Dissertation2008 Pietra

8/10/2019 Dissertation2008 Pietra

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The dissertation entitled Evaluation of pushover procedures for the seismic design of buildings, by

Dario Pietra, has been approved in partial fulfilment of the requirements for the Master Degree inEarthquake Engineering.

Dr. Rui Pinho_____________________


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ABSTRACT

A number of recent studies raised doubts on the effectiveness of conventional pushover methods,whereby a constant incremental force vector is applied to the structure, in estimating the seismicdemand/capacity of framed buildings subjected to earthquake action. The latter motivated the recentdevelopment of the so-called Adaptive Pushover methods whereby the loading vector is updated ateach analysis step, considering one or more response modes, reflecting in this way the effects thatdamage progression have on the response characteristics of structures subjected to increasing loadinglevels. Within such adaptive framework, the application of a displacement incremental loading vectorbecomes not only feasible but also possibly advantageous since it seems to lead to superior responsepredictions, with little or no additional modelling/analysis effort, with respect to conventional pushoverprocedures. In this work, a parametric study, whereby the accuracy of the Displacement-based AdaptivePushover algorithm (DAP) in predicting the seismic response of 3-, 9- and 20-storey high steel buildings

responding in the inelastic range is presented. A large set of natural records is used in the dynamic analyses

that are carried out for comparison. The performance of the adaptive procedures is evaluated in terms of

prediction of the main structural response parameters of interest (interstorey drifts, shears and overturning

moments). Results, expressed as absolute values of these design parameters as well as their ratio between

static- and dynamic-analysis values, are compared with those provided by conventional pushover schemes.

Results show that DAP, compared with non-adaptive procedures, represents an alternative simpler procedure

(involving a single pushover analysis) that allows predicting the response shape of high-rise steel buildingswith an accuracy that is at least as good as that obtained with more complex multiple-pushover procedures.


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ACKNOWLEDGMENTS

The author wishes to thank in particular all those people whose friendly assistance and wise guidancesupported him throughout the duration of this research project. Thanks to the Professors and

Technicians of the Department of Structural Mechanics at the University of Pavia and the Professorsof the Rose School. Thanks particularly to Dr. R. Pinho, for proposing me such an interesting researchtopic and for being always willing and available to help. The author would like to acknowledge theinvaluable assistance of Dr. Mark Aschheim, who kindly supplied data on the structural models andinput motion employed and described in the FEMA-440 report. Finally, the technical support from Dr.Stelios Antoniou is gratefully acknowledged.


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Table of Contents

ABSTRACT........................................................................................................................................................ i

ACKNOWLEDGMENTS............................................................................................................................ iiiTable of Contents.............................................................................................................................................vList of Figures .................................................................................................................................................. ixList of Tables...................................................................................................................................................xv

1 INTRODUCTION.....................................................................................................................................1 1.1 Forward ................................................................................................................................................11.2 Aims of the Work ...............................................................................................................................31.3 Thesis Outline......................................................................................................................................4

2 Nonlinear Static Procedures: Critical Overview......................................................................................52.1 Introduction.........................................................................................................................................5 2.2 NSPs in Earthquake Engineering.....................................................................................................7

2.2.1 The Eurocode 8 (prENV 1998-1, 1994) ................................................................................82.2.2 The Coefficient Method in FEMA 356 (ASCE, 2000) ........................................................92.2.3 The Capacity Spectrum Method in ATC 40 (ATC, 1996) .................................................10

2.3 Pushover Analysis in Earthquake Engineering.............................................................................112.3.1 Non-Adaptive Non-Modal Procedures (NANM Procedures) .........................................132.3.2 Non-Adaptive Modal Procedures (NAM Procedures).......................................................142.3.3 Adaptive Procedures................................................................................................................17

2.3.3.1 Single-run adaptive pushover analyses .............................................................................212.3.3.2 Multiple-run adaptive pushover analyses.........................................................................22

2.3.4 Discussion.................................................................................................................................24 2.3.4.1 Selection of the control node.............................................................................................25

2.3.4.2 Alternative modal combination rules ...............................................................................252.3.4.3 Drawbacks in multiple-run analyses .................................................................................272.3.4.4 Multiple Degrees Of Freedom Effects (MDOFEs).......................................................272.3.4.5 Innovative strategies............................................................................................................32

2.3.5 Adaptive Pushover Algorithm ...............................................................................................343 Case Studies and Analyses Post-Processing...........................................................................................41

3.1 Introduction.......................................................................................................................................41 3.2 Case studies........................................................................................................................................41

3.2.1 First parametric study..............................................................................................................413.2.1.1 3-Storey Frame.....................................................................................................................433.2.1.2 9-Storey Frame.....................................................................................................................44

3.2.2 Second parametric study.........................................................................................................453.2.2.1 20-Storey frame....................................................................................................................463.3 Analytical tool....................................................................................................................................49

3.3.1 Modeling of frames..................................................................................................................493.3.2 Verification of the structural analysis software for steel frames .......................................50

3.3.2.1 Case study.............................................................................................................................503.3.2.2 The model.............................................................................................................................513.3.2.3 Results ...................................................................................................................................533.3.2.4 Conclusions..........................................................................................................................54

3.4 Ground motions................................................................................................................................553.4.1 First parametric study..............................................................................................................55

3.4.2 Second parametric study.........................................................................................................583.5 The Analyses......................................................................................................................................593.5.1 First parametric study..............................................................................................................59


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3.5.2 Second parametric study.........................................................................................................613.6 Response Statistics............................................................................................................................62

3.6.1 First parametric study..............................................................................................................623.6.2 Second parametric study.........................................................................................................64

4 Capacity Curves..........................................................................................................................................654.1 Introduction.......................................................................................................................................65 4.2 Ordinary Ground Motions..............................................................................................................65

4.2.1 IDA curves ............................................................................................................................... 664.2.2 Qualitative evaluation of the accuracy of the static procedures........................................674.2.3 Quantitative evaluation of the accuracy of the static procedures.....................................69

4.3 Near-Field records............................................................................................................................714.3.1 IDA curves ............................................................................................................................... 714.3.2 Qualitative evaluation of the accuracy of the static procedures........................................724.3.3 Quantitative evaluation of the accuracy of the static procedures.....................................73

4.4 Conclusions........................................................................................................................................75

5 Storey Response Parameters .................................................................................................................... 775.1 Introduction.......................................................................................................................................77 5.2 First parametric study.......................................................................................................................77

5.2.1 Overview of FEMA 440 results ............................................................................................ 785.2.1.1 Ordinary Ground Motions ................................................................................................ 785.2.1.2 Near-Field Motions.............................................................................................................795.2.1.3 Conclusions..........................................................................................................................79

5.2.2 Evaluation of results................................................................................................................805.2.3 Ordinary Ground Motions.....................................................................................................80

