion records: SAC Boston design-level soft (SE) soilensemble.
200
0 5 10 15 20 25−0.2
0
0.2
0.4
0.6bs11
acce
lera
tion
(g)
0 5 10 15 20 25−0.4
−0.2
0
0.2
0.4bs12
0 5 10 15 20 25−0.2
−0.1
0
0.1
0.2bs13
acce
lera
tion
(g)
0 5 10 15 20 25−0.2
0
0.2
0.4
0.6bs14
0 10 20 30 40 50−0.4
−0.2
0
0.2
0.4bs15
acce
lera
tion
(g)
0 10 20 30 40 50−0.4
−0.2
0
0.2
0.4bs16
0 10 20 30 40 50−0.2
−0.1
0
0.1
0.2bs17
acce
lera
tion
(g)
0 10 20 30 40 50−0.4
−0.2
0
0.2
0.4bs18
0 10 20 30 40 50−0.4
−0.2
0
0.2
0.4bs19
time (sec)
acce
lera
tion
(g)
0 10 20 30 40 50−0.4
−0.2
0
0.2
0.4bs20
time (sec)
Figure A.7 (continued): Ground motion records: SAC Boston design-level soft(SE) soil ensemble.
201
0 10 20 30−0.5
0
0.5bs21
acce
lera
tion
(g)
0 10 20 30−0.5
0
0.5bs22
0 10 20 30−0.5
0
0.5bs23
acce
lera
tion
(g)
0 10 20 30−0.5
0
0.5bs24
0 10 20 30−0.5
0
0.5bs25
acce
lera
tion
(g)
0 10 20 30−0.5
0
0.5bs26
0 10 20 30−0.5
0
0.5bs27
acce
lera
tion
(g)
0 10 20 30−0.5
0
0.5bs28
0 5 10 15 20−0.5
0
0.5bs29
time (sec)
acce
lera
tion
(g)
0 5 10 15 20−0.5
0
0.5bs30
time (sec)
Figure A.8: Ground motion records: SAC Boston survival-level soft (SE) soilensemble (generated using EERA site response analysis program, Bardet et al.,2000).
202
0 5 10 15 20−0.5
0
0.5bs31
acce
lera
tion
(g)
0 5 10 15 20−0.2
−0.1
0
0.1
0.2bs32
0 5 10 15 20−0.5
0
0.5bs33
acce
lera
tion
(g)
0 5 10 15 20−1
−0.5
0
0.5
1bs34
0 10 20 30−1
−0.5
0
0.5
1bs35
acce
lera
tion
(g)
0 10 20 30−1
−0.5
0
0.5
1bs36
0 10 20 30 40−1
−0.5
0
0.5
1bs37
acce
lera
tion
(g)
0 10 20 30 40−1
−0.5
0
0.5
1bs38
0 10 20 30 40−1
−0.5
0
0.5
1bs39
time (sec)
acce
lera
tion
(g)
0 10 20 30 40−1
−0.5
0
0.5
1bs40
time (sec)
Figure A.8 (continued): Ground motion records: SAC Boston survival-levelsoft (SE) soil ensemble (generated using EERA site response analysis program,Bardet et al., 2000).
203
0 20 40 60−0.5
0
0.5la01
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1la02
0 10 20 30 40−0.5
0
0.5la03
acce
lera
tion
(g)
0 10 20 30 40−0.5
0
0.5la04
0 10 20 30 40−0.5
0
0.5la05
acce
lera
tion
(g)
0 10 20 30 40−0.5
0
0.5la06
0 20 40 60 80 100−0.5
0
0.5la07
acce
lera
tion
(g)
0 20 40 60 80 100−0.5
0
0.5la08
0 20 40 60 80 100−1
−0.5
0
0.5
1la09
time (sec)
acce
lera
tion
(g)
0 20 40 60 80 100−0.5
0
0.5la10
time (sec)
Figure A.9: Ground motion records: SAC Los Angeles design-level stiff (SD)soil ensemble.
204
0 20 40 60−1
−0.5
0
0.5
1la11
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1la12
0 20 40 60−1
−0.5
0
0.5
1la13
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1la14
0 5 10 15−1
−0.5
0
0.5
1la15
acce
lera
tion
(g)
0 5 10 15−1
−0.5
0
0.5
1la16
0 20 40 60−1
−0.5
0
0.5
1la17
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1la18
0 20 40 60−2
−1
0
1
2la19
time (sec)
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1la20
time (sec)
Figure A.9 (continued): Ground motion records: SAC Los Angeles design-level stiff (SD) soil ensemble.
205
0 20 40 60−2
−1
0
1
2la21
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1la22
0 10 20 30−0.5
0
0.5la23
acce
lera
tion
(g)
0 10 20 30−0.5
0
0.5la24
0 5 10 15−1
−0.5
0
0.5
1la25
acce
lera
tion
(g)
0 5 10 15−1
−0.5
0
0.5
1la26
0 20 40 60−1
−0.5
0
0.5
1la27
acce
lera
tion
(g)
0 20 40 60−2
−1
0
1
2la28
0 20 40 60−1
−0.5
0
0.5
1la29
time (sec)
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1la30
time (sec)
Figure A.10: Ground motion records: SAC Los Angeles survival-level stiff(SD) soil ensemble.
206
0 10 20 30−2
−1
0
1
2la31
acce
lera
tion
(g)
0 10 20 30−2
−1
0
1
2la32
0 10 20 30−1
−0.5
0
0.5
1la33
acce
lera
tion
(g)
0 10 20 30−1
−0.5
0
0.5
1la34
0 10 20 30−1
−0.5
0
0.5
1la35
acce
lera
tion
(g)
0 10 20 30−2
−1
0
1
2la36
0 20 40 60−1
−0.5
0
0.5
1la37
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1la38
0 20 40 60−1
−0.5
0
0.5
1la39
time (sec)
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1la40
time (sec)
Figure A.10 (continued): Ground motion records: SAC Los Angeles survival-level stiff (SD) soil ensemble.
207
0 20 40 60 80 100−0.4
−0.2
0
0.2
0.4ls01
acce
lera
tion
(g)
0 20 40 60 80 100−0.4
−0.2
0
0.2
0.4ls02
0 10 20 30 40 50−0.4
−0.2
0
0.2
0.4ls03
acce
lera
tion
(g)
0 10 20 30 40 50−0.4
−0.2
0
0.2
0.4ls04
0 10 20 30 40 50−0.4
−0.2
0
0.2
0.4ls05
acce
lera
tion
(g)
0 10 20 30 40 50−0.4
−0.2
0
0.2
0.4ls06
0 20 40 60 80 100−0.5
0
0.5ls07
acce
lera
tion
(g)
0 20 40 60 80 100−0.4
−0.2
0
0.2
0.4ls08
0 20 40 60 80 100−0.5
0
0.5ls09
time (sec)
acce
lera
tion
(g)
0 20 40 60 80 100−0.4
−0.2
0
0.2
0.4ls10
time (sec)
Figure A.11: Ground motion records: SAC Los Angeles design-level soft (SE)soil ensemble.
208
0 20 40 60 80 100−0.5
0
0.5
1ls11
acce
lera
tion
(g)
0 20 40 60 80 100−0.5
0
0.5ls12
0 20 40 60 80 100−1
−0.5
0
0.5
1ls13
acce
lera
tion
(g)
0 20 40 60 80 100−0.4
−0.2
0
0.2
0.4ls14
0 5 10 15 20 25−1
−0.5
0
0.5ls15
acce
lera
tion
(g)
0 5 10 15 20 25−0.4
−0.2
0
0.2
0.4ls16
0 20 40 60 80 100−1
−0.5
0
0.5ls17
acce
lera
tion
(g)
0 20 40 60 80 100−0.5
0
0.5ls18
0 20 40 60 80 100−1
−0.5
0
0.5ls19
time (sec)
acce
lera
tion
(g)
0 20 40 60 80 100−0.5
0
0.5ls20
time (sec)
Figure A.11 (continued): Ground motion records: SAC Los Angeles design-level soft (SE) soil ensemble.
209
0 20 40 60−2
−1
0
1
2ls21
acce
lera
tion
(g)
0 20 40 60−2
−1
0
1
2ls22
0 10 20 30−0.5
0
0.5ls23
acce
lera
tion
(g)
0 10 20 30−1
−0.5
0
0.5
1ls24
0 5 10 15−1
−0.5
0
0.5
1ls25
acce
lera
tion
(g)
0 5 10 15−2
−1
0
1
2ls26
0 20 40 60−2
−1
0
1
2ls27
acce
lera
tion
(g)
0 20 40 60−2
−1
0
1
2ls28
0 20 40 60−0.5
0
0.5ls29
time (sec)
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1ls30
time (sec)
Figure A.12: Ground motion records: SAC Los Angeles survival-level soft(SE) soil ensemble (generated using EERA site response analysis program,Bardet et al., 2000).
210
0 10 20 30−1
−0.5
0
0.5
1ls31
acce
lera
tion
(g)
0 10 20 30−1
−0.5
0
0.5
1ls32
0 10 20 30−1
−0.5
0
0.5
1ls33
acce
lera
tion
(g)
0 10 20 30−1
−0.5
0
0.5
1ls34
0 10 20 30−2
−1
0
1
2ls35
acce
lera
tion
(g)
0 10 20 30−2
−1
0
1
2ls36
0 20 40 60−1
−0.5
0
0.5
1ls37
acce
lera
tion
(g)
0 20 40 60−2
−1
0
1
2ls38
0 20 40 60−1
−0.5
0
0.5
1ls39
time (sec)
acce
lera
tion
(g)
0 20 40 60−2
−1
0
1
2ls40
time (sec)
Figure A.12 (continued): Ground motion records: SAC Los Angeles survival-level soft (SE) soil ensemble (generated using EERA site response analysisprogram, Bardet et al., 2000).
211
0 20 40 60−1
−0.5
0
0.5
1nf01
acce
lera
tion
(g)
0 10 20 30−1
−0.5
0
0.5
1nf03
0 20 40 60−1
−0.5
0
0.5
1nf05
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1nf07
0 10 20 30−0.5
0
0.5nf09
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1nf11
0 5 10 15−1
−0.5
0
0.5
1nf13
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1nf15
0 20 40 60−2
−1
0
1
2nf17
time (sec)
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1nf19
time (sec)
Figure A.13: Ground motion records: SAC Los Angeles design-level near-field(NF) ensemble.
212
0 20 40 60−1
−0.5
0
0.5
1nf21
acce
lera
tion
(g)
0 20 40 60−2
−1
0
1
2nf23
0 20 40 60−2
−1
0
1
2nf25
acce
lera
tion
(g)
0 20 40 60−1
−0.5
0
0.5
1nf27
0 20 40 60−2
−1
0
1
2nf29
acce
lera
tion
(g)
0 20 40 60 80 100−1
−0.5
0
0.5
1nf31
0 20 40 60 80 100−1
−0.5
0
0.5
1nf33
acce
lera
tion
(g)
0 20 40 60 80 100−1
−0.5
0
0.5
1nf35
0 20 40 60 80 100−1
−0.5
0
0.5
1nf37
time (sec)
acce
lera
tion
(g)
0 20 40 60 80 100−1
−0.5
0
0.5
1nf39
time (sec)
Figure A.13 (continued): Ground motion records: SAC Los Angeles design-level near-field (NF) ensemble.
213
0 0.5 1 1.5 2 2.5 3 3.50
1
2
T (sec)
Sa
(g)
ξ = 0.05
UND SC soil
mean
0 0.5 1 1.5 2 2.5 3 3.50
50
100
T (sec)
Sv
(cm
/s)
ξ = 0.05UND S
C soil
mean
Acceleration Velocity
Figure A.14: Response spectra: University of Notre Dame (UND) very dense(SC) soil ensemble.
Acceleration Velocity
0 0.5 1 1.5 2 2.5 3 3.50
1
2
T (sec)
Sa
(g)
ξ = 0.05
UND SD soil
mean
0 0.5 1 1.5 2 2.5 3 3.50
50
100
T (sec)
Sv
(cm
/s)
ξ = 0.05 UND SD soil
mean
Figure A.15: Response spectra: University of Notre Dame (UND) stiff (SD)soil ensemble.
214
Acceleration Velocity
0 0.5 1 1.5 2 2.5 3 3.50
1
2
T (sec)
Sa
(g)
ξ = 0.05
UND SE soil
mean
0 0.5 1 1.5 2 2.5 3 3.50
50
100
T (sec)
Sv
(cm
/s)
ξ = 0.05UND S
E soil
mean
Figure A.16: Response spectra: University of Notre Dame (UND) soft (SE) soilensemble.
0 0.5 1 1.5 2 2.5 3 3.50
1
2
T (sec)
Sa
(g)
ξ = 0.05
N&K mean
0 0.5 1 1.5 2 2.5 3 3.50
50
100
T (sec)
Sv
(cm
/s)
ξ = 0.05N&K mean
Acceleration Velocity
Figure A.17: Response spectra: Nassar and Krawinkler 15s very dense (SC)soil ensemble.
215
0 0.5 1 1.5 2 2.5 3 3.50
150
300
T (sec)
Sv
(cm
/s)
ξ = 0.05SAC Boston, S
D soil, design level
mean
Acceleration Velocity
0 0.5 1 1.5 2 2.5 3 3.50
2
4
T (sec)
Sa
(g)
ξ = 0.05
SAC Boston, S
D soil, design level
mean
Figure A.18: Response spectra: SAC Boston design-level stiff (SD) soil ensem-ble.
0 0.5 1 1.5 2 2.5 3 3.50
150
300
T (sec)
Sv
(cm
/s)
ξ = 0.05SAC Boston, S
D soil, survival level
mean
Acceleration Velocity
0 0.5 1 1.5 2 2.5 3 3.50
2
4
T (sec)
Sa
(g)
ξ = 0.05
SAC Boston, S
D soil, survival level
mean
Figure A.19: Response spectra: SAC Boston survival-level stiff (SD) soilensemble.
216
0 0.5 1 1.5 2 2.5 3 3.50
150
300
T (sec)
Sv
(cm
/s)
ξ = 0.05SAC Boston, S
E soil, design level
mean
Acceleration Velocity
0 0.5 1 1.5 2 2.5 3 3.50
2
4
T (sec)
Sa
(g)
ξ = 0.05
SAC Boston, S
E soil, design level
mean
Figure A.20: Response spectra: SAC Boston design-level soft (SE) soil ensem-ble.
0 0.5 1 1.5 2 2.5 3 3.50
150
300
T (sec)
Sv
(cm
/s)
ξ = 0.05SAC Boston, S
E soil, survival level
mean
Acceleration Velocity
0 0.5 1 1.5 2 2.5 3 3.50
2
4
T (sec)
Sa
(g)
ξ = 0.05
SAC Boston, S
E soil, survival level
mean
Figure A.21: Response spectra: SAC Boston survival-level soft (SE) soilensemble (generated using EERA site response analysis program, Bardet et al.,2000).
217
0 0.5 1 1.5 2 2.5 3 3.50
150
300
T (sec)
Sv
(cm
/s)
ξ = 0.05SAC Los Angeles, S
D soil, design level
mean
Acceleration Velocity
0 0.5 1 1.5 2 2.5 3 3.50
2
4
T (sec)
Sa
(g)
ξ = 0.05
SAC Los Angeles, S
D soil, design level
mean
Figure A.22: Response spectra: SAC Los Angeles design-level stiff (SD) soilensemble.
