National Taiwan University of Science and

國立台灣科技大學營建工程系

碩士學位論文學號： M9905801

A Pushover Seismic Evaluation Method for Tall and Asymmetric Buildings

研究生

:Yusak Oktavianus (蔡優光 )

指導教授

:歐昱辰

共同指導教授

:蕭輔沛

中華民國一百零一年六月二十九日 JUNE 29, 2012

ACKNOWLEDGEMENT Give thanks to Jesus Christ as my Lord and savior for His blessing so that this thesis can be completed well and on time. This thesis is intended to fulfill one of the requirements set by master degree program, Construction Engineering Department, National Taiwan University of Science and Technology. This thesis cannot be accomplished without support and encourage from any parties. With abundant of honor, I would like to express my sincere gratitude to: 1. Prof. Yu-Chen Ou as my home lecturer and as my supervisor who always give important advices to finish this thesis and encourage me to pour out my best capability. 2. Dr. Fu-Pei Hsiao as my supervisor who always guide me step by step with his knowledge and his patience. His open minded and his willingness to discuss make me can give my best performance. 3. Prof. Jenn-Shin Hwang and Prof. Yin-Nan Huang as my committee members during my oral defense for the positive suggestions related to my research. 4. My lovely family and my girlfriend, Serna Jarny, for their love, spirit, hope, and support that make my life more colorful in pursuing study in NTUST Taiwan. 5. My classmate, my lab mate, and my roommate for the discussion, experience, team work, and time we spend together 6. NTUST for the full scholarship that allow me to pursue my master degree without any funding difficulty. 7. Other parties that cannot be written one by one as the limitation of the space for their help and support while I was studying in NTUST. I realized that nothing perfect, therefore any positive comment and advice are very welcomed. Finally, I hope this thesis can be useful for future study and practice.

Taipei, June 29, 2012

Yusak Oktavianus

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A Pushover Seismic Evaluation Method for Tall and Asymmetric Buildings Graduate Student Thesis Advisor Thesis Co-Advisor

: : :

Yusak Oktavianus Yu-Chen Ou Fu-Pei Hsiao

ABSTRACT Pushover (PO) is a well known and practical analysis for evaluating new or existing buildings. The conventional pushover predicts the seismic demands well in the 1st mode dominant building, and will produce larger error as the higher mode effect contribution increases. This research tries to overcome the weakness based on the observed behavior of the structures in nonlinear dynamic analysis. Two-, 8-, and 20storey RC frame buildings with 0%, 5%, 10%, 15% of eccentricity for each building subject to nonlinear response history analysis (NRHA) are used. An inelastic response spectrum which is used in the extended N2 method is also used in this research. Because of the one way computation, no iteration is needed. Four modifications are made to take account the higher mode effects both in elevation and in plan which are based on the observed behavior. The assumption that higher mode effect will keep in elastic behavior used in the extended N2 method is eliminated. The modifications are verified in a 14-storey building with 10% eccentricity with medium and high inelastic degree of the structure. Displacement and drift at center of mass, and coefficient of torsion are used as the seismic demands parameter. The results show that the proposed method can give better accuracy towards the actual behavior of the structure and keep the simplicity of the PO method.

Keyword: Pushover, Higher mode effect, torsional effect, nonlinear dynamic, extended N2 method

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TABLE OF CONTENTS ACKNOWLEDGEMENT ............................................................................................. i ABSTRACT .................................................................................................................. ii TABLE OF CONTENTS ............................................................................................. iii LIST OF TABLES ........................................................................................................ v LIST OF FIGURES ..................................................................................................... vi 1.

2.

INTRODUCTION ................................................................................................. 1 1.1.

Background and Research Motivation ........................................................... 1

1.2.

Objectives and scopes .................................................................................... 2

1.3.

Outline ............................................................................................................ 3

LITERATURE REVIEW ...................................................................................... 5 2.1.

3.

4.

Previous research............................................................................................ 5

2.1.1.

Modal Pushover Analysis (MPA) ........................................................... 5

2.1.2.

Modified Modal Pushover Analysis (MMPA) ....................................... 6

2.1.3.

Practical Modal Pushover Analysis (PMPA) .......................................... 6

2.1.4.

Method of Modal Combination (MMC) ................................................. 6

2.1.5.

Adaptive Pushover (APO) ...................................................................... 7

2.1.6.

Dynamic pushover with SRM load pattern ............................................. 7

2.2.

ATC-40........................................................................................................... 7

2.3.

FEMA 356 .................................................................................................... 17

2.4.

Basic N2 method .......................................................................................... 23

2.5.

Comparison between ATC-40, FEMA 356, and Basic N2 method ............. 29

2.6.

Extended N2 method .................................................................................... 36

2.7.

Compatible ground motion matching a spectrum ........................................ 37

BUILDING EXAMPLE AND GROUND MOTION ......................................... 39 3.1.

Buildings Example ....................................................................................... 39

3.2.

Ground motion ............................................................................................. 43

NRHA RESULT AND ANALYTICAL STUDY................................................. 45 4.1.

Maximum displacement result ..................................................................... 45

4.2.

Coefficient of torsion result.......................................................................... 52

4.3.

Maximum inter-storey drift result ................................................................ 57

4.4.

Analytical study............................................................................................ 64

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4.5.

Step-by-step procedure of proposed method ................................................ 74

5.

VERIFICATION AND DISCUSSION................................................................ 79

6.

CONCLUSION AND SUGGESTION ................................................................ 91 6.1.

Conclusion .................................................................................................... 91

6.2.

Suggestion .................................................................................................... 92

REFERENCE .............................................................................................................. 93 APPENDIX ................................................................................................................. 95 A.1.

MATLAB Code for extended N2 method for this research ......................... 95

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LIST OF TABLES Table 2.1. Damping modification factor ߢ (Ou, 2012)............................................12 Table 2.2. Structural behavior type (Ou, 2012) .........................................................12 Table 2.3. Near source factors (Ou, 2012) ................................................................13 Table 2.4. Seismic source type (Ou, 2012) ...............................................................14 Table 2.5. Minimum allowable value for ܴܵ‫ ܣ‬and ܴܸܵ ........................................15 Table 2.6. Drift limits (Ou, 2012)..............................................................................17 Table 2.7. Values for modification factor C0 (FEMA-356, 2000) .............................19 Table 2.8. Values for effective mass factor Cm (FEMA-356, 2000)..........................20 Table 2.9. Values for modification factor ‫ܥ‬ଶ (FEMA-356, 2000) ...........................21 Table 2.10. Comparison result of each method ...........................................................34 Table 2.11. Difference action in Basic N2, FEMA 356 AND ATC-40 .......................35 Table 3.1. Details of members for each building ......................................................41 Table 3.2. Natural period of mode n of the building .................................................41 Table 3.3. Effective mass factor in x-direction about mode n ...................................42 Table 3.4. List of earthquake ground motion ............................................................44 Table 5.1. Calculation sheet for defining target top displacement for 14-storey building with 10% eccentricity. ................................................................80 Table 5.2. The displacement error resulted from basic N2, extended N2, and proposed method at center of mass ..........................................................82 Table 5.3. The coefficient of torsion error resulted from basic N2, extended N2, and proposed method................................................................................84 Table 5.4. The inter-storey drift error resulted from basic N2, extended N2, and proposed method at center of mass ..........................................................85 Table 5.5. The displacement error resulted from basic N2, extended N2, and proposed method at flexible edge .............................................................88 Table 5.6. The inter-storey drift error resulted from basic N2, extended N2, and proposed method at flexible edge .............................................................90

