GENERALIZED PUSHOVER ANALYSIS A THESIS SUBMITTED TO …

i

GENERALIZED PUSHOVER ANALYSIS

A THESIS SUBMITTED TO

THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF

MIDDLE EAST TECHNICAL UNIVERSITY

BY

FIRAT SONER ALICI

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR

THE DEGREE OF MASTER OF SCIENCE

IN

CIVIL ENGINEERING

JUNE 2012

ii

Approval of the thesis:


submitted by FIRAT SONER ALICI in partial fulfillment of the requirements for the

degree of Master of Science in Civil Engineering Department, Middle East Technical

University by,

Prof. Dr. Canan Özgen _____________________

Dean, Graduate School of Natural and Applied Sciences

Prof. Dr. Güney Özcebe _____________________

Head of Department, Civil Engineering

Prof. Dr. Haluk Sucuoğlu _____________________

Supervisor, Civil Engineering Dept., METU

Examining Committee Members:

Prof. Dr. Polat Gülkan _____________________

Civil Engineering Dept., Çankaya University

Prof. Dr. Haluk Sucuoğlu _____________________

Civil Engineering Dept., METU

Prof. Dr. Ahmet Yakut _____________________


Assoc. Prof. Dr. Afşin Sarıtaş _____________________


Joseph Kubin, M.Sc. _____________________

Civil Engineer, PROTA

Date: 26.06.2012aaaaa

iii

I hereby declare that all information in this document has been obtained and

presented in accordance with academic rules and ethical conduct. I also declare

that, as required by these rules and conduct, I have fully cited and referenced

all material and results that are not original to this work.

Name, Last name: Fırat Soner ALICI

Signature : __________________

iv

ABSTRACT


Alıcı, Fırat Soner

M.Sc., Department of Civil Engineering

Supervisor: Prof. Dr. Haluk Sucuoğlu

June 2012, 136 pages

Nonlinear response history analysis is considered as the most accurate

analytical tool for estimating seismic response. However, there are several

shortcomings in the application of nonlinear response history analysis, resulting from

its complexity. Accordingly, simpler approximate nonlinear analysis procedures are

preferred in practice. These procedures are called nonlinear static analysis or

pushover analysis in general. The recently developed Generalized Pushover Analysis

(GPA) is one of them. In this thesis study, GPA is presented and evaluated

comparatively with the nonlinear time history analysis and modal pushover analysis.

A generalized pushover analysis procedure was developed for estimating the

inelastic seismic response of structures under earthquake ground excitations

(Sucuoğlu and Günay, 2011). In this procedure, different load vectors are applied

separately to the structure in the incremental form until the predefined seismic

demand is obtained for each force vector. These force vectors are named as

generalized force vectors. A generalized force vector is a combination of modal

forces, and simulates the instantaneous force distribution on the system when a given

response parameter reaches its maximum value during the dynamic response. In this

method, the maximum interstory drift parameters are selected as target demand

v

parameters and used for the derivation of generalized force vectors. The maximum

value of any other response parameter is then obtained from the analysis results of

each generalized force vector. In this way, this procedure does do not suffer from the

statistical combination of inelastic modal responses.

It is further shown in this study that the results obtained by using the mean

spectrum of a set of ground motions are almost identical to the mean of the results

obtained from separate generalized pushover analyses under each ground motion in

the set. These results are also very close to the mean results of nonlinear response

history analyses.

A practical implementation of the proposed generalized pushover analysis is

also developed in this thesis study where the number of pushovers is reduced in view

of the number of significant modes contributing to seismic response. It has been

demonstrated that the reduced generalized pushover analysis is equally successful in

estimating maximum member deformations and member forces as the full GPA

under a ground excitation, and sufficiently accurate with reference to nonlinear

response history analysis.

Keywords: Pushover Analysis, Generalized Force Vectors, Target Drift, Higher

Mode Effects.

vi

ÖZ

GENELLEŞTİRİLMİŞ İTME ANALİZİ

Alıcı, Fırat Soner

Yüksek Lisans, İnşaat Mühendisliği Bölümü

Tez Yöneticisi: Prof. Dr. Haluk Sucuoğlu

Haziran 2012, 136 sayfa

Zaman tanım alanında doğrusal olmayan dinamik analiz sismik davranışın

tahmini için en doğru analitik araç olarak kabul edilir. Bununla birlikte, zaman tanım

alanında doğrusal olmayan dinamik analiz uygulamasında uygulama

karmaşıklığından kaynaklanan çeşitli sorunlar vardır. Bu nedenle uygulamada daha

basit fakat yaklaşık doğrusal olmayan analiz yöntemleri tercih edilmektedir. Bu

yöntemler genel olarak doğrusal olmayan statik analiz veya itme analizi olarak

adlandırılır. Yakın zamanda geliştirilen Genelleştirilmiş İtme Analizi de bunlardan

biridir. Bu tez çalışmasında Genelleştirilmiş İtme Analizi formülasyonu ve

uygulaması sunulmakta ve zaman tanım alanında doğrusal olmayan analiz ve modal

itme analizi ile karşılaştırmalı olarak değerlendirilmektedir.

Genelleştirilmiş itme analizi yöntemi, deprem etkisi altında yapıların doğrusal

olmayan sismik tepkilerini tahmin etmek için geliştirilmiştir (Sucuoğlu ve Günay,

2011). Bu yöntemde, farklı yük vektörleri adım adım artan biçimde önceden

tanımlanmış sismik talebe ulaşıncaya kadar yapıya uygulanır. Bu kuvvet vektörleri

genelleştirilmiş kuvvet vektörleri olarak adlandırılır. Genelleştirilmiş kuvvet vektörü

modal kuvvetlerin kombinasyonudur ve hedef tepki parametresi dinamik davranış

sırasında en yüksek değerine ulaştığında sistem üzerinde bulunan anlık kuvvet

vii

vektörünü temsil eder. Bu metodda, maksimum kat arası ötelenme parametreleri

hedef talep paremetresi olarak seçilir ve genelleştirilmiş kuvvet vektörlerinin

türetilmesinde kullanılır. Daha sonra iç kuvvetler ve deformasyonların maksimum

değeri farklı genelleştirilmiş itme analizlerinin sonuçlarından elde edilir. Böylece, bu

yöntem doğrusal olmayan modal tepkilerin istatiksel kombinasyonlarından zarar

görmemektedir.

Bu çalışmada ayrıca bir grup yer hareketinin ortalamalama spektrumu

kullanılarak elde edilen sonuçların; gruptaki her bir yer hareketi altında ayrı ayrı

yapılan genelleştirilmiş itme analizlerinden elde edilen sonuçların ortalaması ile

hemen hemen aynı olduğu gösterilmektedir. Bu sonuçlar aynı zamanda, zaman tanım

alanında doğrusal olmayan tepki analizi sonuçlarının ortalamalarına çok yakındır.

Sismik tepkiye katkı yapan etkin modların sayısı göz önünde bulundurularak

itme analizi sayısının azaltılması yoluyla elde edilen genelleştirilmiş itme analizinin

pratik uygulaması da bu tez çalısmasında geliştirilmiştir. Genelleştirilmiş itme

analizinin pratik uygulaması, bir yer hareketi altında maksimum eleman

deformasyonları ve eleman kuvvetlerinin tahmininde kapsamlı olarak uygulanan

genelleştirilmiş itme analizi ile aynı başarıyı göstermekte ve zaman tanım alanında

doğrusal olmayan dinamik analiz ile yeterince tutarlı sonuçlar vermektedir.

Anahtar Kelimeler: İtme Analizi, Genelleştirilmiş Kuvvet Vektörleri, Hedef

Öteleme, Yüksek Mod Etkileri.

viii

ACKNOWLEDGEMENTS

I would like to thank gratefully Prof. Dr. Haluk SUCUOĞLU due to his

enthusiastic supervision during my thesis study. This study would be meaningless

without his inspiration and his encouragement. During my thesis study, he showed

great self-sacrifice, and he provided sound advices, and good teaching. I would like

to express my sincere thanks to him.

I would like to thank sincerely my beloved family. Their constant love and

supports are great encouragement in all my life. Their endless courage and

conviction will always inspire me. Their support and patience are thankfully

acknowledged.

I also want to thank my office mates; M. Başar MUTLU, Kaan KAATSIZ,

M. Can YÜCEL, Ahmet KUŞYILMAZ, Alper Ö. GÜR, and Sadun TANIŞER. I will

always remember with pleasure.

My special thanks go to my M. Başar MUTLU for his friendship, support and

for helping me get through difficult times.

ix

TABLE OF CONTENTS

ABSTRACT ................................................................................................................ iv

ÖZ ............................................................................................................................... vi

ACKNOWLEDGMENTS ........................................................................................ viii

TABLE OF CONTENTS ............................................................................................ ix

LIST OF TABLES ..................................................................................................... xii

LIST OF FIGURES .................................................................................................. xiii

CHAPTERS

1 INTRODUCTION .................................................................................................. 1

1.1 Problem Statement ..................................................................................... 1

1.2 Review of Past Studies .............................................................................. 2

1.2.1 Conventional Pushover Analysis ........................................................ 2

1.2.2 Non-adaptive and Adaptive Multi-Mode Pushover Analyses ............ 5

1.2.3 Recent Developments in Nonlinear Static Analysis ......................... 10

1.3 Objective and Scope ................................................................................ 12

2 GENERALIZED PUSHOVER ANALYSIS PROCEDURE ............................... 14

2.1 Generalized Force Vectors....................................................................... 14

2.2 Target Interstory Drift Demand ............................................................... 17

2.3 Generalized Pushover Algorithm............................................................. 18

3 GROUND MOTIONS EMPLOYED IN CASE STUDIES .................................. 20

4 CASE STUDIES ................................................................................................... 24

4.1 Case Study I: Twelve Story RC Frame with Full Capacity Design......... 24

4.1.1 Building Description ........................................................................ 24

4.1.2 Modeling .......................................................................................... 27

4.1.3 Free Vibration Properties ................................................................. 27

x

4.1.4 Presentation of Results ..................................................................... 28

4.1.4.1 Interstory Drift Ratios ................................................................ 29

4.1.4.2 Member End Rotations ............................................................... 30

4.1.4.3 Member Internal Forces ............................................................. 36

4.2 Case Study II: Twelve Story RC Frame with Relaxed Capacity Design . 42


4.2.2 Modeling .......................................................................................... 45






4.3 Case Study III: Twenty Story RC Wall-Frame System with Full Capacity

Design ................................................................................................................ 58


4.3.2 Modeling .......................................................................................... 62






5 GENERALIZED PUSHOVER ANALYSIS WITH THE MEAN SPECTRUM OF

A SET OF GROUND MOTIONS ............................................................................. 91

5.1 Case Study I: Twelve Story RC Frame with Full Capacity Design......... 91

5.2 Case Study II: Twelve Story RC Frame with Relaxed Capacity Design . 95

5.3 Case Study III: Twenty Story RC Wall-Frame System with Full Capacity

Design ................................................................................................................ 98

xi

6 PRACTICAL IMPLEMENTATION OF GENERALIZED PUSHOVER

ANALYSIS .............................................................................................................. 106

6.1 Reduced Generalized Pushover Analysis Procedure ............................. 106

6.2 Comparative Results from Case Studies ................................................ 111

6.2.1 Case Study I: Twelve Story RC Frame with Full Capacity Design 111

6.2.2 Case Study II: Twelve Story RC Frame with Relaxed Capacity

Design... ....................................................................................................... 115

6.2.3 Case Study III: Twenty Story RC Wall-Frame System with Full

Capacity Design ........................................................................................... 121

7 SUMMARY AND CONCLUSIONS ................................................................. 130

7.1 Summary ................................................................................................ 130

7.2 Conclusions ............................................................................................ 131

REFERENCES ......................................................................................................... 133

xii

LIST OF TABLES

TABLES

Table 3.1 Reference ground motion properties .......................................................... 21 Table 4.1 Shear design details of members ................................................................ 27

Table 4.2 Free vibration properties of the twelve story plane model for the first three

modes ......................................................................................................................... 28

Table 4.3 Negative yield moment values for beam ends (tension at top) .................. 36


modes ......................................................................................................................... 45

Table 4.5 Shear design details of columns and beams ............................................... 61

Table 4.6 Free vibration properties of the twenty story plane model for the first four

modes ......................................................................................................................... 63

Table 4.7 Negative yield moment values for beam ends (tension at top) .................. 75

xiii

LIST OF FIGURES

FIGURES

Figure 3.1 Acceleration time histories of the synthetic ground motions ................... 22

Figure 3.2 Acceleration response spectra of synthetic ground motions, mean

acceleration spectrum, and TEC2007 design spectrum ............................................. 23

Figure 4.1 Story plan of the twelve story building with full capacity design ............ 25

Figure 4.2 Plane (2D) model of the twelve story building with full capacity design 25

Figure 4.3 Column and beam section details ............................................................. 26

Figure 4.4 Mode shapes for the first three modes ...................................................... 28

Figure 4.5 Maximum interstory drift ratios obtained under seven ground motions .. 29

Figure 4.6 Maximum values of average beam end plastic rotations under seven

ground motions .......................................................................................................... 31

Figure 4.7 Beam end plastic rotations when bending moments at beam ends reach

their maximum ........................................................................................................... 33

Figure 4.8 Maximum values of average column-end chord rotations under seven

ground motions with yield rotation limits .................................................................. 35

Figure 4.9 Maximum bending moment values of beam ends at 1st, 5

th, and 10

th stories

.................................................................................................................................... 37

Figure 4.10 Maximum shear force values of beam ends at 1st, 5

th, and 10

th stories .. 38

Figure 4.11 Maximum bottom end bending moments along the left exterior column

axis of the inner frame with yield moments ............................................................... 40

Figure 4.12 Maximum shear forces along the left exterior column axis of the inner

frame .......................................................................................................................... 41

Figure 4.13 Plane (2D) model of the twelve story building with relaxed capacity

design ......................................................................................................................... 43

Figure 4.14 Column and beam section details ........................................................... 44

Figure 4.15 Mode shapes for the first three modes .................................................... 46

Figure 4.16 Maximum interstory drift ratios under seven ground motions ............... 47

xiv


ground motions .......................................................................................................... 49

Figure 4.18 Maximum values of average column end chord rotations under seven

ground motions with yield rotation limits .................................................................. 50


th, and 10

th

stories ......................................................................................................................... 52


th, and 10

th stories .. 54


line of the inner frame with yield moments ............................................................... 56

Figure 4.22 Maximum shear forces along the left exterior column line of the inner

frame .......................................................................................................................... 57

Figure 4.23 Story plan of the twenty story building with full capacity design .......... 59

Figure 4.24 Plane (2D) model of the twenty story building with full capacity design

.................................................................................................................................... 59

Figure 4.25 Column and beam section details ........................................................... 60

Figure 4.26 Shear wall section details........................................................................ 61

Figure 4.27 Mode shapes for the first four modes ..................................................... 63

Figure 4.28 Maximum interstory drift ratios obtained under seven ground motions 64

Figure 4.29 Maximum values of average beam-end plastic rotations in Frame A

under seven ground motions ...................................................................................... 67

Figure 4.30 Maximum values of average beam-end plastic rotations in Frame B .... 68

Figure 4.31 Maximum values of average column-end chord rotations in Frame A


Figure 4.32 Maximum values of average column-end chord rotations in Frame B


Figure 4.33 Maximum values of average plastic rotations of the shear wall in Frame

A under seven ground motions .................................................................................. 73


th, and 10

th

stories in Frame A ...................................................................................................... 75


th, and 10

th

stories in Frame B ...................................................................................................... 77


th, and 10

th stories in

Frame A ...................................................................................................................... 79

xv


th, and 10

th stories in

Frame B ...................................................................................................................... 80


axis in Frame A .......................................................................................................... 83


axis in Frame B .......................................................................................................... 84

Figure 4.40 Maximum shear forces along the left exterior column axis in Frame A 85

Figure 4.41 Maximum shear forces along the left exterior column axis in Frame B 87

Figure 4.42 Maximum bending moment values along shear wall in Frame A,

(My=7800 kN.m in Hcr) .............................................................................................. 89

Figure 5.1 Comparison of mean maximum interstory drifts, mean maximum average

plastic rotations of beam ends and mean maximum average chord rotations of

column ends obtained with NRHA, MPA and GPA, with the GPA results obtained

under mean elastic spectrum. ..................................................................................... 92

Figure 5.2 Comparison of maximum shear forces and maximum bottom end bending

moments along the left exterior column axis of the inner frame ............................... 93

Figure 5.3 Comparison of maximum bending moment values of beam ends at the 1st,

5th

, and 10th

stories ..................................................................................................... 93

Figure 5.4 Comparison of maximum shear force values of beam ends at the 1st, 5

th,

and 10th

stories ........................................................................................................... 94

Figure 5.5 Comparison of mean maximum interstory drifts, mean maximum average

plastic rotations of beam ends and mean maximum average chord rotations of

column ends obtained with NRHA, MPA and GPA, with the GPA results obtained

under mean elastic spectrum. ..................................................................................... 96

Figure 5.6 Comparison of maximum shear forces and maximum bottom end bending

moments along the left exterior column axis of the inner frame ............................... 97

Figure 5.7 Comparison of maximum bending moment values of beam ends at the 1st,

5th

, and 10th

stories ..................................................................................................... 97


th,

and 10th

stories ........................................................................................................... 98

Figure 5.9 Comparison of the mean maximum interstory drifts and the mean

maximum average plastic rotations of the shear wall obtained with NRHA, MPA and

xvi

GPA under seven ground motions, with the GPA results obtained under the mean

elastic spectrum of seven ground motions. ................................................................ 99

Figure 5.10 Comparison of the mean maximum average plastic rotations of beam

ends and the mean maximum average chord rotations of column ends in Frame A

obtained with NRHA, MPA and GPA under seven ground motions, with the GPA

results obtained under the mean elastic spectrum of seven ground motions. .......... 100

Figure 5.11 Comparison of the mean maximum average plastic rotations of beam

ends and the mean maximum average chord rotations of column ends in Frame B

obtained with NRHA, MPA and GPA under seven ground motions, with the GPA

results obtained under the mean elastic spectrum of seven ground motions. .......... 100

Figure 5.12 Comparison of maximum shear forces and maximum bottom end

bending moments along the left exterior column axis in Frame A .......................... 101


bending moments along the left exterior column axis in Frame B .......................... 102

Figure 5.14 Comparison of maximum bending moment values of beam ends at the

1st, 5

th, and 10

th stories in Frame A .......................................................................... 102

Figure 5.15 Comparison of maximum bending moment values of beam ends at the

1st, 5

th, and 10

th stories in Frame B ........................................................................... 103


th,

and 10th

stories in Frame A ...................................................................................... 103


th,

and 10th

stories in Frame B ...................................................................................... 104