0 0.5 1 1.5 2 2.5 3 3.50
150
300
T (sec)
Sv
(cm
/s)
ξ = 0.05SAC Los Angeles, S
D soil, survival level
mean
Acceleration Velocity
0 0.5 1 1.5 2 2.5 3 3.50
2
4
T (sec)
Sa
(g)
ξ = 0.05
SAC Los Angeles, S
D soil, survival level
mean
Figure A.23: Response spectra: SAC Los Angeles survival-level stiff (SD) soilensemble.
218
0 0.5 1 1.5 2 2.5 3 3.50
150
300
T (sec)
Sv
(cm
/s)
ξ = 0.05SAC Los Angeles, S
E soil, design level
mean
0 0.5 1 1.5 2 2.5 3 3.50
2
4
T (sec)
Sa
(g)
ξ = 0.05
SAC Los Angeles, S
E soil, design level
mean
Acceleration Velocity
Figure A.24: Response spectra: SAC Los Angeles design-level soft (SE) soilensemble.
0 0.5 1 1.5 2 2.5 3 3.50
150
300
T (sec)
Sv
(cm
/s)
ξ = 0.05SAC Los Angeles, S
E soil, survival level
mean
Acceleration Velocity
0 0.5 1 1.5 2 2.5 3 3.50
2
4
T (sec)
Sa
(g)
ξ = 0.05
SAC Los Angeles, S
E soil, survival level
mean
Figure A.25: Response spectra: SAC Los Angeles survival-level soft (SE) soilensemble (generated using EERA site response analysis program, Bardet et al.,2000).
219
0 0.5 1 1.5 2 2.5 3 3.50
150
300
T (sec)
Sv
(cm
/s)
ξ = 0.05SAC Los Angeles, S
D soil, design level, NF
mean
Acceleration Velocity
0 0.5 1 1.5 2 2.5 3 3.50
2
4
T (sec)
Sa
(g)
ξ = 0.05
SAC Los Angeles, S
D soil, design level, NF
mean
Figure A.26: Response spectra: SAC Los Angeles design-level stiff (SD) soilnear-field (NF) ensemble.
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APPENDIX B
CDSPEC (CAPACITY-DEMANDSPECTRA) PROGRAM LISTING
B.1 CDSPEC.M: Main Program
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [cdspec.m]% SDOF Nonlinear Dynamic Time-History Analysis Program% This program will calculate the response of a range of% specified SDOF oscillators. The program uses:% 1. Preprocessor [cdspec.m]% 2. Ground motion scaling [eqscale.m]% 3. Smooth design spectrum [nehrpdes.m]% 4. Nonlinear dynamic time-history analysis [lnlthist.m]% 5. Postprocessor [cdspecpost.m]% first created: KF 9/27/99% last revised: KF 11/17/00%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clear%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Define variables%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%model=input(’Which hysteresis model? (1-ep, 2-be, 3-bp, 4-sd):’);tfree=5 % length of free vibration (s)tend=25 % length of time history analyses (s)nstruc=30; % number of structural periodsT=[0.1 3.0]; % range of periods (s)aT=exp(log(T(2)/T(1))/(nstruc-1)); % multiplier for structural pointsfor n=1:nstruc
Tj(n)=T(1)*aT^(n-1); % structural periods (s)endg=386.4; % gravitational acceleration (k/in^2)ko=1000 % const. initial stiffness (k/in)mj=Tj.^2*ko/4/pi^2; % structural masses (k-s^2/in)omegaj=2*pi./Tj; % structural frequencies (rad/s)xsi=0.05 % damping ratiocj=2*xsi*mj.*omegaj; % structural damping (k-s/in)posty=input(’What post-yield stiffness ratio? (1-0.00, 2-0.05, 3-0.10):’);if posty==1
alpha=0.00elseif posty==2
alpha=0.05elseif posty==3
alpha=0.1 % initial/post-yield stiffness ratioendR=[1 2:2:8] % strength reduction factorsTjif model==3
br=input(’Which beta_r? (1-1/6, 2-1/3, 3-1/2):’);bs=input(’Which beta_s? (1-3, 2-beta_r):’);if br==1
betar=1/6elseif br==2
betar=1/3elseif br==3
betar=1/2endif bs==1
betas=3elseif bs==2
betas=betarend
elsebetar=0; % hysteretic energy factor, beep modelbetas=0; % stiffness factor, beep model
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Define Earthquakes%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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group=input(’What earthquake group? (1-stiff, 2-medium, 3-soft, 4-near fault, 5-test group, 6-SAC/LA/SD/10in50, 7-SAC/LA/SD/2in50, 8-Nassar & Krawinkler 15s, 9-SAC/BOS/SE(B)/10in50), 10-SAC/LA/SE(B)/10in50), 11-cyclic, 12-SAC/BO/SD/10in50, 13-SAC/BO/SD/2in50, 14-SAC/BO/SE/2in50), 15-SAC/LA/SE/2in50:’);if group==1
gr=’stiff’eqs=[’LPPR’ ’SASJ’ ’MIZI’ ’SFSF’ ’PACH’ ’SFCA’ ’MHGI’ ’CCVA’ ’MILU’ ’MILV’ ’SSNG’ ’SSUC’ ’SSGR’ ’WHMW’ ’LPCO’ ’LPSC’ ’LPCH’ ’LPPH’
’LPRH’ ’LPYB’];elseif group==2
gr=’medium’eqs=[’NONW’ ’LPHO’ ’LAYE’ ’NOSY’ ’SFOR’ ’la02’ ’KCLS’ ’SFFI’ ’SFHO’ ’SFAS’ ’WHTA’ ’WHBT’ ’IVBC’ ’IVJR’ ’IVIV’ ’CCEs’ ’WHLC’ ’LPOA’
’WHAH’ ’WHAD’];elseif group==3
gr=’soft’eqs=[’LPFO’ ’RUBU’ ’MISs’ ’LPTR’ ’EUF1’ ’EUF2’ ’OHW1’ ’OHW2’ ’RUB1’ ’RUB2’ ’TL1s’ ’TL2s’ ’WHA1’ ’WHA2’ ’MIDs’ ’LPOH’ ’MIFs’ ’MIOs’
’LPIA’ ’LPCB’];elseif group==4
gr=’nearfault’eqs=[’nf01’ ’nf03’ ’nf05’ ’nf07’ ’nf09’ ’nf11’ ’nf13’ ’nf15’ ’nf17’ ’nf19’ ’nf21’ ’nf23’ ’nf25’ ’nf27’ ’nf29’ ’nf31’ ’nf33’ ’nf35’ ’nf37’ ’nf39’];
elseif group==5gr=’stiff’eqs=[’LPPR’ ’PACH’];
elseif group==6gr=’SACLASD10in50’eqs=[’la01’ ’la02’ ’la03’ ’la04’ ’la05’ ’la06’ ’la07’ ’la08’ ’la09’ ’la10’ ’la11’ ’la12’ ’la13’ ’la14’ ’la15’ ’la16’ ’la17’ ’la18’ ’la19’ ’la20’];
elseif group==7gr=’SACLASD2in50’eqs=[’la21’ ’la22’ ’la23’ ’la24’ ’la25’ ’la26’ ’la27’ ’la28’ ’la29’ ’la30’ ’la31’ ’la32’ ’la33’ ’la34’ ’la35’ ’la36’ ’la37’ ’la38’ ’la39’ ’la40’];
elseif group==8gr=’n&k15s’eqs=[’eq03’ ’eq04’ ’eq05’ ’eq06’ ’eq07’ ’eq08’ ’eq09’ ’eq10’ ’eq11’ ’eq12’ ’eq14’ ’eq15’ ’eq16’ ’eq17’ ’eq20’];
elseif group==9gr=’boss’eqs=[’BS01’ ’BS02’ ’BS03’ ’BS04’ ’BS05’ ’BS06’ ’BS07’ ’BS08’ ’BS09’ ’BS10’ ’BS11’ ’BS12’ ’BS13’ ’BS14’ ’BS15’ ’BS16’ ’BS17’ ’BS18’ ’BS19’ ’BS20’];
elseif group==10gr=’lass’eqs=[’LS01’ ’LS02’ ’LS03’ ’LS04’ ’LS05’ ’LS06’ ’LS07’ ’LS08’ ’LS09’ ’LS10’ ’LS11’ ’LS12’ ’LS13’ ’LS14’ ’LS15’ ’LS16’ ’LS17’ ’LS18’ ’LS19’ ’LS20’];
elseif group==11gr=’soft’eqs=[’cy05’ ’cy20’];
elseif group==12gr=’SACBOSD10in50’eqs=[’bo01’ ’bo02’ ’bo03’ ’bo04’ ’bo05’ ’bo06’ ’bo07’ ’bo08’ ’bo09’ ’bo10’ ’bo11’ ’bo12’ ’bo13’ ’bo14’ ’bo15’ ’bo16’ ’bo17’ ’bo18’ ’bo19’ ’bo20’];
elseif group==13gr=’SACBOSD2in50’eqs=[’bo21’ ’bo22’ ’bo23’ ’bo24’ ’bo25’ ’bo26’ ’bo27’ ’bo28’ ’bo29’ ’bo30’ ’bo31’ ’bo32’ ’bo33’ ’bo34’ ’bo35’ ’bo36’ ’bo37’ ’bo38’ ’bo39’ ’bo40’];
elseif group==14gr=’SACBOSE2in50’eqs=[’bs21’ ’bs22’ ’bs23’ ’bs24’ ’bs25’ ’bs26’ ’bs27’ ’bs28’ ’bs29’ ’bs30’ ’bs31’ ’bs32’ ’bs33’ ’bs34’ ’bs35’ ’bs36’ ’bs37’ ’bs38’ ’bs39’ ’bs40’];
elseif group==15gr=’SACLASE2in50’eqs=[’ls21’ ’ls22’ ’ls23’ ’ls24’ ’ls25’ ’ls26’ ’ls27’ ’ls28’ ’ls29’ ’ls30’ ’ls31’ ’ls32’ ’ls33’ ’ls34’ ’ls35’ ’ls36’ ’ls37’ ’ls38’ ’ls39’ ’ls40’];
end
neqs=length(eqs)/4; % number of earthquakes in group%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Define and Perform Earthquake Scaling%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%method=input(’What scaling method? (1-PGA, 2-MIV, 3-EPA, 4-EPV, 5-Sa1, 6-SaTe, 7-A95, 8-None): ’);if method==1
mthd=’PGA’elseif method==2
mthd=’MIV’elseif method==3
mthd=’EPA’elseif method==4
mthd=’EPV’elseif method==5
mthd=’Sa1’elseif method==6
mthd=’SaTe’elseif method==7
mthd=’A95’elseif method==8
mthd=’None’end[EQi,ti,Saavg]=eqscale(gr,eqs,neqs,method,Tj,R,alpha,tfree,tend);
subset=input(’Would you like to split up the groundmotions? ’,’s’);if subset==’y’
gmstart=input([’There are currently ’ num2str(neqs) ’ groundmotions--input start number: ’])gmend=input(’Input end number: ’)neqs=gmend-gmstart+1; % redefine number of earthquakeseqs=eqs((gmstart-1)*4+1:(gmend-1)*4+4); % select ground motionsEQi=EQi(gmstart:gmend,:,:,:);
endsaveflag=input(’Would you like to save data periodically?’,’s’);if saveflag==’y’
filename=input([’Enter filename (default is ’ gr ’_’ date ’): ’],’s’)if isempty(filename)==1
filename=[gr ’_’ date]endinterval=input(’Enter interval(default is 5): ’)if isempty(interval)==1
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interval=5end
elsefilename=’Your’;
endns=input(’Additional constant scalar (for MDOF comparison)? ’,’s’);if ns==’y’ newscale=input(’Enter new scale factor: ’) EQi=EQi*newscale;end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Use which reference spectra?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%fel=ones(neqs,1)*100000000; % initialize to infinite strengthanal=input(’(a)verage spectra, (e)lastic response or (n)ehrp spectra?’,’s’);if anal==’n’
stype=input(’soil type? (1-SB, 2-SC, 3-SD, 4-SE, 5-NF):’);perf=input(’performance level? (1-design, 2-survival):’);zone=input(’seismic region? (1-LA, 2-SEA, 3-BOS):’);[felnehrp]=nehrpdes(mj,Tj,neqs,perf,stype,zone);
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nehrp spectra AND scaling?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%if anal==’n’ % nehrp reference spectra?