5.2.3.1 Storey Displacements ......................................................................................................... 825.2.3.2 Interstorey Drifts.................................................................................................................83

5.2.3.3 Interstorey Shears and Moments ...................................................................................... 855.2.4 Near-Field Motions ................................................................................................................. 875.2.4.1 Storey Displacements ......................................................................................................... 875.2.4.2 Interstorey Drifts.................................................................................................................885.2.4.3 Inter- storey Shears and Moments....................................................................................91

5.2.5 Conclusions .............................................................................................................................. 925.3 Second parametric study..................................................................................................................94

5.3.1 Overview of MPA results.......................................................................................................945.3.2 Evaluation of results................................................................................................................965.3.3 Preliminary study ..................................................................................................................... 965.3.4 Extensive study ........................................................................................................................ 99

5.3.4.1 Displacement correction factor method..........................................................................995.3.4.2 Scaling factor method.......................................................................................................1025.3.5 Conclusions ............................................................................................................................105

6 Conclusions...............................................................................................................................................107 6.1. Summary...........................................................................................................................................107 6.2. Future Research ..............................................................................................................................111

References..........................................................................................................................................................113 Appendix A: Prototype Buildings...................................................................................................................119(members section and frame dynamic properties).......................................................................................119

A.1 Section properties ...........................................................................................................................119A.2 Dynamic properties ........................................................................................................................121

Appendix B: First parametric study scaling factors ..................................................................................123Appendix C: First parametric study - Capacity Curves ...............................................................................125C.1 OGMs...............................................................................................................................................125


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C.1.1 Representations of IDA envelopes .....................................................................................125C.1.2 Pushover curves .....................................................................................................................127C.1.3 Mean and maximum values of E1.......................................................................................128C.1.4 E2 .............................................................................................................................................129

C.2 NF records .......................................................................................................................................130C.2.1 3-Storey frame ........................................................................................................................130C.2.2 3-Storey weak frame ..............................................................................................................133C.2.3 9-Storey frame ........................................................................................................................136C.2.4 9-Storey weak frame ..............................................................................................................139

Appendix D: First parametric study - Storey response parameters ...........................................................143D.1 OGMs...............................................................................................................................................143

D.1.1 3-Storey frame ........................................................................................................................143Fig. 104. 3-Storey frame-total drift level 0.5%.....................................................................................144D.1.2 3-Storey weak frame ..............................................................................................................147D.1.3 9-Storey frame ........................................................................................................................150

D.1.4 9-Storey weak frame ..............................................................................................................153D.2 NF records .......................................................................................................................................156D.2.1 3-Storey frame ........................................................................................................................157D.2.2 3-Storey weak frame ..............................................................................................................161D.2.3 9-Storey frame ........................................................................................................................165D.2.4 9-Storey weak frame ..............................................................................................................169

Appendix E: First parametric study - error measurement (OGMs) ..........................................................173E.1 Mean and Maximum E1.................................................................................................................173

E.1.1 3-Storey frame ........................................................................................................................173E.1.2 3-Storey weak frame ..............................................................................................................175E.1.3 9-Storey frame ........................................................................................................................178

E.1.4 9-Storey weak frame ..............................................................................................................180E.2 Values of E1 and E2 for each building model............................................................................182E.2.1 3-Storey frame ........................................................................................................................182E.2.2 3-Storey weak frame ..............................................................................................................186E.2.3 9-Storey frame ........................................................................................................................189E.2.4 9-Storey weak frame ..............................................................................................................192

Appendix F: First parametric study - error measurement (NF records)...................................................195F.1 Mean and maximum E1.................................................................................................................195F.2 3-Storey frame .................................................................................................................................196F.3 3-Storey weak frame .......................................................................................................................202F.4 9-Storey frame .................................................................................................................................207

F.5 9-Storey weak frame .......................................................................................................................212Appendix G: Second parametric study ..........................................................................................................219G.1 Preliminary study.............................................................................................................................219

G.1.1 Storey response parameters..................................................................................................219G.1.2 Bias...........................................................................................................................................221 G.1.3 Dispersion...............................................................................................................................223

G.2 Extensive study Displacement correction factor method ....................................................225G.2.1 Storey response parameters..................................................................................................225G.2.2 Bias...........................................................................................................................................227 G.2.3 Dispersion...............................................................................................................................229

G.3 Extensive study Scaling factor method ...................................................................................231

G.3.1 Storey response parameters..................................................................................................231G.3.2 Bias...........................................................................................................................................233


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List of Figures

Fig. 1. Equivalent Single Degree Of Freedom System ......................................................................................9Fig. 2. Coefficient Method (ASCE, 2000) ........................................................................................................ 10Fig. 3. Capacity Spectrum Method (ATC, 1996) ............................................................................................. 11Fig. 4. Properties of the nthmode inelastic SDOF system derived from the corresponding pushover

curve.............................................................................................................................................................. 16Fig. 5. Periods of vibration of a 4-Storey structure under increasing levels of deformation .................... 18Fig. 6. Interstorey drift profile of a 12-Storey building subjected to increasing levels of deformation... 19Fig. 7. Adaptive pushover: shape of loading vector is updated at each analysis step................................. 19Fig. 8. Bilinear idealization of a modal capacity diagram at the ithpushover step ...................................... 24Fig. 9. Storey force distributions of a 12-Storey building obtained with Displacement-based Adaptive

Pushover as well with standard non-adaptive pushovers...................................................................... 34

Fig. 10. Correlation Coefficient-Frequency Ratio relationship...................................................................... 38Fig. 11. Updating of the loading displacement vector .................................................................................... 39Fig. 12. Incremental Updating strategy ............................................................................................................. 40Fig. 13. 3-Storey (left) and 9-Storey (right) steel frame structures: plane view ........................................... 43Fig. 14. 3-Storey steel frame: vertical view ....................................................................................................... 43Fig. 15. 3 Storey steel frame-1stmode pushover SeismoStruct (left), FEMA 440 (right).......................... 44Fig. 16. 9-Storey steel frame-1stmode pushover SeismoStruct (left), FEMA 440 (right).......................... 44Fig. 17. 9-Storey steel frame: vertical view ....................................................................................................... 45Fig. 18. 20-Storey steel frame- SAC 1stmode (Model M1) and SeismoStruct 1stmode (red) pushover

curves ............................................................................................................................................................ 46Fig. 19. 20-Storey steel frame: plan view and elevation (from Ohtori et al., 2003)...................................................... 47