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LIST OF FIGURES Figure 2.1. Elastic acceleration spectra: (a) Sa-T format; (b) AD format ...................8 Figure 2.2. Pushover analysis of a building (Ou, 2012) ..............................................9 Figure 2.3. Pushover curve (capacity curve) (Ou, 2012) ............................................9 Figure 2.4. Capacity curve: (a) 𝑉 − 𝛿 format; (b) Sa-Sd format (Ou, 2012) ..........10 Figure 2.5. Derivation of damping (Ou, 2012).......................................................... 11 Figure 2.6. Damping modification factor (ATC-40, 1996) .......................................12 Figure 2.7. Reduction of 5% damped spectrum by 𝑆𝑅𝐴 and 𝑆𝑅𝑉 .........................15 Figure 2.8. Procedure to determine performance point (Ou, 2012) ..........................16 Figure 2.9. The acceptance criteria for performance objectives (FEMA-356, 2000) .......................................................................................................17 Figure 2.10 Idealized Force-Displacement Curves (FEMA-356, 2000) ....................18 Figure 2.11. 𝐶1 values (FEMA-356, 2000) ................................................................20 Figure 2.12. 𝐶2 from Table 2.9 (FEMA-356, 2000) ..................................................21 Figure 2.13. 𝐶2 from nonlinear response history analysis (FEMA-356, 2000) ..........22 Figure 2.14. EPP and SD hysteretic models ................................................................22 Figure 2.15. 𝐶3 values (FEMA-356, 2000)................................................................23 Figure 2.16. Building data and Elastic acceleration spectra (Fajfar, 2000).................24 Figure 2.17. Elastic and inelastic response spectra for constant ductility (Fajfar, 1999) .......................................................................................................25 Figure 2.18. Idealized bilinear capacity curve with zero post-yielding stiffness and transformation from base shear-displacement format to Sa-Sd format (Fajfar, 1999) ...............................................................................27 Figure 2.19. Determination of displacement demand, 𝑆𝑑 : (a) 𝑇 ∗ < 𝑇𝐶 ; (b) 𝑇 ∗ ≥ 𝑇𝐶 (Fajfar, 2000) ...........................................................................27 Figure 2.20. Simple version of ductility factor (Fajfar, 1999) ....................................29 Figure 2.21. The plan view of the building model ......................................................30 Figure 2.22. Performance point of ATC-40 .................................................................31 Figure 2.23. Defining the target displacement by FEMA 356 ....................................32 Figure 2.24. Target displacement by basic N2 method ...............................................34 Figure 2.25. The displacement shape of each method ................................................35

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Figure 2.26. Example of correction factor for higher mode effect in elevation, CE (Kreslin & Fajfar, 2011) ..........................................................................36 Figure 2.27. Example of correction factor for higher mode effect in plan, CT (Kreslin & Fajfar, 2011) ..........................................................................37 Figure 2.28. Response spectra: (a) Original response spectra from the ground motions; (b) Compatible response spectra from the compatible ground motions .......................................................................................37 Figure 3.1. Plan view and elevation view: (a) 2-storey (b) 8-storey; (c) 14-storey; (d) 20-storey ............................................................................................40 Figure 3.2. Response spectra: (a) Original response spectra from the ground motions; (b) Compatible response spectra from the compatible ground motions .......................................................................................43 Figure 4.1. Displacement result for 2-storey building: (a) 0% eccentricity 0.1g (𝜇 = 0.46 ) ; (b) 0% eccentricity 0.4g (𝜇 = 1.86 ) ; (c) 5% eccentricity 0.1g 𝜇 = 0.48 ; (d) 5% eccentricity 0.4g (𝜇 = 1.904 ); (e) 10% eccentricity 0.1g (𝜇 = 0.5 ); (f) 10% eccentricity 0.4g (𝜇 = 1.99 ) ; (g) 15% eccentricity 0.1g (𝜇 = 0.52 ) ; (h) 15% eccentricity 0.4g (𝜇 = 2.08 ) .................................................................46 Figure 4.2. Displacement result for 2-storey building: (a) 0% eccentricity 0.6g (𝜇 = 2.78 ); (b) 0% eccentricity 1g (𝜇 = 4.64 ); (c) 5% eccentricity 0.6g 𝜇 = 2.86 ; (d) 5% eccentricity 1g (𝜇 = 4.76 ) ; (e) 10% eccentricity 0.6g (𝜇 = 2.98 ); (f) 10% eccentricity 1g (𝜇 = 4.96 ); (g) 15% eccentricity 0.6g (𝜇 = 3.13 ); (h) 15% eccentricity 1g (𝜇 = 5.21 ) .............................................................................................47 Figure 4.3. Displacement result for 8-storey building: (a) 0% eccentricity 0.1g (𝜇 = 0.85 ) ; (b) 0% eccentricity 0.4g (𝜇 = 3.4 ) ; (c) 5% eccentricity 0.1g 𝜇 = 0.85 ; (d) 5% eccentricity 0.4g (𝜇 = 3.4 ); (e) 10% eccentricity 0.1g (𝜇 = 0.84 ) ; (f) 10% eccentricity 0.4g (𝜇 = 3.36 ) ; (g) 15% eccentricity 0.1g (𝜇 = 0.83 ) ; (h) 15% eccentricity 0.4g (𝜇 = 3.31 ) .................................................................48 Figure 4.4. Displacement result for 8-storey building: (a) 0% eccentricity 0.6g (𝜇 = 5.1 ); (b) 5% eccentricity 0.6g 𝜇 = 5.1 ; (c) 10% eccentricity

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0.6g (𝜇 = 5.04 ); (d) 15% eccentricity 0.6g (𝜇 = 4.96 ) ....................49 Figure 4.5. Displacement result for 20-storey building: (a) 0% eccentricity 0.1g (𝜇 = 0.38 ) ; (b) 0% eccentricity 0.4g (𝜇 = 1.51 ) ; (c) 5% eccentricity 0.1g 𝜇 = 0.38 ; (d) 5% eccentricity 0.4g (𝜇 = 1.51 ); (e) 10% eccentricity 0.1g (𝜇 = 0.38 ); (f) 10% eccentricity 0.4g (𝜇 = 1.5 ) ; (g) 15% eccentricity 0.1g (𝜇 = 0.37 ) ; (h) 15% eccentricity 0.4g (𝜇 = 1.48 ) .................................................................50 Figure 4.6. Displacement result for 20-storey building: (a) 0% eccentricity 0.6g (𝜇 = 2.27 ); (b) 0% eccentricity 1g (𝜇 = 3.79); (c) 5% eccentricity 0.6g 𝜇 = 2.27 ; (d) 5% eccentricity 1g (𝜇 = 3.78 ) ; (e) 10% eccentricity 0.6g (𝜇 = 2.25 ); (f) 10% eccentricity 1g (𝜇 = 3.75 ); (g) 15% eccentricity 0.6g (𝜇 = 2.22 ); (h) 15% eccentricity 1g (𝜇 = 3.71 ) .............................................................................................51 Figure 4.7. Displacement result for 20-storey building: (a) 0% eccentricity 1.4g (μ = 5.3 ); (b) 5% eccentricity 1.4g μ = 5.29 ; (c) 10% eccentricity 1.4g (μ = 5.25 ); (d) 15% eccentricity 1.4g (μ = 5.19 ) .....................51 Figure 4.8. Coefficient of torsion result for 2-storey building: (a) 5% eccentricity 0.1g 𝜇 = 0.48 ; (b) 5% eccentricity 0.4g (𝜇 = 1.904 ); (c) 10% eccentricity 0.1g (𝜇 = 0.5 ); (d) 10% eccentricity 0.4g (𝜇 = 1.99 ); (e) 15% eccentricity 0.1g (𝜇 = 0.52 ); (f) 15% eccentricity 0.4g (𝜇 = 2.08 ) .............................................................................................53 Figure 4.9. Coefficient of torsion result for 2-storey building: (a) 5% eccentricity 0.6g 𝜇 = 2.86 ; (b) 5% eccentricity 1g (𝜇 = 4.76 ) ; (c) 10% eccentricity 0.6g (𝜇 = 2.98 ); (d) 10% eccentricity 1g (𝜇 = 4.96 ); (e) 15% eccentricity 0.6g (𝜇 = 3.13 ); (f) 15% eccentricity 1g (𝜇 = 5.21 ) .............................................................................................53 Figure 4.10. Coefficient of torsion result for 8-storey building: (a) 5% eccentricity 0.1g 𝜇 = 0.85 ; (b) 5% eccentricity 0.4g (𝜇 = 3.4 ) ; (c) 10% eccentricity 0.1g (𝜇 = 0.84 ) ; (d) 10% eccentricity 0.4g (𝜇 = 3.36 ); (e) 15% eccentricity 0.1g (𝜇 = 0.83 ); (f) 15% eccentricity 0.4g (𝜇 = 3.31 ) ....................................................................................54 Figure 4.11. Coefficient of torsion result for 8-storey building: (a) 5% eccentricity