if method==1 % PGA scaling?PGAnhp=max(mean(felnehrp)./mj)/2.5/g;% nehrp pga (g)PGAeqi=max(max(abs(EQi)));% gm pga (g)scalar=PGAnhp/PGAeqi % new scale factorEQi=EQi*scalar; % scale gm’s again
elseif method==5Sanhp=mean(felnehrp)./mj/g;for j=1:nstruc
scalar(j)=Sanhp(j)./Saavg(j,j);EQi(:,:,j)=EQi(:,:,j)*scalar(j);
endend
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Determine Elastic Force Demand/Inelastic Displacement Demand%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%diary([date ’sdofecho.txt’])time=clock;time=time(:,4:end);[filename ’ run has started! Current time is ’ num2str(time)]diary off
for flag=1:2 % indicate elastic/inelastic analysishh = waitbar2(0,[’SDOF Analysis: cycle ’ num2str(flag) ]);for j=1:nstruc
k=ko;c=cj(j);m=mj(j);om=omegaj(j);t=ti;if method==5
EQ=EQi(:,:,j);elseif method==6
EQ(:,:,:)=EQi(:,j,:,:);else
EQ=EQi;endif flag==2
if anal==’e’fel=felasij(:,j);elseif anal==’a’fel=felavg(:,j);elseif anal==’n’fel=felnehrp(:,j);end
endif model==4
[delta,f,a,telapse,vrec,frec,crit,fli,vpli,nyi,nrevi,nzi,vpmi,vri]=lnlthistsd(EQ,neqs,t,k,m,c,om,R,alpha,fel,flag,method,model,betar,betas);else
[delta,f,a,telapse,vrec,frec,crit,fli,vpli,nyi,nrevi,nzi,vpmi]=lnlthist(EQ,neqs,t,k,m,c,om,R,alpha,fel,flag,method,model,betar,betas);endif flag==1
delasij(:,j)=delta;felasij(:,j)=f;telasij(:,j)=telapse;aelasij(:,j)=a;vijelas(:,j,:,:)=vrec;fijelas(:,j,:,:)=frec;nzelasij(:,j)=nzi;
elseif flag==2dinelij(:,j,:)=delta;finelij(:,j,:)=f;tinelij(:,j,:)=telapse;ainelij(:,j,:)=a;vijinel(:,j,:,:)=vrec;fijinel(:,j,:,:)=frec;dpinelij(:,j,:)=vpli;
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dpmaxij(:,j,:)=vpmi;nyinelij(:,j,:)=nyi;nrinelij(:,j,:)=nrevi;nzinelij(:,j,:)=nzi;
endif (saveflag==’y’ & rem(j,interval)==0)
!klog -password sodesnesave(filename)
endwaitbar2(j/nstruc)
endfelavg=ones(neqs,1)*mean(felasij);close(hh)
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Post-process data%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%diary([date ’sdofecho.txt’])time=clock;time=time(:,4:end);[filename ’ run is completed! Current time is ’ num2str(time)]diary off
[mu,mubar,disp,muc,mur,mup,muec]=cdspecpost(delasij,dinelij,felasij,finelij,R,Tj,omegaj,mj,alpha,neqs,nstruc,gr,vijelas,vijinel,fijelas,fijinel,dpinelij,nyinelij,nrinelij,nzelasij,nzinelij,tfree,ti,dpmaxij,tend);
if saveflag==’y’save(filename)
end
figure(1)subplot(2,1,1)title([date ’ Filename: ’ filename ])
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B.2 EQSCALE.M: Ground Motion Scaling Function
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [eqscale.m]% Ground motion scaling program% This program scales earthquakes according to specified% method.% first created: KF 9/27/99% last revised: KF 9/27/00%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [EQi,ti,Saavg] = eqscale(gr,eqs,neqs,method,Tj,R,alpha,tfree,tend)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Define time new time vector%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%dtnew=0.01; % specify same time step (s)if (method==2 | method==7)
dtref=dtnew; % specify time step to revert to for MIV (s)dtnew=0.0025; % specify smaller step for MIV (s)trefn=0:dtref:tend; % reference time vector (s)
endtnewn=0:dtnew:tend; % new time vector (s)ti=tnewn;Ni=length(tnewn); % number of points in ti%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Load earthquake data%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%h = waitbar(0,’Loading and modifying earthquake data...’);for n = 1:neqseqn=[eqs(4*n-3:n+3*n) ’prep’];if method==6
eqspecn=[eqs(4*n-3:n+3*n) ’specl’];else
eqspecn=[eqs(4*n-3:n+3*n) ’spec’];endpath1=[’../groundmotions/final/’ gr ’/’ eqn];path2=[’../groundmotions/final/’ gr ’/’ eqspecn];load(path1,’-ascii’)load(path2,’-ascii’)eval([’tn=’ eqn ’(:,1);’]) % time vector (s)eval([’an=’ eqn ’(:,2);’]) % acceleration vector (g)eval([’Tn=’ eqspecn ’(:,1);’]) % period vector (s)eval([’San=’ eqspecn ’(:,5);’]) % pseudo acceleration spectra (g)eval([’Svn=’ eqspecn ’(:,2);’]) % pseudo velocity spectra (in/s)dt=diff(tn); % time step (s)dt=dt(1);%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Modify earthquakes%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%t0=tn(1); % beginning time (s)tmaxn=max(tn); % check if record is long enoughif tmaxn<tend % if not, make it long enough
tn1=t0:dt:tend;nn=length(tn);n1=length(tn1);an1=zeros(n1,1);nmodif=nn+1:n1;ndiff=length(nmodif);an1(1:nn)=an;an1(nmodif)=zeros(ndiff,1);clear an tnan=an1; % new acceleration vector (g)tn=tn1; % new time vector (s)
endtshiftn=tn-t0; % shift time to relative scale (s)anewn=spline(tshiftn,an,tnewn); % new acceleration vector (s)Ist=find(tnewn==tend-tfree); % start of free vibrationanewn(Ist+1:end)=zeros(length(anewn)-Ist,1); % zero accelSnewn=spline(Tn,San,[Tj]); % new acceleration spectrum vector (g)Tnsec=[Tj , 3:.1:7];Snsec=spline(Tn,San,Tnsec); % new acceleration spectrum vector (g)Svnewn=spline(Tn,Svn,[Tj]); % new velocity spectrum vector (in/s)EQi(n,:)=anewn;Si(n,:)=Snewn;Svi(n,:)=Svnewn;Siorig(n,:)=Snsec;waitbar(n/neqs)endclose(h)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Scale to PGA%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%if method==1 | method==8
amax=max(abs(EQi),[],2) % find max accel’saavg=mean(amax) % find mean of max accel’sif method==1
label=’PGA’;sctype=input(’Scale to (a)verage or (s)pecified PGA? ’,’s’);if sctype==’a’
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sfactor=aavg./amax% determine scale factorelse
aavg=input(’specify PGA (in g): ’);sfactor=aavg./amax
endelse
label=’none’;sfactor=ones(neqs,1)
endfor n=1:neqs % scale records
EQi(n,:)=EQi(n,:)*sfactor(n);Si(n,:)=Si(n,:)*sfactor(n);
endSaavg=mean(Si); % find response spectra avg.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Comparison Plot%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(1)subplot(2,1,1)plot(ti,EQi)hold onplot([0 tend],[aavg aavg],’r--’)plot([0 tend],-[aavg aavg],’r--’)xlabel(’t (sec)’)ylabel(’acceleration (g)’)title([date ’ Groundmotions: ’ gr ’ Scaling: ’ label])hold offsubplot(2,1,2)plot([Tj],Si)hold onplot([Tj],Saavg,’--’,’LineWidth’,5)plot([0 Tj(end)],[aavg aavg],’--’)xlabel(’T (sec)’)ylabel(’S_a (g)’)title(’Response Spectra, \xi=0.05’)hold off
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Scale to MIV%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%elseif method==2
scalar=981; % change units from g->cm/s^2acel=EQi*scalar;N=length(acel);h = waitbar(0,’Scaling using MIV method...’);IV=zeros(neqs,Ni); % initializetIV=zeros(neqs,Ni);for i = 1:neqs
counter=0;for n = 2:N
if (acel(i,n)<=0 & acel(i,n-1)>=0) | (acel(i,n)>=0 & acel(i,n-1)<=0)incross=n;counter=counter+1;for m = incross+1:N
if acel(i,incross)==acel(i,incross-1)breakendif (acel(i,m)<=0 & acel(i,m-1)>=0) | (acel(i,m)>=0 & acel(i,m-1)<=0)outcross=m-1;breakendendIV(i,counter)=sum(acel(i,incross:outcross))*dtnew;tIV(i,counter)=ti(incross);end
end[maxIV(i),I(i)]=max(abs(IV(i,:)));maxt(i)=tIV(i,I(i));
waitbar(i/neqs)endclose(h)maxIV’maxtIVavg=mean(maxIV)sfactor=IVavg./maxIV’ % determine scale factorfor n=1:neqs % scale records
EQin(n,:)=spline(ti,EQi(n,:),trefn)*sfactor(n);Si(n,:)=Si(n,:)*sfactor(n);
endti=trefn;clear EQiEQi=EQin;Saavg=mean(Si); % find response spectra avg.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Comparison Plot%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(1)subplot(2,1,1)plot(ti,EQi)xlabel(’t (sec)’)ylabel(’acceleration (g)’)title([date ’ Groundmotions: ’ gr ’ Scaling: MIV’])
226
subplot(2,1,2)plot([Tj],Si)hold onplot([Tj],Saavg,’--’,’LineWidth’,5)xlabel(’T (sec)’)ylabel(’S_a (g)’)title(’Response Spectra, \xi=0.05’)hold off
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Scale to EPA%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%elseif method==3
Iepa=find((Tj>=.1)&(Tj<=.5)); % define spectral rangeEPA=mean(Si(:,Iepa),2)/2.5EPAavg=mean(EPA)sfactor=EPAavg./EPA % determine scale factorfor n=1:neqs % scale records
EQi(n,:)=EQi(n,:)*sfactor(n);Si(n,:)=Si(n,:)*sfactor(n);
endSaavg=mean(Si); % find response spectra avg.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Comparison Plot%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(1)subplot(2,1,1)plot(ti,EQi)hold onplot([0 tend],[EPAavg EPAavg],’r--’)plot([0 tend],-[EPAavg EPAavg],’r--’)xlabel(’t (sec)’)ylabel(’acceleration (g)’)title([date ’ Groundmotions: ’ gr ’ Scaling: EPA’])hold offsubplot(2,1,2)plot([Tj],Si)hold onplot([Tj],Saavg,’--’,’LineWidth’,5)plot([0 Tj(end)],[EPAavg EPAavg],’--’)xlabel(’T (sec)’)ylabel(’S_a (g)’)title(’Response Spectra, \xi=0.05’)hold off
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Scale to EPV%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%elseif method==4
Iepv=find((Tj>=.8)&(Tj<=1.2)); % define spectral rangeEPV=mean(Svi(:,Iepv),2)/2.5EPVavg=mean(EPV)sfactor=EPVavg./EPV % determine scale factorfor n=1:neqs % scale records
EQi(n,:)=EQi(n,:)*sfactor(n);Si(n,:)=Si(n,:)*sfactor(n);
endSaavg=mean(Si); % find response spectra avg.Svavg=mean(Svi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Comparison Plot%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(1)subplot(2,1,1)plot(ti,EQi)xlabel(’t (sec)’)ylabel(’acceleration (g)’)title([date ’ Groundmotions: ’ gr ’ Scaling: EPV’])subplot(2,1,2)plot([Tj],Svi)hold onplot([Tj],Svavg,’--’,’LineWidth’,5)plot([0 Tj(end)],[EPVavg EPVavg],’--’)xlabel(’T (sec)’)ylabel(’S_v (g)’)title(’Response Spectra, \xi=0.05’)hold off
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Scale to Sa1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%elseif method==5
nstruc=length(Tj); % number of structural periodsh = waitbar(0,’Scaling using Sa1 method...’);for j=1:nstruc
intensity=Si(:,j);intavg=mean(intensity);sfactor=intavg./intensity; % determine scale factorfor n=1:neqs % scale records
EQij(n,:,j)=EQi(n,:)*sfactor(n);Sij(n,:,j)=Si(n,:)*sfactor(n);
endSaavg(j,:)=mean(Sij(:,:,j)); % find response spectra avg.
waitbar(j/nstruc)end
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clear Si EQiEQi=EQij;Si=Sij;close(h)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Comparison Plot%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
nsub=5; % number of subplotsnfigs=nstruc/nsub; % number of figuresif nfigs<1 % if there are less than 5 periods
nfigs=1;nsub=nstruc;
endfor j=1:nfigs
figure(j)for k=1:nsub
subplot(nsub,1,k)strucno=j*k + (j-1)*(nsub-k);plot(Tj,Saavg(strucno,:),’--’,’LineWidth’,5)hold onplot(Tj,Si(:,:,strucno))hold off
endendfigureplot([Tj],Saavg’)xlabel(’T (sec)’)ylabel(’S_a (g)’)title(’Response Spectra, \xi=0.05’)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Scale to SaTsec and plot comparison%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%elseif method==6
methodman=input(’(k)ennedy or (y)our method? ’,’s’);key=2; % preparing to call nkregressioninterval=3; % specify plot intervalnstruc=length(Tj); % number of structural periodsh = waitbar(0,’Scaling using SaTsec method...’);for k=1:length(R)
Rk=R(k); % select R factorfor j=1:nstruc
nsamp=j;% select periodif gr([1:2])==’so’[muk]=nkregression_s(nsamp,alpha,Tj,key,Rk);
elseif gr([1:2])==’ne’ gr
[muk]=nkregression_nf(nsamp,alpha,Tj,key,Rk); elseif gr([1:2])==’me’ gr
[muk]=nkregression_me(nsamp,alpha,Tj,key,Rk); elseif gr([1:2])==’st’ gr
[muk]=nkregression_st(nsamp,alpha,Tj,key,Rk); else
[muk]=nkregression(nsamp,alpha,Tj,key,Rk);% predicted ductility demandendif methodman==’y’Tsec=Tj(j)*sqrt(muk/(muk*alpha+(1-alpha)));elseTsec=Tj(j)*sqrt(muk);end% predicted secant periodjrange=find(Tnsec>=Tj(j) & Tnsec<=Tsec);% det. range of Tintensity=Si(:,j);% intensity at Tintrange=Siorig(:,jrange);% intensity over rangeintrngavg2=mean(intrange);% avg. over rangeintrngavg=mean(intrange,2);% avg. over rangeintavg=mean(intrngavg);% avg. intensitySaavg(k,j)=intavg;sfactor=intavg./intrngavg;% determine scale factorfor n=1:neqs% scale recordsEQijk(n,j,k,:)=EQi(n,:)*sfactor(n);Sijk(n,j,k,:)=Si(n,:)*sfactor(n);Sijkorig(n,j,k,:)=Siorig(n,:)*sfactor(n);endif rem(j,interval)==0figure(j/interval)subplot(length(R),1,k)sa(:,:)=Sijkorig(:,j,k,:);plot(Tnsec,sa)hold onplot(Tnsec(jrange),intrngavg2,’--o’,’Linewidth’,3)hold offend
endwaitbar(k/length(R))
endclear Si EQiEQi=EQijk;Si=Sijk;close(h)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Scale to A95 (Sarma and Yang 1987)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%elseif method==7
scalar=981; % change units from g->cm/s^2acel=EQi*scalar;acel2=acel.^2; % square acceleration record (g^2)Es=sum(acel2,2)*dtnew; % arias intensity (g^2-sec)A95=0.764*Es.^.438 % A95 per Sarma and Yang Regression (g)A95avg=mean(A95); % average A95 (g)sfactor=A95avg./A95 % determine scale factorfor n=1:neqs % scale records
EQin(n,:)=spline(ti,EQi(n,:),trefn)*sfactor(n);Si(n,:)=Si(n,:)*sfactor(n);
endti=trefn;clear EQiEQi=EQin;Saavg=mean(Si); % find response spectra avg.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Comparison Plot%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(1)subplot(2,1,1)plot(ti,EQi)xlabel(’t (sec)’)ylabel(’acceleration (g)’)title([date ’ Groundmotions: ’ gr ’ Scaling: A95’])subplot(2,1,2)plot([Tj],Si)hold onplot([Tj],Saavg,’--’,’LineWidth’,5)xlabel(’T (sec)’)ylabel(’S_a (g)’)title(’Response Spectra, \xi=0.05’)hold off
end
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B.3 NEHRPDES.M: Smooth Design Reference Spectrum Function
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [nehrpdes.m]% Smooth Design Reference Spectrum% This program will calculate the design force for each% SDOF oscillator using smooth design spectra.% first created: KF 11/10/99% last revised: KF 12/21/99%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [fel]=nehrpdes(mj,Tj,neqs,perf,stype,zone)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Load Spectra%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%if perf==1
load(’../groundmotions/final/NEHRPspectra/nehrpdes’)elseif perf==2