Fig. 20. 3-Storey (left) and 9-Storey (right) frame: mode shapes................................................................... 48Fig. 21. 20-Storey frame: mode shapes ............................................................................................................. 48Fig. 22. Quasi-static loading program................................................................................................................ 51Fig. 23. Elevation of the test structure (unit: mm)...................................................................................................... 51Fig. 24. Plan of the test structure (unit: mm) ................................................................................................... 52Fig. 25. Through diaphragm connection (left) and detail of the column base (right) (unit: mm)............ 52Fig. 26. Total base shear vs. total drift...................................................................................................................... 54Fig. 27. Storey shear vs. storey drift 1ststorey .............................................................................................. 55Fig. 28. Storey shear vs. storey drift 2ndstorey ............................................................................................. 55Fig. 29. Ordinary Ground Motions: unscaled acceleration response spectrum.......................................... 56Fig. 30. Ordinary Ground Motions: unscaled displacement response spectrum........................................ 56

Fig. 31. Maximum base shear and top floor displacement values obtained with incremental dynamicanalyses ......................................................................................................................................................... 59Fig. 32. Alternative representation of IDA results .......................................................................................... 67Fig. 33. 3-Storey frame ....................................................................................................................................... 68Fig. 34. 9-Storey weak frame .............................................................................................................................. 68Fig. 35. 9-Storey frame ....................................................................................................................................... 69Fig. 36. 3-Storey frame ....................................................................................................................................... 71Fig. 37. 9-Storey frame ....................................................................................................................................... 71Fig. 38. 3-Storey frame ........................................................................................................................................ 71Fig. 39. 3-Storey frame ........................................................................................................................................ 72Fig. 40. 9-Storey frame ........................................................................................................................................ 73

Fig. 41. 9-Storey frame ........................................................................................................................................ 73Fig. 42. 3-Storey frame (SCHMV1).................................................................................................................. 74Fig. 43. 9-Storey frame (LUCMV1)................................................................................................................... 74Fig. 44. 9-Storey weak frame (LUCMV1)......................................................................................................... 74


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Fig. 89. 3-Storey weak frame-NF records-Capacity Curves SCHMV1................................................... 134Fig. 90. 3-Storey-weak frame-NF records-E1................................................................................................ 135Fig. 91. 3-Storey-weak frame-NF records-E2................................................................................................ 135Fig. 92. 9-Storey frame-NF records-Capacity Curves ERZMV1............................................................. 136

Fig. 93. 9-Storey frame-NF records-Capacity Curves LUCMV11 .......................................................... 136Fig. 94. 9-Storey frame-NF records-Capacity Curves RRSMV1 ............................................................. 137Fig. 95. 9-Storey frame-NF records-Capacity Curves SCHMV1............................................................. 137Fig. 96. 9-Storey frame-NF records-E1 .......................................................................................................... 138Fig. 97. 9-Storey frame-NF records-E2 .......................................................................................................... 138Fig. 98. 9-Storey weak frame-NF records-Capacity Curves ERZMV1 .................................................. 139Fig. 99. 9-Storey weak frame-NF records-Capacity Curves LUCMV1................................................... 139Fig. 100. 9-Storey weak frame-NF records-Capacity Curves RRSMV1 ................................................. 140Fig. 101. 9-Storey weak frame-NF records-Capacity Curves SCHMV1................................................. 140Fig. 102. 9-Storey-weak frame-NF records-E1.............................................................................................. 141Fig. 103. 9-Storey-weak frame-NF records-E2.............................................................................................. 141

Fig. 104. 3-Storey frame-total drift level 0.5%............................................................................................... 144Fig. 105. 3-Storey frame-total drift level 2%.................................................................................................. 145Fig. 106. 3-Storey frame-total drift level 4%.................................................................................................. 146Fig. 107. 3-Storey-weak frame-total drift level 0.5%..................................................................................... 147Fig. 108. 3-Storey-weak frame-total drift level 2%........................................................................................ 148Fig. 109. 3-Storey-weak frame-total drift level 4%........................................................................................ 149Fig. 110. 9-Storey frame-total drift level 0.5%............................................................................................... 150Fig. 111. 9-Storey frame-total drift level 2%.................................................................................................. 151Fig. 112. 9-Storey frame-total drift level 4%.................................................................................................. 152Fig. 113. 9-Storey weak frame-total drift level 0.5%..................................................................................... 153Fig. 114. 9-Storey weak frame-total drift level 2% ........................................................................................ 154

Fig. 115. 9-Storey weak frame-total drift level 2.7%..................................................................................... 155Fig. 116. 3 Storey frame-ERZMV1 ................................................................................................................. 157Fig. 117. 3 Storey frame-LUCMV1 ................................................................................................................. 158Fig. 118. 3 Storey frame-RRSMV1 .................................................................................................................. 159Fig. 119. 3 Storey frame-SCHMV1.................................................................................................................. 160Fig. 120. 3-Storey weak-frame-ERZMV1....................................................................................................... 161Fig. 121. 3-Storey-weak frame-LUCMV1....................................................................................................... 162Fig. 122. 3-Storey-weak frame-RRSMV1........................................................................................................ 163Fig. 123. 3-Storey-weak frame-SCHMV1....................................................................................................... 164Fig. 124. 9-Storey -frame-ERZMV1................................................................................................................ 165Fig. 125. 9-Storey -frame-LUCMV1................................................................................................................ 166

Fig. 126. 9-Storey -frame-RRSMV1................................................................................................................. 167Fig. 127. 9-Storey -frame-SCHMV1................................................................................................................ 168Fig. 128. 9-Storey weak-frame-ERZMV1....................................................................................................... 169Fig. 129. 9-Storey weak-frame-LUCMV1....................................................................................................... 170Fig. 130. 9-Storey weak-frame-RRSMV1........................................................................................................ 171Fig. 131. 9-Storey weak-frame-SCHMV1....................................................................................................... 172Fig. 132. 3-Storey frame-E1 (maximum values) ............................................................................................ 173Fig. 133. 3-Storey frame-E1 (mean values)..................................................................................................... 174Fig. 134. 3-Storey weak frame-E1 (maximum values) .................................................................................. 176Fig. 135. 3-Storey weak frame-E1 (mean values) .......................................................................................... 176Fig. 136. 9-Storey frame-E1 (maximum values) ............................................................................................ 178

Fig. 137. 9-Storey frame-E1 (mean values)..................................................................................................... 179Fig. 138. 9-Storey weak frame-E1 (maximum values) .................................................................................. 180Fig. 139. 9-Storey weak frame-E1 (mean values) ......................................................................................... 181


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Fig. 140. 3-Storey frame-total drift level 0.5%-E1 .........................................................................................183Fig. 141. 3-Storey frame-total drift level 0.5%-E2 .........................................................................................183Fig. 142. 3-Storey frame-total drift level 2%-E1 ............................................................................................184Fig. 143. 3-Storey frame-total drift level 2%-E2 ............................................................................................184