Figure 5.18 Comparison of maximum bending moment values along the shear wall in

Frame A (My=7800 kN.m in Hcr) ............................................................................. 105

Figure 6.1 Drift profile for the first three modes under GM4 .................................. 107

Figure 6.2 Combinations of the first two modes contributing to interstory drift under

GM4 ......................................................................................................................... 107

Figure 6.3 Combinations of the first three modes contributing to interstory drift under

GM4 ......................................................................................................................... 108

Figure 6.4 ETH interstory drift profiles at each story maxima and the related

combination of the first three modes contributing to interstory drift under GM4 ... 109

Figure 6.5 Comparison of SRSS drift profile and combinations of the scaled first

three modal drifts ..................................................................................................... 110

xvii

Figure 6.6 Comparison of maximum interstory drift ratios obtained under seven

ground motions ........................................................................................................ 111

Figure 6.7 Comparison of maximum values of average beam-end plastic rotations

under seven ground motions .................................................................................... 112

Figure 6.8 Comparison of maximum values of average column-end chord rotations


Figure 6.9 Positive drift profile for the first mode, and both positive and negative

drift profiles for the second and third modes under GM4 ........................................ 116

Figure 6.10 Combinations of the first three modes contributing to interstory drifts

under GM4 ............................................................................................................... 116


ground motions ........................................................................................................ 117

Figure 6.12 Comparison of maximum values of average beam-end plastic rotations





drift profiles for the second and third modes under GM4 ........................................ 122

Figure 6.15 Combinations of the first three modes contributing to interstory drift

under GM4 ............................................................................................................... 122


ground motions ........................................................................................................ 123

Figure 6.17 Comparison of maximum values of average beam-end plastic rotations in

Frame A under seven ground motions ..................................................................... 124

Figure 6.18 Comparison of maximum values of average beam-end plastic rotations in

Frame B under seven ground motions ..................................................................... 126


in Frame A under seven ground motions ................................................................. 127


in Frame B under seven ground motions ................................................................. 128

1

CHAPTER 1

INTRODUCTION

1.1 Problem Statement

Nonlinear analysis procedures for determining the seismic performance of

structures can be classified as nonlinear dynamic analysis (nonlinear response

history) and nonlinear static analysis (pushover). Among these, nonlinear dynamic

analysis is accepted as the procedure which provides the most accurate results. This

procedure is very rigorous and includes numerical sophistication. Therefore

nonlinear dynamic analysis is generally used in academic research, but it is not

commonly used in professional civil engineering practice. In addition, this analysis

procedure contains some shortcomings such as stability and convergence. On the

other hand, nonlinear static analysis (pushover analysis) is an alternative procedure

used commonly in earthquake engineering due to the ease of application and

practicality. This procedure depends on the application of a lateral static force vector

to the structure until the desired target response is attained. There are several

alternative ways, but the most common one is the first mode pushover analysis

(conventional pushover analysis). In this method, the force vector is the first mode

force vector which is obtained from the first mode shape of the structure through

classical structural dynamics theory. This method is generally preferable for

structures for which the fundamental mode dominates the deformation shape.

However when the contribution of higher mode effects are significant, this

conventional nonlinear static procedure is not capable of predicting the inelastic

response of structures. Therefore, multi-mode pushover analyses are used to consider

higher mode effects. These procedures can be grouped into two, as adaptive and non-

adaptive. In multi-mode pushover analysis, more than one pushover analysis is

2

conducted for each mode separately, and the modal demands are obtained from the

capacity spectrum method or inelastic SDOF response history analysis of the

associated mode. Finally, the obtained response quantities are combined with

statistical methods. Multi mode pushover analysis is quite simple if it is not adaptive.

Adaptive lateral force schemes are used to overcome the limitations of multi-mode

pushover analysis, but it requires repeating the eigenvalue analysis at every load step

which is onerous. The use of these methods has some shortcomings for calculating

inelastic responses, especially in estimating the member internal forces.

A newly developed pushover analysis, the generalized pushover analysis is

introduced in this study. It considers the contribution of all significant modes to

inelastic response. This method includes conducting several pushover analyses by

using different force vectors, similar to the multi-mode pushover analysis. However,

the main differences of this method from the other multi-mode analysis are the

formulation of force vectors and the use of story drifts as target response parameters.

Each generalized force vector is generated as a combination of modal lateral force

vectors, and these forces simulate the force distribution when the selected story

reaches the maximum interstory drift during the dynamic seismic response.

Therefore, the response parameters are obtained from the pushover results directly,

without use of any statistical modal combination methods. Hence, higher mode

effects are considered effectively, and the application of the algorithm is more

practical.

1.2 Review of Past Studies

Review of past studies is presented here in three parts. In the first part, studies

on the conventional (first mode) pushover analysis are reviewed. In the second part,

studies on non-adaptive and adaptive multi-mode pushover analysis are reviewed. In

the final section, different studies conducted recently on nonlinear static analysis are

presented.

1.2.1 Conventional Pushover Analysis

The idea of nonlinear static analysis (or pushover) was firstly introduced by

Freeman et al. (1975) and Saiidi and Sözen (1981) in order to determine the inelastic

3

force-deformation characteristics and nonlinear response of equivalent SDOF

oscillators. Capacity spectrum method was developed by Freeman et al. (1975) in

order to determine seismic deformation demands under an earthquake ground

excitation or earthquake design spectrum. This was originally a rapid evaluation

procedure for a pilot seismic risk project. Saiidi and Sözen (1981) developed the so

called Q-model for estimating the lateral capacity of a structure. The force-

deformation properties used in this model contains the variation of top story

displacement with the overturning moment under the application of monotonically

increasing lateral force vectors with triangular distribution along height. The

moment-curvature relationship of the individual elements constitutes the variation of

these two parameters. Q-model was modified and utilized to the vertically irregular

buildings by Saiidi and Hudson (1982), Moehle (1984), Moehle and Alarcon (1986).

Fajfar and Fischinger (1987) developed the N2 method, which is more

comprehensive than the Q-model. This extended method includes four main steps. In

the first step, the capacity curve of the considered MDOF system, including the

stiffness, strength and ductility characteristics, is obtained by nonlinear static

analysis under a monotonically increasing lateral force vector. In the following step,

the obtained capacity curve is converted to the equivalent SDOF system. In the third

step, the displacement demand of the SDOF system is determined by using the

nonlinear response history analysis or response spectrum analysis of the inelastic

SDOF system, and then this displacement demand is converted to the top story

displacement of the MDOF system in the final step. Hence, the structural responses

are obtained at the pushover step corresponding to the pre-determined top story

displacement demand. Therefore, it can be suggested that this procedure is applicable

for structures where the fundamental mode dominates inelastic response.

The steps of N2 method are the main steps involved in the seismic assessment

methods called nonlinear static procedure (NSP) in FEMA-356 (2000). The shape of

the lateral force vector may be identified for different versions of the first step (ATC-

40, 1996). Fajfar and Gaspersic (1996) defined a different lateral force vector pattern

obtained from the proportion of the mass matrix multiplied by the assumed

displacement shape vector, considering the post yield mechanism of the structure.

There are different methods for converting the capacity curve to an

equivalent SDOF system. Fajfar and Gaspersic (1996) and Krawinkler and

Seneviratna (1998) used an assumed displacement shape for the conversion, while

4

the first modal mass and participation factor are used in ATC-40. These two different

approaches are equivalent to each other if the first mode shape is employed as the

normalized displacement shape.

The capacity spectrum method (Freeman et al., 1975; ATC-40, 1996) and the

coefficient method (FEMA-356, 2000; FEMA-440, 2005) can be used to obtain the

maximum displacement demand of SDOF system. Both of these methods are

approximate methods to determine the displacement demand and have some

shortcomings. The studies by Chopra and Goel (2000) and Miranda and Akkar

(2002) have mentioned these drawbacks. More accurate methods to determine the

maximum displacement demand of equivalent SDOF systems is nonlinear dynamic

analysis under earthquake ground excitation. Nonlinear analysis of SDOF systems is

simpler when compared to the nonlinear analysis of MDOF systems. On the other

hand, inelastic spectra of ground motions developed in terms of R-μ-T relations can

also be utilized to obtain the displacement demand of equivalent SDOF systems

(Fajfar and Gaspersic, 1996; Fajfar, 2000).

Krawinkler and Seneviranta (1998) investigated the applicability of nonlinear

static analysis (pushover) procedure for the seismic evaluation of existing structures.

According to this study, pushover analysis makes a good estimation of global and

local inelastic deformation demands for structures vibrating in the fundamental

mode. Furthermore, this analysis also expresses the design weaknesses such as story

mechanisms, excessive deformation demands, strength irregularities and overloads

on potentially brittle elements. On the other hand, they also mentioned that the lateral

load pattern used in the analysis is valid if the structural deformation response is not

influenced from the higher mode effects. They also expressed that pushover analysis

may detect only the first local mechanism generated during an earthquake and may

not reflect the other weaknesses that is developed when the dynamic properties of

structures change during actual dynamic response after the formation of the first

local mechanism. By comparing the obtained results, they concluded that pushover

analysis can be utilized for all structures if higher mode effects are not significant.

Tso and Moghadam (1998) discussed pushover analysis procedure for

determining the seismic response of buildings. They expressed significant parameters

for the accuracy of pushover analysis such as mathematical modeling of the

structure, lateral load distribution and target displacement determination. They

concluded that pushover analysis procedure is suitable for regular buildings when the

5

building responds in a single mode, and this procedure provides valuable information

about the inelastic performance of a building during an earthquake.

Mawfy and Elnashai (2001) stated that due to complexity and unsuitability

for practical design application of inelastic dynamic analysis, pushover analysis is a

simple option for estimating the strength capacity in the post-elastic range. They

employed pushover analysis under selected representative ground motions with

different load patterns (the code design lateral pattern, uniform distribution and the

force distribution obtained by combining the external modal forces with SRSS).

They compared the calculated capacity curves with the curves obtained by applying

the ground motions with incrementally increased intensities. In addition, they also

stated that the obtained results are sensitive to the modeling of the structure, selection

of the lateral load pattern, higher mode effects and ground motion record. They

commented that conventional pushover analysis is more suitable for low rise and

short period frame structures, and the difference between the nonlinear static analysis

and the nonlinear dynamic analysis for long period structures can be overcome by

using more than one lateral load pattern.

Bracci et al. (1997) improved pushover analysis by using adaptive force

distributions instead of predetermined distributions. They stated that the capability of

pushover analysis is not sufficient to obtain the inelastic response for the structures

failing from mid-story mechanism, and extended the capacity spectrum methods to

include the effects of potential mid-story mechanisms and story-by-story

performance evaluation using modal superposition. The improved method in their

study includes modification of the lateral forces at each step by considering the story

shear forces at the previous step. They also mentioned that strain rate effects and

system degradation and deterioration cannot be obtained from pushover analysis due

to the static application of lateral loads, which is obvious.

1.2.2 Non-adaptive and Adaptive Multi-Mode Pushover Analyses

Several studies are conducted to improve the conventional (single mode)

pushover analysis by considering the higher mode effects. Considering the simplicity

and conceptual appeal of nonlinear static analysis, multi-mode pushover analysis has

been developed by several researchers. Sasaki et al. (1998) proposed the multi-mode

pushover procedure where a pushover analysis is conducted for each elastic mode

6

separately. They used capacity spectrum method for the modal demand calculations.

They concluded that multi-mode pushover procedure results may more closely match

the actual damage state. However, they did not state any combination procedure for

the obtained modal responses.

Chopra and Goel (2002) improved the multi-mode pushover procedure to

develop the modal pushover analysis. In modal pushover analysis, a pushover

analysis is conducted for each mode separately, similar to the multi-mode pushover

procedure. Inelastic modal demands are calculated by nonlinear response history

analysis of the equivalent SDOF systems. These systems are represented by the

associated modal capacity curves. Finally, the obtained modal response quantities are

combined with SRSS or CQC as in the response spectrum analysis. Hence, in case of

linear elastic response, modal pushover analysis becomes similar to the response

spectrum analysis. They concluded that this improved procedure estimates the

seismic performances of buildings at lower performance levels such as life safety and

collapse prevention. In addition to this, they stated that the procedure is not a good

indicator for member deformation levels. Chopra and Goel (2004) extended the

modal pushover procedure to the unsymmetrical plan buildings. Implementation of

modal pushover analysis is simple and does not require any special programming,

because it is not adaptive. However, the assumption of independent inelastic

response at each mode causes inelastic internal forces exceeding the capacities.

Goel and Chopra (2005) developed a correction procedure for modal

pushover analysis in order to calculate the internal forces that are compatible with the

capacities. Member forces are calculated from the standard modal pushover analysis

in order to obtain the member deformations, and compared with the member

capacities. If the obtained force exceeds the capacity, it is recomputed from the

modal pushover analysis estimation of member deformations by using the member

force-deformation relationships. They observed that this improved procedure gives

good estimates for member deformations for buildings that are not deformed far into

the inelastic range with large degradation in lateral stiffness.

Poursha et al. (2009) proposed the consecutive modal pushover analysis, for

taking into account the higher-mode effects in the pushover analysis of tall buildings

and to improve the estimation of seismic demands, especially for plastic rotations at

member ends. This procedure consists of multi-stage and single-stage pushover

analyses. Multi-stage pushover analysis is carried out in two or three stages

7

depending on the first period of the analyzed structure. The implementation of each

modal pushover stage uses the related modal static force vector in the incremental

form until the roof displacement is equal to the target displacement of the related

stage, which is obtained by multiplying the modal mass participation ratio, αn and the

target displacement of the roof, δt. In multi-stage pushover analysis, initial conditions

of the second stage or third stage are the same as the state at the last step of analysis

in the previous stage. The single-stage pushover analysis is performed with an

inverted triangular load distribution for medium-rise buildings and a uniform lateral

load distribution for high-rise buildings. Then, the final results are obtained by

enveloping the responses of the multi-stage and single stage pushover analyses. After

application of the procedure on four special steel moment-resisting frames, they

observed that the story drifts at upper floors and the plastic rotations at mid and

upper floors are more accurately estimated by the consecutive modal pushover

analysis compared to the modal pushover analysis.

Gupta and Kunnath (2000) developed a procedure which considers the

adaptive lateral load pattern and higher mode effects. By using the instrumented

buildings which experienced strong ground motions during the 1994 Northridge

earthquake, they stated that conventional pushover analysis has limitations to reflect

the seismic response of buildings. They observed the inertia force distributions at the

times of maximum displacement, maximum drifts, maximum base shear and

maximum overturning moments. Then they expressed that higher modes affect the

seismic responses for mid- and high-rise buildings, and these effects are better

reflected by utilizing the inertial force and story drift profiles instead of the

displacement profile. In the developed formulation, response spectrum analysis is

conducted at each step of pushover analysis instead of conventional pushover

analysis by using the modal properties obtained from the eigenvalue analysis of the

structure at the related pushover step. After the application of this method, they

included story drifts and plastic hinge locations as the response parameters. They

pointed out that the uniform and FEMA-273 (ATC, 1997) modal load patterns were

inadequate in capturing the seismic response of high-rise buildings, whereas the

proposed method resulted in predictions close to nonlinear response history analysis.

However, the calculation of target seismic demands was not specified. More

recently, Kalkan and Kunnath (2006) improved the procedure to estimate target

displacements for this method. In the proposed new procedure, a displacement-

8

controlled method is used in which the demand is computed by individual adaptive

pushover analyses utilizing the inertia distribution of each mode which is updated

during the application of loading. The adaptive scheme of Gupta and Kunnath (2000)

has a basis that the contributions of each mode are combined at the end of each step

with SRSS; however in the proposed scheme the total seismic demand is obtained at

the end of each pushover analyses by using the SRSS combination of individual

responses. They stated that by combining the contributions of sufficient number of

modes, estimation of the responses is generally similar to the benchmark results

obtained from nonlinear time history analysis.

Aydınoğlu (2003) found that elastic pseudo accelerations corresponding to

the instantaneous periods in Gupta and Kunnath’s method are not suitable to reflect

the inelastic instantaneous response of the system, and modified their procedure by

considering inelastic spectral displacements at the instantaneous state of the system

instead of the elastic pseudo accelerations. He expressed that use of inelastic spectral

displacements provided applicable estimation of the peak response quantities at the

last step of pushover analysis where the analysis ends upon a control process.

Therefore, the top story displacement obtained at the last step is the global

displacement demand. He showed that the obtained results were sufficiently close to

the nonlinear response history analysis results, but the procedure required

considerable computational work. Aydınoğlu also developed a practical version of

his method for smooth elastic response spectrum by utilizing the equal displacement

rule principle for calculating modal demands.

Antoniou and Pinho (2004a) developed a procedure which is similar to that of

Gupta and Kunnath (2000) to take into account higher mode effects and variant force

distributions. They expressed that equilibrium in the Gupta and Kunnath method

cannot be satisfied at each step of pushover analysis due to use of combination rules

(SRSS or CQC) for modal response parameters. On the other hand, Antoniou and

Pinho combined the external modal forces with modal combination rules at each step

of pushover analysis in order to compute the lateral load distribution to be applied at

the corresponding step. Alternatively, they combined the modal static forces. These

methods were called as “with spectral amplification” and “without spectral

amplification” respectively. They stated that the combination of modal external

forces should be conducted by vectorial summation to obtain a better match of

nonlinear dynamic response. In addition to “with spectral amplification” and

9

“without spectral amplification” cases, they also utilized “total updating” and

“incremental updating” alternatives for the lateral forces applied at each step. In the

“total updating” alternative, the lateral force computed as mentioned above is applied

as the external force, whereas in the “incremental updating”, it is added to the

external force with proper scaling at the previous step. They noted that the method

“with spectral amplification” and “incremental updating” is preferable when

considering the numerical stability and the accuracy of the results. After evaluation

of the proposed method with regular, irregular and dual system using 2D models and

four ground motions, they concluded that the proposed adaptive pushover method

provides a relatively minor advantage over the conventional pushover analysis (using

uniform and triangular distributions) when compared with the nonlinear dynamic

analysis results.

In the other studies of Antoniou and Pinho (2004b), they developed a

displacement-based adaptive pushover procedure in which a displacement vector is

updated according to the stiffness of the structure and imposed to the structure at

each step of the analysis. The displacement vector is combined by employing the

combination rules (SRSS or CQC) in order to include all significant modes. Similar

to the companion paper (2004a), “with spectral amplification”,” without spectral

amplification”, “incremental updating” and “total updating” alternatives were

mentioned in this study. Similarly, a global displacement demand is not computed in

this method. They used the same models and the ground motions as in the companion

paper. They concluded that the proposed method improved the estimation of the

responses throughout the deformation range in comparison to the conventional

pushover analysis.