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0.6g 𝜇 = 5.1 ; (b) 10% eccentricity 0.6g (𝜇 = 5.04 ); (c) 15% eccentricity 0.6g (𝜇 = 4.96 ) .................................................................55 Figure 4.12. Coefficient of torsion result for 20-storey building: (a) 5% eccentricity 0.1g 𝜇 = 0.38 ; (b) 5% eccentricity 0.4g (𝜇 = 1.51 ); (c) 10% eccentricity 0.1g (𝜇 = 0.38 ); (d) 10% eccentricity 0.4g (𝜇 = 1.5 ) ; (e) 15% eccentricity 0.1g (𝜇 = 0.37 ) ; (f) 15% eccentricity 0.4g (𝜇 = 1.48 ) .................................................................56 Figure 4.13. Coefficient of torsion result for 20-storey building: (a) 5% eccentricity 0.6g (𝜇 = 2.27); (b) 5% eccentricity 1g (𝜇 = 3.78 ); (c) 10% eccentricity 0.6g (𝜇 = 2.25 ); (d) 10% eccentricity 1g (𝜇 = 3.75 ) ; (e) 15% eccentricity 0.6g (𝜇 = 2.22 ) ; (f) 15% eccentricity 1g (𝜇 = 3.71 ) ....................................................................57 Figure 4.14. Coefficient of torsion result for 20-storey building: (a) 5% eccentricity 1.4g (𝜇 = 5.29) ; (b) 10% eccentricity 1.4g (𝜇 = 5.25 ); (c) 15% eccentricity 1.4g (𝜇 = 5.19 )....................................................57 Figure 4.15. Drift result for 2-storey building: (a) 0% eccentricity 0.1g (𝜇 = 0.46 ) ; (b) 0% eccentricity 0.4g (𝜇 = 1.86 ) ; (c) 5% eccentricity 0.1g 𝜇 = 0.48 ; (d) 5% eccentricity 0.4g (𝜇 = 1.904 ); (e) 10% eccentricity 0.1g (𝜇 = 0.5 ); (f) 10% eccentricity 0.4g (𝜇 = 1.99 ) ; (g) 15% eccentricity 0.1g (𝜇 = 0.52 ) ; (h) 15% eccentricity 0.4g (𝜇 = 2.08 ) .................................................................58 Figure 4.16. Drift result for 2-storey building: (a) 0% eccentricity 0.6g (𝜇 = 2.78 ); (b) 0% eccentricity 1g (𝜇 = 4.64 ); (c) 5% eccentricity 0.6g (𝜇 = 2.86) ; (d) 5% eccentricity 1g (𝜇 = 4.76 ) ; (e) 10% eccentricity 0.6g (𝜇 = 2.98 ); (f) 10% eccentricity 1g (𝜇 = 4.96 ); (g) 15% eccentricity 0.6g (𝜇 = 3.13 ); (h) 15% eccentricity 1g (𝜇 = 5.21 ) ..............................................................................................59 Figure 4.17. Drift result for 8-storey building: (a) 0% eccentricity 0.1g (𝜇 = 0.85 ); (b) 0% eccentricity 0.4g (𝜇 = 3.4 ); (c) 5% eccentricity 0.1g 𝜇 = 0.85 ; (d) 5% eccentricity 0.4g (𝜇 = 3.4 ) ; (e) 10% eccentricity 0.1g (𝜇 = 0.84 ); (f) 10% eccentricity 0.4g (𝜇 = 3.36 ); (g) 15% eccentricity 0.1g (𝜇 = 0.83 ); (h) 15% eccentricity 0.4g

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(𝜇 = 3.31 ) ..............................................................................................60 Figure 4.18. Drift result for 8-storey building: (a) 0% eccentricity 0.6g (𝜇 = 5.1 ); (b) 5% eccentricity 0.6g ( 𝜇 = 5.1 ) ; (c) 10% eccentricity 0.6g (𝜇 = 5.04 ); (d) 15% eccentricity 0.6g (𝜇 = 4.96 ) ...............................61 Figure 4.19. Drift result for 20-storey building: (a) 0% eccentricity 0.1g (𝜇 = 0.38 ) ; (b) 0% eccentricity 0.4g (𝜇 = 1.51 ) ; (c) 5% eccentricity 0.1g (𝜇 = 0.38) ; (d) 5% eccentricity 0.4g (𝜇 = 1.51 ); (e) 10% eccentricity 0.1g (𝜇 = 0.38 ); (f) 10% eccentricity 0.4g (𝜇 = 1.5 ) ; (g) 15% eccentricity 0.1g (𝜇 = 0.37 ) ; (h) 15% eccentricity 0.4g (𝜇 = 1.48 ) .................................................................62 Figure 4.20. Drift result for 20-storey building: (a) 0% eccentricity 0.6g (𝜇 = 2.27 ); (b) 0% eccentricity 1g (𝜇 = 3.79); (c) 5% eccentricity 0.6g (𝜇 = 2.27) ; (d) 5% eccentricity 1g (𝜇 = 3.78 ) ; (e) 10% eccentricity 0.6g (𝜇 = 2.25 ); (f) 10% eccentricity 1g (𝜇 = 3.75 ); (g) 15% eccentricity 0.6g (𝜇 = 2.22 ); (h) 15% eccentricity 1g (𝜇 = 3.71 ) ..............................................................................................63 Figure 4.21. Drift result for 20-storey building: (a) 0% eccentricity 1.4g (𝜇 = 5.3 ) ; (b) 5% eccentricity 1.4g (𝜇 = 5.29 ); (c) 10% eccentricity 1.4g (𝜇 = 5.25 ); (d) 15% eccentricity 1.4g (𝜇 = 5.19 ) ..64 Figure 4.22. Linear assumed displacement vs proposed displacement shape in 20-storey building 5% eccentricity: (a) load pattern; (b) assumed displacement shape .................................................................................66 Figure 4.23. Reduction factor to the target top displacement considering the torsion resulted by NRHA ......................................................................68 Figure 4.24. The multiplier or weight factor of CT value resulted from NRHA: (a) CTt PO multiplier; (b) CTtRSA multiplier................................................69 Figure 4.25. The proposed multiplier of PO and RSA to calculate the final coefficient of torsion ...............................................................................70 Figure 4.26. The “whip effect” on chi-chi earthquake: 20-storey 1.4g μ ≈ 5............71 Figure 4.27. The max storey displacement versus the max drift displacement in 0% eccentricity building: (a) 2-storey 0.4g; (b) 8-storey 0.4g; (c) 20-storey 0.4g .........................................................................................71

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Figure 4.28. The value of max drift displacement divided by the max storey displacement in the top floor: (a) NRHA result; (b) proposed γr ...........73 Figure 4.29. The value of decomposed max drift displacement divided by the max storey displacement: (a) 2-storey by ground motion; (b) 2-storey by proposed coefficient,γj ;(c) 8-storey by ground motion; (d) 8-storey by proposed coefficient, γj ;(e) 20-storey by ground motion; (f) 20-storey by proposed coefficient,γj .......................................................74 Figure 4.30. Elastic acceleration spectra: (a) Sa-T format; (b) AD format .................75 Figure 4.31. Flowchart of the proposed method .........................................................78 Figure 5.1. (a) Normalized lateral force pattern 𝐹𝑗 ; (b) assumed displacement shape ∅𝑗 .................................................................................................80 Figure 5.2. Graphical way to obtain the target top displacement of SDOF of 14-storey building, 10% eccentricity, 0.6g resulted by 1st modification of proposed method. ..........................................................80 Figure 5.3. Displacement result at center of mass for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g ........................................................................82 Figure 5.4. Coefficient of torsion result for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g ..............................................................................83 Figure 5.5. Inter-storey

drift

result

at

center

of

mass

for

14-storey

10%-eccentricity: (a) pga=0.6g; (b) pga=1g ...........................................85 Figure 5.6. Displacement result at flexible edge for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g ........................................................................87 Figure 5.7. Inter-storey

drift

result

at

flexible

edge

for

14-storey

10%-eccentricity: (a) pga=0.6g; (b) pga=1g ...........................................89

Figure A.1. Extended N2 result from the MATLAB code .......................................105

