sesmic analysis project

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SEISMIC PERFORMANCE EVALUATION AND

FRAGILITY ANALYSIS OF REINFORCED CONCRETE

BUILDINGS

Submitted in partial fulfillment of the requirements

of the degree of

Master of Technology

in

Structural Engineering

by

A.Pavan Kumar

(08304022)

Under the supervisionof

Prof. Ravi Sinha

CIVIL ENGINEERING DEPARTMENT

INDIAN INSTITUTE OF TECHNOLOGY BOMBAY

POWAI, MUMBAI 400076

JUNE 2010


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Acknowledgement

I express a deep sense of sincere gratitude to Prof. Ravi Sinha for his constant, encouraging

and inspiring guidance and support throughout this study. It was great experience working

under Prof. Ravi Sinhas supervision, which helped me to achieve in depth insight in this

field.

A.Pavan Kumar

08304022


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Acceptance certificate

This M.Tech project thesis titled Seismic Performance Evaluation and Fragility Analysis

of Reinforced Concrete Buildingssubmitted by A.Pavan Kumar (Roll No: 08304022) is

approved for the degree ofMaster of Technology in Structural Engineering.

Examiners

..

..

Supervisor

..

Chairman

.

Date: 2nd July, 2010

Place: IIT Bombay


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Abstract

Earthquakes are one of the most destructive calamities and cause a lot of casualties, injuries

and economic losses leaving behind a trail of panic. Earthquake risk assessment is needed for

disaster mitigation, disaster management, and emergency preparedness. Vulnerability of

building is one of the major factors contributing to earthquake risk. The vulnerability

functions developed for specific building or building class is input parameter for loss

estimation.

In this report procedure for seismic performance evaluation of reinforced concrete buildings is

studied. Three methods namely capacity spectrum method (CSM), displacement coefficient

method (DCM), modal pushover analysis (MPA) are discussed for estimating seismic

inelastic displacement. Using these methods seismic performance of reinforced concrete

buildings is evaluated. The validation of these methods has been done with models reported in

literature. Two example problems have been evaluated using the methods mentioned.

Procedure for developing vulnerability or fragility curves of specific building or generic

building type is discussed. Applicability of HAZUS drift ratio based damage state thresholds

for building designed as per IS 456-2000 code is studied. Comparison of fragility curves

developed using different procedures is studied and applicability is discussed. Seismic

fragility curves were developed and damage probability indices has been constructed for the

chosen example problems. Influence of structural parameters on vulnerability of commercial

building class is studied and band of fragility curves representing medium rise commercial

reinforced building class is developed.