All multi-mode pushover analyses in the literature have two common

disadvantages. First, they are adaptive except MPA. Adaptive algorithms require

eigenvalue analysis at each loading increment which requires significant

computational work. Besides, adaptive algorithms cannot be applied with

conventional software analysis programs, and they require an extra algorithm for the

adaptive parameter. Therefore, they lose practical appeal in engineering practice. In

this respect, MPA is simpler to apply when compared with the adaptive methods.

Second, all developed procedures combine modal responses by statistical rules

(SRSS or CQC), which are approximate rules and developed for combining linear

10

elastic modal responses. Therefore, they possess shortcomings for combining

inelastic modal responses, especially in member internal forces and deformations.

Although multi-mode pushover analysis procedures improve the nonlinear

static procedures to predict the dynamic responses, they are not capable of capturing

the response parameters during the nonlinear dynamic response due to the change in

modal interaction under the earthquake excitation. Vamvatsikos and Cornell (2002)

developed the (single-record) incremental dynamic analysis procedure (IDA) for

obtaining intensity-demand diagrams similar to the pushover curves in order to

capture variations in the structural response under a specific ground motion

excitation. Generation of a single IDA curve requires several nonlinear dynamic

analysis of the structural system under scaled ground motions. Hence, the application

of the method is far from being practical. Aschheim et al. (2007) prepared a scaled

nonlinear dynamic procedure (NDP) to improve the nonlinear dynamic procedure. In

this procedure, several nonlinear dynamic analyses of the considered structure are

conducted under the scaled ground motions to match the target demand obtained

from nonlinear static analysis. Then the results obtained are evaluated statistically to

measure levels of confidence for the corresponding response parameters. The

purpose of IDA and NDP is to improve nonlinear static procedure, but they lose

computational simplicity.

1.2.3 Recent Developments in Nonlinear Static Analysis

Different approaches have been considered and studied in order to improve

disadvantages of nonlinear static analysis, especially for reducing computational

effort and considering higher mode effects more effectively. In this sense, Sucuoğlu

and Günay (2011) developed a procedure, named as generalized pushover analysis

(GPA), for estimating the inelastic seismic response of structures under earthquake

ground excitations. In this procedure, different load vectors are applied separately to

the structure in the incremental form until the predefined seismic demand is obtained

for each force vector. These force vectors are named as generalized force vectors. A

generalized force vector is a combination of modal forces, and simulates the

instantaneous force distribution on the system when a given response parameter

reaches its maximum value during the dynamic response. In this method, the

maximum interstory drift parameters are selected as target demand parameters and

11

used for derivation of generalized force vectors. The maximum value of any other

response parameter is then obtained from analysis results of each generalized force

vector. In this way, this procedure does do not suffer from the statistical combination

of inelastic modal responses. They stated that each conducted nonlinear static

analysis under a generalized force vector initiates multi-degree of freedom effects

simultaneously. Moreover, they pointed out that interstory drift values at different

stories are stronger representatives of local maximum response parameters for the

corresponding stories, and takes into account contributions of higher mode effects for

local responses. The developed procedure requires less computational effort when

compared to adaptive multi-mode pushover analysis and considers higher mode

effects more effectively than modal pushover analysis. They concluded that the

generalized pushover analysis gives good estimates for seismic demands when

compared to nonlinear dynamic analysis especially for member deformations and

forces. The success of the method for estimation of member demands provides an

improvement for nonlinear static analyses.

Jerez and Mebarki (2011) proposed a procedure called the pseudo-adaptive

uncoupled modal response history analysis (PSA). This method considers the modal

properties after yielding. In this sense, modal properties after yielding are derived

from the pushover analysis on the basis of story displacements. This derivation is

based on the assumption that the modal shapes are proportional to the story

displacements produced by modal load profiles. After finding the modified modal

properties, the equivalent displacement value is computed, and then base shear vs.

equivalent displacement curve is constructed and idealized as a bilinear curve. In the

next step, this idealized curve is converted to the force-deformation curve of the

equivalent SDOF system in order to conduct nonlinear time history analysis of this

equivalent system. They used the assumption that modal superposition principle is

still valid, and the general framework of classic modal analysis remains applicable

even for inelastic systems. Thus, the peak modal responses in this method are

obtained from maximum absolute values of total responses. They pointed out that the

number of modes required for obtaining an appropriate accuracy is two or three when

the higher mode effects become dominant. After applying the method to six different

concrete moment resisting frames with a set of six ground motion records, they

observed that PSA method succeeds in producing acceptable estimates of inelastic

responses for low levels of deformation with respect to the time history analysis

12

results. However, they also expressed that for higher levels of inelastic response, the

developed method provides better, or at least similar, results than other widely used

nonlinear static analysis methods. PSA method produces an improvement for

nonlinear static analysis by using modal properties after yielding. On the other hand,

this method is very sensitive to particular features of ground motion records and

model complexity regarding computational cost and general accuracy.

Kreslin and Fajfar (2012) developed the extended N2 method for taking into

account higher mode effects in elevation. The extension is based on the assumption

that the structure remains in the elastic range when vibrating in higher modes. This

developed method is based on three main steps. In the first step, the basic N2 is

performed. Then, the standard elastic modal analysis is conducted considering all

relevant modes as the second step. In the last step, the envelope results in the first

and second steps are determined. In this sense, correction factors for each storey,

which are defined as the ratio between the results obtained by elastic modal analysis

and the results obtained by pushover analysis, are determined. If the obtained ratios

is larger than 1.0, the correction factor is equal to this ratio, otherwise it is taken as

1.0. Then, the resulting storey drifts are computed by multiplying the results of

pushover analysis with the related correction factors. In this study, the developed

method is tested for a 3-storey, 9-storey and 20-storey steel frame buildings under

the two different ground motion sets, which are grouped with respect to ground

motion intensities. The mean results of extended N2 method are compared with

respect to NRHA and the other nonlinear simplified methods, MPA and MMPA.

They stated that the extended N2 method usually provides slightly larger estimations

when compared to MPA and MMPA procedures, but it is conservative in comparison

with the mean values of NRHA results. Similar to the other nonlinear simplified

methods, the height of structures, ground motion intensity and the spectral shape of

ground motion affects the accuracy of the proposed method. Finally, they concluded

that the proposed approach extends the applicability of N2 method at least to

medium-rise buildings subjected to realistic intensities of ground motion, and the

method is relatively simple to be implemented in most commercial computer

program.

13

1.3 Objective and Scope

Generalized Pushover Analysis (GPA) is presented for the estimation of

inelastic seismic responses of MDOF systems and compared with the results of

nonlinear response history analysis (NRHA) and modal pushover procedure (MPA)

in this study. The procedure is applied to three different reinforced concrete

symmetric-plan buildings, which are a twelve story frame with full capacity design, a

twelve story frame with relaxed capacity design, and a twenty story wall-frame

system with full capacity design. The results obtained from GPA are compared with

the results of NRHA and MPA by using a set of 7 ground motions. In comparative

evaluations, interstory drift ratios, member-end plastic rotations and member end

forces are employed. In addition to these comparisons, mean results of 7 ground

motions from three analysis procedures are compared to the GPA results conducted

by using the mean spectrum of ground motions in order to examine the applicability

and accuracy of GPA when it is used with design (code) spectra. Moreover, in order

to reduce the computational effort of GPA, a reduced procedure is developed based

on the mathematical background of modal expansions of story drifts. The results of

three analyses procedures are also compared with the results from the newly

developed reduced GPA procedure.

Main objective of this study is to examine and extend the practical

implementation of GPA for inelastic seismic response prediction of symmetric

MDOF systems. The success of the procedure is tested mainly by comparing the

results with the results of nonlinear response history analysis.

14

CHAPTER 2

GENERALIZED PUSHOVER ANALYSIS PROCEDURE

The generalized pushover analysis (GPA) procedure developed by Sucuoğlu

and Günay (2011) is described in detail in the following sections. Formulation of

generalized force vectors and target drifts are also presented.

2.1 Generalized Force Vectors

The GPA procedure is based on an effective force vector acting on the system

when a specific response parameter reaches its maximum value. This effective force

vector is a generalized force vector, since it includes contributions from all modal

forces at the time of maximum response of the selected response parameter. If this

force vector is defined, it can be applied to the system as a static force in order to

produce the maximum value of this response parameter. In GPA, generalized

effective force vectors are derived from the dynamic response of linear elastic

MDOF systems to earthquake ground excitations by using the modal superposition

procedure.

The effective force vector f(tmax) at time tmax is expressed as the modal

superposition of modal forces fn (tmax) :

(1)

The nth mode effective force in Equation (1) at time tmax is defined by:

(2)

15

Here,

; is the nth mode

shape; is the mass matrix, is the influence vector, and is represented as

(3)

where is the nth mode vibration frequency, and is the modal

displacement amplitude at tmax which satisfies the equation of motion of the SDOF

system representing the nth mode under ground motion excitation, .

satisfies the equation of motion presented in Equation (4).

( )

cannot be found out if tmax is not known. tmax is the time when the selected

response parameter attains its maximum value.

In this method, the target response parameter is selected as the interstory drift

∆j at the jth story. Then,

(5)

The modal expansion of the interstory drift at time is

( )

where is the jth element of the nth mode shape vector, . By dividing both

sides with , Equation (6) can be normalized.

(7)

Equation (7) expresses the contribution of nth mode to the maximum interstory drift

at the jth story in a normalized form.

The maximum value of interstory drift at the jth story in Equation (5) can also

be estimated by response spectrum analysis (RSA) through SRSS of the related

spectral modal responses.

16

( )

in Equation (8) is the spectral displacement of the nth mode, and obtained from

the displacement response spectrum of the ground motion excitation, .

Similarly, Equation (8) can also be normalized by dividing both sides with

.

( )

The respective terms on the right hand sides of Equation (7) and (9) are the

normalized contributions of the nth mode to the maximum interstory drift at the jth

story. Equating right hand sides of Equation (7) and (9), and considering Equation

(5) leads to

( )

The term in the parentheses in Equation (10) is equal to from Equation (8). Then

the above equation becomes

( )

In Equation (11), is the nth mode contribution to the maximum interstory drift

of the jth story determined from RSA, and is the quadratic combination of the

terms according to Equation (8).

The generalized force vector expression is obtained by using the derived

equations. In this sense, firstly is calculated by substituting from

Equation (11) into Equation (3). Then, the obtained is substituted from

Equation (3) into Equation (2). Finally, substituting from Equation (2) into

Equation (1) gives the generalized force vector expression.

17

( 2)

is the pseudo-spectral acceleration of the nth mode in Equation (12), and obtained

from Equation (3).

The generalized force vector acts on the system when the interstory drift at

the jth story becomes maximum. Therefore, the generalized force vector expression

in Equation (12) will be identified with the subscript j.

( 3)

2.2 Target Interstory Drift Demand

The maximum value of the interstory drift at the jth story during dynamic

response was expressed by Equation (6) at the previous section. Target interstory

drift demand can be obtained consistently with the generalized force vector by

substituting from Equation (11) into Equation (6) and taking into account

Equation (5).

( 4)

in Equation (14) is equal to the interstory drift at the jth story calculated from

response spectrum analysis (RSA) if the response is linear elastic. In order to

improve the prediction of Equation (14), the first mode linear elastic spectral

displacement demand can be replaced with the first mode inelastic spectral

displacement demand . This operation requires conducting an ‘a priori’ first mode

pushover analysis. Then, can be estimated from the nonlinear response history

analysis of the equivalent SDOF system representing the first mode behavior, or

from the associated R-μ-T relation.

GPA uses the higher-order interstory drift parameter as target demand rather

than a story (roof) displacement. Accordingly, when the associated generalized force

vector pushes the system to the target drift , the system adopts itself in the

18

inelastic deformation range, and the higher-order deformation parameters (rotations,

curvatures) and force parameters (moments, shears) take their inelastic values with

more effective contributions from higher modes. On the other hand, if story (roof)

displacement is used as target demand, the contribution of higher modes becomes

less significant. If different local response parameters are primary considerations

during inelastic dynamic response, interstory drift values are more effective

representatives of local maximum response parameters, because they are well

synchronized with the local response parameters.

2.3 Generalized Pushover Algorithm

The GPA algorithm contains the following six basic steps. These steps can be

summarized as below:

1. Eigenvalue analysis: Natural frequencies (natural periods Tn), modal

shape vectors , and modal participation factors are obtained from the

eigenvalue analysis.

2. Response spectrum analysis (RSA): Modal spectral amplitudes and are

determined from elastic spectra of the corresponding ground motion. Modal

interstory drift ratios of the jth story, and the maximum interstory drift

ratio of the jth story, are obtained from RSA.

3. Generalized force vectors: Generalized force vectors , which act on the

system when the interstory drift at the jth story becomes maximum, are

computed from Equation (13).

4. Target interstory drift demand: Target interstory drift demands for each story

are calculated from Equation (14). If the first mode inelastic spectral

displacement demand is utilized instead of the elastic demand in order

to improve the accuracy, a first mode pushover analysis is conducted to

determine the first mode capacity curve. After approximation of the capacity

curve with a bi-linear curve and converting it to the acceleration-

displacement spectrum format, nonlinear dynamic analysis or inelastic

response spectrum analysis of the equivalent bi-linear SDOF system can be

19

conducted in order to obtain . On the other hand, values for n = 2-N, are

taken from the elastic response spectrum of the corresponding ground motion.

5. Generalized pushover analysis: N number of GPA’s are conducted

sequentially. In the jth GPA (j=1-N), the structural system is pushed

incrementally in the lateral direction with the force distribution proportional

to the corresponding generalized force vector, At the end of each loading

increment i during the pushover analysis, the obtained interstory drift value

at the jth story is compared with the target interstory drift computed

from Equation (14). Displacement-controlled pushover analysis is conducted

until reaches .

6. Maximum response values: All member deformations and member internal

forces are directly obtained from the jth GPA at the target interstory drift

demand . After completing all GPA’s for j=1-N, member deformations and

member internal forces are determined by obtaining the envelopes of the

related GPA results, and these envelope values are registered as the

maximum seismic response values.

20

CHAPTER 3

GROUND MOTIONS EMPLOYED IN CASE STUDIES

The ground motion set employed in the case studies contains seven different

ground motions, including pulse types and ordinary types. Acceleration records of

these seven ground motions were generated from the selected reference data set of

ground motions which have similar properties (Hancock et al., 2006; Hancock and

Bommer, 2007). The spectrum of each ground motion is adjusted with the Turkish

Earthquake Code (TEC 2007) design spectrum (A0 = 0.4, I = 1.0, Z3). These

reference ground motions were selected to be capable of generating higher mode

effects on the structural systems, and downloaded from the PEER strong motion

database. Important features of reference ground motions are presented in Table 3.1.

Figure 3.1 shows the acceleration time histories of the generated ground motions.

The derivation of synthetic ground motions were conducted by using

RSPMatch (2005) software program. The derivation process is based on changing

the frequency range of the reference ground motion in order to match the original

acceleration spectrum with the given target acceleration spectrum. Therefore, the

obtained synthetic ground motion properties (PGA, PGV, and PGD) do not show

significant variations from the original ground motion data. After derivation of the

ground motions that will be employed in case studies, their properties were checked

with the reference properties. Figure 3.2 shows the acceleration spectra of synthetic

ground motions, their mean spectrum and TEC (2007) design spectrum.

21

#G

M C

ode

Ea

rthqua

ke (

Mw

)S

tati

on-C

om

po

nent

CD

(km

)

Sit

e

Geo

l.

PG

A

(g)

PG

V

(cm

/s)

PG

D

(cm

)

GM

Ty

pe

1C

LS

090

Lo

ma

Priet

a, 1

0/1

8/8

9 (

7)

Co

rral

itos-

090

3.9

A0

.479

45.2

11.3

Pul

se

2L

EX

000

Lo

ma

Priet

a, 1

0/1

8/8

9 (

7)

Lo

s G

at.

- L

ex.

Dam

-000

5.0

A0

.420

73.5

20.0

Pul

se

3P

CD

254

San

Fer

., 0

2/0

9/7

1 (

6.6

)P

aco

ima

Dam

-254

2.8

B1

.160

54.1

11.8

Pul

se

4C

HY

006

-EC

hi-C

hi,

09

/20/9

9 (

7.6

)C

HY

006

-E9

.8B

0.3

64

55.4

25.6

Pul

se

5B

olu

000

Duz

ce,

11

/12/9

9 (

7.1

)B

olu

-000

12.0

D0

.728

56.4

23.1

Ord

inar

y

6O

RR

090

No

rthr

idge

, 01

/17/9

4 (

6.7

)C

ast.-O

ld R

dg

Ro

ute-

090

20.7

B0

.568

51.8

9.0

Ord

inar

y

7E

RZ

-EW

Erz

inca

n,03

/13/9

2 (

6.9

)E

rzin

can-

EW

4.4

D0

.496

64.3

21.9

Pul

se

Tab

le 3

.1 R

efer

ence

gro

und

mo

tion

pro

per

ties

A=

Ro

ck, B

=S

hal

low

(st

iff)

soil

, D

=D

eep b

road

soil

CD

: C

lose

st d

ista

nce

to f

ault

rap

ture

22

Figure 3.1 Acceleration time histories of the synthetic ground motions

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 10 20 30 40

Acc

eler

atio

n (

g)

Time (sec)

GM1

-0.6

-0.4

-0.2

0.0

0.2

0.4

0 10 20 30 40

Acc

eler

atio

n (

g)

Time (sec)

GM2

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0 10 20 30 40

Acc

eler

atio

n (

g)

Time (sec)

GM3

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 10 20 30 40 50

Acc

eler

atio

n (

g)

Time (sec)

GM4

-0.6

-0.4

-0.2

0.0

0.2

0.4

0 10 20 30 40 50 60

Acc

eler

atio

n (

g)

Time (sec)

GM5

-0.4

-0.2

0.0

0.2

0.4

0.6

0 10 20 30 40

Acc

eler

atio

n (

g)

Time (sec)

GM6

-0.6

-0.4

-0.2

0.0

0.2

0.4

0 5 10 15 20

Acc

eler

atio

n (

g)

Time (sec)

GM7

23

Figure 3.2 Acceleration response spectra of synthetic ground motions, mean acceleration

spectrum, and TEC2007 design spectrum

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Acc

eler

atio

n (g

)

Period (sec)

GM1

GM2

GM3

GM4

GM5

GM6

GM7

Mean Spectrum

TEC2007 Design

Spectrum, Z3

24

CHAPTER 4

CASE STUDIES

The GPA procedure is tested for three different cases. In the first and second

case studies, a twelve story RC symmetric plan building with moment resisting

frames, with different design principles in each case, is employed. In the third case

study, a twenty story RC symmetric plan building with wall-frame system is utilized.