load(’../groundmotions/final/NEHRPspectra/nehrpsur’)end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Choose spectra and calculate design forces%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%g=386.4; % gravity (in/sec^2)sa(1,:)=Sa(stype,zone,:); % spectrumsa=spline(Torig,sa,Tj); % resamplefel=ones(neqs,1)*(sa.*mj*g) ; % design forcefigure(1)subplot(2,1,2)hold onplot(Tj,sa,’r’,’LineWidth’,3)bar(Tj,sa,’r’)hold off
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B.4 LNLTHIST.M, LNLTHISTSD.M: Nonlinear Dynamic Time-History Analysis Functions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [lnlthist.m]% Linear Acceleration Program% This program calculates the response of a SDOF oscillator using the% linear acceleration method.% first created: KF 9/27/99% last revised: KF 9/26/00%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [vmaxi,fmaxi,amaxi,telapsei,vrec,frec,crit,fl,vpli,nyi,nrevi,nzi,vpmi]=lnlthist(EQi,neqs,ti,ko,m,co,om,R,alpha,fel,flag,method,model,betar,betas)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Initialize%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%scalar=386.4; % change g->in/sec^2niter=6; % number of maximum iterationsalphao=alpha; % original post-yield stiffnessif flag==1 % determine if elastic or inelastic
rstart=1;rend=1;tol=0.01 % error tolerance
elseif flag==2rstart=2;rend=length(R);tol=0.01 % error tolerance
endif method==6
EQorig=EQi;clear EQi
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% reduction factor loop%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%for kk=rstart:rend
Ro=R(kk) % select reduction factorif method==6
EQi(:,:)=EQorig(:,kk,:);end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ground motion loop%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for e=1:neqsfy=fel(e)/Ro; % determine yield strength (k)vy=fy/ko; % determine yield displacement (in)vy1=0;vy2=0;P=EQi(e,:); % select earthquake (g)timeo=ti; % time (sec)ho=diff(timeo); % time increment,dt (s)ho=ho(1);No=length(timeo); % no. of pointspo=-m*P*scalar; % force vector (k)vmax=0; % initializefmax=0;amax=0;telapse=0;tstart=zeros(size(clock));tend=zeros(size(clock));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% iteration loop%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for l=2:niter % iteration loopM=2^(l-2)% interpolationh=ho/M;% modified dt (s)p=interp(po,M);% modified load vectortime=interp(timeo,M);% modified time vectorN=No*M;% modified no of pointsv=0;% init dispa=0;% init acelvdi=0;% init velf=0;% init spring forcek=ko;% initialize stiffnessc=co;% iintialize dampingfu=fy;% set yield surfaceflag=0;% init reversal codefl=0;% init reversal codecrit=0;% init yield criteriavd=0;% init velvpl=0;% init plastic defovpm=[0 0];% init max plastic defonrev=0;% init total yield reversalsny=0;% init yield eventsnzero=0;% init zero crossingsr=0;% init yield reversal coderev=0;% init yield reversal codeif model==3alpha2=0.00001;% post-yield stiffness, epif (betar==betas | Ro==1)k1o=ko/(1+betas);% elastic stiffness, bealpha=alphao*(1+betas);% modified post-yield stiffness
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elsek1o=ko/(1+betar);% elastic stiffness, bealpha=alphao*(1+betar);% modified post-yield stiffnessendk2o=betas*k1o;% elastic stiffness, epk1=k1o;% initialize stiffness, bek2=k2o;% initialize stiffness, epk=k1+k2;% total stiffness, beepflag1=0;% init reversal code, beflag2=0;% init reversal code, epf1=0;f2=0;r1=0;r2=0;rev1=0;rev2=0;vy1=fy/(1+betar)/k1;% yield displacement, bevy2=fy/(1+betar)/k2*betar;% yield displacement, ependai=(p(1)-k*v(1)-c*vdi)/m;% accelerationkhat=6*m/h^2 + 3*c/h +k;% pseudo stiffnesststart=clock;% start timehh = waitbar1(0,[’Calculating response: EQ ’ num2str(e) ’, Iteration ’ num2str(l-1)]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% linear acceleration loop%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i=2:Ndp=p(i)-p(i-1);% change in loading forcekhat=6*m/h^2 + 3*c/h +k;% pseudo stiffnessdphat=dp+(6*m/h + 3*c)*vdi+(3*m + c*h/2)*ai;% change in pseudo forcedv=dphat/khat;% change in displacementdvd=ai*h + 3*(dv-vdi*h-ai*h^2/2)/h;% change in velocityda=(dv-vdi*h-ai*h^2/2)*6/h^2;% change in acceleration
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Update variables%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
a(i)=ai;ai=ai+da;% accelerationv(i)=v(i-1)+dv;% displacementvdi=vdi+dvd;% velocityvd(i)=vdi;df=k*dv;% change in spring forceif i==2fu=-sign(v(i))*vy;fu2=-sign(v(i))*vy2;flag=0;vd(1)=sign(v(i))*.000000001;end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Check yielding and reversals%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if model==1% bilinear elasto-plastic[df,fu,k,flag,dpl,y,r,rev]=epdisp(flag,fu,vd(i),vd(i-1),ko,alpha,vy,v(i),dv,r,rev);elseif model==2% bilinear elastic[df,k,flag,dpl,y,r,rev]=bedisp2(flag,vd(i),vd(i-1),ko,alpha,vy,v(i),dv,r,rev);elseif model==3% bilinear elasto-plastic[df1,k1,flag1,dpl,y,r1,rev1]=bedisp2(flag1,vd(i),vd(i-1),k1o,alpha,vy1,v(i),dv,r1,rev1);[df2,fu2,k2,flag2,dpl,y,r2,rev2]=epdisp(flag2,fu2,vd(i),vd(i-1),k2o,alpha2,vy2,v(i),dv,r2,rev2);f1(i)=f1(i-1)+df1;f2(i)=f2(i-1)+df2;df=df1+df2;k=k1+k2;fu=fu2;flag=flag2;rev=rev2;rev2=0;endvpl=vpl+dpl;% accumulative plastic defony=ny+y;% number of yield eventsvpm(ny+1)=vpm(ny+1)+dpl;% plastic defo per excursionvpm(ny+2)=[0];nrev=nrev+rev;% number of total yield reversalsrev=0;% reset counterf(i)=f(i-1)+df;% spring forceif (sign(f(i))~=sign(f(i-1)))nzero=nzero+1;% zero crossingendfl(i)=flag;crit(i)=fu;if (fl(i)>fl(i-1) & model~=3)ai=(p(i)-f(i)-c*vdi)/m;% correct accelerationendif rem(i,100)==0waitbar1(i/N)endendclose(hh)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Check error%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
tend=clock;% end timetelapse=etime(tend,tstart);% elapsed time
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vmax(l)=max(abs(v));% maximum displacement (absolute)fmax=max(abs(f));% maximum forcevpmax=max(abs(vpm));% maximum plastic excursionat=a-p/m;% total accelerationamax=max(abs(at));% maximum accelerationif Ro~=0err=(vmax(l)-vmax(l-1))/vmax(l)% errorelsevcheck(:,l)=spline(time’,v’,ti’);err=max(abs((vcheck(:,l)-vcheck(:,l-1)))./vmax(l))% errorendif abs(err)<tol% check if error is within tolerancebreakend
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Store final values%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if l>=niterdiary([date ’sdofecho.txt’])’CHECK CONVERGENCE’[’error= ’ num2str(err) ’ iter= ’ num2str(l) ’ delta= ’ num2str(vmax(l)) ’ eq= ’ num2str(e)]diary off
endtelapsei(e,1,kk)=telapse;vmaxi(e,1,kk)=vmax(l);fmaxi(e,1,kk)=fmax;amaxi(e,1,kk)=amax;vpli(e,1,kk)=vpl;vpmi(e,1,kk)=vpmax;nyi(e,1,kk)=ny;nrevi(e,1,kk)=nrev;nzi(e,1,kk)=nzero;vrec(e,1,kk,:)=spline(time,v,ti);frec(e,1,kk,:)=spline(time,f,ti);
endend
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [lnlthistsd.m]% Linear Acceleration Prog% This program calculates the response of a SDOF oscillator using the% linear acceleration method.% first created: KF 9/27/99% last revised: KF 9/26/00%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [vmaxi,fmaxi,amaxi,telapsei,vrec,frec,crit,fl,vpli,nyi,nrevi,nzi,vpmi,vri]=lnlthistsd(EQi,neqs,ti,ko,m,co,om,R,alpha,fel,flagg,method,model,betar,betas)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Initialize%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%scalar=386.4; % change g->in/sec^2tol=0.01 % error toleranceniter=6; % number of maximum iterationsalphao=alpha; % original post-yield stiffnessif flagg==1 % determine if elastic or inelastic
rstart=1;rend=1;
elseif flagg==2rstart=2;rend=length(R);
endif method==6
EQorig=EQi;clear EQi
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% reduction factor loop%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%for kk=rstart:rend
Ro=R(kk) % select reduction factorif method==6
EQi(:,:)=EQorig(:,kk,:);end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ground motion loop%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for e=1:neqsfy=fel(e)/Ro; % determine yield strength (k)vy=fy/ko; % determine yield displacement (in)vy1=0;vy2=0;P=EQi(e,:); % select earthquake (g)timeo=ti; % time (sec)ho=diff(timeo); % time increment,dt (s)ho=ho(1);No=length(timeo); % no. of pointspo=-m*P*scalar; % force vector (k)vmax=0; % initializefmax=0;amax=0;telapse=0;
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tstart=zeros(size(clock));tend=zeros(size(clock));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% iteration loop%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for l=2:niter % iteration loopM=2^(l-2)% interpolationh=ho/M;% modified dt (s)p=interp(po,M);% modified load vectortime=interp(timeo,M);% modified time vectorN=No*M;% modified no of pointsv=0;% init dispa=0;% init acelvdi=0;% init velf=0;% init spring forcek=ko;% initialize stiffnessc=co;% iintialize dampingfu=fy;% set yield surfacefsh=[-fy fy];% init shoot-through forcevsh=[-vy vy];% init shoot-through displflag=0;% init reversal codefl=0;% init reversal codecrit=0;% init yield criteriavd=0;% init velvpl=0;% init plastic defovr=0;% init residual dispvpm=[0 0];% init max plastic defonrev=0;% init total yield reversalsny=0;% init yield eventsnzero=0;% init zero crossingsr=0;% init yield reversal coderev=0;% init yield reversal codeif model==3alpha2=0.00001;% post-yield stiffness, epif (betar==betas | Ro==1)k1o=ko/(1+betas);% elastic stiffness, bealpha=alphao*(1+betas);% modified post-yield stiffnesselsek1o=ko/(1+betar);% elastic stiffness, bealpha=alphao*(1+betar);% modified post-yield stiffnessendk2o=betas*k1o;% elastic stiffness, epk1=k1o;% initialize stiffness, bek2=k2o;% initialize stiffness, epk=k1+k2;% total stiffness, beepflag1=0;% init reversal code, beflag2=0;% init reversal code, epf1=0;f2=0;r1=0;r2=0;rev1=0;rev2=0;vy1=fy/(1+betar)/k1;% yield displacement, bevy2=fy/(1+betar)/k2*betar;% yield displacement, ependai=(p(1)-k*v(1)-c*vdi)/m;% accelerationkhat=6*m/h^2 + 3*c/h +k;% pseudo stiffnesststart=clock;% start timehh = waitbar1(0,[’Calculating response: EQ ’ num2str(e) ’, Iteration ’ num2str(l-1)]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% linear acceleration loop%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i=2:Ndp=p(i)-p(i-1);% change in loading forcekhat=6*m/h^2 + 3*c/h +k;% pseudo stiffnessdphat=dp+(6*m/h + 3*c)*vdi+(3*m + c*h/2)*ai;% change in pseudo forcedv=dphat/khat;% change in displacementdvd=ai*h + 3*(dv-vdi*h-ai*h^2/2)/h;% change in velocityda=(dv-vdi*h-ai*h^2/2)*6/h^2;% change in acceleration
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Update variables%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
a(i)=ai;ai=ai+da;% accelerationv(i)=v(i-1)+dv;% displacementvdi=vdi+dvd;% velocityvd(i)=vdi;df=k*dv;% change in spring forcef(i)=f(i-1)+df;% spring forceif i==2fu=-sign(v(i))*vy;fu2=-sign(v(i))*vy2;fsh=sign(v(i))*fsh;vsh=sign(v(i))*vsh;flag=0;vd(1)=sign(v(i))*.000000001;end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Check yielding and reversals%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if model==1% stiffness-degrading
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[df,fu,k,flag,dpl,y,r,rev,fsh,vsh]=sddisppl(flag,fu,vd(i),vd(i-1),ko,alpha,vy,v(i),dv,r,rev,f(i),f(i-1),fsh,vsh,k);
elseif model==2% bilinear elastic[df,k,flag,dpl,y,r,rev]=bedisp2(flag,vd(i),vd(i-1),ko,alpha,vy,v(i),dv,r,rev);elseif model==3% bilinear elasto-plastic[df1,k1,flag1,dpl,y,r1,rev1]=bedisp2(flag1,vd(i),vd(i-1),k1o,alpha,vy1,v(i),dv,r1,rev1);[df2,fu2,k2,flag2,dpl,y,r2,rev2]=epdisp(flag2,fu2,vd(i),vd(i-1),k2o,alpha2,vy2,v(i),dv,r2,rev2);f1(i)=f1(i-1)+df1;f2(i)=f2(i-1)+df2;df=df1+df2;k=k1+k2;fu=fu2;flag=flag2;rev=rev2;rev2=0;endvpl=vpl+abs(dpl);% accumulative plastic defovr=vr+dpl;% residual dispny=ny+y;% number of yield eventsvpm(ny+1)=vpm(ny+1)+abs(dpl);% plastic defo per excursionvpm(ny+2)=[0];nrev=nrev+rev;% number of total yield reversalsrev=0;% reset counterf(i)=f(i-1)+df;% spring forceif (sign(f(i))~=sign(f(i-1)))nzero=nzero+1;% zero crossingendfl(i)=flag;flag;crit(i)=fu;if (fl(i)>fl(i-1) & model~=3)ai=(p(i)-f(i)-c*vdi)/m;% correct accelerationendif rem(i,100)==0waitbar1(i/N)endendclose(hh)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Check error%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
tend=clock;% end timetelapse=etime(tend,tstart);% elapsed timevmax(l)=max(abs(v));% maximum displacement (absolute)fmax=max(abs(f));% maximum forcevpmax=max(abs(vpm));% maximum plastic excursionat=a-p/m;% total accelerationamax=max(abs(at));% maximum accelerationif Ro~=0err=(vmax(l)-vmax(l-1))/vmax(l)% errorelsevcheck(:,l)=spline(time’,v’,ti’);err=max(abs((vcheck(:,l)-vcheck(:,l-1)))./vmax(l))% errorendif abs(err)<tol% check if error is within tolerancebreakend
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Store final values%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if l>=niterdiary([date ’sdofecho.txt’])’CHECK CONVERGENCE’[’error= ’ num2str(err) ’ iter= ’ num2str(l) ’ delta= ’ num2str(vmax(l)) ’ eq= ’ num2str(e)]diary off
endtelapsei(e,1,kk)=telapse;vmaxi(e,1,kk)=vmax(l);fmaxi(e,1,kk)=fmax;amaxi(e,1,kk)=amax;vpli(e,1,kk)=vpl;vri(e,1,kk)=vr;vpmi(e,1,kk)=vpmax;nyi(e,1,kk)=ny;nrevi(e,1,kk)=nrev;nzi(e,1,kk)=nzero;vrec(e,1,kk,:)=spline(time,v,ti);frec(e,1,kk,:)=spline(time,f,ti);
% endend
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B.5 CDSPECPOST.M: Post-Processing Function