Fig. 144. 3-Storey frame-total drift level 4%-E1 ............................................................................................185Fig. 145. 3-Storey frame-total drift level 4%-E2 ............................................................................................185Fig. 146. 3-Storey weak-frame-total drift level 0.5%-E1...............................................................................186Fig. 147. 3-Storey weak-frame-total drift level 0.5%-E2...............................................................................186Fig. 148. 3-Storey weak-frame-total drift level 2%-E1..................................................................................187Fig. 149. 3-Storey weak-frame-total drift level 2 %-E2.................................................................................187Fig. 150. 3-Storey weak-frame-total drift level 4%-E1..................................................................................188Fig. 151. 3-Storey weak-frame-total drift level 4%-E2..................................................................................188Fig. 152. 9-Storey frame-total drift level 0.5%-E1 .........................................................................................189Fig. 153. 9-Storey frame-total drift level 0.5%-E2 .........................................................................................189Fig. 154. 9-Storey frame-total drift level 2%-E1 ............................................................................................190

Fig. 155. 9-Storey frame-total drift level 2%-E2 ............................................................................................190Fig. 156. 9-Storey frame-total drift level 4%-E1 ............................................................................................191Fig. 157. 9-Storey frame-total drift level 4%-E2 ............................................................................................191Fig. 158. 9-Storey weak-frame-total drift level 0.5%-E1...............................................................................192Fig. 159. 9-Storey weak-frame-total drift level 0.5%-E2...............................................................................192Fig. 160. 9-Storey weak-frame-total drift level 2%-E1..................................................................................193Fig. 161. 9-Storey weak-frame-total drift level 2%-E2..................................................................................193Fig. 162. 9-Storey weak-frame-total drift level 2.7%-E1...............................................................................194Fig. 163. 9-Storey weak-frame-total drift level 2.7%-E2...............................................................................194Fig. 164. 3-Storey frame.....................................................................................................................................195Fig. 165. 3-Storey weak frame...........................................................................................................................195

Fig. 166. 9-Storey frame.....................................................................................................................................196Fig. 167. 9-Storey weak frame...........................................................................................................................196Fig. 168. 3-Storey frame-E1-Floor Displacement..........................................................................................197Fig. 169. 3-Storey frame-E1-Interstorey Drift................................................................................................197Fig. 170. 3-Storey frame-E1-Interstorey Shear...............................................................................................198Fig. 171. 3-Storey frame-E1-Interstorey Moment..........................................................................................199Fig. 172. 3-Storey frame-E2-Floor Displacement..........................................................................................199Fig. 173. 3-Storey frame-E2-Interstorey Drift................................................................................................200Fig. 174. 3-Storey frame-E1-Interstorey Shear...............................................................................................201Fig. 175. 3-Storey frame-E1-Interstorey Moment..........................................................................................201Fig. 176. 3-Storey-weak frame-E1-Floor Displacement................................................................................202

Fig. 177. 3-Storey-weak frame-E1-Interstorey Drift......................................................................................203Fig. 178. 3-Storey-weak frame-E1-Interstorey Shear.....................................................................................203Fig. 179. 3-Storey-weak frame-E1-Interstorey Moment ...............................................................................204Fig. 180. 3-Storey-weak frame-E2-Floor Displacement................................................................................205Fig. 181. 3-Storey-weak frame-E2-Interstorey Drift......................................................................................205Fig. 182. 3-Storey-weak frame-E1-Interstorey Shear.....................................................................................206Fig. 183. 3-Storey-weak frame-E1-Interstorey Moment ...............................................................................207Fig. 184. 9-Storey frame-E1-Floor Displacement..........................................................................................207Fig. 185. 9-Storey frame-E1-Interstorey Drift................................................................................................208Fig. 186. 9-Storey frame-E1-Interstorey Shear...............................................................................................209Fig. 187. 9-Storey frame-E1-Interstorey Moment..........................................................................................209

Fig. 188. 9-Storey frame-E2-Floor Displacement..........................................................................................210Fig. 189. 9-Storey frame-E2-Interstorey Drift................................................................................................211Fig. 190. 9-Storey frame-E1-Interstorey Shear...............................................................................................211


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List of Tables

Table 1. Nonlinear Static Procedures in earthquake engineering.....................................................................8Table 2. FEMA 356: lateral load distribution requirements .......................................................................... 30Table 3. Member properties (from Nakashima, 2006) ................................................................................... 53Table 4. Material properties after testing .......................................................................................................... 53Table 5. 9-Ground Motions characteristics...................................................................................................... 57Table 6. Near-Field records: total drift recorded in dynamic analyses......................................................... 58Table 7. SAC joint venture records: PGA and duration................................................................................. 58Table 8. Correction factor................................................................................................................................. 103Table 9. Steel section properties....................................................................................................................... 119Table 10. Building models: first three periods of vibration ......................................................................... 121Table 11. 3-Storey frame: mode shapes. ......................................................................................................... 121

Table 12. 9-Storey frame: mode shapes. ......................................................................................................... 121Table 13. 20-Storey frame: mode shapes. ....................................................................................................... 122Table 14. 3-Storey frame: scaling factors for OGM...................................................................................... 123Table 15. 3-Storey weak-frame: scaling factors for OGM ........................................................................... 123Table 16. 9-Storey frame: scaling factors for OGM...................................................................................... 123Table 17. 9-Storey weak-frame: scaling factors for OGM ........................................................................... 124Table 18. Total drift recorded in the analyse [%] .......................................................................................... 156Table 19. 3-Storey frame-drift level 0.5%-E1 (mean)................................................................................... 174Table 20. 3-Storey frame-drift level 0.5%-E1 (standard deviation) ............................................................ 174Table 21. 3-Storey frame-drift level 2%-E1 (mean) ...................................................................................... 175Table 22. 3-Storey frame-drift level 2%-E1 (standard deviation) ............................................................... 175

Table 23. 3-Storey frame-drift level 4%-E1 (mean) ...................................................................................... 175Table 24. 3-Storey frame-drift level 4%-E1 (standard deviation) ............................................................... 175Table 25. 3-Storey weak frame-drift level 0.5%-E1 (mean) ......................................................................... 176Table 26. 3-Storey weak frame-drift level 0.5%-E1 (standard deviation) .................................................. 177Table 27. 3-Storey weak frame-drift level 2%-E1 (mean) ............................................................................ 177Table 28. 3-Storey weak frame-drift level 2%-E1 (standard deviation) ..................................................... 177Table 29. 3-Storey weak frame-drift level 4%-E1 (mean) ............................................................................ 177Table 30. 3-Storey weak frame-drift level 4%-E1 (standard deviation) ..................................................... 177Table 31. 9-Storey frame-drift level 0.5%-E1 (mean)................................................................................... 179Table 32. 9-Storey frame-drift level 0.5%-E1 (standard deviation) ............................................................ 179Table 33. 9-Storey frame-drift level 2%-E1 (mean) ...................................................................................... 179