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1. INTRODUCTION 1.1.Background and Research Motivation Nowadays, seismic design criteria tend to shift from the force-based procedure to performance-based procedure both for design and evaluation purpose. Pushover analysis becomes very well known method in determining seismic demand because of its simplicity and accuracy for short and symmetric or 1st mode dominant building. Pushover analysis estimates seismic demands directly from the earthquake design spectrum and capacity curve, excluding the complications to select and to scale the ground motions. In the other side, Nonlinear Response History Analysis (NRHA) typically demands high computational resources. For tall and asymmetric building, traditional pushover analysis cannot come up with higher mode effect both in elevation and in plan (torsional effect), respectively. Moreover, traditional pushover (ATC-40 and FEMA 356) requires an iteration process to get the performance point or target displacement. A lot of researches have been done to make the traditional pushover can give better result and retain its simplicity. (Chopra & Goel, 2002, 2004; Chopra, Goel, & Chintanapakdee, 2004; Fajfar, 1999, 2000; Kreslin & Fajfar, 2011; Kunnath, 2004; Marus̆ić & Fajfar, 2005; R. Rofooei, K. Attari, Rasekh, & Shodja, 2006; Reyes & Chopra, 2011; Rofooei, Attari, Rasekh, & Shodja, 2007). The previous methods are time consuming and there is an assumption that higher mode effect will remain in elastic behavior. In the other side, N2 method has a simple way (no need iteration) to get the target displacement which uses the inelastic response spectra and the capacity curve to get the target displacement (Fajfar, 1999, 2000). The extended N2 method has the same way as the basic N2 to calculate the target displacement. The extension is the usage of response spectrum analysis (RSA) as the correction to the seismic demands which assume that the higher mode effect will keep in the elastic behavior (Kreslin & Fajfar, 2011; Marus̆ić & Fajfar, 2005). Though N2 method has its simplicity and has been extended, but it usually still conservative in determining the coefficient of torsion in large earthquake and in some cases result unconservatism in determining

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the drift. By explanation explained above, a proposed method is introduced to come up with the higher mode effect problems using the pushover analysis. The proposed method also uses the inelastic response spectra like the N2 method does to keep the simplicity, i.e. no need iteration in obtaining the target displacement.Additional 4 modifications are made to improve the PO such the seismic demands will approach the real behavior. The modifications are made to consider the higher mode effect and based on the behavior of the real building. NRHA is assumed to be similar with the real behavior of the building. Three different buildings which are 2-storey, 8-storey and 20-storey reinforced concrete frame building with 0%, 5%, 10%, and 15% eccentricity and with several pga value are taken as the source of database for finding the behavior. By these modifications, there is no need to assume that the higher mode effect will keep in elastic behavior. A 14-storey reinforced concrete frame building with NRHA, basic N2, extended N2, and the proposed method is established to check the competence of the proposed method. Several seismic demands are taken into account, which are displacement and drift at center of mass and at the flexible edge 1.2.Objectives and scopes The objective of this research is to find the simplest and the most accurate method to approach the real behavior. This objective is accomplished by several small modifications: 1. New lateral load pattern which takes account the contribution of higher mode effect in elevation 2. Adjusted target displacement which considers the higher mode effect in plan (torsional effect) based on the real behavior 3. Adjusted coefficient of torsion based on the real behavior 4. Adjusted inter-storey displacement to calculate inter-storey drift based on the real behavior Three different buildings which are 2-storey, 8-storey and 20-storey reinforced concrete frame building with 0%, 5%, 10%, and 15% eccentricity and with several pga value of ground motions are taken as the source of database for finding the real

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behavior. 1.3. Outline This thesis is divided into 6 chapters as follows: 

Chapter 1 gives general introduction about this research, including background and research motivation, objectives and scopes, and outline.



Chapter 2 describes briefly literatures which are related to this research.



Chapter 3 shows buildings example and ground motions which are used in analytical study and verification study.



Chapter 4 describes the result of NRHA, basic N2, and extended N2, including the modifications that should be applied based on the analytical study and step by step procedure.



Chapter 5 contains verification of proposed method in a 14-storey building and the comparison with the NRHA result, basic N2, and extended N2 method.



Chapter 6 describes conclusions and suggestions of this study.

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2. LITERATURE REVIEW 2.1. Previous research There are a lot of researches done to modify the pushover such approach the NRHA result. Some of them are described as follows: 2.1.1. Modal Pushover Analysis (MPA) Modal Pushover Analysis (MPA) combines the pushover analysis with uncouple modal response history analysis. Several dominant modes are used separately as the lateral load for the pushover analysis with target displacement defined from the uncouple response history analysis result to an inelastic SDOF system (Chopra & Goel, 2002, 2004). This method gives good result compare with NRHA, but it is a time consuming method because need to run the uncouple modal response history analysis in order to get the target displacement for each mode. The short description of this method is described as follows: The governing differential equations of the response of MDOF system subjected to horizontal ground motion 𝑢𝑔 (𝑡) are presented in Eqs.(2.1) to (2.3). 𝑚𝑢 + 𝑐𝑢 + 𝑘𝑢 = −𝑚𝑖𝑢𝑔 𝑡 𝑁

𝑚𝑖 =

𝑁

𝑠𝑛 = 𝑛 =1

𝛤𝑛 =

(2.1)

𝛤𝑛 𝑚𝜙𝑛

(2.2)

𝑛=1

𝐿𝑛 ; 𝐿𝑛 = 𝜙𝑛𝑇 𝑚𝑖 ; 𝑀𝑛 = 𝜙𝑛𝑇 𝑚𝜙𝑛 𝑀𝑛

(2.3)

where𝑢 is lateral floor displacement relative to the ground; 𝑚, 𝑐, 𝑘 are the mass, damping, and stiffness matrices, respectively; 𝑖 is influence vector of each element which equal to 1; ϕ𝑛 is the structural natural vibration of nth mode. For „exact‟ NRHA, because history of the displacement controls the next displacement, thus the relation between the lateral force 𝑓𝑠 at the N floor levels and the lateral displacements 𝑢 are not single-valued. Therefore, Eq.(2.1) becomes: 𝑚𝑢 + 𝑐𝑢 + 𝑓𝑠 𝑢, 𝑠𝑖𝑔𝑛𝑢 = −𝑚𝑖𝑢𝑔 𝑡

(2.4)

By neglecting the coupling of the N equations, Eqs.(2.5) to (2.7) will become the governing equation of uncoupled modal response history analysis (UMRHA). 5

𝑚𝑢 + 𝑐𝑢 + 𝑓𝑠 𝑢, 𝑠𝑖𝑔𝑛𝑢 = −𝑠𝑛 𝑢𝑔 𝑡 𝐷𝑛 + 2𝜁𝑛 𝜔𝑛 𝐷𝑛 +

𝐹𝑠𝑛 = −𝑢𝑔 𝑡 𝐿𝑛

𝐹𝑠𝑛 = 𝐹𝑠𝑛 𝐷𝑛 , 𝑠𝑖𝑔𝑛𝐷𝑛 = 𝜙𝑛𝑇 𝑓𝑠 𝐷𝑛 , 𝑠𝑖𝑔𝑛𝐷𝑛

(2.5) (2.6) (2.7)

2.1.2. Modified Modal Pushover Analysis (MMPA) Modified Modal Pushover Analysis (MMPA) takes account the higher mode effect by assuming that the behaviour of higher mode will remain in elastic state. By this assumption, clasical modal analysis for linear system is used to take account the higher mode effect. Therefore no need to perform the pushover analysis for higher mode. NRHA is used to get the target displacement of the inelastic first mode which in turn will be used in the pushover analysis. This method is simpler than MPA, but will result larger error in larger degree of inelastic action(Chopra et al., 2004) 2.1.3. Practical Modal Pushover Analysis (PMPA) Practical Modal Pushover Analysis (PMPA) is similar with the MMPA with additional simplification to determine the target displacement of inelastic first mode. The target displacement of inelastic first mode is obtained by multiplying the median target displacement of linear system with inelastic deformation ratio (Reyes & Chopra, 2011). This method has good prediction of the seismic demand and will be similar as RSA result in linear system, and no need to run the NRHA, but still need to select the ground motions and run many of linear dynamic analysis to get the median elastic target displacement. 2.1.4. Method of Modal Combination (MMC) Method of Modal Combination (MMC) tries to combine several modes by adding or reducing the contribution of different mode in determining the lateral forces. This method will have so many alternative lateral force patterns and will consume much time (Kunnath, 2004).