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [cdspecpost.m]% SDOF Post-processing Program% This program plots results of the SDOF analyses.% method.% first created: KF 10/25/99% last revised: KF 9/29/00%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [mu,mubar,mud,muc,mur,mup,muec] =cdspecpost(delasij,dinelij,felasij,finelij,R,Tj,omegaj,mj,alpha,neqs,nstruc,gr,vijelas,vijinel,fijelas,fijinel,dpinelij,nyinelij,nrinelij,nzelasij,nzinelij,tfree,ti,dpmaxij,tend)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Plotting options%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%supress=1; % supress plots (1=y, 0=n)range1=[0 3.5 0 14]; % specify plotting rangesrange2=[0 3.5 0 1.6];range3=[0 3.5 0 1.6];range4=[0 3.5 0 4];range5=[0 3.5 0 100];range6=[0 3.5 0 1.2];range7=[0 3.5 0 50];%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Determine R-L-T relationship%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%mur2=zeros(neqs,nstruc,length(R)); % initialize matrixintog=386.4; % conversion factordinelij(:,:,1)=delasij; % set elastic displacementfinelij(:,:,1)=felasij; % set elastic forcenzinelij(:,:,1)=nzelasij; % set elastic zero crossingsumax=max(vijinel,[],4); % maximum displacements (in)umin=min(vijinel,[],4); % minimum displacements (in)fmax=max(fijinel,[],4); % maximum force (k)fmin=min(fijinel,[],4); % minimum force (k)umax(:,:,1)=max(vijelas,[],3); % include max elastic displacements (in)umin(:,:,1)=min(vijelas,[],3); % include min elastic displacements (in)fmax(:,:,1)=max(fijelas,[],3); % include max elastic force (in)fmin(:,:,1)=min(fijelas,[],3); % include min elastic force (in)Ist=find(ti==tend-tfree); % find point where free vibration startsures=abs(mean(vijinel(:,:,:,Ist+1:end),4)); % residual displacement (in)for k=1:length(R)
dy(:,:,k)=delasij./R(k); % yield displacementsfy(:,:,k)=felasij./R(k); % yield forceRbar(:,k)=ones(nstruc,1)*R(k);for j=1:nstruc
mu(:,j,k)=dinelij(:,j,k)./dy(:,j,k);% displacement ductility ratioainelij(:,j,k)=finelij(:,j,k)./mj(j)/intog;% accel spectramuc(:,j,k)=(-umin(:,j,k)+umax(:,j,k))./dy(:,j,k) - 1;% cyclic displacement ductilitymur(:,j,k)=ures(:,j,k)./dy(:,j,k);% residual displacement ductilitymup(:,j,k)=dpinelij(:,j,k)./dy(:,j,k) + 1;% cumulative plastic deformation ductilitymuec(:,j,k)=dpmaxij(:,j,k).*fy(:,j,k)*2./(.5.*umax(:,j,k).*fmax(:,j,k)+.5.*umin(:,j,k).*fmin(:,j,k));% max cyclic energyI1=find(alpha.*(mu(:,j,k)-1)<1);% residual disp condition 1I2=find(alpha.*(mu(:,j,k)-1)>=1);% residual disp condition 2mur2(I1,j,k)=abs(ures(I1,j,k)./((mu(I1,j,k)-1).*(1-alpha).*dy(I1,j,k)));mur2(I2,j,k)=abs(ures(I2,j,k)./((1-alpha).*dy(I2,j,k)./alpha));
endend%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Calculate mean and dispersion measures%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%mubar(:,:)=mean(mu); % mean displacement ductilitymucbar(:,:)=mean(muc); % mean cyclic ductilitymurbar(:,:)=mean(mur); % mean residual displacement ductilitymur2bar(:,:)=mean(mur2); % mean residual displacement ductilitymupbar(:,:)=mean(mup); % mean cumulative plastic deformation ductilitymuecbar(:,:)=mean(muec); % mean maximum cyclic energy ductilitydeltabar(:,:)=mean(dinelij); % mean displacementacelbar(:,:)=mean(ainelij); % mean accelerationnybar(:,:)=mean(nyinelij); % mean no. of yield eventsnrbar(:,:)=mean(nrinelij); % mean no. of full yield reversalsnzbar(:,:)=mean(nzinelij); % mean no. of zero crossings%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%musig(:,:)=std(mu); % std. dev. of displacement ductilitymucsig(:,:)=std(muc); % std. dev. of cyclic ductilitymursig(:,:)=std(mur); % std. dev. of residual displacement ductilitymur2sig(:,:)=std(mur2); % std. dev. of residual displacement ductilitymupsig(:,:)=std(mup); % std. dev. of cumulative plastic deformation ductilitymuecsig(:,:)=std(muec); % std. dev. of max cyclic energy ductilitydeltasig(:,:)=std(dinelij); % std. dev. of displacementacelsig(:,:)=std(ainelij); % std. dev. of accelerationnysig(:,:)=std(nyinelij); % std. dev. of no. of yield eventsnrsig(:,:)=std(nrinelij); % std. dev. of no. of full yield reversalsnzsig(:,:)=std(nzinelij); % std. dev. of no. of zero crossings%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%for k=1:length(R)
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mucov(:,k)=musig(:,k)./mubar(:,k); % COV of displacement ductilitymuccov(:,k)=mucsig(:,k)./mucbar(:,k); % COV of cyclic ductilitymurcov(:,k)=mursig(:,k)./murbar(:,k); % COV of residual displacement ductilitymur2cov(:,k)=mur2sig(:,k)./mur2bar(:,k); % COV of residual displacement ductilitymupcov(:,k)=mupsig(:,k)./mupbar(:,k); % COV of cumulative plastic deformation ductilitymueccov(:,k)=muecsig(:,k)./muecbar(:,k); % COV of max cyclic energy ductilitydeltacov(:,k)=deltasig(:,k)./deltabar(:,k); % COV of displacementacelcov(:,k)=acelsig(:,k)./acelbar(:,k); % COV of accelerationnycov(:,k)=nysig(:,k)./nybar(:,k); % COV of no. of yield eventsnrcov(:,k)=nrsig(:,k)./nrbar(:,k); % COV of no. of full yield reversalsnzcov(:,k)=nzsig(:,k)./nzbar(:,k); % COV of no. of zero crossings
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%mud(:,:)=std(log(mu)); % dispersion of displacement ductilitymucd(:,:)=std(log(muc)); % dispersion of cyclic ductilitymurd(:,:)=std(log(mur)); % dispersion of residual displacement ductilitymur2d(:,:)=std(log(mur2)); % dispersion of residual displacement ductilitymupd(:,:)=std(log(mup)); % dispersion of cumulative plastic deformation ductilitymuecd(:,:)=std(log(muec)); % dispersion of max cyclic energy ductilitydeltad(:,:)=std(log(dinelij)); % dispersion of displacementaceld(:,:)=std(log(ainelij)); % dispersion of accelerationnyd(:,:)=std(log(nyinelij)); % dispersion of no. of yield eventsnrd(:,:)=std(log(nrinelij)); % dispersion of no. of full yield reversalsnzd(:,:)=std(log(nzinelij)); % dispersion of no. of zero crossings%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Nassar & Krawinkler regression comparison%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%nsamp1=1; % where to sample 1st periodnsamp2=4; % where to sample 2nd periodmunk=0:.1:10;key=1; % prepare to retrieve reduction factorsT1=Tj(nsamp1);T2=Tj(nsamp2);[Rnk1]=nkregression(nsamp1,alpha,Tj,key,R); % 1st R factor[Rnk2]=nkregression(nsamp2,alpha,Tj,key,R); % 2nd R factorRbar1=Rbar(nsamp1,:); % 1st R factor, analRbar2=Rbar(nsamp2,:); % 2nd R factor, analmubar1=mubar(nsamp1,:); % 1st mu factor, analmubar2=mubar(nsamp2,:); % 2nd mu factor, analmusig1=musig(nsamp1,:); % 1st mu factor, analmusig2=musig(nsamp2,:); % 2nd mu factor, anal%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Plot R-L-T relationships%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%if supress==0
lastfig=gcf; % determine last figurefor i=1:neqs
figure(lastfig+i)for k=1:length(R)
plot(Tj,mu(i,:,k),’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’\mu’)title([date ’ Ductility Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)hold off
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Plot Response Spectra%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%lastfig=gcf;for i=1:neqs
figure(lastfig+i)for k=1:length(R)
plot(Tj,ainelij(i,:,k),’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’S_a (g)’)title([date ’ Response Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)hold off
end
lastfig=gcf;for i=1:neqs
figure(lastfig+i)for k=1:length(R)
plot(Tj,dinelij(i,:,k),’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’S_d (in) ’)title([date ’ Displacement Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)hold off
end
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end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%lastfig=gcf;figure(lastfig+1)subplot(3,1,1)for k=1:length(R)
plot(Tj,mubar(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’\mu’)title([date ’ Displacement Ductility Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range1)hold(’off’)
subplot(3,1,2)for k=1:length(R)
plot(Tj,mud(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’†elta’)title([date ’ Dispersion Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)
subplot(3,1,3)for k=1:length(R)
plot(Tj,mucov(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’Coefficient of Variation’)title([date ’ COV Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%lastfig=gcf;figure(lastfig+1)subplot(3,1,1)for k=1:length(R)
plot(Tj,mucbar(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’\mu_c’)title([date ’ Cyclic Displacement Ductility Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range1)hold(’off’)
subplot(3,1,2)for k=1:length(R)
plot(Tj,mucd(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’†elta_c’)title([date ’ Dispersion Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)
subplot(3,1,3)for k=1:length(R)
plot(Tj,muccov(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’COV_c’)title([date ’ Coefficient of Variation Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%lastfig=gcf;figure(lastfig+1)subplot(3,1,1)for k=1:length(R)
plot(Tj,murbar(:,k),’b’,’LineWidth’,k)hold on
end
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grid onxlabel(’T (sec)’)ylabel(’\mu_r’)title([date ’ Residual Displacement Ductility Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range4)hold(’off’)
subplot(3,1,2)for k=1:length(R)
plot(Tj,murd(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’†elta_r’)title([date ’ Dispersion Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)
subplot(3,1,3)for k=1:length(R)
plot(Tj,murcov(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’COV_r’)title([date ’ Coefficient of Variation Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%lastfig=gcf;figure(lastfig+1)subplot(3,1,1)for k=1:length(R)
plot(Tj,mur2bar(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’\mu_r_2’)title([date ’ Residual Displacement Ductility Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range6)hold(’off’)
subplot(3,1,2)for k=1:length(R)
plot(Tj,mur2d(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’†elta_r_2’)title([date ’ Dispersion Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)
subplot(3,1,3)for k=1:length(R)
plot(Tj,mur2cov(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’COV_r_2’)title([date ’ Coefficient of Variation Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%lastfig=gcf;figure(lastfig+1)subplot(3,1,1)for k=1:length(R)
plot(Tj,mupbar(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’\mu_p’)title([date ’ Cumulative Plastic Deformation Ductility Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range5)hold(’off’)
subplot(3,1,2)for k=1:length(R)
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plot(Tj,mupd(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’†elta_p’)title([date ’ Dispersion Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)
subplot(3,1,3)for k=1:length(R)
plot(Tj,mupcov(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’COV_p’)title([date ’ Coefficient of Variation Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%lastfig=gcf;figure(lastfig+1)subplot(3,1,1)for k=1:length(R)
plot(Tj,muecbar(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’\mu_e_c’)title([date ’ Maximum Cyclic Energy Ductility Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range4)hold(’off’)
subplot(3,1,2)for k=1:length(R)
plot(Tj,muecd(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’†elta_e_c’)title([date ’ Dispersion Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)
subplot(3,1,3)for k=1:length(R)
plot(Tj,mueccov(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’COV_e_c’)title([date ’ Coefficient of Variation Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%lastfig=gcf;figure(lastfig+1)subplot(3,1,1)for k=1:length(R)
plot(Tj,deltabar(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’‡elta (in)’)title([date ’ Displacement Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range1)hold(’off’)
subplot(3,1,2)for k=1:length(R)
plot(Tj,deltad(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’†elta_‡elta (in)’)title([date ’ Dispersion Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)
240
subplot(3,1,3)for k=1:length(R)
plot(Tj,deltacov(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’COV_‡elta’)title([date ’ Coefficient of Variation Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%lastfig=gcf;figure(lastfig+1)subplot(3,1,1)for k=1:length(R)
plot(Tj,acelbar(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’a (g)’)title([date ’ Acceleration Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range6)hold(’off’)
subplot(3,1,2)for k=1:length(R)
plot(Tj,aceld(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’†elta_a (g)’)title([date ’ Dispersion Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)
subplot(3,1,3)for k=1:length(R)
plot(Tj,acelcov(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’COV_a’)title([date ’ Coefficient of Variation Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%lastfig=gcf;figure(lastfig+1)subplot(3,1,1)for k=1:length(R)
plot(Tj,nybar(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’n_y’)title([date ’ Number of Inelastic Excursions Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range7)hold(’off’)
subplot(3,1,2)for k=1:length(R)
plot(Tj,nyd(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’†elta_n’)title([date ’ Dispersion Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)
subplot(3,1,3)for k=1:length(R)
plot(Tj,nycov(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’COV_n’)title([date ’ Coefficient of Variation Spectra for ’ gr])
241
legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%lastfig=gcf;figure(lastfig+1)subplot(3,1,1)for k=1:length(R)
plot(Tj,nrbar(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’n_r’)title([date ’ Number of Yield Reversals Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range7)hold(’off’)
subplot(3,1,2)for k=1:length(R)
plot(Tj,nrd(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’†elta_n’)title([date ’ Dispersion Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)
subplot(3,1,3)for k=1:length(R)
plot(Tj,nrcov(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’COV_n’)title([date ’ Coefficient of Variation Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%lastfig=gcf;figure(lastfig+1)subplot(3,1,1)for k=1:length(R)
plot(Tj,nzbar(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’n_z’)title([date ’ Number of Zero Crossings Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range5)hold(’off’)
subplot(3,1,2)for k=1:length(R)
plot(Tj,nzd(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’†elta_n’)title([date ’ Dispersion Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)
subplot(3,1,3)for k=1:length(R)
plot(Tj,nzcov(:,k),’b’,’LineWidth’,k)hold on
endgrid onxlabel(’T (sec)’)ylabel(’COV_n’)title([date ’ Coefficient of Variation Spectra for ’ gr])legend(’R=1’,’R=2’,’R=4’,’R=6’,’R=8’)axis(range2)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Plot Against Nassar & Krawinkler Regression%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%lastfig=gcf;figure(lastfig+1)subplot(2,1,1)plot(mubar1,Rbar1)