4.1 Case Study I: Twelve Story RC Frame with Full Capacity Design

4.1.1 Building Description

The first case study employed in this thesis study is a twelve story reinforced

concrete symmetric-plan building. The floor plan is shown in Figure 4.1. This

structure is designed according to TS 500 (2000) and TEC (2007) by considering

capacity design principles. Hence it is called “full capacity design”. An enhanced

ductility level is employed. The properties selected in TEC (2007) can be listed as

seismic zone 1, soil type Z3 and residential use. The member dimensions for beams

are 30x55 cm2

for the first four stories, 30x50 cm2

for the second four stories, and

30x45 cm2

for the last four stories. The columns dimensions are 50x50 cm2, 45x45

cm2, and 40x40 cm

2 in the first four, the second four, and the last four stories,

respectively. The story heights are 4 m for the first story, and 3.2 m for all other

stories. There is no basement, and story levels start from the ground level. The

frames used in the analytical models are shown in Figure 4.2. Section details of

beams and columns are given in Figure 4.3. Table 4.1 includes shear design details of

each beam and column section. Concrete and steel characteristic strengths are 25

MPa and 420 MPa, respectively.

25

Figure 4.1 Story plan of the twelve story building with full capacity design

Figure 4.2 Plane (2D) model of the twelve story building with full capacity design

6 m 4 m 6 m

5 m

5 m

5 m

Frame A

Frame B

Frame A Frame B

6 m 6 m 4 m 6 m 6 m 4 m

4 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

26

a) Column section details

b) Beam section details at the support region

c) Beam section details at the span region

Figure 4.3 Column and beam section details

8Ф28

50 cm

50 c

m

8Ф26

40 c

m

40 cm

45 c

m

8Ф28

45 cm

4Ф18

45 c

m

30 cm

2Ф18

30 cm

2Ф22

55 c

m

4Ф22

30 cm

50 c

m

2Ф20

4Ф20

50 c

m

30 cm

4Ф20

2Ф20

30 cm

4Ф22

55 c

m

2Ф22 2Ф18

45 c

m

30 cm

4Ф18

27

Table 4.1 Shear design details of members

Along End Region Along Span Region

Columns

50x50 cm2 Ф8 / 10 cm Ф8 / 15 cm

45x45 cm2 Ф8 / 10 cm Ф8 / 15 cm

40x40 cm2 Ф8 / 10 cm Ф8 / 15 cm

Beams

55x30 cm2 Ф8 / 12 cm Ф8 / 18 cm

50x30 cm2 Ф8 / 10 cm Ф8 / 15 cm

45x30 cm2 Ф8 / 10 cm Ф8 / 15 cm

4.1.2 Modeling

Analytical model of the twelve story frame is generated by using the

OpenSees software (2005). All nonlinear and linear analyses are conducted with this

program. Structural frame members are modeled with “Beam with Hinges” element.

In this sense, plastic hinge lengths are defined at both ends of all columns and beams.

For beam sections, bi-linear moment curvature relationships are defined along the

hinge lengths by considering the designed section properties of each beam type. On

the other hand, column sections are defined by utilizing fiber sections along the

hinge lengths. The remaining part in between the plastic hinges of a member is

defined with elastic cracked section properties. In order to describe cracked section

stiffness for each member, the gross moment of inertia values are multiplied by 0.4

and 0.6 for beams and columns respectively. Rigid diaphragms are assigned to each

story level, and P-∆ effects are taken into account in the model. In time history

analyses, Rayleigh damping is computed by considering 5% damping in the 1st and

3rd

modes.

4.1.3 Free Vibration Properties

Free vibration properties of the twelve story frame model are obtained from

the elastic model with cracked section properties. Free vibration properties for the

first three modes are presented in Table 4.2. The modal mass ratios and the scaled

top story amplitudes of the mode shape vectors in Table 4.2 indicate the effects of

28

second and third modes of the system. Figure 4.4 shows the first three mode vectors

of the model, normalized with respect to mass.


modes

(*)Product of modal participation factor and amplitude of the mode vector at the roof

Figure 4.4 Mode shapes for the first three modes

4.1.4 Presentation of Results

Maximum interstory drifts, maximum average plastic rotations of beam ends

in a story, maximum average chord rotations of column ends in a story, beam end

moments and shear forces, column end moments and shear forces are obtained under

the ground motion set given in Chapter 3. Maximum average plastic or chord

0

1

2

3

4

5

6

7

8

9

10

11

12

-0.1 -0.05 0 0.05 0.1

Sto

ry #

Mode Vector Amplitude

Mode 1

Mode 2

Mode 3

Mode T (sec) Effective Modal

Mass (ton)

Effective Modal

Mass Ratio Γnφnr

(*)

1 2.39 434.34 0.79 1.32

2 0.82 66.62 0.12 -0.50

3 0.48 21.75 0.04 0.29

29

rotations at each story level are calculated by first averaging the member end

rotations at the corresponding story at each load (or time) step, and then taking the

maximum of these values as the story maximum. The results of nonlinear response

history analysis (NRHA), modal pushover analysis (MPA) and generalized pushover

analysis (GPA) are compared with each other for each ground motion in the set. The

target drift in GPA is determined by employing the inelastic maximum displacement

amplitude for the first mode in Equation (14).

4.1.4.1 Interstory Drift Ratios

Maximum interstory drift ratios obtained from the three procedures (NRHA,

MPA and GPA) are compared for each ground motion in Figure 4.5. Interstory drift

profiles given in Figure 4.5 indicate that higher modes contribute significantly to the

total response of the system. MPA underestimates higher mode effects at the upper

story levels. Generally, MPA results are well synchronized with the first mode

response due to the adopted combination rule SRSS. GPA shows improvement in the

estimation of drift values at upper stories except for GM3. GPA and MPA results are

almost close to each other for GM3, and GPA cannot catch the deformation demand

at the 9th

and 10th

stories due to peculiar distribution of plastic deformations at the

upper stories during NRHA.

Figure 4.5 Maximum interstory drift ratios obtained under seven ground

motions

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020

Sto

ry #

Interstory Drift Ratio

GM1

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020 0.025

Sto

ry #


GM2

30

Figure 4.5 (Continued)

4.1.4.2 Member End Rotations

Maximum values of average beam-end plastic rotations, plastic rotations of

beam ends at the 1st, 5

th and 10

th story levels when the beam end moments reach their

maximum, and maximum values of average column-end chord rotations under each

ground motion are presented in Figure 4.6, Figure 4.7 and Figure 4.8, respectively.

The aim of using chord rotations for column ends instead of plastic rotations is the

fact that plastic rotations are too small to be compared accurately. Therefore,

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM4

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM6

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020 0.025

Sto

ry #


NRHA GPA MPA

GM7

31

comparing chord rotations for columns is more suitable for performance test of MPA

and GPA with respect to NRHA. In Figure 4.8, yield rotation limits for columns are

also shown. Yield rotation values were calculated from the section analysis of each

column section. The axial load values were taken from the gravity load analysis of

the system. It is evident from Figure 4.8 that plastic deformations do not occur in

columns.

Larger plastic deformations are generally accumulated between the 7th

and

11th

stories and between the 2nd

and 5th

stories according to Figure 4.6. Beam-end

plastic rotations at lower stories are estimated well by MPA and GPA. However

MPA gives lower plastic rotation demands for upper stories with respect to NRHA

results. On the other hand, GPA gives more realistic demands for the upper stories

than MPA. GPA results are well synchronized with NRHA results at the lower

stories, and improve MPA results for upper stories. GPA cannot show significant

improvement for GM3, similar to interstory drift ratio results. The special plastic

deformation distribution between 7th

and 11th

stories due to the ground motion

characteristic in GM3 cannot be predicted by GPA, and also by MPA.


ground motions

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020

Sto

ry #

Plastic Rotation (rad)

GM1

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.010 0.020 0.030

Sto

ry #

Pastic Rotation (rad)

GM2

32


Maximum beam-end plastic rotations at the 1st, 5

th, and 10

th stories, when

bending moments at beam ends reach their maximum, are presented in Figure 4.7.

Generally it can be observed GPA predicts NRHA results reasonably well.

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015

Sto

ry #


GM4

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015

Sto

ry #


GM6

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


NRHA GPA MPA

GM7

33

1st Story 5

th Story 10

th Story

Figure 4.7 Beam end plastic rotations when bending moments at beam ends reach their

maximum

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Pla

stic

Ro

tati

on

(ra

d)

GM1

NRHA

GPA

MPA

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

0.012

Pla

stic

Ro

tati

on

(ra

d)

GM1

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

Pla

stic

Ro

tati

on

(ra

d)

GM1

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

Pla

stic

Ro

tati

on

(ra

d)

GM2

6 m 4 m 6 m

0.000

0.005

0.010

0.015

0.020

Pla

stic

Ro

tati

on

(ra

d)

GM2

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

0.012

Pla

stic

Ro

tati

on

(ra

d)

GM2

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

Pla

stic

Ro

tati

on

(ra

d)

GM3

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

Pla

stic

Ro

tati

on

(ra

d)

GM3

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

0.012

Pla

stic

Ro

tati

on

(rad

)

GM3

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

0.012

Pla

stic

Ro

tati

on

(ra

d)

GM4

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

0.012

Pla

stic

Ro

tati

on

(ra

d)

GM4

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

Pla

stic

Ro

tati

on

(ra

d)

GM4

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

0.012

Pla

stic

Ro

tati

on

(ra

d)

GM5

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

Pla

stic

Ro

tati

on

(ra

d)

GM5

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

0.012

Pla

stic

Ro

tati

on

(ra

d)

GM5

6 m 4 m 6 m

34

It can be seen from Figure 4.8 that maximum average chord rotations of

columns over the first story levels do not exceed the yield rotation limits, and plastic

actions do not develop at that column ends, which conforms to the capacity design

principle. The results of MPA and GPA are generally close to each other. It can be

concluded that GPA and MPA as approximate methods predict column chord

rotation demands within acceptable tolerance limit.


0.000

0.002

0.004

0.006

0.008

0.010

Pla

stic

Ro

tati

on

(ra

d)

GM6

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

Pla

stic

Ro

tati

on

(ra

d)

GM6

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

0.012

Pla

stic

Ro

tati

on

(ra

d)

GM6

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Pla

stic

Ro

tati

on

(ra

d)

GM7

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

Pla

stic

Ro

tati

on

(ra

d)

GM7

6 m 4 m 6 m

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Pla

stic

Ro

tati

on

(ra

d)

GM7

6 m 4 m 6 m

35

Figure 4.8 Maximum values of average column-end chord rotations under

seven ground motions with yield rotation limits

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010

Sto

ry #

Chord Rotation (rad)

GM1

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010

Sto

ry #


GM2

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010

Sto

ry #


GM4

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010

Sto

ry #


GM6

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010

Sto

ry #


NRHA GPA MPA Yield Rotation

GM7

36

4.1.4.3 Member Internal Forces

Maximum values of bending moments and shear forces at the ends of 1st

story, 5th

story and 10th

story beams are shown in Figure 4.9 and Figure 4.10

respectively, for each analysis case. The results of MPA and GPA are the envelopes

of the results obtained by pushing the system in both directions. Table 4.3 shows the

yield moment values for each beam type. The maximum shear forces and bending

moments for columns are obtained along the left exterior column axis. Similarly,

they are the envelopes of results obtained by pushing the system in both directions.

Column forces are presented in Figure 4.11 and Figure 4.12. Yield moment values

are also plotted in Figure 4.11 with the column bottom end moments.

Table 4.3 Negative yield moment values for beam ends (tension at top)

It is revealed by Figure 4.9 and Figure 4.10 that GPA exactly predicts the

beam end forces computed by NRHA. On the other hand, MPA overestimates shear

forces and bending moments at beam ends, especially for the 1st and 10

th stories. The

error of MPA in predicting the member internal forces varies from 6% to 50% with

respect to the NRHA results. However the maximum GPA error is about 6%. MPA

results are not consistent with the member capacities given in Table 4.3. MPA

requires the correction of internal forces suggested by Goel and Chopra, (2005).

MPA is close to NRHA at the 5th

story, because second mode contribution to

member forces at this middle story is negligible.

Story Range Beam Dimension Yield Moment, My (kN.m)

1st-4

th 30x55 cm2 231.5

5th

-8th 30x50 cm

2 230.5 9

th-12

th 30x45 cm2 170.0

37

1st Story 5

th Story 10

th Story

Figure 4.9 Maximum bending moment values of beam ends at 1st, 5th, and 10th stories

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM1

NRHA

GPA

MPA

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM1

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM1

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM5

6 m 4 m 6 m

38


1st Story 5

th Story 10

th Story

Figure 4.10 Maximum shear force values of beam ends at 1st, 5th, and 10th stories

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM1

NRHA

GPA

MPA

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM1

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM1

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce (

kN

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce (

kN

)

GM3

6 m 4 m 6 m

39

Bottom-end moments and shear forces along the left exterior column axis is

presented in Figure 4.11 and Figure 4.12 respectively. For bending moments, MPA

and GPA show the same accuracy in estimating the NRHA results. The difference

between the NRHA results and MPA and GPA estimations become larger between

the 5th

and 9th

stories. The error of GPA for the first story moment is about 8%, and

this error becomes 16% for MPA. It is observed from Figure 4.12 that GPA generally

predicts the NRHA shear force distribution quite well. GPA is also quite accurate in

estimating the shear forces at the upper story levels.


0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce (

kN

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM7

6 m 4 m 6 m

40

Figure 4.11 Maximum bottom end bending moments along the left exterior

column axis of the inner frame with yield moments

0 1 2 3 4 5 6 7 8 9

10 11 12

0 100 200 300 400 500 600

Sto

ry

#

Moment (kN.m)

GM1

0 1 2 3 4 5 6 7 8 9

10 11 12

0 100 200 300 400 500 600 700

Sto

ry

#

Moment (kN.m)

GM2

0 1 2 3 4 5 6 7 8 9

10 11 12

0 100 200 300 400 500 600 700

Sto

ry

#

Moment (kN.m)

GM3

0 1 2 3 4 5 6 7 8 9

10 11 12

0 100 200 300 400 500 600

Sto

ry

#

Moment (kN.m)

GM4

0 1 2 3 4 5 6 7 8 9

10 11 12

0 100 200 300 400 500 600

Sto

ry

#

Moment (kN.m)

GM5

0 1 2 3 4 5 6 7 8 9

10 11 12

0 100 200 300 400 500 600 700

Sto

ry

#

Moment (kN.m)

GM6

0 1 2 3 4 5 6 7 8 9

10 11 12

0 100 200 300 400 500 600

Sto

ry

#

Moment (kN.m)

NRHA GPA

MPA Yield Moment

GM7

41

Figure 4.12 Maximum shear forces along the left exterior column axis of the

inner frame

0 1 2 3 4 5 6 7 8 9

10 11 12

0 50 100 150

Sto

ry

#

Shear Force (kN)

GM1

0 1 2 3 4 5 6 7 8 9

10 11 12

0 50 100 150

Sto

ry

#

Shear Force (kN)

GM2

0 1 2 3 4 5 6 7 8 9

10 11 12

0 50 100 150

Sto

ry

#

Shear Force (kN)

GM3

0 1 2 3 4 5 6 7 8 9

10 11 12

0 50 100 150

Sto

ry

#

Shear Force (kN)

GM4

0 1 2 3 4 5 6 7 8 9

10 11 12

0 50 100 150 200

Sto

ry

#

Shear Force (kN)

GM5

0 1 2 3 4 5 6 7 8 9

10 11 12

0 50 100 150

Sto

ry

#

Shear Force (kN)

GM6

0 1 2 3 4 5 6 7 8 9

10 11 12

0 50 100 150

Sto

ry

#

Shear Force (kN)

NRHA GPA MPA

GM7

42

4.2 Case Study II: Twelve Story RC Frame with Relaxed Capacity

Design


In the second case study, the building plan geometry is similar to the building

in the first case study. The floor plan is given in Figure 4.1. The difference between

the two case studies is the design principle utilized in each case. In case study II,

relaxed design principle is used. In this sense, the first story height is changed as 5

meters, and the longitudinal reinforcement areas in columns are reduced to 80% of

the reinforcement areas used in the full capacity design. Hence, strong-column-weak

beam principle is violated at some of the beam-column joints. Similar to the previous

case, the properties selected in accordance with TEC (2007) can be specified as

seismic zone 1, soil type Z3 and residential use. The member dimensions are the

same with the full capacity design model (Section 4.1.1). Similar to the first case,

there is no basement floor, and story levels start from the ground level. Figure 4.13

shows the frames employed in the analytical models. Section details of beams and

columns are given in Figure 4.14. Shear design details are the same with the plane

model used in case study I and given in Table 4.1. Concrete and steel characteristic

strengths are 25 MPa and 420 MPa, respectively.

43

Figure 4.13 Plane (2D) model of the twelve story building with relaxed capacity design

5 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

Frame A

6 m 6 m 4 m

Frame B

6 m 6 m 4 m

44





4Ф18

45 c

m

30 cm

2Ф18

30 cm

2Ф22

55

cm

4Ф22

30 cm 50 c

m

2Ф20

4Ф20

30 cm

4Ф20

2Ф20

30 cm

4Ф22

55 c

m

2Ф22

8Ф26

50 cm

50 c

m

8Ф24

40 c

m

40 cm

45 c

m

8Ф26

45 cm

45 c

m

30 cm

4Ф18

2Ф18

45

4.2.2 Modeling

Analytical model of the twelve story frame is generated by using the

OpenSees software (2005) similar to the case study I, and all nonlinear and linear

analyses are conducted with this program. The modeling details are the same with

the first case (Section 4.1.2). Bi-linear moment curvature relationships are used for

beams along defined plastic hinge lengths. Fiber sections are defined for columns

along the defined plastic hinges at both ends. Similar cracked section properties with

the first case study are used for the remaining part in between the plastic hinges of a

member. Rigid diaphragms are assigned to each story level and P-∆ effects are taken

into account in the model. In time history analyses the same parameters utilized in

the full capacity model are used for Rayleigh damping computation.