Table 34. 9-Storey frame-drift level 2%-E1 (standard deviation) ............................................................... 179Table 35. 9-Storey frame-drift level 4%-E1 (mean) ...................................................................................... 180Table 36. 9-Storey frame-drift level 4%-E1 (standard deviation) ............................................................... 180Table 37. 9-Storey weak frame-drift level 0.5%-E1 (mean) ......................................................................... 181Table 38. 9-Storey weak frame-drift level 0.5%-E1 (standard deviation) .................................................. 181Table 39. 9-Storey weak frame-drift level 2%-E1 (mean) ............................................................................ 182Table 40. 9-Storey weak frame-drift level 2%-E1 (standard deviation) ..................................................... 182Table 41. 9-Storey weak frame-drift level 2.7%-E1 (mean) ......................................................................... 182Table 42. 9-Storey weak frame-drift level 2.7%-E1 (standard deviation) .................................................. 182


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Evaluation of Pushover Procedures for the Seismic Design of Buildings 2

means of a pushover analysis is a fundamental element, and thus the call for further improvements in

this field has been increasing in the last few years.

In a pushover analysis, a mathematical model of the building, that includes all significant lateral force

resisting members, is subjected to a monotonically increasing invariant (or adaptive) lateral force (or

displacement) pattern until a pre-determined target displacement is reached or the building is on the

verge of incipient collapse.

Due to the static nature of the analysis the overall response of the system cannot be reliably estimated

principally due to (i) higher mode effects and/or (ii) high ductility demand, which are the main issues

investigated in the present work. In particular, the higher mode contributions are typically difficult toidentify, and the spreading of inelastic deformations among the structural members leads to

degradation and softening of the system resulting in period elongation and change of modal shape

characteristics, not accounted for in traditional pushover schemes. Moreover, pushover procedures, due

to their static nature, are unable to reproduce peculiar dynamic effects, such as sources of energy

dissipation (kinetic energy and viscous damping) as well as duration effects, and account for a site-

specific response by considering both the actual dynamic properties of the system and the frequency

content of the seismic motion. Three-dimensional effects are also difficult to incorporate, whereas the

effects of cyclic earthquake loading cannot be modelled.

Conventional pushovers consist in the application and monotonic increase of a predefined lateral force

pattern, kept constant throughout the analysis. The lateral load pattern should approximate the inertial

forces expected in the building during an earthquake. However, the inertia force distribution will vary

with the severity of the earthquake and with time, due to changes in the contribution of different

modes and also as a consequence of the spread of inelastic deformations into the system. Thus the

adoption of an invariant load pattern is an approximation that is likely to yield accurate predictions only

for low to medium-rise framed structures, where the system behaviour is dominated by a single mode.

With the assumption of an adaptive, instead of invariant, lateral load pattern (eitherforcesor displacements),

updated knowing the current dynamic properties of the system during the analysis, alteration of the

local resistance and modal characteristics of the structure can be accounted for. In this way the stiffness

degradation and period elongation induced by the progressive accumulation of damage can be taken

into account during the pushover analysis. Moreover, the contribution of several modes can beaccounted for, through an appropriate combination rule of their respective shapes. In this way, the


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effect of higher modes can be considered, and if the modal contributions are weighted according to a

selected response spectrum the attainment of site-specific results is also possible.

Among the adaptive procedures the Displacement-based Adaptive Pushover (DAP) (Antoniou and

Pinho 2004(b)) represents an appealing alternative to traditional invariant load shape procedures. The

lateral load distribution is continuously updated according to the modal shapes and participation factors

derived by eigenvalue analysis carried out at each analysis step. DAP manages to provide greatly

improved predictions throughout the entire deformation range, even if an exact reproduction of the

dynamic response cannot be achieved (Antoniou and Pinho, 2004(b)).

The present work fits within the above described research field, with the main objective to furtherassess the accuracy of different pushover analysis procedures in predicting the dynamic response of

multi-storey structures at different ductility (i.e. drift) demand levels. Traditional (invariant) load shape

schemes as well as the recently proposed DAP are considered and their performance compared with

respect to nonlinear dynamic analyses.

1.2 Aims of the Work

Nonlinear static pushover analysis, even if fundamentally limited due to its static nature, represents the

most attractive alternative to nonlinear response history analysis tools. The main limitations of these

analysis procedures are essentially related to the development of higher mode-dominated responses,

and the increase of the inelastic deformation demand throughout the structure. Therefore, the main

objectives of the present work will be to:

Evaluate and compare performances of traditional pushover schemes with those obtained with the

more recently proposed Displacement-based Adaptive Pushover (Antoniou and Pinho, 2004(b)).

The accuracy of pushover methods will be assessed in predicting (i) the global response, through acomparison of pushover curves with Incremental Dynamic Analysis (IDA) envelopes (Hamburger

et al., 2000; Vamvatsikos and Cornell, 2002; Mwafy and Elnashai, 2000), as well as (ii) local response

quantities, such as storey displacements, interstorey drifts, interstorey shears and moments.

Verify the applicability of an alternative adaptive scheme which makes use, instead of a record-

specific spectrum, of an average design spectrum, as commonly adopted in the current design

practice.

Further assess (with respect to Antoniou and Pinho (2004) and Lpez-Menjivar (2004)) theeffectiveness of the three plot options (maximum total drift vs. correspondent base shear,


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correspondent total drift vs. maximum base shear, maximum total drift vs. maximum base shear) of

IDA envelopes.

Assess the reliability of DAP for assessment of steel-frame structures, something that has so far not

been carried out.

1.3 Thesis Outline

The second chapter presents the current state-of-art of Nonlinear Static Procedures in earthquake

engineering: the most updated code provisions are briefly presented so as to highlight the role played

by pushover analysis in the current design practice. Then, current pushover procedures are carefully

discussed, starting from conventional techniques to the most recent multimodal and adaptive schemes.Moreover, the main drawbacks in current pushover procedures, together with questions regarding the

dynamic response of Multiple Degrees Of Freedom Systems (MDOFSs), are pointed out and

discussed, in order to highlight the main objectives of the present work.

The third chapter is constituted by an overall summary of the performed study. Most important

features and assumptions in prototype buildings modeling are described, together with the selection of

the employed input ground motions and the fibre-modeling finite element program in which all the

analyses have been performed is introduced. Finally, analyses performed and procedures for post-

processing of results obtained are presented.

In chapter four results obtained under the point of view of the global response prediction are

presented. Pushover curves provided by each pushover method are compared with IDA envelopes, and

their performance evaluated. Moreover, the effectiveness of the three plot options (maximum total drift

vs. correspondent base shear, correspondent total drift vs. maximum base shear, maximum total drift

vs. maximum base shear) of IDA envelopes is further assessed.