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hold onplot(mubar1+musig1,Rbar1,’--’)plot(mubar1-musig1,Rbar1,’--’)plot(munk,Rnk1,’r-.’,’Linewidth’,2.5)grid onxlabel(’\mu’)ylabel(’R’)title([date ’ Comparison with N&K for T=’ num2str(T1) ’ sec; \alpha=’ num2str(alpha)])legend(’\mu_a_v_g’,’\mu+\sigma’,’\mu-\sigma’,’N&K’)axis([0 10 0 8])hold off
subplot(2,1,2)plot(mubar2,Rbar2)hold onplot(mubar2+musig2,Rbar2,’--’)plot(mubar2-musig2,Rbar2,’--’)plot(munk,Rnk2,’r-.’,’Linewidth’,2.5)grid onxlabel(’\mu’)ylabel(’R’)title([date ’ Comparison with N&K for T=’ num2str(T2) ’ sec; \alpha=’ num2str(alpha)])legend(’\mu_a_v_g’,’\mu+\sigma’,’\mu-\sigma’,’N&K’)axis([0 10 0 8])hold off
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B.6 SDREG.M:R-µ-T Nonlinear Regression Program
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [sdofreg.m]% SDOF Results Plotting Program% This program plots results of the SDOF analyses.%% first created: KF 01/03/01% last revised: KF 02/09/01%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Initialize%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%path(path,’/afs/nd.edu/user30/kfarrow1/m-files’)keepold=input(’Would you like to keep the old normalizing data?’,’s’);if keepold==’n’
clearcfile=input(’Which comparison data file?’,’s’);nfile=input(’Which normalizing data file?’,’s’);’LOADING NORMALIZING DATA’load(nfile)filenameo=filename;gro=gr;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Determine R-L-T relationship%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clear dy fy mu mur mup’CALCULATING DEMAND INDICES’
ures=zeros(neqs,nstruc,length(R));% initialize matrixintog=386.4; % conversion factordinelij(:,:,1)=delasij; % set elastic displacementfinelij(:,:,1)=felasij; % set elastic forceIst=find(ti==tend-tfree); % find point where free vibration startsif model~=2
ures=abs(mean(vijinel(:,:,:,Ist+1:end),4));% residual displacement (in)endif anal==’a’ % effective R using smooth spectra
Reff=felasij./felavg;elseif anal==’n’
Reff=felasij./felnehrp;else
Reff=ones(size(felasij));endInot=find(Reff<1);elas=ones(size(Inot));Reff(Inot)=elas;hh = waitbar2(0,[’Calculating demand indicies...’]);for k=1:length(R)
dy(:,:,k)=delasij./Reff./R(k);% yield displacementsfy(:,:,k)=felasij./Reff./R(k);% yield forcefor j=1:nstruc
mu(:,j,k)=dinelij(:,j,k)./dy(:,j,k);% displacement ductility ratioainelij(:,j,k)=finelij(:,j,k)./mj(j)/intog;% accel spectramur(:,j,k)=ures(:,j,k)./dy(:,j,k);% residual displacement ductilitymup(:,j,k)=dpinelij(:,j,k)./dy(:,j,k) + 1;% cumulative plastic deformation ductility
endwaitbar2(k/length(R))
endclose(hh)
mu(:,:,1)=ones(neqs,nstruc);%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Calculate mean and dispersion measures%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%mubaro(:,:)=mean(mu); % mean displacement ductilitymurbaro(:,:)=mean(mur); % mean residual displacement ductilitymupbaro(:,:)=mean(mup); % mean cumulative plastic deformation ductilitydeltabaro(:,:)=mean(dinelij); % mean displacementacelbaro(:,:)=mean(ainelij); % mean accelerationnybaro(:,:)=mean(nyinelij); % mean no. of yield events%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%mudo(:,:)=std(log(mu)); % dispersion of displacement ductilitymurdo(:,:)=std(log(mur)); % dispersion of residual displacement ductilitymupdo(:,:)=std(log(mup)); % dispersion of cumulative plastic deformation ductilitydeltado(:,:)=std(log(dinelij)); % dispersion of displacementaceldo(:,:)=std(log(ainelij)); % dispersion of accelerationnydo(:,:)=std(log(nyinelij)); % dispersion of no. of yield events%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%musigo(:,:)=std(mu); % std. dev. of displacement ductilitymursigo(:,:)=std(mur); % std. dev. of residual displacement ductilitymupsigo(:,:)=std(mup); % std. dev. of cumulative plastic deformation ductilitydeltasigo(:,:)=std(dinelij); % std. dev. of displacementacelsigo(:,:)=std(ainelij); % std. dev. of accelerationnysigo(:,:)=std(nyinelij); % std. dev. of no. of yield events%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%for k=1:length(R)
mucovo(:,k)=musigo(:,k)./mubaro(:,k); % COV of displacement ductilitymurcovo(:,k)=mursigo(:,k)./murbaro(:,k); % COV of residual displacement ductilitymupcovo(:,k)=mupsigo(:,k)./mupbaro(:,k); % COV of cumulative plastic deformation ductilitydeltacovo(:,k)=deltasigo(:,k)./deltabaro(:,k); % COV of displacement
244
acelcovo(:,k)=acelsigo(:,k)./acelbaro(:,k); % COV of accelerationnycovo(:,k)=nysigo(:,k)./nybaro(:,k); % COV of no. of yield events
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%muo=mu;muro=mur;mupo=mup;nyo=nyinelij;else
cfile=input(’Which comparison data file?’,’s’);save plottemp *o cfileclearload plottemp
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Load comparison data%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%’LOADING COMPARISON DATA’load(cfile)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Determine R-L-T relationship%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clear dy fy mu mur mup’CALCULATING DEMAND INDICES’
ures=zeros(neqs,nstruc,length(R));% initialize matrixintog=386.4; % conversion factordinelij(:,:,1)=delasij; % set elastic displacementfinelij(:,:,1)=felasij; % set elastic forceIst=find(ti==tend-tfree); % find point where free vibration startsif model~=2
ures=abs(mean(vijinel(:,:,:,Ist+1:end),4));% residual displacement (in)endif anal==’a’ % effective R using smooth spectra
Reff=felasij./felavg;elseif anal==’n’
Reff=felasij./felnehrp;else
Reff=ones(size(felasij));endInot=find(Reff<1);elas=ones(size(Inot));Reff(Inot)=elas;hh = waitbar2(0,[’Calculating demand indicies...’]);for k=1:length(R)
dy(:,:,k)=delasij./Reff./R(k);% yield displacementsfy(:,:,k)=felasij./Reff./R(k);% yield forcefor j=1:nstruc
mu(:,j,k)=dinelij(:,j,k)./dy(:,j,k);% displacement ductility ratioainelij(:,j,k)=finelij(:,j,k)./mj(j)/intog;% accel spectramur(:,j,k)=ures(:,j,k)./dy(:,j,k);% residual displacement ductilitymup(:,j,k)=dpinelij(:,j,k)./dy(:,j,k) + 1;% cumulative plastic deformation ductilitygammamui(:,j,k)=mu(:,j,k)./muo(:,j,k);gammamuri(:,j,k)=mur(:,j,k)./muro(:,j,k);gammamupi(:,j,k)=mup(:,j,k)./mupo(:,j,k);gammanyi(:,j,k)=nyinelij(:,j,k)./nyo(:,j,k);
endwaitbar2(k/length(R))
endclose(hh)
mu(:,:,1)=ones(neqs,nstruc);%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Calculate mean and dispersion measures%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%mubar(:,:)=mean(mu); % mean displacement ductilitymurbar(:,:)=mean(mur); % mean residual displacement ductilitymupbar(:,:)=mean(mup); % mean cumulative plastic deformation ductilitydeltabar(:,:)=mean(dinelij); % mean displacementacelbar(:,:)=mean(ainelij); % mean accelerationnybar(:,:)=mean(nyinelij); % mean no. of yield eventsgammamu(:,:)=mean(gammamui);gammamur(:,:)=mean(gammamuri);gammamup(:,:)=mean(gammamupi);gammany(:,:)=mean(gammanyi);%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%mud(:,:)=std(log(mu)); % dispersion of displacement ductilitymurd(:,:)=std(log(mur)); % dispersion of residual displacement ductilitymupd(:,:)=std(log(mup)); % dispersion of cumulative plastic deformation ductilitydeltad(:,:)=std(log(dinelij)); % dispersion of displacementaceld(:,:)=std(log(ainelij)); % dispersion of accelerationnyd(:,:)=std(log(nyinelij)); % dispersion of no. of yield events%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%musig(:,:)=std(mu); % std. dev. of displacement ductilitymursig(:,:)=std(mur); % std. dev. of residual displacement ductilitymupsig(:,:)=std(mup); % std. dev. of cumulative plastic deformation ductilitydeltasig(:,:)=std(dinelij); % std. dev. of displacementacelsig(:,:)=std(ainelij); % std. dev. of accelerationnysig(:,:)=std(nyinelij); % std. dev. of no. of yield events%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%for k=1:length(R)
mucov(:,k)=musig(:,k)./mubar(:,k); % COV of displacement ductilitymurcov(:,k)=mursig(:,k)./murbar(:,k); % COV of residual displacement ductilitymupcov(:,k)=mupsig(:,k)./mupbar(:,k); % COV of cumulative plastic deformation ductility
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deltacov(:,k)=deltasig(:,k)./deltabar(:,k); % COV of displacementacelcov(:,k)=acelsig(:,k)./acelbar(:,k); % COV of accelerationnycov(:,k)=nysig(:,k)./nybar(:,k); % COV of no. of yield events
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Calculate normalized spectra%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%if (size(gro)==size(gr) & sum(gro==gr)==length(gr))
gratio=’individual’else
gammamu=mubar./mubaro;gammamur=murbar./murbaro;gammamup=mupbar./mupbaro;gammany=nybar./nybaro;gratio=’mean’
endgammadel=deltacov./deltacovo;gammaacc=acelcov./acelcovo;gammamud=mucov./mucovo;%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Plot R-L-T relationships%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%rangemu=[0 3.5 0 14]; % specify rangesrangemup=[0 3.5 0 100];rangemur=[0 3.5 0 4];rangeny=[0 3.5 0 50];rangedd=[0 2 0 2];rangega=[0 3.5 0 4];rangecov=[0 2 0 1.5];close allfilenamefilenameorename=input(’change these names? ’,’s’)if rename==’y’
filename=input(’enter comparison filename: ’,’s’)filenameo=input(’enter normalization filename: ’,’s’)
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%figure(1)subplot(2,1,1)plot(0.1,0.1,’w:’)hold onfor k=1:length(R)
plot(Tj,mubaro(:,k),’-’,’LineWidth’,k)hold onplot(Tj,mubar(:,k),’r--’,’LineWidth’,k)
endgrid onxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\mu’,’FontName’,’Times’)title([filenameo ’ (solid) vs. ’ filename ’ (dashed)’],’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)axis(rangemu)set(gca,’FontName’,’times’)hold off%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%subplot(2,1,2)plot(0.1,0.1,’w:’)hold onfor k=1:length(R)
plot(Tj,mupbaro(:,k),’-’,’LineWidth’,k)hold onplot(Tj,mupbar(:,k),’r--’,’LineWidth’,k)
endgrid onxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\it\mu_pm’,’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)axis(rangemup)set(gca,’FontName’,’times’)hold off%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%figure(2)subplot(2,1,1)plot(0.1,0.1,’w:’)hold onfor k=1:length(R)
plot(Tj,murbaro(:,k),’-’,’LineWidth’,k)hold onplot(Tj,murbar(:,k),’r--’,’LineWidth’,k)
endgrid onxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\it\mu_r
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m’,’FontName’,’Times’)title([filenameo ’ (solid) vs. ’ filename ’ (dashed)’],’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)axis(rangemur)set(gca,’FontName’,’times’)hold off%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%subplot(2,1,2)plot(0.1,0.1,’w:’)hold onfor k=1:length(R)
plot(Tj,nybaro(:,k),’-’,’LineWidth’,k)hold onplot(Tj,nybar(:,k),’r--’,’LineWidth’,k)
endgrid onxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\itn_ym’,’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)axis(rangeny)set(gca,’FontName’,’times’)hold off%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Plot R-g-T relationships%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%baseline=ones(size(Tj));lastfig=gcf;figure(3)subplot(2,1,1)plot(0.1,0.1,’w:’)hold onplot(Tj,baseline)hold onfor k=2:length(R)
plot(Tj,gammamu(:,k),’r-’,’LineWidth’,k)hold on
endxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\it„amma_\mum’,’FontName’,’Times’)title([’„amma = ’ filename ’/’ filenameo],’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)axis(rangega)set(gca,’FontName’,’times’)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%subplot(2,1,2)plot(0.1,0.1,’w:’)hold onplot(Tj,baseline)hold onfor k=2:length(R)
plot(Tj,gammamup(:,k),’r-’,’LineWidth’,k)hold on
endxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\it„amma_\mu_pm’,’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)axis(rangega)set(gca,’FontName’,’times’)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%figure(4)subplot(2,1,1)plot(0.1,0.1,’w:’)hold onplot(Tj,baseline)hold onfor k=2:length(R)
plot(Tj,gammamur(:,k),’r-’,’LineWidth’,k)hold on
endxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\it„amma_\mu_rm’,’FontName’,’Times’)title([’„amma = ’ filename ’/’ filenameo],’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thin
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ightarrow thick lines)’)axis(rangega)set(gca,’FontName’,’times’)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%subplot(2,1,2)plot(0.1,0.1,’w:’)hold onplot(Tj,baseline)hold onfor k=2:length(R)
plot(Tj,gammany(:,k),’r-’,’LineWidth’,k)hold on
endxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\it„amma_n_ym’,’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)axis(rangega)set(gca,’FontName’,’times’)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Calculate Regression Constants Using N&K Model%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%figure(5)subplot(2,1,1)plot(0.1,0.1,’w:’)hold onbetao=[1]; % initial guess for c constantRj=ones(nstruc,1)*R; % R matrixtstart=15; % index of control periodstend=30;for j=tstart:tend
X=Rj(j,:);Y=mubar(j,:);cfit(j)=nlinfit(X,Y,’muhat’,betao);% calculate c const.
endbetao=[0.8 0.29]; % initial guess for a & b constantsX=Tj(tstart:tend);Y=cfit(tstart:tend);betahat=nlinfit(X,Y,’chat’,betao);% calculate reg consts.for k=1:length(R)
X=[Tj’ Rj(:,k)]; % regression indep varsmufit(:,k)=farrow(betahat,X);plot(Tj,mubar(:,k),’-’,’Linewidth’,k)hold onplot(Tj,mufit(:,k),’r--’,’Linewidth’,k)
endaxis(rangemu)grid onxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\it\mum’,’FontName’,’Times’)title([ ’Regression for ’ filename ’: a = ’ num2str(betahat(1)) ’, b = ’ num2str(betahat(2))],’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)set(gca,’FontName’,’times’)hold offbetahati=betahat%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%subplot(2,1,2)plot(0.1,0.1,’w:’)hold onbetao=[1]; % initial guess for c constantfor j=tstart:tend
X=Rj(j,:);Y=mubaro(j,:);cfit(j)=nlinfit(X,Y,’muhat’,betao);% calculate c const.