6

2.1.5. Adaptive Pushover (APO) Adaptive Pushover (APO) tries to change the load pattern in every step by following the displacement pattern from the previous step. The load pattern will be changed as many as the step required in pushover analysis. This cause the method also becomes time consuming (Rofooei et al., 2007). 2.1.6. Dynamic pushover with SRM load pattern Combination of effective modal load pattern and NRHA in SDOF system to get the target displacement (Dynamic pushover with SRM load pattern) tries to take account the higher mode effect in the loading pattern by including the higher mode with effective modal mass as the multiplication factor, and use the NRHA to get the target displacement (R. Rofooei et al., 2006) 2.2. ATC-40 ATC-40 (ATC-40, 1996) uses peak roof displacement of the building to determine the performance of the building subjected to earthquake ground motion. Combination between capacity curve and demand response spectrum with some iteration process are used to get the performance point or target displacement. ATC-40, called Capacity Spectrum method, requires AD format for both of capacity curve and demand spectra. The steps of ATC-40 are described as follows: 1. Convert a demand response spectrum found in the building codes from the standard

Sa

(Spectra Acceleration) - T (Period) format to AD format as shown

in Figure 2.1and Eq. (2.8) 𝑆𝑑 =

𝑇2 𝑆 4𝜋 2 𝑎

7

(2.8)

1.2

Spectral acceleration (g)

Spectral acceleration (g)

1.2

0.8

0.4

0

0

1

2 3 Period (s)

4

0.8

0.4

0

5

0

20 40 60 80 Spectral displacement (cm)

(a)

(b)

Figure 2.1. Elastic acceleration spectra: (a) Sa-T format; (b) AD format

2. Perform pushover analysis and generate the relationship between roof displacement 

and base shear

V

(pushover capacity curve). This is

illustrated in Figure 2.2 and Figure 2.3. The lateral story forces applied to the structure is in proportion to the product of the mass and first mode shape which is described in Eq. (2.9). Alternatively, Eq.(2.10) can be used to determine the lateral story force pattern. Gravity loads should be included in this analysis. Subsequently, convert the base shear-displacement format to the Sa-Sd format. 𝐹𝑥 =

𝑤𝑥 ϕ𝑥 𝑛 𝑗 =1 𝑤𝑗 ϕ𝑗

𝑤𝑥 𝑕𝑥𝑘 𝐹𝑥 = 𝑛 𝑘𝑉 𝑗 =1 𝑤𝑗 𝑕𝑗

(2.9) (2.10)

where 𝐹𝑥 is the lateral force at level x; 𝑤𝑥 or 𝑤𝑗 is the weight at level x or j; and ϕ𝑥 orϕ𝑗 is the displacement at level x or j corresponding to first mode shape of the structure;𝑕𝑥 or 𝑕𝑗 is the height from the base to level x or j; For structures having a period of 0.5 seconds or less, k = 1; For structures having a period of 2.5 seconds or more, k = 2; For structures having a period between 0.5 and 2.5 seconds, k can be determined by linear interpolation between 1 and 2 or may be taken equal to 2.

8

F4

Level 4

F3

Level 3

F2

Level 2

F1

Level 1

Figure 2.2. Pushover analysis of a building (Ou, 2012) 200

Base shear (tf)

160 120 80 40 0 0

10 20 30 Roof displacement (cm)

40

Figure 2.3. Pushover curve (capacity curve) (Ou, 2012) (a) Convert points 

(roof displacement) into

Sd

Eq.(2.11) is used to convert the roof displacement 

into the

Sd

coordinate. 𝑆𝑑 =

𝛿 Γ1 ∙ 𝜙𝑟𝑜𝑜𝑓 ,1

(2.11)

where  1 is modal participation factor for the first mode and is defined by Eq. (2.12) Γ1 = (b) Convert points

V

(base shear) into

𝑁 𝑗 =1 𝜙𝑗 1 𝑁 2 𝑗 =1 𝜙𝑗 1

∙ 𝑚𝑗

Sa

Eqs.(2.13) and (2.14) is used to convert the base shear coordinate.

9

(2.12)

∙ 𝑚𝑗

V

into the

Sa

𝑆𝑎 =

(2.13)

2 𝑁 𝑗 =1 𝜙𝑗 1 ∙ 𝑚𝑗 𝑁 2 𝑗 =1 𝜙𝑗 1 ∙ 𝑚𝑗

𝛼1 =

200

0.5

160

0.4

120

0.3

Sa (g)

Base shear (tf)

𝑉𝑏 𝛼1

80 40

(2.14)

0.2 0.1

0

0 0

10 20 30 Roof displacement (cm)

40

0

5

10 15 Sd (cm)

(a)

20

(b)

Figure 2.4. Capacity curve: (a) 𝑉 − 𝛿 format; (b) Sa-Sd format (Ou, 2012)

3. Estimation of damping and reduction of 5 percent damped response spectrum The damping that occurs when an earthquake ground motion shakes the structure to the inelastic range can be defined as a combination of hysteretic damping and viscous damping that is natural in the structure. Hysteretic damping is defined as the area inside the loops that are formed when the base shear is plotted in opposition to the structure displacement as shown in Figure 2.5. The equivalent viscous damping, 𝛽𝑒𝑞 , correlated with a maximum displacement of𝑑𝑝 can be estimated from the Eqs. (2.15) to (2.18) 𝛽𝑒𝑞 = 𝛽0 + 0.05 𝛽0 =

1 𝐸𝐷 4𝜋 𝐸𝑆0

(2.15) (2.16)

𝐸𝐷 = 4(𝑎𝑦 𝑑𝑝 − 𝑎𝑝 𝑑𝑦 )

(2.17)

𝐸𝑆0 = 𝑎𝑝 𝑑𝑝 /2

(2.18)

where  0 is hysteretic damping represented as equivalent viscous damping; and 0.05 is 5% viscous damping which is natural in the structure (assumed to be constant); 𝐸𝐷 is energy dissipated by damping; and 𝐸𝑆0 is maximum strain 10

25

energy as shown in Figure2.5.

Figure 2.5. Derivation of damping (Ou, 2012)

Figure 2.5 shows an idealized hysteresis loop which is reasonable for a ductile detailed building subjected to relatively short duration ground motion (not enough cycles to extensively degrade elements) and with equivalent viscous damping less than approximately 30%. For non ductile buildings, calculation of the equivalent viscous damping using Eq. (2.15)and the idealized hysteresis loop in Figure 2.5will overestimate the realistic value of damping. In order to consider imperfect hysteresis loops (loops reduced in area), a damping modification factor,𝜅, is used in Eq. (2.19) 𝛽𝑒𝑓𝑓 = 𝜅𝛽0 + 0.05

(2.19)

The 𝜅-factor is listed in Table 2.1 and Table 2.2. Moreover, it is shown in Figure 2.6.The 𝜅-factor depends on the structural behavior of the building, i.e. the quality of the seismic resisting system and the duration of ground shaking.