Free vibration properties of the relaxed design plane model are obtained from

the model generated with the elastic cracked section properties. Table 4.4 presents

free vibration properties of the first three modes. Similar to the case study I model, it

can be observed that modal mass ratios and the scaled top story amplitudes of the

mode shape vectors indicates contribution of second and third modes to the system

response. In Figure 4.15, the first three mode vectors of the model, normalized with

respect to mass are presented.


modes


Mass (ton)

Effective Modal

Mass Ratio Γnφnr

(*)

1 2.50 453.1 0.82 1.32

2 0.86 64.0 0.12 -0.48

3 0.50 17.2 0.03 0.25


46

Figure 4.15 Mode shapes for the first three modes


Similar to the previous case study, maximum interstory drifts, maximum

average plastic rotations of beam-ends in a story, maximum average chord rotations

of column ends in a story, beam end moments and shear forces, column end

moments and shear forces are obtained under the ground motion set given in Chapter

3. Maximum member end rotations are computed according to Section 4.1.4. Results

of nonlinear response history analysis (NRHA), modal pushover analysis (MPA) and

generalized pushover analysis (GPA) are compared with each other for each ground

motion in the set. The target drift in GPA is determined by employing the inelastic

maximum displacement amplitude for the first mode in Equation (14).


Figure 4.16 shows the comparison of maximum interstory drift ratios

obtained from three procedures (NRHA, MPA and GPA). Interstory drift ratio

profiles in Figure 4.16 resemble the profiles obtained in the previous case study. The

difference is expectedly at the first story drift ratios of the relaxed capacity design

0

1

2

3

4

5

6

7

8

9

10

11

12

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

Sto

ry #


Mode 1

Mode 2

Mode 3

47

which are larger than results of full capacity design due to reduction in lateral

stiffness of this story. GPA predicts NRHA results with higher accuracy. GPA

catches NRHA drift profiles almost exactly for GM1, GM2, GM4 and GM7.

However, MPA underestimates higher mode effects at the upper story levels as in the

first case study due to the dominance of first mode response in the adopted

combination rule SRSS. GPA shows improvement in the estimation of drift values at

upper stories except for GM3. GPA and MPA results are almost close to each other

for GM3, and GPA cannot catch the deformation demand between the 8th

and 10th

stories due to peculiar distribution of plastic deformations at the upper stories during

NRHA, which is the similar situation expressed in the first case study.

Figure 4.16 Maximum interstory drift ratios under seven ground motions

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM1

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.010 0.020 0.030

Sto

ry #


GM2

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM4

48



Maximum values of average beam-end plastic rotations and maximum values

of average column-end chord rotations for each ground motion are presented in

Figure 4.17 and Figure 4.18 respectively. Chord rotations are utilized for column

ends instead of plastic rotations because plastic rotations are also too small to be

compared accurately. In Figure 4.18, yield rotation limits for columns are also

shown. Yield rotation capacities for columns were computed from the section

analysis of each column section according to calculated axial loads obtained from

gravity analysis, similar to the previous case study.

Although the building employed in case study II does not satisfy the capacity

design requirements of TEC 2007, the plastic deformations obtained at beam ends

are not too large. However, plastic deformations at the first story level become larger

when compared to the previous model values due to the increased height of that

story. GPA estimates NRHA plastic rotations reasonably well for all ground motions.

However, MPA gives lower plastic rotation demands for upper stories with respect to

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015

Sto

ry #


GM6

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.010 0.020 0.030

Sto

ry #


NRHA GPA MPA

GM7

49

NRHA results. For GM1, GM2, GM4 and GM7, GPA accurately matches NRHA

demands. However, increased plastic rotations at the base influence the distribution

of plastic deformations at the upper stories during NRHA. For GM3, this situation is

clearly observed where the nonlinear vibration properties change significantly during

the ground motion excitation. Therefore, GPA cannot estimate NRHA results

accurately for GM3.

Figure 4.17 Maximum values of average beam end plastic rotations under

seven ground motions

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015

Sto

ry #


GM1

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020 S

tory

#


GM2

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015

Sto

ry #


GM4

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015

Sto

ry #


GM6

50


It can be observed from Figure 4.18 that chord rotations of column ends at

the first story level become larger due to reduced stiffness of that story when

compared to rotation values in case study I. Although the first story plastic rotation

values for each ground motion slightly exceed yield rotation limits, significant plastic

actions do not develop at column ends. Reduction in column stiffness especially for

the first story columns may violate the strong column-weak beam principle, and

columns at this story under GM1, GM2 and GM7 are close to the yield state.

Generally, GPA and MPA predict NRHA results quite well. However, GPA

improves the estimation of the first story and the top stories rotations especially for

GM2, GM6 and GM7. It can be suggested that GPA and MPA as approximate

methods predict elastic column chord rotation demands successfully for second and

upper stories whereas GPA shows some improvement for the predictions of the

inelastic rotations at the first story.

Figure 4.18 Maximum values of average column end chord rotations under

seven ground motions with yield rotation limits

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


NRHA GPA MPA

GM7

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010

Sto

ry #


GM1

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010

Sto

ry #


GM2

51



Internal forces of beams and columns are the envelopes of the results

obtained by pushing the system in both directions. Maximum values of bending

moments and shear forces at the ends of 1st story, 5

th story and 10

th story beams are

shown in Figure 4.19 and Figure 4.20 respectively, for each analysis case. Yield

moment values for each beam type are the same with beams in the full capacity

model, and they are given in Table 4.3. Maximum shear forces and bending moments

for columns are obtained along the left exterior column axis of the inner frame of the

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010

Sto

ry #


GM4

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010

Sto

ry #


GM6

0 1 2 3 4 5 6 7 8 9

10 11 12

0.000 0.005 0.010

Sto

ry #



GM7

52

plane model and they are shown respectively in Figure 4.21 and Figure 4.22. Yield

moment values are also plotted in Figure 4.21 with the column bottom end moments.

It is observed from Figure 4.19 and Figure 4.20 that GPA estimates beam end

forces computed by NRHA almost exactly. On the contrary, MPA overestimates

shear forces and bending moments at beam ends, especially for the 1st and 10

th

stories. Errors of MPA results are very high especially for the 10th

story responses,

and show significant changes for bending moments and shear forces. The error of

MPA in predicting the member internal forces varies from 8% to 60% with respect to

NRHA results. However, the maximum error between GPA results and NRHA

results is about 5%. The results of MPA are not consistent with the given member

capacities in Table 4.2; therefore, MPA requires the correction of internal forces

suggested by Goel and Chopra, (2005).

1st Story 5

th Story 10

th Story

Figure 4.19 Maximum bending moment values of beam ends at 1st, 5th, and 10th stories

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM1

NRHA

GPA

MPA

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM1

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM1

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM3

6 m 4 m 6 m

53


0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM7

6 m 4 m 6 m

54

1st Story 5

th Story 10

th Story

Figure 4.20 Maximum shear force values of beam ends at 1st, 5th, and 10th stories

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM1

NRHA

GPA

MPA

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM1

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM1

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM5

6 m 4 m 6 m

55

Bottom-end bending moments of the left exterior column axis is presented in

Figure 4.21. GPA and MPA predictions are close to each other, and they estimate

NRHA results quite well especially for lower story levels. The difference between

the NRHA results and MPA and GPA estimations become larger between 5th

and 9th

stories, however the moments are small. On the other hand, GPA improves MPA

results for the first story moment values because MPA overestimates the related

NRHA values for GM2, GM4 and GM7. The error of GPA for the first story moment

is about 10%, and this error becomes 20% for MPA. For shear forces along the left

column axis given in Figure 4.22, the accuracy of GPA and MPA results are almost

the same for the first five stories. However, for upper stories GPA is quite accurate in

the estimation of shear forces. It is observed from Figure 4.22 that GPA generally

predicts the NRHA shear force distribution quite well.


0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

GM7

6 m 4 m 6 m

56

Figure 4.21 Maximum bottom end bending moments along the left exterior

column line of the inner frame with yield moments

0 1 2 3 4 5 6 7 8 9

10 11 12

0 200 400 600

Sto

ry

#

Moment (kN.m)

GM1

0 1 2 3 4 5 6 7 8 9

10 11 12

0 200 400 600

Sto

ry

#

Moment (kN.m)

GM2

0 1 2 3 4 5 6 7 8 9

10 11 12

0 200 400 600

Sto

ry

#

Moment (kN.m)

GM3

0 1 2 3 4 5 6 7 8 9

10 11 12

0 200 400 600

Sto

ry

#

Moment (kN.m)

GM4

0 1 2 3 4 5 6 7 8 9

10 11 12

0 200 400 600

Sto

ry

#

Moment (kN.m)

GM5

0 1 2 3 4 5 6 7 8 9

10 11 12

0 200 400 600

Sto

ry

#

Moment (kN.m)

GM6

0 1 2 3 4 5 6 7 8 9

10 11 12

0 200 400 600

Sto

ry

#

Moment (kN.m)

NRHA GPA

MPA Yield Moment

GM7

57

Figure 4.22 Maximum shear forces along the left exterior column line of the

inner frame

0 1 2 3 4 5 6 7 8 9

10 11 12

0 50 100 150 200

Sto

ry

#

Shear Force (kN)

GM1

0 1 2 3 4 5 6 7 8 9

10 11 12

0 50 100 150 200

Sto

ry

#

Shear Force (kN)

GM2

0 1 2 3 4 5 6 7 8 9

10 11 12

0 50 100 150 200

Sto

ry

#

Shear Force (kN)

GM3

0 1 2 3 4 5 6 7 8 9

10 11 12

0 50 100 150 200

Sto

ry

#

Shear Force (kN)

GM4

0 1 2 3 4 5 6 7 8 9

10 11 12

0 50 100 150 200

Sto

ry

#

Shear Force (kN)

GM5

0 1 2 3 4 5 6 7 8 9

10 11 12

0 50 100 150 200

Sto

ry

#

Shear Force (kN)

GM6

0 1 2 3 4 5 6 7 8 9

10 11 12

0 50 100 150 200

Sto

ry

#

Shear Force (kN)

NRHA GPA MPA

GM7

58

4.3 Case Study III: Twenty Story RC Wall-Frame System with Full

Capacity Design


In the third case study, a twenty story reinforced concrete wall-frame building

with symmetrical plan is utilized. The floor plan is identical to the plan of twelve

story buildings employed in the previous case studies. Two shear walls with 30 cm

thickness are placed along the middle bays of interior frames in the direction of

ground motion excitation. Reinforcement details of shear walls are conducted with

respect to TEC (2007). Figure 4.23 shows the floor plan of the building. Similar to

the first case study, this structure is designed according to TS500 (2000) and TEC

(2007) by considering the capacity design principles. Therefore, it is called “full

capacity design” and an enhanced ductility level is employed. In a similar way, the

design parameters selected in view of TEC (2007) can be listed as seismic zone 1,

soil type Z3 and residential use. The member dimensions for beams are 30x55 cm2

for the first six stories, 30x50 cm2

for the second six stories, and 30x45 cm2

for the

last eight stories. The columns dimensions are 55x55 cm2, 50x50 cm

2, and 45x45

cm2 in the first six, the second six, and the last eight stories, respectively. The story

heights are 4 m for the first story, and 3.2 m for all other stories, which is similar to

the building in the first case study. There is no basement, and story levels start from

the ground level. The frames used in the analytical models are shown in Figure 4.24.

Section details of beams and columns are presented in Figure 4.25, and section

details of shear walls are shown in Figure 4.26. Shear design details of each beam

and column section are given in Table 4.5. Concrete and steel characteristic strengths

are 35 MPa and 420 MPa, respectively.

59

Figure 4.23 Story plan of the twenty story building with full capacity design

Figure 4.24 Plane (2D) model of the twenty story building with full capacity design

6 m 4 m 6 m

5 m

5 m

5 m

Frame B

Frame A

Frame A Frame B

6 m 6 m 4 m 6 m 6 m 4 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

3.2 m

4.0 m

60





4Ф20

45 c

m

30 cm

2Ф20

30 cm

2Ф22

55 c

m

4Ф22

30 cm

50 c

m

2Ф20

4Ф20

30 cm

4Ф22

55 c

m

2Ф22

50 c

m

8Ф24

50 cm

45 c

m

8Ф24

45 cm

8Ф24

55 c

m

55 cm

45 c

m

30 cm

4Ф20

2Ф20

30 cm

4Ф20

2Ф20

50 c

m

61

a) For H < Hcr (Hcr =7.2 m) b) For H > Hcr

Figure 4.26 Shear wall section details

Table 4.5 Shear design details of columns and beams

Along End Region Along Span Region

Columns

55x55 cm2 Ф8 / 10 cm Ф8 / 20 cm

50x50 cm2 Ф8 / 10 cm Ф8 / 20 cm

45x45 cm2 Ф8 / 10 cm Ф8 / 20 cm

Beams

55x30 cm2 Ф8 / 10 cm Ф8 / 15 cm

50x30 cm2 Ф8 / 10 cm Ф8 / 15 cm

45x30 cm2 Ф8 / 10 cm Ф8 / 15 cm

30 cm

400 c

m

80 c

m

80 c

m

240 c

m

2 x

6Ф

16/1

2 c

m

2 x

6Ф

16/1

2 c

m

2 x

12Ф

14/2

0 c

m

30 cm

400 c

m

40 c

m

40 c

m

320 c

m

2 x

4Ф

16/1

2 c

m

2 x

4Ф

16/1

2 c

m

2 x

16Ф

14/2

0 c

m

62

4.3.2 Modeling

Analytical model of the twenty story frame is generated by using the

OpenSees software (2005). All nonlinear and linear analyses are conducted with this

program. Structural frame members are modeled with “Beam with Hinges” element.

In this sense, plastic hinge lengths are defined at both ends of all columns, beams and

shear wall story segments. For beam sections, bi-linear moment curvature

relationships are defined along the hinge lengths by considering the designed section

properties of each beam type. On the other hand, column sections are defined by

utilizing fiber sections along the hinge lengths. The remaining part in between the

plastic hinges of columns and beams is defined with elastic cracked section

properties. For shear wall ends within Hcr, extra nodes are defined at the midpoint of

story segments of the wall in order to reflect the potential plastic behavior, and

plastic hinges are specified along the distance between successive nodes within Hcr.

For shear wall story segments above Hcr, plastic hinges are identified with very small

lengths and the remaining parts between plastic hinges are defined with cracked

section properties of the shear wall. Along plastic hinges of shear wall ends, bi-linear

moment curvature relationship with shear aggregator is defined. In addition to that,

rigid links are defined on the top end of each shear wall story segment. In order to

describe cracked section stiffness for each member, the gross moment of inertia

values are multiplied by 0.4, 0.6, and 0.8 for beams, columns and shear wall

respectively. Rigid diaphragms are assigned to each story level, and P-∆ effects are

taken into account in the model. In time history analyses, Rayleigh damping is

computed by considering 5% damping in the 1st and 4

th modes.


Free vibration properties of the twenty story frame model are obtained from

the elastic model with cracked section properties. Free vibration properties of the

first four modes are given in Table 4.6. The modal mass ratios and the scaled top

story amplitudes of the mode shape vectors in Table 4.6 indicate the effects of

second, third and fourth modes of the system. Figure 4.27 shows the first four mode

vectors of the model, normalized with respect to mass.

63

Table 4.6 Free vibration properties of the twenty story plane model for the first four

modes


Figure 4.27 Mode shapes for the first four modes


In addition to the results presented in the previous case studies for the ground

motion set given in Chapter 3, maximum average plastic rotations of shear wall ends

at each story level, and maximum average bending moments along the shear wall are

also presented. The utilized procedure in the previous cases for obtaining maximum

average plastic rotations of beams and maximum average chord rotations of columns

is also conducted for shear wall plastic rotations. Accordingly, maximum average

plastic rotations of shear wall ends at each story level are calculated by first

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

-0.06 -0.04 -0.02 0 0.02 0.04

Sto

ry #


Mode 1

Mode 2

Mode 3

Mode 4


Mass (ton)

Effective Modal

Mass Ratio Γn φnr

(*)

1 2.60 1127.1 0.70 1.44

2 0.72 233.2 0.14 -0.70

3 0.31 99.6 0.06 0.43

4 0.17 51.0 0.03 -0.29

64

averaging end rotations at the corresponding story, and then taking the maximum of

these values as the story maximum. The results of nonlinear response history analysis

(NRHA), modal pushover analysis (MPA) and generalized pushover analysis (GPA)

are also compared with each other for each ground motion in the set. The target drift

in GPA is determined by employing the inelastic maximum displacement amplitude

for the first mode in Equation (14).


Maximum interstory drift ratios obtained from the three procedures (NRHA,

MPA and GPA) are presented comparatively for each ground motion in Figure 4.28.

Utilizing shear walls with code based design are very effective in reducing the drift

demands when compared to the previous frame models. The maximum drift ratio

value is close to 1% for each ground motion, and the interaction between frame and

shear wall systems can be seen easily from the given profiles. From the results

obtained from three procedures, it can be observed that GPA predicts NRHA results

almost exactly for all ground motions. Pushing the system to roof displacement in

MPA does not lead to good estimations of interstory drifts. Considering all

significant modes in the response provides improved accuracy to GPA for estimating

NRHA results.

Figure 4.28 Maximum interstory drift ratios obtained under seven ground

motions

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


GM2

65


0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010

Sto

ry #


GM4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010

Sto

ry #


GM6

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


NRHA GPA MPA

GM7

66


Maximum average beam-end plastic rotations, maximum average column-end

chord rotations and maximum average plastic rotations at the shear wall for each

ground motion are given in this section. For beams and columns, results are

presented separately for each frame (Figure 4.24). Figure 4.29 and Figure 4.30

presents maximum average beam-end plastic rotations for Frame A and Frame B

respectively. Maximum average column-end chord rotations are then shown in

Figure 4.31 and Figure 4.32 respectively for each frame. Maximum average plastic

rotations along the shear wall are given only for Frame A in Figure 4.33. Similarly,

chord rotations are calculated for column ends instead of plastic rotations for

comparison, because plastic rotations are too small to be compared accurately. Yield

rotation values for columns are also shown with the results obtained in Figure 4.31

and Figure 4.32. In yield rotation calculations for columns, axial loads required were

taken from the gravity load analysis of the system. It can be observed that rotation

values at column ends become smaller due to the presence of shear wall.

It is expected that maximum average beam-end plastic rotations in Frame A

are larger than Frame B due to effect of the shear wall behavior on beam rotations in

Frame A. In Figure 4.29 and Figure 4.30, maximum plastic rotation values for beams

in Frame A are about 0.013 radian; however, for Frame B maximum plastic rotation

values are about 0.08 radian. Generally, beam plastic rotations obtained from NRHA

are estimated well by GPA for Frame A and Frame B. MPA gives reasonable

demands only for lower stories for both frames with respect to NRHA results.

However, MPA cannot predict maximum rotation values at upper stories, especially

after the 8th

story, and gives underestimated values. GPA gives underestimated

results only for GM1, but the difference between GPA and NRHA results is in the

acceptable tolerance limit. Maximum error of GPA is about 12%.