endbetao=[0.8 0.29]; % initial guess for a & b constantsX=Tj(tstart:tend);Y=cfit(tstart:tend);betahat=nlinfit(X,Y,’chat’,betao);% calculate reg consts.for k=1:length(R)
X=[Tj’ Rj(:,k)]; % regression indep varsmufit(:,k)=farrow(betahat,X);plot(Tj,mubaro(:,k),’-’,’Linewidth’,k)hold onplot(Tj,mufit(:,k),’r--’,’Linewidth’,k)
endaxis(rangemu)grid onxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\it\mum’,’FontName’,’Times’)title([ ’Regression for ’ filenameo ’: a = ’ num2str(betahat(1)) ’, b = ’ num2str(betahat(2))],’FontName’,’Times’)
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legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)set(gca,’FontName’,’times’)hold offbetahato=betahat;%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Compare Scatter%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%figure(6)subplot(2,1,1)plot(0.1,0.1,’w:’)hold onfor k=1:length(R)
plot(Tj,acelsigo(:,k),’-’,’LineWidth’,k)hold onplot(Tj,acelsig(:,k),’r--’,’LineWidth’,k)
endxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\it\sigma(S_d)m’,’FontName’,’Times’)title([filenameo ’ (solid) vs. ’ filename ’ (dashed)’],’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)%axis(rangega)set(gca,’FontName’,’times’)hold off%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%subplot(2,1,2)plot(0.1,0.1,’w:’)hold onfor k=1:length(R)
plot(Tj,musigo(:,k),’-’,’LineWidth’,k)hold onplot(Tj,musig(:,k),’r--’,’LineWidth’,k)
endxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\it\sigma(\mu)m’,’FontName’,’Times’)title([’„amma = ’ filename ’/’ filenameo],’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)axis(rangemu)set(gca,’FontName’,’times’)hold off%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%figure(7)subplot(2,1,1)plot(0.1,0.1,’w:’)hold onfor k=1:length(R)
plot(Tj,acelcovo(:,k),’-’,’LineWidth’,k)hold onplot(Tj,acelcov(:,k),’r--’,’LineWidth’,k)
endxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\itCOV(S_d)m’,’FontName’,’Times’)title([filenameo ’ (solid) vs. ’ filename ’ (dashed)’],’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)axis(rangecov)set(gca,’FontName’,’times’)hold off%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%subplot(2,1,2)plot(0.1,0.1,’w:’)hold onplot(Tj,baseline)hold onfor k=1:length(R)
plot(Tj,gammaacc(:,k),’r-’,’LineWidth’,k)hold on
endgrid onxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\it„amma_C_O_V_(_S_d_)m’,’FontName’,’Times’)title([’„amma = ’ filename ’/’ filenameo],’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)axis(rangedd)set(gca,’FontName’,’times’)hold(’off’)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%figure(8)subplot(2,1,1)plot(0.1,0.1,’w:’)hold onfor k=1:length(R)
plot(Tj,mucovo(:,k),’-’,’LineWidth’,k)hold onplot(Tj,mucov(:,k),’r--’,’LineWidth’,k)
endxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\itCOV(\mu)m’,’FontName’,’Times’)title([filenameo ’ (solid) vs. ’ filename ’ (dashed)’],’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)axis(rangecov)set(gca,’FontName’,’times’)hold off%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%subplot(2,1,2)plot(0.1,0.1,’w:’)hold onplot(Tj,baseline)hold onfor k=2:length(R)
plot(Tj,gammamud(:,k),’r-’,’LineWidth’,k)hold on
endgrid onxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\it„amma_C_O_V_(_\mu_)m’,’FontName’,’Times’)title([’„amma = ’ filename ’/’ filenameo],’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)axis(rangedd)set(gca,’FontName’,’times’)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Demand Index Cross-correlations%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%N=0; % set counterdeg=2; % degree of polynomial fitnx=30; % number of discrete pointsmumax=10; % maximum controllable mulabels=[’\mu \mu_r\mu_pn_y ’];% matrix titlesTmax=24; % maximum period for residual disp[Imax]=find(mubar<mumax);submubar=mubar(1:Tmax,:);submurbar=murbar(1:Tmax,:);submupbar=mupbar(1:Tmax,:);subnybar=nybar(1:Tmax,:);[Jmax]=find(submubar<mumax);lambda(:,:,1)=mubar; % set up demand index matrixlambda(:,:,2)=murbar;lambda(:,:,3)=mupbar;lambda(:,:,4)=nybar;sublambda(:,:,1)=submubar; % set up demand index matrixsublambda(:,:,2)=submurbar;sublambda(:,:,3)=submupbar;sublambda(:,:,4)=subnybar;figure(9)for i=1:4
for j=1:4N=N+1;subplot(4,4,N)lambda1=lambda(:,:,i);lambda2=lambda(:,:,j);sublambda1=sublambda(:,:,i);sublambda2=sublambda(:,:,j);if i==j
axis([0 2 0 2])axis offtext(1,1,[’\fontname{times}\it’ labels(5*i-4:5*i)])
elseif (i==2 | j==2)lambda1=sublambda1(Jmax);lambda2=sublambda2(Jmax);elselambda1=lambda1(Imax);lambda2=lambda2(Imax);endplot(lambda2,lambda1,’.’)hold onP=polyfit(lambda2,lambda1,deg);% find coeff’smi=min(lambda2);ma=max(lambda2);
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dx=(ma-mi)/nx;xfit=[mi:dx:ma];yfit=polyval(P,xfit);coeffs(i,j,:)=P;plot(xfit,yfit,’r’)set(gca,’FontName’,’times’)hold off
endend
endsuptitle([’\fontname{times}Cross-correlations, ’ filename ’: data points and regression lines’])%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%[Imax]=find(mubaro<mumax);submubar=mubaro(1:Tmax,:);submurbar=murbaro(1:Tmax,:);submupbar=mupbaro(1:Tmax,:);subnybar=nybaro(1:Tmax,:);[Jmax]=find(submubar<mumax);lambda(:,:,1)=mubaro; % set up demand index matrixlambda(:,:,2)=murbaro;lambda(:,:,3)=mupbaro;lambda(:,:,4)=nybaro;sublambda(:,:,1)=submubar; % set up demand index matrixsublambda(:,:,2)=submurbar;sublambda(:,:,3)=submupbar;sublambda(:,:,4)=subnybar;N=0; % set counterfigure(10)for i=1:4
for j=1:4N=N+1;subplot(4,4,N)lambda1=lambda(:,:,i);lambda2=lambda(:,:,j);sublambda1=sublambda(:,:,i);sublambda2=sublambda(:,:,j);if i==j
axis([0 2 0 2])axis offtext(1,1,[’\fontname{times}\it’ labels(5*i-4:5*i)])
elseif (i==2 | j==2)lambda1=sublambda1(Jmax);lambda2=sublambda2(Jmax);elselambda1=lambda1(Imax);lambda2=lambda2(Imax);endplot(lambda2,lambda1,’.’)hold onP=polyfit(lambda2,lambda1,deg);% find coeff’smi=min(lambda2);ma=max(lambda2);dx=(ma-mi)/nx;xfit=[mi:dx:ma];yfit=polyval(P,xfit);coeffso(i,j,:)=P;plot(xfit,yfit,’r’)set(gca,’FontName’,’times’)hold off
endend
endsuptitle([’\fontname{times}Cross-correlations, ’ filenameo ’: data points and regression lines’])%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Compare Scatter, part II%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%figure(11)subplot(2,1,1)plot(0.1,0.1,’w:’)hold onfor k=1:length(R)
plot(Tj,deltacovo(:,k),’-’,’LineWidth’,k)hold onplot(Tj,deltacov(:,k),’r--’,’LineWidth’,k)
endxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\itCOV(S_d)m’,’FontName’,’Times’)title([filenameo ’ (solid) vs. ’ filename ’ (dashed)’],’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)axis(rangecov)set(gca,’FontName’,’times’)hold off%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%subplot(2,1,2)plot(0.1,0.1,’w:’)hold onplot(Tj,baseline)
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hold onfor k=1:length(R)
plot(Tj,gammadel(:,k),’r-’,’LineWidth’,k)hold on
endgrid onxlabel(’\itT_om (sec)’,’FontName’,’Times’)ylabel(’\it„amma_C_O_V_(_S_d_)m’,’FontName’,’Times’)title([’„amma = ’ filename ’/’ filenameo],’FontName’,’Times’)legend(’\fontname{times}\itRm = 1, 2, 4, 6, 8 (thinightarrow thick lines)’)axis(rangedd)set(gca,’FontName’,’times’)hold(’off’)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Save Regression Constants?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%compile=input(’save these regression constants? ’,’s’)if compile==’y’
save([’./constants/’ filename],’betahati’,’coeffs’)save([’./constants/’ filenameo],’betahato’,’coeffso’)
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Save Figures?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%hardcopy=input(’save these plots? ’,’s’)if hardcopy==’y’
check=ls([’./figures/’ filename ’*’])if (check(1:2)~=’No’)
addon=input([filename ’ exists. Please enter modifier text or press CTRL-C:’],’s’)filename=[filename addon]
endfor f=1:11
figure(f)orient tallprint(’-depsc’,[’./figures/’ filename num2str(f)])
endend
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B.7 DEMANDREG.M: Cross-Correlation Nonlinear Regression Program
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [demandreg.m]% SDOF Regression Program% This program performs nonlinear regression analyses for the demand indices.%% first created: KF 06/19/01% last revised: KF 06/19/01%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Initialize%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%path(path,’/afs/nd.edu/user30/kfarrow1/m-files’)init=input(’Load new data?’,’s’)if init==’y’
clearnfile=input(’Which data file?’,’s’);’LOADING DATA’load(nfile)filenameo=filename;gro=gr;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Determine R-L-T relationship%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clear dy fy mu mur mup’CALCULATING DEMAND INDICES’ures=zeros(neqs,nstruc,length(R)); % initialize matrixintog=386.4; % conversion factordinelij(:,:,1)=delasij; % set elastic displacementfinelij(:,:,1)=felasij; % set elastic forceIst=find(ti==tend-tfree); % find point where free vibration startsif model~=2
ures=abs(mean(vijinel(:,:,:,Ist+1:end),4)); % residual displacement (in)endif anal==’a’ % effective R using smooth spectra
Reff=felasij./felavg;elseif anal==’n’
Reff=felasij./felnehrp;else
Reff=ones(size(felasij));endInot=find(Reff<1);elas=ones(size(Inot));Reff(Inot)=elas;hh = waitbar2(0,[’Calculating demand indicies...’]);for k=1:length(R)
dy(:,:,k)=delasij./Reff./R(k); % yield displacementsfy(:,:,k)=felasij./Reff./R(k); % yield forcefor j=1:nstruc
mu(:,j,k)=dinelij(:,j,k)./dy(:,j,k);% displacement ductility ratioainelij(:,j,k)=finelij(:,j,k)./mj(j)/intog;% accel spectramur(:,j,k)=ures(:,j,k)./dy(:,j,k);% residual displacement ductilitymup(:,j,k)=dpinelij(:,j,k)./dy(:,j,k) + 1;% cumulative plastic deformation ductility
endwaitbar2(k/length(R))
endclose(hh)mu(:,:,1)=ones(neqs,nstruc);%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Calculate mean and dispersion measures%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%mubaro(:,:)=mean(mu); % mean displacement ductilitymurbaro(:,:)=mean(mur); % mean residual displacement ductilitymupbaro(:,:)=mean(mup); % mean cumulative plastic deformation ductilitynybaro(:,:)=mean(nyinelij); % mean no. of yield eventsmupbaro=mupbaro-1; % mean cumulative plastic deformation ductility%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%mudo(:,:)=std(log(mu)); % dispersion of displacement ductilitymurdo(:,:)=std(log(mur)); % dispersion of residual displacement ductilitymupdo(:,:)=std(log(mup)); % dispersion of cumulative plastic deformation ductilitynydo(:,:)=std(log(nyinelij)); % dispersion of no. of yield events%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%musigo(:,:)=std(mu); % std. dev. of displacement ductilitymursigo(:,:)=std(mur); % std. dev. of residual displacement ductilitymupsigo(:,:)=std(mup); % std. dev. of cumulative plastic deformation ductilitynysigo(:,:)=std(nyinelij); % std. dev. of no. of yield events%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%for k=1:length(R)
mucovo(:,k)=musigo(:,k)./mubaro(:,k); % COV of displacement ductilitymurcovo(:,k)=mursigo(:,k)./murbaro(:,k); % COV of residual displacement ductilitymupcovo(:,k)=mupsigo(:,k)./mupbaro(:,k); % COV of cumulative plastic deformation ductilitynycovo(:,k)=nysigo(:,k)./nybaro(:,k); % COV of no. of yield events
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%else
save plottemp *oclearload plottemp
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end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Plot R-L-T relationships%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ranges=[10 4 50 25]; % specify rangesclose allfilenameorename=input(’change this name? ’,’s’)if rename==’y’
filenameo=input(’enter filename: ’,’s’)end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Demand Index Cross-correlations%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%N=0; % set counterdeg=2; % degree of polynomial fitnx=30; % number of discrete pointsmumax=10; % maximum controllable mulabels=[’\mu \mu_r\mu_pn_y ’]; % matrix titlesTmin=15; % minimum periodTmax=24; % maximum period (residual disp)[Imax]=find(mubaro<mumax);submubar=mubaro(Tmin:Tmax,:);submurbar=murbaro(Tmin:Tmax,:);submupbar=mupbaro(Tmin:Tmax,:);subnybar=nybaro(Tmin:Tmax,:);[Jmax]=find(submubar<mumax);lambda(:,:,1)=mubaro; % set up demand index matrixlambda(:,:,2)=murbaro;lambda(:,:,3)=mupbaro;lambda(:,:,4)=nybaro;sublambda(:,:,1)=submubar; % set up demand index matrixsublambda(:,:,2)=submurbar;sublambda(:,:,3)=submupbar;sublambda(:,:,4)=subnybar;betao=[0.8 0.29]; % initial guess for a & b constantsf=25figure(f)for j=1:4
for i=j:4N=4*i+j-4;M=4*j+i-4;lambda1=lambda(:,:,i);lambda2=lambda(:,:,j);sublambda1=sublambda(:,:,i);sublambda2=sublambda(:,:,j);if i==j
subplot(4,4,N)axis([0 2 0 2])axis offtext(1,1,[’\fontname{times}\it’ labels(5*i-4:5*i)])
elselambda1=lambda1(Imax);lambda2=lambda2(Imax);subplot(4,4,N)plot(lambda2,lambda1,’.’)hold onsubplot(4,4,M)plot(lambda1,lambda2,’.’)hold onmi2=min(lambda2);ma2=max(lambda2);dx2=(ma2-mi2)/nx;xfit2=[mi2:dx2:ma2]’;mi1=min(lambda1);ma1=max(lambda1);dx1=(ma1-mi1)/nx;xfit1=[mi1:dx1:ma1]’;if (j==1 | i==1)keys=[1 3];[betahat2,r2]=nlinfit(lambda2,lambda1,’lambdahat1’,betao);% calculate reg consts.[betahat1,r1]=nlinfit(lambda1,lambda2,’lambdahat3’,betao);% calculate reg consts.elsekeys=[2 4];[betahat2,r2]=nlinfit(lambda2,lambda1,’lambdahat2’,betao);% calculate reg consts.[betahat1,r1]=nlinfit(lambda1,lambda2,’lambdahat4’,betao);% calculate reg consts.endbetahati=[betahat2 betahat1];resi=sum(abs([r2 r1]));[Yk,Ik]=min(resi);betahat=betahati(:,Ik);coeffs2(i,j,:)=betahat;coeffs2(j,i,:)=betahat;res(i,j)=resi(Ik);res(j,i)=resi(Ik);subplot(4,4,N)xfit=xfit2;yfit=lambdahat(betahat,xfit,keys(1));plot(xfit,yfit,’r’)axis([0 ranges(j) 0 ranges(i)])set(gca,’FontName’,’times’)hold off