For easiness, ATC-40 defines three categories of structural behavior. Structural behavior Type A represents stable, reasonably full hysteresis loops

11

most similar to Figure 2.5, and is assigned a 𝜅 of 1.0 (except at higher damping values). Type B represents a moderate reduction of area and is assigned a basic 𝜅 of 2/3 (𝜅 is also reduced at higher values of 𝛽𝑒𝑞 to be consistent with the Type A relationships). Type C represents poor hysteretic behavior with a substantial reduction of loop area (severely pinched) and is assigned a 𝜅 of 1/3. Table 2.1. Damping modification factor 𝜅 (Ou, 2012) Structural behavior type Type A

0



 1 6 .2 5

1.0

 1 6 .2 5

1.13-

0 .5 1  a y d

 d ya p

p

a pd

 25

Type B

 25

p

0.67 0.845-

0 .4 4 6  a y d

Any value

 d ya p

p

a pd

Type C





p

0.33

Figure 2.6. Damping modification factor (ATC-40, 1996) Table 2.2. Structural behavior type (Ou, 2012) Shaking duration(a) Short Long

Essentially new building(b) Type A Type B 12

Average existing building(c) Type B Type C

Poor existing building(d) Type C Type C

(a) Shaking duration  Sites with a near-source factor,

N 

1.2 (see Table 2.3 and Table 2.4),

may be assumed to have short-duration ground shaking. (sites near a seismic source (fault), a relatively short duration of very strong shaking would be expected )  Sites located in seismic zone 3 should be assumed to have long duration ground shaking. (sites far from fault rupture, a much longer duration of ground shaking would be expected at the level of response described by the site spectrum. Although, ground shaking is not as strong the previous case, longer duration of shaking increases the potential for degradation of the structural system)  Sites located in seismic zone 4 (with a near-source factor of N mean + σ

(5.1)

%error =

(mean − σ) − Y ∙ 100% (mean − σ)

for Y < mean − σ

(5.2)

where Y is the seismic demand (displacement, drift, or coefficient of torsion) resulted both from extended N2 method or proposed method; mean is the average seismic demand from all of the maximum value of the ground motions‟ response; σ is the standard deviation. Based on the Eqs. (5.1) and (5.2), the negative error represents the conservative result, while the positive error represents the unconservative result. The lateral force patterns and assumed displacement shape for 14-storey building both in extended N2 and proposed method are described in Figure 5.1. The difference of the lateral load pattern becomes clearer when the contribution of higher mode in elevation is larger. Or in the other word, the difference between extended N2 method and the proposed method will be clearer in high rise building. The target displacement can be determined either by calculation sheet in Table 5.1or by graphical way in Figure 5.2. In Table 5.1, It can be concluded that target top displacement resulted by extended N2 method is larger than the proposed method. This occurrence happens especially when the higher mode effect becomes dominant.

79

(a)

(b)

Figure 5.1. (a) Normalized lateral force pattern 𝐹𝑗 ; (b) assumed displacement shape ∅𝑗

Figure 5.2.Graphical way to obtain the target top displacement of SDOF of 14-storey building, 10% eccentricity, 0.6g resulted by 1st modification of proposed method. Table 5.1. Calculation sheet for defining target top displacement for 14-storey building with 10% eccentricity. Basic N2 and Extended N2 method ∗ 2 5026 m (kg ∙ sec /cm)

Note Bilinear approximation of MDOF and SDOF

Dy

Proposed method 5716

ɼ

1.46

1.32

(cm)

25.97

25.97

15432

15432

Fy = Fu (kN)

80

D∗y (cm)

17.83

19.61

Fu∗

10597

11655

T (s)

1.81

1.93

Say (g)

0.219

0.212

yield pga (g)

-

0.273

-

0.033

Sde (cm)

67.5

71.95

Sae (g)

0.829

0.777

Fy∗

=

(kN)

∗

n

𝑇𝐸𝑀𝐹𝑖 i=1

Calculate target displacement in 1g of pga

Calculate target displacement in 0.6g of pga

Rμ

3.784

3.669

d∗tti (cm)

67.5

71.95

dtti (cm)

98.3

95.32

dtt (cm)

98.3

92.52

Sde (cm)

40.48

43.17

Sae (g)

0.497

0.466

Rμ

2.27

2.2

d∗tti (cm)

40.48

43.17

dtti (cm)

58.98

57.19

dtt (cm)

58.98

55.54

The displacement result at the center of mass, coefficient of torsion result, and inter-storey drift result at the center of mass are graphed in Figure 5.3, Figure 5.4, and Figure 5.5, respectively. In those figure, the basic N2 method, extended N2 method, proposed method, and the ground motions are compared. The ground motions results are summarized become three value which are average from all of the maximum value of each ground motions, and plus minus one times standard deviation. From Table 5.2, the maximum absolute error of displacement induced by both basic N2 and extended N2 is 5.06% and 6.07% for pga equal to 0.6 and 1g, respectively. The error is induced from the linear assumption of the displacement shape and the ignorance of the reduction effect in target displacement in CM caused by torsion (described previously in 1st and 2nd modification). The basic N2 and extended N2 have the same value, because pushover result dominates the target displacement instead of RSA result. In the other hand, the maximum absolute error of displacement induced by proposed method is 0.47% and 0.39% for pga equal to 0.6 and 1g, respectively. The small error (less than 5% for displacement) resulted from proposed method shows that by 1st and 2nd modifications, the proposed method can capture the effect of the 81

higher mode effect both in elevation and in plan toward the target displacement. 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

Average Ave+Stdev Ave-Stdev Basic N2 Extended N2 Proposed method

0

15 30 45 Displacement (cm)

60

(a)

(b) Figure 5.3.Displacement result at center of mass for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g

Table 5.2. The displacement error resulted from basic N2, extended N2, and proposed method at center of mass Error percentage (%) Height (cm)

5200 4850

0.6g (=2.2) 1g (=3.7) Floor level basic N2 Extended Proposed basic N2 Extended Proposed N2 method method N2 method method method method 14 13

-0.85 -1.12

-0.85 -1.12

0.00 0.00 82

-4.81 -5.08

-4.81 -5.08

0.00 0.00

4500 4150 3800 3450 3100 2750 2400 2050 1700 1350 1000

12 11 10 9 8 7 6 5 4 3 2

-1.48 -1.98 -2.73 -3.67 -4.58 -5.02 -5.06 -4.55 -3.94 -3.28 -2.60

-1.48 -1.98 -2.73 -3.67 -4.58 -5.02 -5.06 -4.55 -3.94 -3.28 -2.60

0.00 0.00 0.00 0.00 0.00 -0.16 -0.47 -0.21 0.00 0.00 0.00

-5.54 -6.00 -6.07 -5.85 -5.36 -4.52 -3.39 -2.20 -1.06 -0.20 0.00

-5.54 -6.00 -6.07 -5.85 -5.36 -4.52 -3.39 -2.20 -1.06 -0.20 0.00

0.00 0.00 -0.25 -0.39 -0.28 0.00 0.00 0.00 0.00 0.00 0.00

500

1

-1.56

-1.56

0.00

0.00

0.00

0.00

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 1.04 1.08 1.12 1.16 1.2 1.24 Coefficient of torsion, CT


(a) 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 1.04


1.08 1.12 1.16 1.2 Coefficient of torsion, CT

(b) Figure 5.4.Coefficient of torsion result for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g

83

Table 5.3. The coefficient of torsion error resulted from basic N2, extended N2, and proposed method Error percentage (%) 0.6g (=2.2) 1g (=3.7) Height Floor (cm) level basic N2 Extended Proposed basic N2 Extended Proposed method N2 method method method N2 method method 5200 4850 4500 4150 3800 3450 3100 2750 2400 2050 1700 1350 1000

14 13 12 11 10 9 8 7 6 5 4 3 2

3.65 4.01 4.32 3.98 3.87 3.89 3.99 3.87 3.66 3.24 3.18 3.18 3.19

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.11 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1.94 1.62 1.44 1.37 1.48 1.61 1.82 2.06 2.40 2.69 2.77 2.76 2.88

-4.96 -4.94 -4.92 -4.99 -5.59 -6.09 -6.13 -6.14 -6.20 -6.13 -5.69 -5.20 -5.07

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

500

1

3.24

0.00

0.00

3.15

-4.75

0.00

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0


0

0.4

0.8 1.2 Drift (%)

1.6

(a)

84

2

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0


0

1

2

3

Drift (%)

(b) Figure 5.5.Inter-storey drift result at center of mass for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g

Table 5.4. The inter-storey drift error resulted from basic N2, extended N2, and proposed method at center of mass Error percentage (%) Height (cm)

Floor level

5200-4850 4850-4500 4500-4150 4150-3800 3800-3450 3450-3100 3100-2750 2750-2400 2400-2050 2050-1700 1700-1350 1350-1000 1000-500 500-0

0.6g (=2.2)