67

Figure 4.29 Maximum values of average beam-end plastic rotations in Frame A

under seven ground motions

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


GM2

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


GM4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


GM6

68


Figure 4.30 Maximum values of average beam-end plastic rotations in Frame B

under seven ground motions

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015 S

tory

#


NRHA GPA MPA

GM7

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006 0.008

Sto

ry #


GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006 0.008 0.010

Sto

ry #


GM2

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006 0.008

Sto

ry #


GM4

69


It can be observed from Figure 4.31 and Figure 4.32 that maximum average

chord rotations of columns are very small, and do not exceed yield rotation limits.

Therefore, plastic actions do not develop at column ends. This condition is generally

expected for wall-frame systems. Generally, GPA and MPA estimate NRHA results

with high accuracy. Estimated values for the first story, middle stories and upper

stories are close to the obtained demands from NRHA. In Figure 4.32 presenting

Frame B results, MPA gives slightly underestimated results for upper stories with

respect NRHA results. GPA predictions are accurate for those stories in Frame B.

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006

Sto

ry #


GM6

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006 0.008

Sto

ry #


NRHA GPA MPA

GM7

70

Figure 4.31 Maximum values of average column-end chord rotations in Frame

A under seven ground motions

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003 0.004 0.005

Sto

ry #


GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003 0.004 0.005

Sto

ry #


GM2

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003 0.004 0.005

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003 0.004 0.005

Sto

ry #


GM4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003 0.004 0.005

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003 0.004 0.005

Sto

ry #


GM6

71


Figure 4.32 Maximum values of average column-end chord rotations in Frame

B under seven ground motions

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006 S

tory

#



GM7

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003 0.004 0.005

Sto

ry #


GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003 0.004 0.005

Sto

ry #


GM2

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003 0.004 0.005

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003 0.004 0.005

Sto

ry #


GM4

72


It can be expected for plastic rotations along the shear wall story segments

that they are generally accumulated within the Hcr region, which is the most critical

part for ductile shear wall behavior. In Figure 4.33, the maximum rotations are at the

first story level, and rotation values of the upper stories are smaller when compared

to the first story. Formation of plastic rotations in upper stories in NRHA results is

due to the instant response of the system under a ground motion excitation, and they

slightly affect behavior of the shear wall during an earthquake. Therefore,

performance test of GPA and MPA with respect to NRHA can be conducted for the

first story estimation. GPA predictions for the first story rotations are quite well. The

difference between GPA and NRHA results is larger in GM1 than for the other

ground motion results. MPA gives underestimated demands for the first story

rotations, and the accuracy of MPA is lower. Neither GPA nor MPA can capture the

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003 0.004 0.005

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003 0.004 0.005

Sto

ry #


GM6

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006

Sto

ry #



GM7

73

plastic rotation profiles along the upper stories. GPA captures formation of plastic

rotations at the upper stories under GM1 and GM6.

Figure 4.33 Maximum values of average plastic rotations of the shear wall in

Frame A under seven ground motions

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.0000 0.0005 0.0010 0.0015 0.0020

Sto

ry #


GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.0000 0.0005 0.0010 0.0015 0.0020

Sto

ry #


GM2

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.0000 0.0005 0.0010 0.0015

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.0000 0.0005 0.0010 0.0015 0.0020

Sto

ry #


GM4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.0000 0.0005 0.0010 0.0015

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.0000 0.0005 0.0010 0.0015

Sto

ry #


GM6

74



Maximum values of bending moments at the ends of 1st story, 5

th story and

10th

story beams in Frame A and Frame B are shown in Figure 4.34 and Figure 4.35,

respectively. In addition to beam end moments, maximum values of shear forces at

beam ends for the same stories in Frame A and Frame B are shown in Figure 4.36

and Figure 4.37. Table 4.7 shows yield moment values for each beam type. The

maximum shear forces and bending moments for columns are obtained along the left

exterior column axes of Frame A and Frame B. Maximum moments along column

lines for Frame A and Frame B are presented in Figure 4.38 and Figure 4.39,

respectively. Yield moment values of columns are also plotted in these figures.

Maximum shear forces along column lines are given in Figure 4.40 and Figure 4.41

for Frame A and Frame B. Similarly, the maximum bending moments and along the

shear wall line in Frame A are given in Figure 4.42. Similar to the previous case

studies, member internal force results of MPA and GPA are the envelopes of the

results obtained by pushing the system in both directions.

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.0000 0.0005 0.0010 0.0015 0.0020 S

tory

#


NRHA GPA MPA

GM7

75

Table 4.7 Negative yield moment values for beam ends (tension at top)

Maximum beam end forces at three different story levels are presented

separately for each frame. It is expected that maximum forces attained at beam ends,

which are connected to the shear wall in Frame A, are larger than forces obtained for

other ends. According to maximum bending moment values of beam ends given in

Figure 4.34 and Figure 4.35, GPA estimates NRHA results reasonably well for both

frames. However, MPA gives overestimated demands for Frame A, and estimates

NRHA results for Frame B slightly better than Frame A predictions. Maximum

average shear force values at beam ends are also given in Figure 4.36 and Figure

4.37 for each frame. Similar to the bending moment results, GPA estimates

maximum shear forces at the beam ends almost exactly for both frames with respect

to NRHA results. MPA again shows different estimation accuracy for maximum

beam end shear forces in each frame. Estimation accuracy of MPA is higher for

Frame B, and Frame A results are overestimated. It can be observed that MPA results

for beam end forces in Frame A are not consistent with member capacities.

Therefore, MPA requires the correction for beam forces in Frame A suggested by

Goel and Chopra, (2005).

Story Range Beam Dimension Yield Moment, My (kN.m)

1st-6

th 30x55 cm2 231.5

7th

-12th 30x50 cm

2 230.5 13

th-20

th 30x45 cm2 204.1

1st Story 5

th Story 10

th Story

Figure 4.34 Maximum bending moment values of beam ends at 1st, 5th, and 10th stories in

Frame A

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM1

NRHAGPAMPA

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM1

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM1

6 m 4 m 6 m

76


0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM6

6 m 4 m 6 m

77


1st Story 5

th Story 10

th Story

Figure 4.35 Maximum bending moment values of beam ends at 1st, 5th, and 10th stories in

Frame B

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM1

NRHAGPAMPA

6 m 4 m 6 m

0

50

100

150

200

250

300M

om

ent (

kN

.m)

GM1

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM1

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM3

6 m 4 m 6 m

78


0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

300

Mo

men

t (k

N.m

)

GM7

6 m 4 m 6 m

79

1st Story 5

th Story 10

th Story

Figure 4.36 Maximum shear force values of beam ends at 1st, 5th, and 10th stories in

Frame A

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM1

NRHAGPAMPA

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM1

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM1

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM2

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM3

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM4

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM4

6 m 4 m 6 m

80


1st Story 5

th Story 10

th Story

Figure 4.37 Maximum shear force values of beam ends at 1st, 5th, and 10th stories in

Frame B

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM5

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM6

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM7

6 m 4 m 6 m

0

50

100

150

200

250

300

Sh

ear

(kN

)

GM7

66 m 44 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM1

NRHAGPAMPA

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM1

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM1

6 m 4 m 6 m

81


0

50

100

150

200

Sh

ear

(kN

)

GM2

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM2

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM2

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM3

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM3

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM3

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM4

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM4

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM4

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM5

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM5

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM5

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM6

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM6

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM6

6 m 4 m 6 m

82

Maximum bottom end bending moments and maximum shear forces along

the left exterior column axis of each frame are presented separately for the

comparison of maximum column internal forces. Maximum bottom end moments

along the left exterior column axis in Frame A and Frame B are given in Figure 4.38

and Figure 4.39, respectively. Yield moment values are also plotted in the figures.

GPA captures NRHA profiles almost exactly for each frame. However, MPA

predictions only for lower stories in both frames, up to the 7th

story level, are in

acceptable tolerance limit. MPA gives underestimated values for the upper stories

when compared to NRHA results. For the first story column moments, GPA

generally gives closely estimated moment values; however, MPA generally

underestimates the demand.

It can be observed from Figure 4.40 and Figure 4.41 that GPA gives accurate

estimations of NRHA results of maximum shear forces along the column line in both

frames. Although GPA estimates the first story shear force demands with lower

accuracy for GM1, GM2 and GM4, it can be said that NRHA shear force profile

along the column axis are captured almost exactly by GPA. In shear force

comparison of columns, MPA cannot give reasonable demands for any story levels,

and these demands are generally underestimated. It is clear that GPA shows great

improvement for the estimation of column internal forces with respect to MPA.


0

50

100

150

200

Sh

ear

(kN

)

GM7

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM7

6 m 4 m 6 m

0

50

100

150

200

Sh

ear

(kN

)

GM7

6 m 4 m 6 m

83


axis in Frame A

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800

Sto

ry

#

Moment (kN.m)

GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800

Sto

ry

#

Moment (kN.m)

GM2

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800

Sto

ry

#

Moment (kN.m)

GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800

Sto

ry

#

Moment (kN.m)

GM4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800

Sto

ry

#

Moment (kN.m)

GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800

Sto

ry

#

Moment (kN.m)

GM6

84



axis in Frame B

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800 S

tory

#

Moment (kN.m)

NRHA GPA

MPA Yield Moment

GM7

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800

Sto

ry

#

Moment (kN.m)

GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800

Sto

ry

#

Moment (kN.m)

GM2

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800

Sto

ry

#

Moment (kN.m)

GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800

Sto

ry

#

Moment (kN.m)

GM4

85


Figure 4.40 Maximum shear forces along the left exterior column axis in Frame A

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800

Sto

ry

#

Moment (kN.m)

GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800

Sto

ry

#

Moment (kN.m)

GM6

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400 500 600 700 800

Sto

ry

#

Moment (kN.m)

NRHA GPA

MPA Yield Moment

GM7

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 50 100 150 200

Sto

ry

#

Shear (kN)

GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 50 100 150

Sto

ry

#

Shear (kN)

GM2

86


0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 50 100 150

Sto

ry

#

Shear (kN)

GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 50 100 150

Sto

ry

#

Shear (kN)

GM4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 50 100 150

Sto

ry

#

Shear (kN)

GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 50 100 150

Sto

ry

#

Shear (kN)

GM6

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 50 100 150

Sto

ry

#

Shear (kN)

NRHA GPA MPA

GM7

87

Figure 4.41 Maximum shear forces along the left exterior column axis in Frame B

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200

Sto

ry

#

Shear (kN)

GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200

Sto

ry

#

Shear (kN)

GM2

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200

Sto

ry

#

Shear (kN)

GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200

Sto

ry

#

Shear (kN)

GM4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200

Sto

ry

#

Shear (kN)

GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200

Sto

ry

#

Shear (kN)

GM6

88


According to the maximum bending moment distribution along the shear wall

axis in Frame A presented in Figure 4.42, estimation accuracies of GPA and MPA

are close to each other, excepting the segments along the critical height. GPA

estimates NRHA results reasonably well along the critical height, which includes the

first two stories. MPA gives overestimated demands for this region. GPA estimates

NRHA result almost exactly in the critical region and the maximum error is about

4%. Plastic deformation information of the shear wall in Frame A is described in

Section 4.3.4.2. Due to plastic deformation accumulation along the critical region

(Figure 4.33), capturing maximum moment values in this region is important the

design purposes. Therefore, it can be suggested that GPA estimations for maximum

plastic deformations and moments for shear wall are consistent and accurate.

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 50 100 150 S

tory

#

Shear (kN)

NRHA GPA MPA

GM7

89

Figure 4.42 Maximum bending moment values along shear wall in Frame A,

(My=7800 kN.m in Hcr)

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 5000 10000 15000 20000

Sto

ry

#

Moment (kN.m)

GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 5000 10000 15000 20000

Sto

ry

#

Moment (kN.m)

GM2

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 5000 10000 15000

Sto

ry

#

Moment (kN.m)

GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 5000 10000 15000 20000

Sto

ry

#

Moment (kN.m)

GM4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 5000 10000 15000 20000

Sto

ry

#

Moment (kN.m)

GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 5000 10000 15000

Sto

ry

#

Moment (kN.m)

GM6

90


0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 5000 10000 15000 20000

Sto

ry

#

Moment (kN.m)

NRHA GPA MPA

GM7

91

CHAPTER 5

GENERALIZED PUSHOVER ANALYSIS WITH THE MEAN

SPECTRUM OF A SET OF GROUND MOTIONS

In this chapter, the GPA procedure is examined in a different way. For each

case study, GPA is conducted by using the mean spectrum of the ground motion set.

These results are then compared with the mean results of NRHA, GPA and MPA,

respectively, obtained under each ground motion separately for each case study.

5.1 Case Study I: Twelve Story RC Frame with Full Capacity Design

Detailed description of the model employed in this case study was given in

the previous Chapter (Section 4.1). The responses obtained for each ground motion

in Chapter 4 for Case Study I are used to obtain the mean values of responses under

the seven ground motions. Maximum interstory drift ratios, maximum average

plastic rotations of beam ends in a story, maximum average chord rotations of

column ends in a story, maximum beam end moments and shear forces, maximum

column end moments and shear forces were already presented in the previous chapter

for each ground motion in the set. In this section, the mean values of NRHA, GPA

and MPA results are calculated and compared with the results of GPA obtained

under the mean spectrum. In the target drift calculation under mean spectrum, elastic

spectral displacement amplitude is used for the first mode in Equation (14) since

the mean spectrum can be obtained for the elastic spectra of seven ground motions.

Mean responses from NRHA, GPA and MPA and the responses obtained

from GPA under the mean spectrum are presented comparatively in Figure 5.1 for

interstory drift ratios, beam end plastic rotations and column end chord rotations. It is

observed that the GPA results obtained under mean elastic spectrum are very close to

92

the mean of GPA results obtained under seven ground motions. These results predict

the mean NRHA results quite well. Slight differences between mean GPA results and

GPA results obtained under mean elastic spectrum is due to using different

displacement amplitudes in target drift estimations. Similar to the GPA results

presented in the previous chapter, GPA results obtained under mean spectrum exhibit

great improvement in predicting the NRHA demands at the upper stories. Generally,

it can be suggested that GPA results obtained for mean elastic spectrum are well

synchronized with the mean GPA and mean NRHA results for interstory drift ratios

and member end rotations.

Figure 5.1 Comparison of mean maximum interstory drifts, mean maximum

average plastic rotations of beam ends and mean maximum average chord

rotations of column ends obtained with NRHA, MPA and GPA, with the GPA

results obtained under mean elastic spectrum.

Mean values of member internal forces obtained from NRHA, MPA and

GPA, and the GPA results obtained under the mean elastic spectrum are presented in

Figure 5.2, Figure 5.3, and Figure 5.4 for shear forces and bending moments in

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015

Sto

ry #

Beam-End Plastic Rotations (rad)

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.002 0.004 0.006 0.008

Sto

ry #

Column End Chord Rotations (rad)

Mean NRHA Mean GPA

Mean MPA GPA MEAN SPEC

93

columns and beams, respectively. Similar to the previous comparisons, GPA under

mean elastic spectrum captures mean GPA values almost exactly. In addition, both

GPA results for column forces are well synchronized with the mean NRHA values

and they exactly estimate the mean NRHA results obtained for beam end forces.

Mean MPA results for column bottom end moments and beam end forces at the 5th

story are in acceptable tolerances. Mean MPA results cannot provide reasonable

estimates of member internal forces at the lower and upper stories where the second

mode effect is significant. For column shear forces, mean GPA results and the GPA

results obtained under mean elastic spectrum show significant improvement for

upper stories compared to MPA.


bending moments along the left exterior column axis of the inner frame

1st Story 5

th Story

Figure 5.3 Comparison of maximum bending moment values of beam ends at

the 1st, 5th, and 10th stories

0

1

2

3

4

5

6

7

8

9

10

11

12

0 50 100 150

Sto

ry

#

Shear Force (kN)

0

1

2

3

4

5

6

7

8

9

10

11

12

0 200 400 600

Sto

ry

#

Moment (kN.m)

NRHA

GPA

MPA

GPA MEAN SPEC

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

6 m 4 m 6 m

94

10th

Story


1st Story 5

th Story

10th

Story

Figure 5.4 Comparison of maximum shear force values of beam ends at the 1st,

5th, and 10th stories

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

Mean NRHA

Mean GPA

Mean MPA

Mean Spec GPA

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

Mean NRHA

Mean GPA

Mean MPA

Mean Spec GPA

6 m 4 m 6 m

95

5.2 Case Study II: Twelve Story RC Frame with Relaxed Capacity

Design

Description and details of the model employed in case study II were given in

Section 4.2. The results obtained for each ground motion in Chapter 4 are used to

compute mean values of responses under the seven ground motions. Mean values of

NRHA, GPA and MPA results for interstory drifts, member end rotations and

member internal forces are calculated and compared to the GPA results obtained

under the mean spectrum. As indicated previously, in the target drift calculation GPA

under mean spectrum, elastic modal displacement amplitude is used in Equation

(14) since the mean spectrum can be obtained for the elastic spectra of seven ground

motions.

Mean responses of NRHA, GPA and MPA for interstory drift ratios, beam

end plastic rotations and column end chord rotations are compared with the

responses obtained from GPA under the mean spectrum in Figure 5.5. As in the

previous case, the GPA results obtained under mean elastic spectrum and the mean

values of GPA results for the seven ground motions are almost equal to each other.

Furthermore, both of them capture mean NRHA results quite well. There is a slight

difference between the two GPA results due to using different displacement

amplitudes in target drift estimations. Similar to the GPA results obtained in the

previous chapter, GPA target demands obtained under mean spectrum show great

improvement in predicting the NRHA demands at the upper stories. Mostly, it can be

seen that GPA results obtained under mean elastic spectrum are well representing the

mean values of GPA and NRHA results for interstory drift ratios and member end

rotations.

96

Figure 5.5 Comparison of mean maximum interstory drifts, mean maximum

average plastic rotations of beam ends and mean maximum average chord

rotations of column ends obtained with NRHA, MPA and GPA, with the GPA

results obtained under mean elastic spectrum.

Mean values of member internal forces obtained from NRHA, MPA and GPA

under seven ground motions, and the GPA results obtained under the mean elastic

spectrum are shown comparatively in Figure 5.6, Figure 5.7 and Figure 5.8. Similar

to comparisons in the previous case, GPA under mean elastic spectrum and mean

values of GPA results are almost equal to each other. Both GPA results are well

synchronized with the mean NRHA forces. For bending moments along the column

axis, mean values of MPA and GPA results, and GPA results under the mean elastic

spectrum exhibit the same accuracy for estimating the mean NRHA values.