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subplot(4,4,M)xfit=xfit1;yfit=lambdahat(betahat,xfit,keys(2));plot(xfit,yfit,’r’)axis([0 ranges(i) 0 ranges(j)])set(gca,’FontName’,’times’)hold off
endend
endsuptitle([’\fontname{times}Cross-correlations, ’ filenameo ’: data points and regression lines’])%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Save Regression Constants?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%coeffs2rescompile=input(’save these regression constants? ’,’s’)if compile==’y’
save([’./constants/’ filenameo],’coeffs2’,’-append’)end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Save Figures?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%hardcopy=input(’save these plots? ’,’s’)if hardcopy==’y’
check=ls([’./figures/’ filenameo ’*’])if (check(1:2)~=’No’)
addon=input([filenameo ’ exists. Please enter modifier text or press CTRL-C:’],’s’)filename=[filenameo addon]
endfigure(f)orient tallprint(’-depsc’,[’./figures/’ filenameo num2str(f)])
end
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B.8 NKREGRESSION.M, NKREGRESSION_ST.M, NKREGRESSION_ME.M,NKREGRESSION_SO.M, NKREGRESSION_NF.M: Regression Functions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [nkregression.m]% Nassar & Krawinkler R-mu-T regression% This program calculates the R or mu for a given T depending% on the user specification.% first created: KF 3/10/00% last revised: KF 3/10/00%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [nkout] = nkregression(nsamp,alpha,Tj,key,R)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Nassar & Krawinkler regression%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%aldata=[0 .02 .1]; % strain hardening ptsadata=[1 1.01 .8]; % a coeff. databdata=[.42 .37 .29]; % b coeff. dataalnk=0:.01:.1; % strain hardening fitank=spline(aldata,adata,alnk); % a coeff. fitbnk=spline(aldata,bdata,alnk); % b coeff. fitI=find(alnk==alpha); % pick pointcnk=Tj.^ank(I)./(Tj.^ank(I)+1)+bnk(I)./Tj;T=Tj(nsamp); % sampled periodcnk=cnk(nsamp); % c parameterif key==1
munk=0:.1:10;Rnk=(cnk*(munk-1)+1).^(1/cnk); % R factornkout=Rnk; % output R
elseif key==2Rnk=R;munk=(Rnk.^cnk-1)./cnk+1; % mu factornkout=munk; % output mu
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [nkregression_st.m]% Nassar & Krawinkler R-mu-T regression% This program calculates the R or mu for a given T depending% on the user specification.% This was modified for the UND very dense soil ensemble results%% first created: KF 2/22/01% last revised: KF 2/22/01%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [nkout] = nkregression_st(nsamp,alpha,Tj,key,R)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Nassar & Krawinkler regression%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ank=1.4628;bnk=0.5778;cnk=Tj.^ank./(Tj.^ank+1)+bnk./Tj;T=Tj(nsamp); % sampled periodcnk=cnk(nsamp); % c parameterif key==1
munk=0:.1:10;Rnk=(cnk*(munk-1)+1).^(1/cnk); % R factornkout=Rnk; % output R
elseif key==2Rnk=R;munk=(Rnk.^cnk-1)./cnk+1; % mu factornkout=munk; % output mu
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [nkregression_me.m]% Nassar & Krawinkler R-mu-T regression% This program calculates the R or mu for a given T depending% on the user specification.% This was modified for the UND stiff ensemble results%% first created: KF 2/22/01% last revised: KF 2/22/01%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [nkout] = nkregression_me(nsamp,alpha,Tj,key,R)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Nassar & Krawinkler regression%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ank=1.4925;bnk=0.4619;cnk=Tj.^ank./(Tj.^ank+1)+bnk./Tj;T=Tj(nsamp); % sampled periodcnk=cnk(nsamp); % c parameterif key==1
munk=0:.1:10;Rnk=(cnk*(munk-1)+1).^(1/cnk); % R factornkout=Rnk; % output R
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elseif key==2Rnk=R;munk=(Rnk.^cnk-1)./cnk+1; % mu factornkout=munk; % output mu
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [nkregression_so.m]% Nassar & Krawinkler R-mu-T regression% This program calculates the R or mu for a given T depending% on the user specification.% This was modified for the UND soft soil ensemble results%% first created: KF 2/22/01% last revised: KF 2/22/01%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [nkout] = nkregression_so(nsamp,alpha,Tj,key,R)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Nassar & Krawinkler regression%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ank=-0.40898;bnk=0.94534;cnk=Tj.^ank./(Tj.^ank+1)+bnk./Tj;T=Tj(nsamp); % sampled periodcnk=cnk(nsamp); % c parameterif key==1
munk=0:.1:10;Rnk=(cnk*(munk-1)+1).^(1/cnk); % R factornkout=Rnk; % output R
elseif key==2Rnk=R;munk=(Rnk.^cnk-1)./cnk+1; % mu factornkout=munk; % output mu
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [nkregression_nf.m]% Nassar & Krawinkler R-mu-T regression% This program calculates the R or mu for a given T depending% on the user specification.% This was modified for the SAC near field ensemble results%% first created: KF 2/22/01% last revised: KF 2/22/01%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [nkout] = nkregression_nf(nsamp,alpha,Tj,key,R)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Nassar & Krawinkler regression%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ank=0.3490;bnk=0.8949;cnk=Tj.^ank./(Tj.^ank+1)+bnk./Tj;T=Tj(nsamp); % sampled periodcnk=cnk(nsamp); % c parameterif key==1
munk=0:.1:10;Rnk=(cnk*(munk-1)+1).^(1/cnk); % R factornkout=Rnk; % output R
elseif key==2Rnk=R;munk=(Rnk.^cnk-1)./cnk+1; % mu factornkout=munk; % output mu
end
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B.9 EPDISP.M, BEDISP.M, SDDISP.M: Hysteretic Rule Functions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [epdisp.m]% Bilinear Elasto-Plastic Hysteretic Rule% This program determines the yielding rules for a SDOF oscillator.%% first created: KF 9/18/00% last revised: KF 9/26/00%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [df,fu,k,flag,dpl,y,r,rev]=epdisp(flag,fu,vdi,vdo,ko,alpha,vy,vi,dv,r,rev)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Yielding condition%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%if flag==1 % yielding
y=0; % yield counterif (sign(vdi)~=sign(vdo)) % reversal
flag=0; % unloaddf=alpha*ko*dv; % unloading force (k)fu=vi; % redefine yield surface (in)k=ko; % unloading stiffness (k/in)dpl=0; % no plastic defor=1; % full reversal counter
else % inelasticdf=alpha*ko*dv; % incremental force (k)fu=vi; % redefine yield surface (in)k=alpha*ko; % secondary stiffness (k/in)dpl=abs(dv); % plastic defo (in)r=0; % full reversal counter
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Non-yielding condition%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%elseif flag==0 % elastic
if abs(vi-(fu))>=2*vy % check for yieldingy=1; % yield counterdvo=abs(vi-(fu))-2*vy; % abs plastic disp (in)pct=dvo/abs(dv); % plastic percentagedv1=dv*pct; % plastic disp (in)dv2=dv-dv1; % elastic disp (in)df=ko*dv2+alpha*ko*dv1; % incremental force (k)flag=1; % yieldingfu=vi; % redefine yield surface (in)k=alpha*ko; % secondary stiffness (k/in)dpl=abs(dv1); % plastic defo (in)if r==1 % check for full reversal
rev=1;endr=0; % full reversal counter
else % elasticy=0; % yield counterk=ko; % elastic stiffness (k/in)df=ko*dv; % incremental force (k)dpl=0; % no plastic defoif (sign(vdi)~=sign(vdo)) % reversal
if sign(fu)==sign(fu-sign(vdi)*2*vy)mult2=sign(vdi);elsemult2=sign(fu);endfu=fu-mult2*2*vy;% redefine yield surface (in)r=0;% full reversal code
endend
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [bedisp.m]% Bilinear Elastic Hysteretic Rule% This program determines the yielding rules for a SDOF oscillator.%% first created: KF 9/19/00% last revised: KF 9/26/00%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [df,k,flag,dpl,y,r,rev]=bedisp(flag,vdi,vdo,ko,alpha,vy,vi,dv,r,rev)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Yielding condition%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%if flag==1 % yielding
y=0; % yield countervo=vi-dv; % disp before yieldingif (sign(vo)~=sign(vi) & abs(dv)>=2*vy)
dv2=dv-sign(dv)*2*vy; % plastic disp (in)dv1=dv-dv2; % elastic disp (in)flag=0; % unloaddf=ko*dv1+alpha*ko*dv2; % incremental force (k)k=ko; % elastic stiffness (k/in)
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dpl=abs(dv2); % plastic defo (in)r=1; % full reversal counter
elseif abs(vi)<=vy % elasticflag=0; % unloaddv1=vi-sign(vo)*vy; % elastic disp (in)dv2=dv-dv1; % plastic disp (in)df=ko*dv1+alpha*ko*dv2; % incremental force (k)k=ko; % elastic stiffness (k/in)dpl=abs(dv2); % plastic defo (in)r=1; % full reversal counter
else % inelastick=alpha*ko; % secondary stiffness (k/in)df=alpha*ko*dv; % incremental force (k)dpl=abs(dv); % plastic defo (in)r=0; % full reversal counter
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Non-yielding condition%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%elseif flag==0 % elastic
if abs(vi)>=vy % yieldingy=1; % yield counterflag=1; % yieldingdvo=abs(vi)-vy; % abs plastic disp (in)pct=dvo/abs(dv); % plastic percentagedv1=dv*pct; % plastic disp (in)dv2=dv-dv1; % elastic disp (in)df=ko*dv2+alpha*ko*dv1; % incremental force (k)k=alpha*ko; % secondary stiffness (k/in)dpl=abs(dv1); % plastic defo (in)if r==1 % check for full reversal
rev=1;endr=0; % full reversal counter
else % elasticy=0; % yield counterk=ko; % elastic stiffness (k/in)df=ko*dv; % incremental force (k)dpl=0; % no plastic defoif (sign(vdi)~=sign(vdo)) % reversal
r=0;% full reversal codeend
endend
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [sddisp.m]% Stiffness-Degrading Hysteretic Rule% This program determines the yielding rules for a SDOF oscillator.%% first created: KF 3/14/01% last revised: KF 3/14/01%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [df,fu,k,flag,dpl,y,r,rev,fsh,vsh]=sddisp(flag,fu,vdi,vdo,ko,alpha,vy,vi,dv,r,rev,fi,fo,fsh,vsh,k)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Yielding condition%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%if flag==1 % yielding
y=0; % yield counterif (sign(vdi)~=sign(vdo)) % reversal
flag=0; % unloaddf=alpha*ko*dv; % unloading force (k)fu=vi; % redefine yield surface (in)k=ko; % unloading stiffness (k/in)dpl=0; % no plastic defor=1; % full reversal counterfsh=fliplr(fsh); % reverse shoot-through force vectorvsh=fliplr(vsh); % reverse shoot-through displ vector
else % inelasticdf=k*dv; % incremental force (k)fu=vi; % redefine yield surface (in)dpl=dv; % plastic defo (in)r=0; % full reversal countervsel=[vi,vsh(2)]; % candidates for shoot-through pointfsel=[fi,fsh(2)];[Ysh,Ish]=max(sign(dv)*vsel);% select farthest pointvsh(2)=vsel(Ish); % shoot-through displacementfsh(2)=fsel(Ish); % shoot-through force
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Non-yielding condition%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%elseif (flag==0 | flag==3) % elastic
if abs(vi-(fu))>=2*vy % check for yieldingy=1; % yield counterdvo=abs(vi-(fu))-2*vy; % abs plastic disp (in)pct=dvo/abs(dv); % plastic percentagedv1=dv*pct; % plastic disp (in)dv2=dv-dv1; % elastic disp (in)if flag==3
k=abs((fsh(2)-(fo+k*dv2))/(vsh(2)-(vi-dv1)));% shoot-through stiffness (k)
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flag=2;else
k=alpha*ko;flag=1;
enddf=ko*dv2+k*dv1; % incremental force (k)fu=vi; % redefine yield surface (in)dpl=dv1; % plastic defo (in)if r==1 % check for full reversal
rev=1;endr=0; % full reversal counter
elseif (abs(vsh(1))~=abs(vsh(2)) & sign(fi)~=sign(fo))% check for shoot-throughflag=2; % shoot-throughy=0; % yield counterdfo=fi-fo; % initial incremental force (k)pct=abs(fo/dfo); % elastic percentagedv1=dv*pct; % elastic disp (in)dv2=dv-dv1; % plastic disp (in)k=abs(fsh(2)/(vsh(2)-(vi-dv2)));% shoot-through stiffness (k)df=ko*dv1+k*dv2; % incremental forcer=0; % full reversal counterdpl=dv2; % plastic defo (in)
else % elasticy=0; % yield counterk=ko; % elastic stiffness (k/in)df=ko*dv; % incremental force (k)dpl=0; % no plastic defoif (sign(vdi)~=sign(vdo)) % reversal
if sign(fu)==sign(fu-sign(vdi)*2*vy)mult2=sign(vdi);elsemult2=sign(fu);endfu=fu-mult2*2*vy;% redefine yield surface (in)r=0;% full reversal codefsh=fliplr(fsh);% reverse shoot-through force vectorvsh=fliplr(vsh);% reverse shoot-through displ vector
endend
elseif flag==2if abs(fi)>=abs(fsh(2))
y=1; % yield counterdvo=abs(vi-vsh(2)); % abs plastic disp (in)pct=dvo/abs(dv); % plastic percentagedv1=dv*pct; % plastic disp (in)dv2=dv-dv1; % elastic disp (in)df=k*dv2+alpha*ko*dv1; % incremental force (k)flag=1; % yieldingfu=vi; % redefine yield surface (in)k=alpha*ko; % secondary stiffness (k/in)dpl=dv1; % plastic defo (in)vsh(2)=vi; % shoot-through displacementfsh(2)=df+fo; % shoot-through force
elsey=0; % yield counterdf=k*dv; % incremental force (k)dpl=dv; % plastic defoif (sign(vdi)~=sign(vdo)) % reversal
fu=vi;% redefine yield surface (in)r=0;% full reversal codefsh=fliplr(fsh);% reverse shoot-through force vectorvsh=fliplr(vsh);% reverse shoot-through displ vectork=ko;% back to elastic stiffnessflag=3;% elastic unloading on shoot-through
endend
end
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STRUCTURAL ENGINEERING RESEARCH REPORT SERIESLIST OF TECHNICAL REPORTS
NDSE-01-01 “Design of Rectangular Openings in Unbonded Post-Tensioned Precast ConcreteWalls,” by M. Allen and Y.C. Kurama, April 2001, 142 pp. (this report may bedownloaded from http://www.nd.edu/~concrete/)
NDSE-01-02 “Capacity-Demand Index Relationships for Performance-Based Seismic Design,”by K.T. Farrow and Y.C. Kurama, November 2001, 260 pp. (this report may bedownloaded from http://www.nd.edu/~concrete/)