1g (=3.7)

basic N2 method

Extended N2 method

Proposed method

basic N2 Extended Proposed method N2 method method

14-13 13-12 12-11 11-10 10-9 9-8 8-7 7-6 6-5 5-4 4-3 3-2 2-1

33.97 29.88 18.83 0.39 0.00 0.00 0.00 0.00 0.00 -2.25 -5.13 -5.04 -3.52

-7.46 -10.79 -13.31 -5.96 0.00 0.00 0.00 0.00 0.00 -2.25 -5.13 -5.04 -3.52

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.66 -4.07 -5.37 -1.31 0.00

30.37 24.13 8.99 0.00 0.00 0.00 -0.85 -4.72 -6.09 -5.20 -2.91 -1.74 -0.60

-25.34 -29.65 -33.08 -8.34 -4.55 -5.39 -6.62 -7.35 -6.25 -5.20 -2.91 -1.74 -0.60

-0.41 0.00 -2.18 0.00 0.00 -3.43 -7.15 -9.64 -9.58 -7.12 -3.28 0.00 0.00

1-0

-1.56

-1.56

0.00

0.00

0.00

0.00

From Figure 5.4, the coefficient of torsion resulted by NRHA is smaller when the inelastic degree is larger. This indicates that the extended N2 method will result larger 85

degree of conservatism in larger ductility. The value of coefficient of torsion in extended N2 method is relatively the same in all intensity because the RSA result which has the same result in arbitrary pga usually takes control. In the other hand, by giving weight factor described in 3rd modification, the proposed method reduces the level of conservatism. From Table 5.3, basic N2 usually underestimate the result, while the extended N2 overestimate it. Both extended N2 and proposed method give very good result in pga equal to 0.6g, which are 0% and 0.1% error for extended N2 and proposed method, respectively. When the pga equal to 1g, the maximum error of CT generated by basic N2 and extended N2 method is 3.15% and 6.2%, respectively. In the other hand the proposed method still gives good result which is zero error. This coefficient of torsion actually is not the final result of seismic demand. The coefficient of torsion should be multiplied by the displacement and drift at center of mass to define the displacement and drift at flexible edge. The comparison of displacement and drift at flexible edge are described in Figure 5.6 and Figure 5.7, respectively. Moreover, the comparisons of those errors are described in Table 5.5 and Table 5.6. From Figure 5.5, it is drawn that all of the method gives small error in the middle of the building. The proposed method becomes better than basic N2 and extended N2 method in the upper level of the building. From Table 5.4, the maximum absolute error of inter-storey drift produced by basic N2 method is 33.97% and 30.37% for pga equal to 0.6 and 1g, respectively. The maximum absolute error of inter-storey drift induced by extended N2 is 13.31% and 33.08% for pga equal to 0.6 and 1g, respectively. The maximum absolute error of inter-storey drift produced by proposed method is 5.37% and 9.64% for pga equal to 0.6 and 1g, respectively. The error produced by extended N2 method becomes larger in larger inelastic degree of a structure. This is caused by the assumption that the higher mode effect keep in elastic state. In the other hand, because the proposed method is made based on the real behavior of the building, the error generated is below 10% which is very small.

86

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0


0

15 30 45 60 Displacement (cm)

75

(a)

(b) Figure 5.6.Displacement result at flexible edge for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g

From Figure 5.6 and Table 5.5, it is shown that the error from basic N2 is reduced compare with the error produced in center of mass, while the extended N2 method produce larger error in flexible edge compare with the error produced in center of mass. It is because the coefficient of torsion (CT) value resulted by basic N2 is unconservative and the displacement at the CM is overestimated, hence make the result in flexible edge eventually become better. In the other side, both CT value and displacement at CM resulted by the extended N2 are overestimated, hence make the result in flexible edge become much more overestimated. The proposed method gives better result both in center of mass and in flexible edge because the CT value and displacement at center of mass resulted by proposed method have no error and small 87

error, respectively. From Table 5.5, the maximum absolute error of displacement at flexible edge produced by basic N2 method is 0% and 2.13% for pga equal to 0.6 and 1g, respectively. The maximum absolute error of displacement at flexible edge induced by extended N2 is 8.12% and 13.51% for pga equal to 0.6 and 1g, respectively. The maximum absolute error of displacement at flexible edge produced by proposed method is 0% and 0.83% for pga equal to 0.6 and 1g, respectively.

Table 5.5. The displacement error resulted from basic N2, extended N2, and proposed method at flexible edge Error percentage (%) Height (cm)

0.6g (=2.2) 1g (=3.7) Floor level basic N2 Extended Proposed basic N2 Extended Proposed N2 N2 method method method method method method

5200 4850 4500 4150 3800 3450 3100 2750 2400 2050 1700 1350 1000

14 13 12 11 10 9 8 7 6 5 4 3 2

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-4.13 -4.38 -4.66 -5.08 -5.81 -6.71 -7.34 -7.79 -8.10 -8.12 -7.56 -6.76 -5.91

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-0.32 -0.56 -1.03 -1.58 -2.02 -2.13 -1.63 -0.81 0.00 0.00 0.00 0.00 0.00

-11.40 -11.68 -12.18 -12.76 -13.30 -13.51 -13.05 -12.25 -11.19 -10.05 -8.96 -8.21 -7.53

0.00 0.00 0.00 0.00 -0.33 -0.83 -0.72 -0.26 0.00 0.00 0.00 0.00 0.00

500

1

0.00

-4.56

0.00

0.00

-6.21

0.00

88

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0


0

0.4

0.8 1.2 1.6 Drift (%)

2

2.4

(a) 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0


0

1

2 Drift (%)

3

4

(b) Figure 5.7.Inter-storey drift result at flexible edge for 14-storey 10%-eccentricity: (a) pga=0.6g; (b) pga=1g

From Figure 5.7, it is drawn that all of the method gives small error in the middle of the building. The proposed method becomes better than basic N2 and extended N2 method in the upper level of the building. From Table 5.6, the maximum absolute error of inter-storey drift produced by basic N2 method is 38.51% and 39.57% for pga equal to 0.6 and 1g, respectively. The maximum absolute error of inter-storey drift induced by extended N2 is 14.88% and 33.96% for pga equal to 0.6 and 1g, respectively. The maximum absolute error of inter-storey drift produced by proposed method is 5.71% and 10.73% for pga equal to 0.6 and 1g, respectively. The basic N2 produces large error both in pga equal to 0.6g and 1g. This phenomenon is explained in the 4th modification. The error produced by extended N2 method becomes larger in 89

larger inelastic degree of a structure. This is caused by the assumption that the higher mode effect keep in elastic state. In the other hand, because the proposed method is made based on the real behavior of the building, the maximum error generated is only 10% which is very small. Table 5.6. The inter-storey drift error resulted from basic N2, extended N2, and proposed method at flexible edge Error percentage (%) Height (cm)

0.6g (=2.2) 1g (=3.7) Floor level basic N2 Extended Proposed basic N2 Extended Proposed method N2 method method method N2 method method

5200-4850 4850-4500 4500-4150 4150-3800 3800-3450 3450-3100 3100-2750 2750-2400 2400-2050 2050-1700 1700-1350 1350-1000 1000-500 500-0

14-13 13-12 12-11 11-10 10-9 9-8 8-7 7-6 6-5 5-4 4-3 3-2 2-1 1-0

38.51 34.96 24.52 5.07 0.00 0.00 0.00 0.00 0.00 0.00 -0.68 -0.68 0.00 0.00

-9.09 -12.39 -14.88 -7.04 0.00 0.00 0.00 -1.07 -3.64 -7.53 -9.98 -10.01 -8.26 -4.56

0.00 1.33 0.00 0.00 0.00 0.00 0.00 0.00 -1.70 -4.99 -5.71 -1.74 0.00 0.00

90

39.57 32.73 18.82 0.00 0.00 0.00 0.00 -0.78 -2.87 -2.28 -0.04 0.00 0.00 0.00

-20.96 -27.16 -33.96 -11.62 -9.61 -11.21 -13.34 -15.04 -14.82 -14.13 -11.77 -10.68 -9.68 -6.21

0.00 0.00 0.00 0.00 0.00 -2.22 -6.63 -9.92 -10.73 -8.58 -4.73 0.00 0.00 0.00

6. CONCLUSION AND SUGGESTION 6.1. Conclusion Several main conclusions of this research are gathered as follows: 1. Compare with the conventional pushover, the proposed method does not need the iteration process. This is because the proposed method adopts the N2 method that use the inelastic spectra thus does not require the iteration process. 2. In the first modification, the effective mass factor (EMF) included in determining the lateral load pattern effectively contributes to give a better result of target displacement, particularly in high rise symmetric buildings. 3. In the second modification, the reduction factor toward the displacement which includes the torsional effective mass factor (TEMF) successfully gives a better result of the target displacement, particularly in the torsional dominated building. 4. By combining the 1st and 2nd modification, an asymmetric high rise building can be well predicted in target displacement. 5. In the third modification, the ductility or degree of inelasticity, 𝜇, is used as the weight factor towards the coefficient of torsion of the Pushover (PO) result and Response Spectra Analysis (RSA) result. Pushover (PO) result represents the nonlinear behavior, while Response Spectra Analysis (RSA) result represents the linear behavior. This third modification gives better result of the Coefficient of Torsion (CT) in reducing the level of conservatism caused by the elastic behavior assumption of the higher modes. 6. In the fourth modification, the multiplication factor towards the target displacement to get the maximum inter-storey drift can predict the actual inter-storey drift. This factor allows nonlinear behavior of higher modes. 7. The proposed method eliminates the assumption that the higher mode effects keep in the elastic behavior and hence reduces the level of either conservatism or unconservatism resulted by the elastic higher mode assumption.