However, mean values of MPA results for column shear forces are not accurate. It

can be seen that both GPA demands show improvement for upper story predictions

for column forces. GPA under mean elastic spectrum and mean values of GPA

results for beam end forces estimate mean NRHA demands exactly according to

0

1

2

3

4

5

6

7

8

9

10

11

12

0.0000 0.0050 0.0100 0.0150 0.0200

Sto

ry #


0

1

2

3

4

5

6

7

8

9

10

11

12

0.0000 0.0050 0.0100 0.0150

Sto

ry #

Beam End Plastic Rotations (rad)

0

1

2

3

4

5

6

7

8

9

10

11

12

0.0000 0.0020 0.0040 0.0060 0.0080

Sto

ry #


NRHA GPA

MPA GPA MEAN SPEC

97

Figure 5.7 and Figure 5.8. As stated previously, MPA results for beam end forces

require correction according to member capacities (Goel and Chopra, 2005) .


bending moments along the left exterior column axis of the inner frame

1st Story 5

th Story

10th

Story


the 1st, 5th, and 10th stories

0

1

2

3

4

5

6

7

8

9

10

11

12

0 50 100 150

Sto

ry

#

Shear Force (kN)

0

1

2

3

4

5

6

7

8

9

10

11

12

0 200 400 600

Sto

ry

#

Moment (kN.m)

NRHA

GPA

MPA

GPA MEAN SPEC

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

Mean NRHA

Mean GPA

Mean MPA

Mean Spec GPA

6 m 4 m 6 m

98

1st Story 5

th Story

10th

Story

Figure 5.8 Comparison of maximum shear force values of beam ends at the 1st,

5th, and 10th stories

5.3 Case Study III: Twenty Story RC Wall-Frame System with Full

Capacity Design

Detailed description of the twenty story building model employed in the third

Case Study was given in Chapter 4 (Section 4.3). Results obtained for each ground

motion in Chapter 4 are utilized to calculate the mean values of results under the

seven ground motions. Maximum interstory drift ratios, maximum average plastic

rotations of beam ends in a story, maximum average chord rotations of column ends

in a story, maximum plastic rotations along the shear wall, maximum beam end

moments and shear forces, maximum column end moments and shear forces, and

maximum bending moments along the shear wall were already presented for this

case study in the previous chapter. In this section, mean values of NRHA, GPA and

MPA responses obtained under seven ground motions are compared with the results

of GPA obtained under the mean spectrum. In the target drift calculation under mean

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce (

kN

)

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

Mean NRHA

Mean GPA

Mean MPA

Mean Spec GPA

6 m 4 m 6 m

99

elastic spectrum, elastic spectral displacement amplitude is used for the first

mode in Equation (14) since the mean spectrum is obtained from the elastic spectra

of seven ground motions.

Mean values of NRHA, GPA and MPA results under seven ground motions

and GPA results under the mean spectrum of seven ground motions are presented

comparatively in Figure 5.9 for interstory drift ratios and plastic rotations at shear

wall ends. Additionally, mean beam-end plastic rotations and column end chord

rotations are presented for Frame A and Frame B in Figure 5.10, and Figure 5.11,

respectively. It can be observed that GPA results under the mean spectrum are almost

equal to the mean values of GPA results, and both GPA results estimate the mean

NRHA values quite well. As in previous cases, there is a slight difference between

mean GPA results and the GPA results under the mean spectrum This difference

arises from using different displacement amplitudes (inelastic and linear elastic) for

the first mode in target drift calculations. Similar to the GPA results for Case Study

III in the previous chapter, GPA results under the mean spectrum show great

improvement in estimating the NRHA results at the upper stories compared to MPA.

It can be expressed generally that GPA results obtained under mean elastic spectrums

are well synchronized with mean GPA and mean NRHA results for interstory drift

ratios and member end rotations.

Figure 5.9 Comparison of the mean maximum interstory drifts and the mean maximum

average plastic rotations of the shear wall obtained with NRHA, MPA and GPA under

seven ground motions, with the GPA results obtained under the mean elastic spectrum

of seven ground motions.

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006 0.008 0.010

Sto

ry #


0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.0000 0.0005 0.0010 0.0015

Sto

ry #

Shear Wall Ends Plastic Rotations (rad)

NRHA

GPA

MPA

Mean Spec GPA

100

Figure 5.10 Comparison of the mean maximum average plastic rotations of

beam ends and the mean maximum average chord rotations of column ends in

Frame A obtained with NRHA, MPA and GPA under seven ground motions,

with the GPA results obtained under the mean elastic spectrum of seven ground

motions.

Figure 5.11 Comparison of the mean maximum average plastic rotations of

beam ends and the mean maximum average chord rotations of column ends in

Frame B obtained with NRHA, MPA and GPA under seven ground motions,

with the GPA results obtained under the mean elastic spectrum of seven ground

motions.

Mean values of member internal forces obtained from NRHA, GPA, MPA

under seven ground motions and the GPA results obtained under the mean elastic

spectrum of seven ground motions are presented comparatively for each frame

separately. Mean values of maximum shear forces and bending moments along the

left exterior column line are shown in Figure 5.12 and Figure 5.13 for Frame A and

Frame B respectively. Mean values of maximum beam end moments and shear

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006 0.008 0.010

Sto

ry #


0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003

Sto

ry #


NRHA

GPA

MPA

Mean Spec GPA

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006

Sto

ry #


0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003

Sto

ry #


NRHA

GPA

MPA

Mean Spec GPA

101

forces for each frame are given in Figure 5.14, Figure 5.15, Figure 5.16 and Figure

5.17. Mean maximum bending moment values along the shear wall axis in Frame A

are also given in Figure 5.18. It is obvious from these figures that GPA results with

mean spectrum and the mean GPA results under seven ground motions give almost

the same values. Column forces and beam end forces from mean GPA and GPA

results obtained under mean elastic spectrum capture exactly mean NRHA demands.

The maximum difference between them is about 6%. In addition, for moment values

along the shear wall, both GPA results are equal to each other, and they estimate

mean NRHA results reasonably well. On the other hand, mean MPA results for

member internal forces cannot estimate mean NRHA demands within a reasonable

accuracy especially at the upper stories. As stated in the previous case studies, MPA

results require correction according to member capacities for beam end forces.


bending moments along the left exterior column axis in Frame A

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 50 100 150

Sto

ry

#

Shear Force (kN)

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 200 400 600

Sto

ry

#

Moment (kN.m)

102


bending moments along the left exterior column axis in Frame B

1st Story 5

th Story

10th

Story


the 1st, 5th, and 10th stories in Frame A

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 50 100 150

Sto

ry

#

Shear Force (kN)

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 100 200 300 400

Sto

ry

#

Moment (kN.m)

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

Mean NRHA

Mean GPA

Mean MPA

Mean Spec GPA

6 m 4 m 6 m

103

1st Story 5

th Story

10th

Story


the 1st, 5th, and 10th stories in Frame B

1st Story 5

th Story

Figure 5.16 Comparison of maximum shear force values of beam ends at the

1st, 5th, and 10th stories in Frame A

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

6 m 4 m 6 m

0

50

100

150

200

250

300

350

400

Mo

men

t (k

N.m

)

Mean NRHA

Mean GPA

Mean MPA

Mean Spec GPA

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

6 m 4 m 6 m

104

10th

Story


1st Story 5

th Story

10th

Story

Figure 5.17 Comparison of maximum shear force values of beam ends at the

1st, 5th, and 10th stories in Frame B

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

Mean NRHA

Mean GPA

Mean MPA

Mean Spec GPA

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

6 m 4 m 6 m

0

50

100

150

200

250

Sh

ear

Fo

rce

(kN

)

Mean NRHA

Mean GPA

Mean MPA

Mean Spec GPA

6 m 4 m 6 m

105

Figure 5.18 Comparison of maximum bending moment values along the shear

wall in Frame A (My=7800 kN.m in Hcr)

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 5000 10000 15000 S

tory

#

Moment (kN.m)

106

CHAPTER 6

PRACTICAL IMPLEMENTATION OF


Practical implementation of generalized pushover analysis (GPA) is

developed in this chapter which requires significantly less computational effort. GPA

and its practical implementation (RGPA) are compared with the benchmark

nonlinear response history analysis, based on the results obtained from three case

studies.

6.1 Reduced Generalized Pushover Analysis Procedure

Generalized pushover analysis (GPA) is based on the general assumption that

the interstory drift ratios, occur independently at each story (j=1-N) at different

instants tj, max, j=1-N. However, if there are n modes contributing significantly to the

total dynamic response (n<N), then there are only 2(n-1)

possible combinations of the

n modes leading to the maximum values of interstory drifts at specific stories. Hence

there are 2(n-1)

independent instants tmax for calculating and in an N DOF

system.

Let’s consider the contribution of first three modes to the linear elastic drift

profiles of the 12-story RC frame of the first case study under GM4, shown in Figure

6.1.

107

Figure 6.1 Drift profile for the first three modes under GM4

Here,

( )

is the n’th mode contribution to the maximum total drift ratio at the j’th story where

is obtained from the linear elastic response spectrum. There are 2(n-1)

combinations for n number of significant modes contributing to interstory drifts. For

n=2 and n=3, possible combinations are shown in Figure 6.2 and Figure 6.3,

respectively. It should be noted that these are the absolute maximum combinations of

modal drifts. During actual dynamic response, certainly assumes lower values

than the spectral values. However this situation does not affect the approach

presented.

According to Figure 6.2, the combinations Δ1+Δ2 and Δ1-Δ2 control the

system in two ranges along its height, the lower 1st-5

th stories and the upper 6

th-12

th

stories, respectively.

0 1 2 3 4 5 6 7 8 9

10 11 12

0 0.02 0.04 0.06

Sto

ry #

Interstory Drift (m)

Δ1

0 1 2 3 4 5 6 7 8 9

10 11 12

-0.03 0 0.03


Δ2

0 1 2 3 4 5 6 7 8 9

10 11 12

-0.03 0 0.03 Interstory Drift (m)

Δ3

Figure 6.2 Combinations of the first two modes contributing to interstory drift

under GM4

0123456789

101112

0 0.02 0.04 0.06 0.08

Sto

ry #


Δ1-Δ2

Δ1

Range 2

0123456789

101112

-0.02 -1E-16 0.02 0.04 0.06 0.08 0.1

Sto

ry #


Δ1+Δ2

Δ1

Range 1

108

On the other hand, according to Figure 6.3, different combinations of the first

three modes control the system in four different ranges. Figure 6.3 reveals that

Δ1-Δ2+Δ3 combination controls the upper (9th

-12th

) stories whereas Δ1+Δ2+Δ3 controls

the lower (1st-3

rd) stories; however, for the middle stories, Δ1+Δ2-Δ3 and Δ1-Δ2-Δ3

combinations control the interstory drift maxima of the 4th

-5th

, and 6th

-8th

stories,

respectively.

Figure 6.3 Combinations of the first three modes contributing to interstory drift

under GM4

This condition is also established with the elastic time history (ETH) results

and SRSS combination of the first three modes under GM4 as shown in Figure 6.4

and Figure 6.5 respectively. In Figure 6.4, drift profiles of each story level when the

story attains its maximum drift during elastic dynamic response analysis under GM4

are plotted along with the related combination of the first three modes. It is obvious

from the figure that drift profile groups on each graph in Figure 6.4 attain their

maxima at the time steps close to each other and confirm the system responses in

four ranges given in Figure 6.3. Moreover, they are consistent with the related drift

0123456789

101112

-0.02 -1E-16 0.02 0.04 0.06 0.08 0.1

Sto

ry #


Δ1+Δ2+Δ3

Δ1+Δ2

Range 1

0123456789

101112

-0.02 0 0.02 0.04 0.06 0.08 0.1

Sto

ry #


Δ1+Δ2-Δ3

Δ1+Δ2

Range 2

0123456789

101112

0 0.02 0.04 0.06 0.08 0.1

Sto

ry #


Δ1-Δ2-Δ3

Δ1-Δ2

Range 3

0123456789

101112

0 0.02 0.04 0.06 0.08 0.1

Sto

ry #


Δ1-Δ2+Δ3

Δ1-Δ2

Range 4

109

profile of the first three mode combinations. Story 12 is an exception here since the

fourth mode contribution is significant for the top interstory drift. It should be

considered again that the modal combinations shown in Figure 6.4 (black lines) are

those of maximum (spectral) modal drifts, not the actual modal drifts which occur

during dynamic response. The intention for such comparison is to show that the

interstory drift distributions are similar; hence considering reduced number of

pushovers with the related force vectors would be sufficient to capture the maximum

dynamic response of the system under a ground motion.

Figure 6.4 ETH interstory drift profiles at each story maxima and the related

combination of the first three modes contributing to interstory drift under GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

-0.02 0.03 0.08

Sto

ry #


Max Δ1, t=37.234

Max Δ2, t=37.230

Max Δ3, t=37.146

Max Δ4, t=37.126

Δ1+Δ2+Δ3

Range 1

0

1

2

3

4

5

6

7

8

9

10

11

12

-0.02 0 0.02 0.04 0.06 0.08

Stor

y #


Max Δ5, t=35.810

Δ1+Δ2-Δ3

Range 2

0

1

2

3

4

5

6

7

8

9

10

11

12

0 0.02 0.04 0.06 0.08

Stor

y #


Max Δ6, t=36.318

Max Δ7, t=36.290

Max Δ8, t=36.286

Δ1-Δ2-Δ3

Range 3

0

1

2

3

4

5

6

7

8

9

10

11

12

0 0.02 0.04 0.06 0.08

Stor

y #


Max Δ9, t=36.850

Max Δ10, t=36.818

Max Δ11, t=36.774

Max Δ12, t=34.334

Δ1-Δ2+Δ3

Range 4

110

Each drift profile from three mode combinations is also scaled in Figure 6.5

with the ratio of story range maxima to the related story drift value in the SRSS

profile, and compared with the SRSS drift profile in order to demonstrate the validity

of regions controlled by the combined drift profiles. It is observed from Figure 6.5

that the drift distributions in each range obtained from the combined drift profiles are

also compatible with the SRSS drift profile. Consequently, one story level from each

story range can be selected, and the associated four force vectors can be employed in

GPA instead of N numbers of force vectors, i.e., N number of generalized pushovers.

For the 12-story frame in the first case study, considering the combinations of the

first three modes, 2nd

, 5th

, 7th

and 11th

stories are selected from each story range, and

only four generalized pushover analyses are conducted by applying f2, f5, f7 and f11 in

accordance with Equation (13). Finally, the envelopes of these four generalized

pushover analyses are employed for calculating the maximum response parameters.

Thus, the computation effort in GPA is reasonably reduced from 12 to 4 pushovers.

This procedure is called the “reduced GPA” (RGPA). For comparison, MPA carries

out three pushovers for the three significant modes. Hence, their computational

efforts are similar although the accuracy of results is different.

The upper bound modal combinations introduced in Figure 6.2 and Figure 6.3

for interstory drifts were previously employed by Matsumori et al. (1999), Kunnath

(2004) and Jan et al. (2004) for calculating the lateral force distributions in pushover

analysis to account for the higher mode effects.

Figure 6.5 Comparison of SRSS drift profile and combinations of the scaled first

three modal drifts

0

1

2

3

4

5

6

7

8

9

10

11

12

-0.02 0 0.02 0.04 0.06 0.08

Sto

ry #


Δ1+Δ2+Δ3

Δ1-Δ2+Δ3

Δ1+Δ2-Δ3

Δ1-Δ2-Δ3

SRSS

Range 1

Range 2

Range 3

Range 4

111

6.2 Comparative Results from Case Studies

6.2.1 Case Study I: Twelve Story RC Frame with Full Capacity Design

The reduced generalized pushover analysis (RGPA) results for the maximum

interstory drift ratios, maximum average beam-end plastic rotations and maximum

average column-end chord rotations are compared with the results of NRHA, GPA

and MPA in Figure 6.6, Figure 6.7 and Figure 6.8, respectively. It can be observed

from these figures that the results of RGPA are very close to those of GPA and

sufficiently close to the benchmark NRHA results. RGPA and GPA give almost

equal results for column-end chord rotations.


ground motions

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020 0.025

Sto

ry #


GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM3

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM4

112


Figure 6.7 Comparison of maximum values of average beam-end plastic

rotations under seven ground motions

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM6

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020 0.025

Sto

ry #


NRHA GPA MPA RGPA

GM7

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020 0.025

Sto

ry #


GM2

113


0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015

Sto

ry #


GM3

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015

Sto

ry #


GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015

Sto

ry #


GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015

Sto

ry #


GM6

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020 0.025

Sto

ry #


NRHA GPA MPA RGPA

GM7

114

Figure 6.8 Comparison of maximum values of average column-end chord


0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.002 0.004 0.006 0.008

Sto

ry #


GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010

Sto

ry #


GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.002 0.004 0.006

Sto

ry #


GM3

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.002 0.004 0.006 0.008

Sto

ry #


GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.002 0.004 0.006 0.008

Sto

ry #


GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.002 0.004 0.006

Sto

ry #


GM6

115


6.2.2 Case Study II: Twelve Story RC Frame with Relaxed Capacity Design

The positive drift profile for the first mode and both positive and negative

drift profiles for the second and third modes for the 12-story RC frame with relaxed

capacity design, employed in the second case study, are shown in Figure 6.9 under

GM4. The combination of these modes is also shown in Figure 6.10. It is observed

from Figure 6.10 that Δ1-Δ2+Δ3 combination controls the upper (9th

-12th

) stories

whereas Δ1+Δ2+Δ3 controls the lower (1st-3

rd ) stories. In addition to that, for the

middle stories, Δ1+Δ2-Δ3 and Δ1-Δ2-Δ3 combinations control the story maxima of the

4th

, and 5th

-8th

stories, respectively. Similar to the previous case study, one story level

can be selected from each story group, and the related 4 force vectors can be

employed in GPA instead of N numbers of force vectors. For the 12-story frame in

the second case study, 1st, 4

th, 7

th and 11

th stories were selected from each story

group, and only four generalized pushover analysis were conducted by applying f1,

f4, f7 and f11 in accordance with Equation (13). As expressed previously, the

envelopes of these generalized pushover analysis are taken as the maximum response

parameters.