The fifth chapter shows results obtained under the perspective of local response prediction. Predictions

of storey displacements, interstorey drifts, interstorey shears and moments given by static procedures,

are compared with nonlinear dynamic analyses responses, and the effectiveness of every pushover

method evaluated.

Finally, in chapter six concluding remarks are presented and unsolved issues, which might constituteobjectives for future research, highlighted.


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2 Nonlinear Static Procedures: Critical Overview

2.1 Introduction

In the last few years with the development of performance-based design procedures, the demand for

the definition of simplified methods to estimate, with an adequate level of confidence, the seismic

demand for structures has increased.

For the seismic evaluation of yielded systems, the consideration of inelastic displacements rather than

elastic forces should be a more rational approach, because as the structure starts responding in inelastic

manner the displacements keep increasing at relatively constant levels of lateral forces. Previous results(Priestley, 1993) have shown that traditional Force-Based Design procedures (FBD) are clearly flawed.

Some of the major drawbacks are that (i) they do not account for force redistribution following

yielding, and (ii) they do not consider potential failure modes that arise from mid and upper storey

mechanisms caused by the influence of higher modes.

The application of Performance-Based Design principles (PBD) thus requires the definition of analysis

procedures able to provide an adequate prediction of such inelastic mechanisms avoiding an excessive

computational effort.

It is unquestionable that nonlinear dynamic analysis is the most accurate method for assessing the

response of structures subjected to earthquake action. Indeed, any type of static analysis will always be

inherently flawed, given the conspicuous absence of time-dependent effects. However, as noted by

Goel and Chopra (2005(a)), amongst others, such type of analysis is not without its difficulties or

drawbacks, particularly for what concerns application within a design office environment.

Firstly, in order to employ dynamic analysis for seismic design/assessment of structures, an ensembleof site-specific ground motions compatible with the seismic hazard spectrum for the site must be


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of qualitative data that is always most informative and useful, within a design application, even when

time-history analysis is then employed for the definitive verifications.

The above constitute, in the opinion of the author, strong reasons for nonlinear static analysis methods

to continue to be developed and improved (as recently proposed documents, such as the FEMA 440,

also confirm), so that these tools can become even more reliable and useful when employed either as a

replacement to time-history analysis in the seismic design/assessment of relatively simple non-critical

structures, or as a complement to dynamic analysis of more complex/critical facilities. It is, therefore,

within this framework of warranted development of nonlinear static analysis procedures that the

current endeavour finds its justification and rationale.

2.2 NSPs in Earthquake Engineering

Nonlinear Static Procedures appear as one of the most attractive analysis tool due to their ease of use

and also because they provide a simple and effective graphical representation of the structural response,

by means of the so called Pushover Curve. The latter relates directly the capacity of the system, usually

in terms of base shear, with the response of a significant structural node (control node): this kind of

representation of the overall response allows for a direct idealization of the system as a Single Degree

Of Freedom System (SDOFS) that greatly simplifies the design (or assessment) procedure.

The majority of the Nonlinear Static procedures follow the same basic principles:

1. A pushover analysis is performed.

2. An equivalent SDOFS, based on the pushover curve, obtained throughout a static pushover

analysis, is defined.

3. The maximum global displacement demand is estimated, according to a selected design response

spectrum.

4. The SDOFS response and the actual response of the structure are related by means of a shape

coefficient, typically identified in the first mode participation factor.5. Finally, the response parameters, storey drift and forces on each structural member, can be

evaluated, knowing the global demand, through the pushover curve (or capacity curve) of the

system.

Due to the simplified nature of such methods, they involve many unsolved issues regarding both the

capability to capture the dynamic response by means of a pushover analysis as well as the effectiveness

of the SDOFS idealization.


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Fig. 1. Equivalent Single Degree Of Freedom System

2.2.2 The Coefficient Method in FEMA 356 (ASCE, 2000)

The NSP adopted in FEMA 356, the coefficient method, consists in the definition of an equivalent

linear SDOFS considering an effective period Tegenerated from the initial period Ti, accounting for

some loss of stiffness in the transition from the elastic to inelastic behaviour. This procedure estimates

the total maximum displacement of the SDOF oscillator by multiplying the elastic SDOFS response

(assuming the initial linear properties, stiffness and damping) by one or more coefficients empirically

derived (Fig. 2). These coefficients accounts for (i) the SDOF idealization (a shape factor scales the

SDOFS response to the roof displacement of the building), (ii) the linear response assumed(conventionally characterized in terms of strength, ductility and period (R--T relationships)), (iii)

stiffness and strength degradation, and (iv) the dynamic amplification of the response due to P-

effects. It might be noted that the design displacement is defined by means of an iterative procedure

until convergence of the linear SDOFS displacement amplitude to the response spectrum ordinate.


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Fig. 2. Coefficient Method (ASCE, 2000)

2.2.3 The Capacity Spectrum Method in ATC 40 (ATC, 1996)

The NSP adopted in this code is the capacity spectrum method proposed by Freeman (1994). This

technique, following an equivalent linearization approach, estimates the maximum global displacement

of the structure by means of an iterative graphical procedure (Fig. 3). The basic assumption is that the

maximum inelastic deformation of a nonlinear SDOFS can be approximated from the maximum

deformation of a linear elastic SDOFS with a larger period and damping ratio than the initial values of

the inelastic one.

According to this procedure the capacity curve is converted into an equivalent SDOFS pushover

response and plotted on the same axes as the seismic ground motion demand in the Acceleration-

Displacement Response Spectrum (ADRS) format, assuming a trial damping ratio. The secant period at

the interception identifies the equivalent period of the elastic SDOFS with an equivalent viscous

damping ratio proportional to the area enclosed by the capacity curve of the equivalent nonlinear

SDOFS. Since both the period and damping are function of the displacement, the procedure requires

iterations until the assumed damping is equal to the value computed at the design displacement.


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Fig. 3. Capacity Spectrum Method (ATC, 1996)

Recently, the ATC-55 projects (ATC, 2005) demonstrate that the Coefficient Method (as proposed in

FEMA 356) as well as the CSM (as proposed in ATC-40) show some inconsistencies in the prediction

of the displacement demand. Thus they propose a new formulation of both design approaches. The

updated design methods obtained lead to approximately the same results with a significant

improvement in the prediction of the displacement demand when compared to response historyanalysis results.

As it might be concluded from the discussion reported above, these procedures differ only in the

approach used to estimate the global displacement demand (global response parameter, i.e. top floor

displacement or the equivalent SDOFS displacement demand); instead, the pushover method adopted

will affect not only the global response demand but also the local response parameters of interest,

because both are related to the capacity curve obtained. For this reason a more accurate prediction of

the dynamic response by means of a pushover analysis is a fundamental element, and thus the call for

further improvements in this field was increase in the last few years.