91

6.2. Suggestion Based on this research, several suggestions to the future studies are proposed as follows: 1. Since this research only focuses on medium and long period structures, the future studies can observe the validity of this proposed method in short and very long period structures. 2. Include the effect of the strength and stiffness degradation by using the appropriate hysteretic models such as the Takeda model.

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REFERENCE ATC-40. (1996). Seismic Evaluation and Retrofit of Concrete Buildings. Washington, DC: Applied Technology Council. Causevic, M., & Mitrovic, S. (2011). Comparison between non-linear dynamic and static seismic analysis of structures according to European and US provisions. Bulletin

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10.1007/s10518-010-9199-1 Chopra, A. K., & Goel, R. K. (2002). A modal pushover analysis procedure for estimating seismic demands for buildings. Earthquake Engineering and Structural Dynamics, 31(3), 561-582. doi: 10.1002/eqe.144 Chopra, A. K., & Goel, R. K. (2004). A modal pushover analysis procedure to estimate seismic demands for unsymmetric-plan buildings. Earthquake Engineering & Structural Dynamics, 33(8), 903-927. doi: 10.1002/eqe.380 Chopra, A. K., Goel, R. K., & Chintanapakdee, C. (2004). Evaluation of a Modified MPA Procedure Assuming Higher Modes as Elastic to Estimate Seismic Demands. Earthquake Spectra, 20(3), 757-778. CSI. (2008). PERFORM-3D Nonlinear Analysis and Performance Assessment 3D Structures (Version 4.0.4). Berkeley, California: Computers and Structures, Inc. Fajfar, P. (1999). Capacity spectrum method based on inelastic demand spectra. Earthquake Engineering & Structural Dynamics, 28(9), 979-993. doi: 10.1002/(sici)1096-9845(199909)28:93.0.co;2-1 Fajfar, P. (2000). A Nonlinear Analysis Method for Performance-Based Seismic Design. Earthquake Spectra, 16(3), 573-592. FEMA-356. (2000). Pre standard and Commentary for the Seismic Rehabilitation of Buildings-FEMA 356. Washington, DC: Federal Emergency Management Agency. Hancock, J., Jennie Watson-Lamprey, Abrahamson, N. A., Bommer, J. J., Markatis, A., Mccoy, E., & Mendis, R. (2006). An Improved Method of Matching Response Spectra of Recorded Earthquake Ground Motion using Wavelets. Journal of Earthquake Engineering, 10(1), 1-23. Kreslin, M., & Fajfar, P. (2011). The extended N2 method considering higher mode 93

effects in both plan and elevation. Bulletin of Earthquake Engineering, 10(2), 695-715. doi: 10.1007/s10518-011-9319-6 Kunnath, S. K. (2004). Identification of modal combinations for nonlinear static analysis of building structures. Computer-Aided Civil and Infrastructure Engineering, 19(4), 246-259. doi: 10.1111/j.1467-8667.2004.00352.x Marus̆ ić, D., & Fajfar, P. (2005). On the inelastic seismic response of asymmetric buildings under bi-axial excitation. Earthquake Engineering & Structural Dynamics, 34(8), 943-963. doi: 10.1002/eqe.463 Ou, Y.-C. (2012). Seismic Resistant Design. Lecture note. Construction Engineering. National Taiwan University of Science and Technology. R. Rofooei, F., K. Attari, N., Rasekh, A., & Shodja, A. H. (2006). Comparison of Static and Dynamic Pushover Analysis in Assessment of the Target Displacement. [Research Paper]. International Journal of Civil Engineering, 4(3), 212-225. Reyes, J. C., & Chopra, A. K. (2011). Three-dimensional modal pushover analysis of buildings subjected to two components of ground motion, including its evaluation for tall buildings. Earthquake Engineering & Structural Dynamics, 40(7), 789-806. doi: 10.1002/eqe.1060 Rofooei, F. R., Attari, A., Rasekh, A., & Shodja, A. H. (2007). Adaptive Pushover Analysis. Asian Journal of Civil Engineering (Building and Housing), 8(2007), 343-358. The MathWorks, I. (2009). MATLAB The Language of Technical Computing (Version 7.8.0.347 (R2009a)): The MathWorks, Inc.

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APPENDIX A.1. MATLAB Code for extended N2 method for this research %% input by user n=101; aaa=linspace(0,1,n)'; for zzz=1:n pga=aaa(zzz); coef1=2.5; coef2=0.4; coef3=0.6; Sds=coef1*pga; % Sd1=pga/coef2*coef3; T0=0.1; Ts=Sd1/Sds; TL=3; Tc=0.6; %% Pick-up the data from txt file load 'building data.txt'; load 'PO output.txt'; load 'RSA result.txt'; load 'PO displacement history.txt'; stheight=building_data(:,1); height=cumsum(stheight); ads=building_data(:,2); mass=building_data(:,3); drift=PO_output(:,1); bshear=PO_output(:,2); cmRSA=RSA_result(:,1); f1RSA=RSA_result(:,2); f4RSA=RSA_result(:,3); %% Lateral force distribution & modal participation factor [a b]=size(mass); lfd=zeros(a,b); for i=1:a lfd(i,b)=ads(i,b).*mass(i,b)./mass(a,b); end Ln=mass.*ads; Mstar=mass.*ads.^2; MPF=sum(Ln)/sum(Mstar); %% Target displacement displ=drift*height(a); bshear=bshear*9.8145/1000; % convert kg to KN

95

maxdispl=max(displ); [c d]=size(bshear); area_po=zeros(c,d); area_po(1,1)=0; for i=2:c area_po(i,d)=area_po(i-1,d)+.5*(displ(i,d)-displ(i-1,d))*(bshear(i,d)-bshear(i-1,d))+(d ispl(i,d)-displ(i-1,d))*min(bshear(i,d),bshear(i-1,d)); end max_area_po=max(area_po); x_seed=rand(1,1)*maxdispl; x1=x_seed; diffx_60=1; while diffx_60>10^-10 y1=(max_area_po)/(maxdispl-.5*x1); x_60=0.6*x1; y_60=0.6*y1; cc=y_60-bshear; [e f]=size(cc); dd=zeros(e-1,1); for i=1:e-1 dd=cc(i,f)*cc(i+1,f); end [dd_sort ind]=sort(dd,'ascend'); ind1=ind(1,1); ind2=ind(1,1)+1; x_60_new=interpolate(bshear(ind1),displ(ind1),bshear(ind2),displ(ind2),y_60); diffx_60=abs(x_60-x_60_new); x1=1/0.6*x_60_new; end Dtstar=maxdispl/MPF; Dystar=x1/MPF; Fystar=y1/MPF; Fustar=Fystar; Tstar=2*pi*(sum(Ln)*Dystar*981.45/100/(Fystar*1000))^.5; Say=Fystar*1000/sum(Ln)/9.8145/981.45; if Tstar > 0 && Tstar < T0 Sae=(Sds-pga)/T0*Tstar+pga; elseif Tstar >= T0 && Tstar Ts && Tstar need check ii=1; while yield_displ>X2(ii,1) ii=ii+1; end pga_elastic = interpolate(X2(ii-1,1),Y2(ii-1,1),X2(ii,1),Y2(ii,1),yield_displ); X1mod=max(X2); ii=1; while X1mod>X1(ii,1) && X1(ii,1)