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.002 0.004 0.006 0.008

Sto

ry #


NRHA GPA MPA RGPA

GM7

116


drift profiles for the second and third modes under GM4

Figure 6.10 Combinations of the first three modes contributing to interstory drifts under

GM4

Maximum interstory drift ratios, maximum average beam-end plastic

rotations and maximum average column-end chord rotations of RGPA are compared

with NRHA, MPA and GPA results in Figure 6.11, Figure 6.12 and Figure 6.13,

0 1 2 3 4 5 6 7 8 9

10 11 12

0 0.02 0.04 0.06

Sto

ry #

Drift Value (m)

Δ1

0 1 2 3 4 5 6 7 8 9

10 11 12

-0.04 -0.02 0 0.02 0.04

Sto

ry #

Drift Value (m)

Δ2 - Δ2

0 1 2 3 4 5 6 7 8 9

10 11 12

-0.02 -0.01 0 0.01 0.02

Sto

ry #

Drift Value (m)

Δ3 - Δ3

0

1

2

3

4

5

6

7

8

9

10

11

12

-0.02 0 0.02 0.04 0.06 0.08 0.1

Sto

ry #

Drift Value (m)

Δ1+Δ2+Δ3

Δ1-Δ2+Δ3

Δ1+Δ2-Δ3

Δ1-Δ2-Δ3

117

respectively. It is obvious for interstory drifts and beam-end plastic rotations that the

results of RGPA are well synchronized with GPA results for all ground motions in

the set except GM2, and sufficiently close to the benchmark results of NRHA. The

difference between RGPA and GPA in GM2 is due to the fact that the 4th

and 7th

story force vectors employed in RGPA procedure cannot control the demands

between 2nd

and 7th

stories, and using different force vectors for the related story

groups improve the obtained results for GM2. For column end chord rotations from

Figure 6.13, RGPA and GPA results are almost equal to each other, and very close to

the NRHA results. The difference between RGPA and GPA in column chord

rotations under GM2 is too small when compared to story drift and beam-end plastic

rotation comparisons.

Figure 6.11 Comparison of maximum interstory drift ratios obtained under


0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.010 0.020 0.030

Sto

ry #


GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM3

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM4

118




0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015

Sto

ry #


GM6

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020 0.025

Sto

ry #


NRHA GPA MPA RGPA

GM7

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015

Sto

ry #


GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


GM2

119


0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015

Sto

ry #


GM3

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015

Sto

ry #


GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015

Sto

ry #


GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015

Sto

ry #


GM6

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015 0.020

Sto

ry #


NRHA GPA MPA RGPA

GM7

120



0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.002 0.004 0.006 0.008

Sto

ry #


GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010 0.015

Sto

ry #


GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.002 0.004 0.006

Sto

ry #


GM3

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.002 0.004 0.006 0.008

Sto

ry #


GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.002 0.004 0.006 0.008

Sto

ry #


GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.002 0.004 0.006

Sto

ry #


GM6

121


6.2.3 Case Study III: Twenty Story RC Wall-Frame System with Full Capacity

Design

The positive drift profile of the first mode and both positive and negative drift

profiles of the second and third modes of the 20-story RC wall frame system with

full capacity design, utilized in the third case study, are presented in Figure 6.14

under GM4. Combinations of modal drifts are also shown in Figure 6.15. It can be

observed from Figure 6.15 that Δ1-Δ2+Δ3 combination controls the upper (15th

-20th

)

stories whereas Δ1+Δ2+Δ3 controls the lower (1st-6

th ) stories. In addition, Δ1+Δ2-Δ3

and Δ1-Δ2-Δ3 combinations control the story maxima of the 6th

-9th

and 10th

-14th

stories, respectively. Similarly, one story level is selected from each story group in

order to conduct GPA with four force vectors. In this sense, 3rd

, 8th

, 12th

, and 17th

stories are selected from each story group for the 20-story wall-frame system, and

only f3, f8, f12 and f17 force vectors are used in GPA in accordance with Equation

(13). The envelopes of these four generalized pushover analyses are taken as the

maximum responses, and compared with the benchmark NRHA results.

0

1

2

3

4

5

6

7

8

9

10

11

12

0.000 0.005 0.010

Sto

ry #


NRHA GPA MPA RGPA

GM7

122

Figure 6.14 Positive drift profile for the first mode, and both positive and

negative drift profiles for the second and third modes under GM4

Figure 6.15 Combinations of the first three modes contributing to interstory drift under

GM4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0 0.01 0.02 0.03

Sto

ry #

Drift Value (m)

Δ1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

-0.02 -0.01 0 0.01 0.02

Sto

ry #

Drift Value (m)

Δ2 - Δ2

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

-0.005 0 0.005

Sto

ry #

Drift Value (m)

Δ3 - Δ3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

-0.01 0 0.01 0.02 0.03 0.04 0.05

Sto

ry #

Drift Value (m)

Δ1+Δ2+Δ3

Δ1-Δ2+Δ3

Δ1+Δ2-Δ3

Δ1-Δ2-Δ3

123

Maximum interstory drift ratios obtained from RGPA for the 20-story RC

wall frame system under seven ground motions are given in Figure 6.16. In addition,

maximum average beam-end plastic rotations and maximum average column-end

chord rotations from RGPA are presented for each frame separately in Figure 6.17,

Figure 6.18, Figure 6.19 and Figure 6.20. It can be inferred from interstory drift

ratios and beam-end plastic rotations that RGPA results are almost equal to GPA

results, and both GPA results are well synchronized with the NRHA values for all

ground motions in the set. On the other hand, the successful match of RGPA results

with GPA values are also observed for column-end chord rotation presented in

Figure 6.19 and Figure 6.20, and these results estimates reasonably the NRHA

benchmark results.

Figure 6.16 Comparison of maximum interstory drift ratios obtained under


0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


GM2

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010

Sto

ry #


GM4

124



rotations in Frame A under seven ground motions

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010

Sto

ry #


GM6

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


NRHA GPA

MPA RGPA

GM7

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


GM2

125


0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


GM4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010

Sto

ry #


GM6

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010 0.015

Sto

ry #


NRHA GPA

MPA RGPA

GM7

126


rotations in Frame B under seven ground motions

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006 0.008

Sto

ry #


GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.005 0.010

Sto

ry #


GM2

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006 0.008

Sto

ry #


GM4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006

Sto

ry #


GM6

127



rotations in Frame A under seven ground motions

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.002 0.004 0.006 0.008 S

tory

#


NRHA GPA

MPA RGPA

GM7

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003 0.004

Sto

ry #


GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003

Sto

ry #


GM2

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003

Sto

ry #


GM4

128



rotations in Frame B under seven ground motions

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002

Sto

ry #


GM6

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003

Sto

ry #


NRHA GPA

MPA RGPA

GM7

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003 0.004

Sto

ry #


GM1

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003

Sto

ry #


GM2

129


0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002

Sto

ry #


GM3

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003

Sto

ry #


GM4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002

Sto

ry #


GM5

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002

Sto

ry #


GM6

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.000 0.001 0.002 0.003

Sto

ry #


NRHA GPA

MPA RGPA

GM7

130

CHAPTER 7

SUMMARY AND CONCLUSIONS

7.1 Summary

Generalized Pushover Analysis (GPA) procedure (Sucuoğlu and Günay,

2011), summarized in this study, was based on the generalized force vector concept.

This procedure consists of applying different generalized force vectors separately to

the structure, one for each story, in an incremental form until the target interstory

drift value at the corresponding story is achieved. Using interstory drift as a target

response parameter in GPA instead of the roof story displacement provides more

accurate estimation of local maximum response parameters and more effective

contribution of higher mode effects. GPA requires N numbers of pushover analyses

for an N story building. GPA is not adaptive, and it can be implemented with any

conventional software program capable of performing displacement-controlled

nonlinear static incremental analysis. The maximum value of response parameters

are determined directly by taking the envelopes of the related GPA results, and these

envelope values are registered as the maximum seismic response values. In order to

test the performance of GPA, interstory drifts, member end rotations and member

internal forces obtained from GPA are compared with the results of NRHA and MPA

for three different case studies. It is observed that GPA predicts NRHA results quite

reasonably for member deformations and member internal forces.

GPA is also examined herein by using mean spectrum of a set of ground

motions. In this context, the mean values of NRHA, GPA and MPA results in each

case study, obtained for each ground motion in the set, are calculated and compared

with the results of GPA obtained under the mean spectrum. In this way, practicality

of GPA for different cases is examined.

131

A practical implementation of the GPA procedure is also developed in this

study in order to reduce the number of pushover analyses in GPA. This newly

developed procedure is based on the premise that if there are n modes contributing

significantly to the total dynamic response (n<N), then there are only 2(n-1)

possible

combinations of the n modes leading to the maximum values of interstory drifts at

specific stories. According to such profiles, each modal drift combination controls a

different story group for maximum response. Consequently, one story level from

each story group can be selected, and the related force vectors can be employed in

GPA. Using reduced numbers of generalized force vectors decrease computational

effort in GPA. Reduced GPA results are compared with NRHA, GPA and MPA

results for each case study in order to examine the estimation accuracy of the

procedure.

7.2 Conclusions

According to the results obtained in this study, the following conclusions are

reached:

GPA is successful in predicting the maximum responses of interstory drifts,

member end rotations and member internal forces obtained from NRHA. GPA

considers higher mode effects efficiently. Additionally, GPA does not suffer from

shortcoming of statistically combined inelastic modal responses, because internal

forces and deformations are directly obtained from GPA at the target drift

demand.

GPA results obtained under the mean spectrum of a ground motion set

successfully estimates the mean results of NRHA and GPA achieved under each

ground motion separately for each case study. Hence, GPA can be employed

effectively under any code or design spectrum which represents the statistical

average of several ground motions.

The reduced GPA procedure (RGPA) exhibit same accuracy with the original

GPA results. This indicates that RGPA procedure takes into account higher mode

effects effectively, similar to the original GPA. Reduced number of generalized

force vectors, hence reduced number of pushovers in RGPA decrease

132

computational time significantly, and make RGPA a more practical procedure. It

is important to note that using only two force vectors give sufficiently accurate

results with respect to the original GPA and NRHA for tall buildings.

133

REFERENCES

American Society of Civil Engineers (ASCE), 2000, ‘Prestandard and Commentary

for the Seismic Rehabilitation of Buildings’, Report No. FEMA-356, Washington,

D.C.

Antoniou S. and Pinho R., 2004a, ‘Advantages and Limitations of Adaptive and

Non-Adaptive Force-Based Pushover Procedures’, Journal of Earthquake

Engineering, Volume 8, No. 4, pp. 497-522.

Antoniou S. and Pinho R., 2004b, ‘Development and Verification of a Displacement-

Based Adaptive Pushover Procedure’, Journal of Earthquake Engineering, Volume 8,

No. 5, pp. 643-661.

Applied Technology Council (ATC), 199 , ‘Seismic Evaluation and Retrofit of

Concrete Buildings’, Report No. ATC-40, Volume 1-2, Redwood City, California.

Applied Technology Council (ATC), 2005, ‘Improvement of Nonlinear Static

Seismic Analysis Procedures’, Report No. FEMA-440, Washington, D.C.

Applied Technology Council (ATC), 1997, ‘NEHRP Guidelines for the Seismic

Rehabilitation of Buildings’ Report No. FEMA-273, Washington, D.C.

Aschheim M., Tjhinb T., Comartin C., Hamburger R. and Inel M., 2007, ‘The Scaled

Nonlinear Dynamic Procedure’, Engineering Structures, Volume 29, No. 7, pp.

1422-1441.

Aydinoglu N.M., 2003, ‘An Incremental Response Spectrum Analysis Procedure

Based on Inelastic Spectral Displacements for Multi-Mode Seismic Performance

Evaluation’, Bulletin of Earthquake Engineering, Volume 1, pp.3–36.

Bracci J. M., Kunnath S. K. and Reinhorn A. M., 1997, ‘Seismic Performance and

Retrofit Evaluation of Reinforced Concrete Structures’, Journal of Structural

Engineering, ASCE, Volume 123, No. 1, pp. 3-10.

134

Chopra A.K. and Goel R.K., 2000, ‘Evaluation of NSP to Estimate Seismic

Deformation: SDF Systems’, Journal of Structural Engineering, ASCE, Volume 12 ,

No. 4, pp. 482-490.

Chopra A.K. and Goel R.K., 2002, ‘A Modal Pushover Analysis Procedure for

Estimating Seismic Demands for Buildings’, Earthquake Engineering and Structural

Dynamics, Volume 31, pp. 561-582.

Chopra A. K. and Goel R. K., 2004, ‘A Modal Pushover Analysis Procedure to

Estimate Seismic Demands for Unsymmetrical-Plan Buildings’, Earthquake

Engineering and Structural Dynamics, Volume 33, pp. 903–927.

Fajfar P. and Fischinger M., 1987, ‘Nonlinear Seismic Analysis of RC Buildings:

Implications of a Case Study’, European Earthquake Engineering, Volume 1, pp. 31-

43.

Fajfar P. and Gaspersic P., 199 , ‘The N2 Method for the Seismic Damage Analysis

of RC Buildings’, Earthquake Engineering and Structural Dynamics, Volume 25, pp.

31-46.

Fajfar P., 2000, ‘A Nonlinear Analysis Method for Performance-Based Seismic

Design’, Earthquake Spectra, Volume 1 , No. 3, pp. 573-592.

Freeman S.A., Nicoletti J.P. and Tyrell J.V., 1975, ‘Evaluation of Existing Buildings

for Seismic Risk - A Case Study of Puget Sound Naval Shipyard, Bremerton,

Washington’, Proceedings of the First U.S. National Conference on Earthquake

Engineering, Berkeley, California.

Goel R. K. and Chopra A. K., 2005, ‘Extension of Modal Pushover Analysis

Procedure to Compute Member Forces’, Earthquake Spectra, Volume 21(1), pp. 125-

139.

Gupta B. and Kunnath S. K., 2000, ‘Adaptive Spectra-Based Pushover Procedure for

Seismic Evaluation of Structures’, Earthquake Spectra, Volume 1 , No. 2, pp. 367–

391.

Hancock, J. and Bommer, J.J., 200 , ‘An Improved Method of Matching Response

Spectra of Recorded Earthquake Ground Motion Using Wavelets’, Journal of

Earthquake Engineering, Volume 10,No.1, pp. 67-89.

Hancock, J. and Bommer, J.J., 2007, ‘Using Spectral Matched Records to Explore

The Influence of Strong-Motion Duration on Inelastic Structural Response’, Soil

Dynamics and Earthquake Engineering, Volume 27, pp. 291-299.

135

Jerez S. and Mebarki A., 2011, ‘Seismic Assessment of Framed Buildings: A

Pseudo-Adaptive Uncoupled Modal Response Analysis’, Journal of Earthquake


Jan T.S., Liu M.W. and Kao, Y.C., 2004, ‘An Upper-Bound Pushover Analysis

Procedure for Estimating The Seismic Demands of High-Rise Buildings’,

Engineering Structures, Volume 26, No. 1, pp. 117-128.

Kalkan E. and Kunnath S.K., 200 , ‘Adaptive Modal Combination Procedure For

Nonlinear Static Analysis of Building Structures’, ASCE, Journal of Structural


Krawinkler H. and Seneviratna G.D.P.K., 1998, ‘Pros and Cons of a Pushover

Analysis of Seismic Performance Evaluation’, Engineering Structures, Volume 20,

pp. 452-464.

Kreslin M. and Fajfar R., 2012, ‘The Extended N2 Method Taking into Account

Higher Mode Effects in Elevation’, Earthquake Engineering and Structural

Dynamics, Volume 40, pp.1571-1589.

Kunnath, S.K., 2004, ‘Identification of Modal Combinations for Nonlinear Static

Analysis of Building Structures’, Computer Aided Civil Infrastructure Engineering,

Volume 19, No. 4, pp. 246-259.

Matsumori, T., Otani, S., Shiohara, H. and Kabayesawa, T., 1999, ‘Earthquake

Member Deformation Demands in Reinforced Concrete Frame Structures’, Proc. US-

Japan Workshop on Performance Based Earthquake Engineering, PEER Report,

University of California, Berkeley.

Ministry of Public Works and Settlement, 2007, Turkish Earthquake Code:

Specifications for the Buildings in Disaster Areas, Ankara, Turkey.

Miranda E., and Akkar S., 2002, ‘Evaluation of Approximate Methods to Estimate

Target Displacements in Nonlinear Static Procedures’, Report No. PEER-2002/21;

Proceedings of the 4th U.S.-Japan Workshop on Performance-Based Earthquake

Engineering Methodology for Reinforced Concrete Building Structures, Pacific

Earthquake Engineering Research Center, California, pp. 75–86.

Moehle J.P., 1984, ‘Seismic Analysis of R/C Frame-Wall Structures’, Journal of

Structural Engineering, ASCE, Vol. 110, No. 11, pp. 2619-2634.

Moehle J.P. and Alarcon I.F., 198 , ‘Seismic Analysis Methods For Irregular

Buildings’, Journal of Structural Engineering, ASCE, Vol. 112, No. 1, pp. 35-52.

136

Mwafy A.M., and Elnashai A.S., 2001, ‘Static Pushover versus Dynamic Collapse

Analysis of RC Buildings’, Engineering Structures, Volume 23, pp. 407-424.

OpenSees, 2005, ‘Open System for Earthquake Engineering Simulation’,

‘http://opensees.berkeley.edu’, November 15, 2011.

PEER Strong Motion Database (2010). Available from:

http://peer.berkeley.edu/smcat.

Poursha M., Khoshnoudian F. and Moghadam A.S., 2009, ‘A Consecutive Modal

Pushover Procedure for Estimating The Seismic Demands of Tall Buildings’,

Engineering Structures, Volume 31, pp. 591-599.

Saiidi M. and Sözen M.A., 1981, ‘Simple Nonlinear Seismic Analysis of R/C

Structures’, Journal of the Structural Division, ASCE, Vol. 107, No. ST5, pp. 937-

952.

Saiidi M. and Hudson K.E.Jr., 1982, ‘Analytical Study of Irregular R/C Structures

Subjected to In-Plane Earthquake Loads’, College of Engineering, Report No. 59,

University of Nevada, Reno.

Sasaki F., Freeman S. and Paret T., 1998, ‘Multi-Mode Pushover Procedure (MMP)-

A Method to Identify the Effect of Higher Modes in a Pushover Analysis’,

Proceedings of the 6th U.S. National Conference on Earthquake Engineering, Seattle,

CD-ROM, EERI, Oakland.

Sucuoğlu H. and Günay M. S., 2011, ‘Generalized Force Vectors for Multi-Mode

Pushover Analysis’, Earthquake Engineering and Structural Dynamics, Volume 40,

pp. 55-74.

Tso W. K. and Moghadam A.S., 1998, ‘Pushover Procedure for Seismic Analysis of

Buildings’, Progress in Structural Engineering and Materials, Vol I(3), pp:337-344

Vamvatsikos D. and Cornell C. A., 2002, ‘Incremental Dynamic Analysis’,

Earthquake Engineering and Structural Dynamics, Volume 31, No. 3, pp. 491–514.

Documents

GENERALIZED PUSHOVER ANALYSIS A THESIS SUBMITTED TO …