2.3 Pushover Analysis in Earthquake Engineering

Pushover analysis, which represents the fundamental analytical tool in all the NSPs quoted above, is a

static method that directly incorporates nonlinear material characteristics. A mathematical model of the

building, that includes all significant lateral force resisting members, is subjected to a monotonically


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increasing invariant (or adaptive) lateral force (or displacement) pattern until a pre-determined target

displacement is reached or the building is on the verge of incipient collapse.

The termpushover analysisdescribes a modern variation of the classical collapse analysismethod, as fittingly

described by Kunnath (2004). It refers to an analysis procedure whereby an incremental-iterative

solution of the static equilibrium equations is carried out to obtain the response of a structure subjected

to monotonically increasing lateral load pattern. The structural resistance is evaluated and the stiffness

matrix is updated at each increment of the forcing function, up to convergence. The solution proceeds

until (i) a predefined performance limit state is reached, (ii) structural collapse is incipient or (iii) the

program fails to converge. In this manner, each point in the resulting displacement vs. base shear

capacity curve represents an effective and equilibrated stress state of the structure, i.e. a state of

deformation that bears a direct correspondence to the applied external force vector.

Even if representing a simplified analytical tool, with respect to nonlinear dynamic analyses, they may

provide important structural response information such as:

Identify the progress of the overall capacity curve of the structure.

Identify critical regions, where large inelastic deformations may occur.

Identify strength irregularities in plan and elevation that might cause important changes in the

inelastic dynamic response characteristics (e.g. Krawinkler and Seneviratna, 1998). Estimate the force demand in potentially brittle elements.

Predict the sequence of yielding and/or failure of structural members.

Many authors (Lawson et al., 1994; Krawinkler and Seneviratna, 1998; Kim and DAmore, 1999; Naeim

and Lobo, 1999; Antoniou and Pinho, 2004(a)) have however noted that nonlinear static methods

suffer from a number of limitations, which stem essentially from their inherently static nature. Such

limitations become particularly evident when high-rise flexible frames, whose response may be heavily

influenced by higher modes, are being assessed.

The most important limitations in the applicability of current NSPs are due to the static nature of this

kind of analysis. In fact NSPs assume that all the structural response quantities (displacements, internal

forces, plastic deformations etc.) can be estimated by means of those recorded in a pushover analysis at

the design displacement level.

Due to the static nature of the analysis, the overall response of the system cannot be reliably estimated

because of an inaccurate deformation prediction, principally due to (i) higher mode effects and/or (ii)

high ductility demand. Both these contributions, which are the main issues investigated in the present

work, lead to a redistribution of the internal forces with respect to the commonly assumed 1stmode


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response, which results in a concentration of deformations at the location of damage. In particular, the

higher mode contributions are typically difficult to identify, and the spreading of inelastic deformations

among the structural members leads to degradation and softening of the system resulting in period

elongation and change of modal shape characteristics.

Moreover, pushover procedures are unable to reproduce peculiar dynamic effects, such as: (1) consider

not only the strain energy, but also other sources of energy dissipation (kinetic energy and viscous

damping) as well as duration effects; (2) account for a site-specific response by considering both the

actual dynamic properties of the system and the frequency content of the seismic motion.

Three-dimensional effects are also difficult to incorporate, whereas the effects of cyclic earthquake

loading cannot be modelled.

With the assumption of an adaptive, instead of an invariant, lateral load pattern (either forces or

displacements), updated knowing the current dynamic properties of the system during the analysis,

alteration of the local resistance and modal characteristics of the structure can be accounted for. In this

way the stiffness degradation and period elongation induced by the progressive accumulation of

damage can be taken into account during the pushover analysis.

Further improvements, as discussed later, can be achieved considering the contribution of several

modes by combining their respective responses with an appropriate combination rule. In this way the

effect of higher modes can be considered, and if the modal contributions are weighted according with a

selected response spectrum site-specific results can also be obtained.

Finally, an important issue, currently ignored in all code implementations, regards the definition of a

displacement load vector rather than lateral forces. Although it cannot constitute an acceptable approach

using an invariant vector shape since it leads to a predefined and often unreliable failure mode (i.e.

deformed shape), it allows for a better understanding of the deformation of the structure in the inelastic

regime if an adaptive approach is used (Antoniou and Pinho, 2004(b)).

2.3.1 Non-Adaptive Non-Modal Procedures (NANM Procedures)

Conventional pushover consists in the application and monotonic increase of a predefined lateral force

pattern, kept constant throughout the analysis.

The lateral load pattern should approximate the inertial forces expected in the building during an

earthquake. However, the inertia force distribution will vary with the severity of the earthquake and

with time, due to changes in the contribution of different modes in the elastic range and also as a

consequence of the spread of inelastic deformations into the system. Thus, the adoption of an invariant


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load pattern is an approximation that is likely to yield accurate predictions only for low to medium-rise

framed structures, where the system behaviour is dominated by a single mode.

Usually the following NANM lateral load force vectors are adopted:

1stMode: forces are proportional to the amplitude of the elastic first mode shape and mass at each

floor.

Inverted Triangular: forces are linearly increasing with height, in order to approximate a 1 stmode

shape vector for a frame structure with regular geometric and mass distribution.

Uniform (or Rectangular): the force pattern is uniform with the building height. It is an

approximate representation of the inertial forces distribution in the inelastic range assuming the

structure has formed a soft storey.

Code distribution: code-specified distribution that usually varies from an inverted triangular to a

parabolic shape.

Many researchers have analysed the performances and defined the limitations of using this kind of

force-patterns. Among these Mwafy and Elnashai (2000), and Gupta and Kunnath (2000), have found

that whereas in the elastic range force distributions with a triangular or a trapezoidal shape provides a

better fit to dynamic analysis results, at large deformations, after that the structure has sustainedsignificant damage at a particular storey level, the dynamic envelopes are closer to the uniformly

distributed force solutions.

2.3.2 Non-Adaptive Modal Procedures (NAM Procedures)

This kind of procedures try to improve the performances of the static methods by focusing the

attention on the higher modes effects, by considering several modes contribution in the definition of

the applied lateral load pattern.

Among these, an SRSS (Square Root of the Sum of the Squares) load vector is considered in the ATC-

55 project (ATC, 2005). The lateral force vector is defined as the vector that produces an interstorey

shear distribution correspondent to the SRSS combination of the storey shears obtained with different

modal force distributions. As shown in FEMA 440 (ATC, 2005), however, it leads to inconsistent

improvements with respect to other NANM load vectors (e.g. 1 stMode vector) requiring also greater

computational effort.


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Recent years have also witnessed the development and introduction of an alternative type of nonlinear

static analysis, which involves running multiple pushover analyses separately, each of which

corresponding to a given modal distribution, and then estimating the structural response by combining

the action effects derived from each of the modal responses (i.e. each displacement-force pair derived

from such procedures does not actually correspond to an equilibrated structural stress state).

Paret et al. (1996) and Sasaki et al. (1998) suggested the Multi-Mode Pushover analysis (MMP). A MMP

analysis consists in performing several pushover analyses under forcing vectors representing the various

modes deemed to be excited in the dynamic response. For each pushover curve obtained, the Capacity

Spectrum Method (ATC, 1996) is utilized to compare the structural capacity with the earthquake

demand. This method leads only to the identification of the critical mode but does not provide anyresults on the overall response of the system since it does not consider any kind of modal combination

rule.

A refinement of the MMP procedure is given by the Pushover Results Combination (PRC), which was

proposed by Moghadam and Tso (2002). According to this method, firstly the structural response

parameters related to each mode contribution have to be found according with the MMP approach,

and then the overall response is calculated by combining them according with the modal participating

factor of each mode. Usually the first three or four modes are considered.

Recently, a Modal Pushover Analysis (MPA) procedure has been proposed by Chopra and Goel (2002).

Pushover analyses are conducted independently in each mode, using lateral force profiles that represent

the response in each of the modes considered. The pushover curve associated with each modal

pushover analysis is idealised as the response curve of a bilinear SDOF system (Fig. 4) and response

values are determined at the target displacement (computed by a dynamic analysis of the nonlinear

SDOFS, or by applying displacement modification or equivalent linearization procedures). Finally

responses are combined together using the SRSS method in order to define the overall response.

The present methodology has been further improved including P-effects due to gravity loads in the

pushover analysis for all modes and computing the plastic hinge rotation directly from the total storey

drift values (Goel and Chopra, 2004).


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Fig. 4. Properties of the nthmode inelastic SDOF system derived from the corresponding pushover curve

(after Chopra and Goel, 2001)

The main approximations of this method are that (i) the coupling among modal coordinates due to

yielding of the system is neglected (i.e. it is assumed that when the excitation is proportional to the j th

mode, the response in that mode is still predominant), and that (ii) the SRSS combination rule is

adopted to find the structural response and thus equilibrium cannot be provided. Whilst the first source

of approximation relates with the level of the inelastic demand on the system, the second dependsmainly on how much the higher modes contribute to the overall response. Moreover, further

limitations consist in the fact that (iii) the modal responses that are being combined correspond to

different stress states of the same structure since different levels of inelasticity are attained in each

mode, and that (iv) the equivalent SDOF system properties depends on the estimated displacement

demand and thus an iterative strategy is required to avoid further sources of uncertainties.

Chopra and Goel (2002) found that this approach leads to good estimates of displacements and drifts

but did not satisfactorily estimate plastic hinge rotations for some high-rise structures (e.g. the 9-storeysteel frame considered in this work). Moreover, this method is not able to provide a reasonable

estimate of drifts for the upper storeys of tall frames or at large degrees of inelasticity.

Further researches (Chintanapakdee and Chopra, 2003), considering several buildings with different

strength levels (correspondent to different strength reduction factors for the equivalent SDOFS, i.e.

different ductility levels), shown that MPA fails to predict the drifts profile, in particular at the upper

storey locations, as the building height as well as the ductility level increases.

Goel and Chopra (2004) further developed and tested this procedure for six SAC buildings subjected to

ensembles of 20 ground motions. Results indicate that, if the response of the structure is nearly elastic

then the main source of error is the SRSS combination rule adopted, whilst, with higher level of


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ductility demand, also a higher contribution of P-effects due to gravity loads and a large coupling of

the modal responses further affects the degree of accuracy of the pushover prediction. For these

reasons, even if the MPA estimates of demand were much better than those provided by the force

distribution specified in FEMA-356, they cannot be considered satisfactory for buildings that are

deformed well into the inelastic range with significant degradation in lateral capacity. For such cases a

nonlinear response history analysis is recommended by Chopra and Goel (2004).

In the recently proposed FEMA 440 document (ATC, 2005) this method has been compared with

other types of pushover schemes. Results shown that, although often improved over the simple mode

load vectors, estimates of interstorey drift over the full height of the buildings may not be consistently

reliable. The MPA procedure appear fundamentally limited because its accuracy depends upon the

parameter of interest (forces, deformations, plastic hinge rotations, etc.), the characteristics of thestructure, and the details of the specific procedure (as other researches confirm (e.g. Yu et al. (2001),

Lpez-Menjivar (2004)) and thus it is suggested only as an alternative method to obtain results to

compare.

For the prototype frame models considered in the ATC-55 project (ATC, 2005) a modified version of

the MPA has been adopted, where the first three modes contribution are considered, but the 2ndand 3rd

mode equivalent SDOFSs are assumed linear because of the reversal of some higher mode capacity

curves. This fact, as pointed out also in (Hernndez-Montes et al., 2004; Goel and Chopra, 2005(b))

represents one of the main drawbacks in the MPA procedure, which thus remains limited and it does

not allow an easy application within a design office environment.

The present methodology has been further improved for the calculation of member forces (Goel and

Chopra, 2006) and also extended to the case of 3D pushover analysis (Chopra and Goel, 2006).

However, as discussed by the authors themselves, the method does not always provide satisfactory

results for high-rise steel frames.

The above multi-modal procedures constitute a significant improvement with respect to conventional

pushover analysis since they explicitly consider the response of more than one vibration mode and the

influence of the expected ground motion. However, none of the invariant force distributions accounts

for the change in the modal contributions as well as the redistribution of inertia forces due to structural

yielding. To overcome these limitations, adaptive procedures, whereby the load vector shape is updated

at every single analysis step to reflect the actual dynamic properties of the system (i.e. stiffness state),

will represent an attractive alternative.

2.3.3 Adaptive Procedures


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0

2

4

6

8

10

12

0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0%

top drift = 0.5%

top drift = 1.0%

top drift = 1.5%

top drift = 2.5%

Fig. 6. Interstorey drift profile of a 12-Storey building subjected to increasing levels of deformation

Because of the aforementioned limitations, recent years have witnessed the development and

introduction of so-called Adaptive Pushover methods whereby the loading vector is updated at each

analysis step, reflecting the progressive stiffness degradation of the structure induced by the penetration

in the inelastic range, as schematically shown in Fig. 7, below; it might be noted that in adaptive

pushover the response of the structure is computed in incremental fashion, through a piecewise

linearization procedure (as described in 2.3.5), hence rendering it possible to use the tangent stiffness

at the start of each increment, together with the mass of the system, to compute modal responsecharacteristics of each incremental pseudo-system through elastic eigenvalue analysis, and use such

modal quantities to congruently update (i.e. increment) the pushover loading vector.

.

0% 1% 2% 3%total drift

0

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Dissertation2008 Pietra