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Table of Contents Page

NoAbstract iv

List of figures viii

List of tables xii

Chapter 1 1

Introduction 1

1.1 Background 1

1.2 Scope of the Dissertation 2

1.3 Organization of the Report 3

Chapter 2 4

Seismic Performance Evaluation of Buildings 4

2.1 Seismic performance based evaluation and design 4

2.1.1 Performance objectives 5

2.1.2 Performance levels 5

2.2 Earthquake ground motion 6

2.3 Basic safety objective 8

2.4 Preliminary evaluation of structure 8

2.5 Retrofit strategy and retrofit method 8

2.6 Methods of evaluation of seismic performance of buildings 9

2.6.1 Elastic method of analysis 102.6.1.1 Seismic coefficient method 10

2.6.1.2 Linear elastic dynamic analysis 10

2.6.2 Inelastic method of analysis 10

2.6.2.1 Inelastic time history analysis 10

2.6.2.2 Inelastic static analysis or pushover analysis 11

2.6.2.2.1 Background to pushover analysis 12

2.6.2.2.2 Implementation of pushover analysis 14

2.6.2.2.3 Limitations of pushover analysis 15

2.7 Estimation of inelastic displacement demand 16

2.7.1 Capacity spectrum method (CSM) 16

2.7.1.1 Design demand spectrum 18

2.7.1.2 Conversion of design demand spectrum into ADRS format 19

2.7.1.3 Conversion of capac ity curve into capacity spectrum 19

2.7.1.4 Bilinear representation of capacity spectrum 21

2.7.1.5 Calculation of spectral reduction factors 21

2.7.2 Displacement coefficient method-FEMA 273,1997 21

2.7.3 Modal pushover analysis 23

2.8 Acceptance criteria for different performance levels 26


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2.9 Discussions 26

Chapter 3 27

Seismic Vulnerability and Fragility Analysis of Buildings 27

3.1 Seismic vulnerability of building 273.2 Fragility curves of building 27

3.2.1 Procedure to develop DPM of building for given level of earthquake 28

3.2.1.1 Building type and classification 28

3.2.1.2 Seismic design levels and quality of construction 29

3.2.1.3 Damage states 29

3.2.1.4 Calculation of cumulative damage probability of particular damage

state

30

3.2.1.5 Calculation of discrete damage probability of damage states 31

3.2.1.6 Median spectral displacements for different damage states 31

3.2.1.7 Development of damage state variability 36

3.3 Discussions 36

Chapter 4 37

Modeling of Reinforced Concrete Buildings 37

4.1 Introduction 37

4.2 Modeling of nonlinearity in beams and columns 37

4.3 Bilinear representation of nonlinear curve 37

4.4 Procedure to develop moment-curvature relationship 384.4.1 Conversion of moment-curvature into moment-rotation 40

4.4.2 Validation of program for moment-curvature relationship 41

4.5 Modeling of nonlinearity in infill walls 43

4.6 Modeling of nonlinearity in shear wall 44

4.7 Discussions 47

Chapter 5 45

Analysis and Evaluation of Reinforced Concrete Buildings 45

5.1 Analysis and validation of 2D frame (Inel,2006) 45

5.1.1 Modal analysis of 2D frame 45

5.1.2 Nonlinear static analysis of 2D frame 46

5.1.3 Seismic performance evaluation of 2D frame -CSM 48

5.1.3.1 Manual calculation of performance point of 2D frame 48

5.1.3.2 SAP 2000 calculation of performance point of 2D frame 52

5.1.4 Discussions 56

5.2 Analysis and validation of 3D frame (Fajfar,1996) 56

5.2.1 Modal analysis of 3D frame 58

5.2.2 Nonlinear static analysis of 3D frame 58

5.2.3 Seismic performance evaluation of 3D frame-CSM 59


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5.3 Analysis and evaluation of 3 storey buildings (Irtem,2007 ) 61

5.3.1 Modal analysis of 3 storey buildings 61

5.3.2 Pushover analysis of 3 storey buildings with and without infill walls 625.3.3 Seismic performance evaluation of 3 story bare frame (3SBF)-DCM 64

5.3.4 Seismic fragility analysis 3 story bare frame 66

5.3.5 Probability of different damage states of 3SBF for E4 level of earthquake 68


5.4 Analysis and evaluation of example building no1 70

5.4.1 Linear static analysis of example building no1 70

5.4.2 Modal analysis of example building no1 70

5.4.3 Nonlinear static analysis of example building no1 72

5.4.4 Performance evaluation of example building no1 using SCM 73

5.4.5 Performance evaluation of example building no1 using CSM 74

5.4.5.1 Performance evaluation of example building no1 for zone III, MCE 75

5.4.5.2 Performance evaluation of example building no1 for zone IV, MCE 76

5.4.5.3 Performance evaluation of example building no1 for zone V, MCE 76


5.5 Analysis and evaluation of example building no 2 79

5.5.1 Modal analysis of original and designed example building no 2 80

5.5.2 Pushover analysis of original and designed example building no 2 81

5.5.3 Seismic fragility analysis of original and designed example building no 2 82

5.5.4 Seismic performance of original example building no 2 for DBE,MCE 87

5.5.4.1 Seismic performance evaluation for design basis earthquake 87

5.5.4.2 Seismic performance evaluation for maximum considered earthquake 90

5.5.5 Probability of damage states of original example building no 2 for DBE,

and MCE

93


5.6 Influence of structural parameters on vulnerability of reinforced concrete 95commercial buildings

5.6.1 Discussions 1005.7 Discussions 100

Chapter 6 103

Discussions and Conclusions 103

6.1 Discussions 103

6.2 Conclusions 104

6.3 Scope for future work 105

References

Appendix 1Appendix 2

106

109113


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List of Figures

Fig.

No

Description PageNo

2.1 Performance levels on pushover curve and load deformation curve (Irtem,2007)

7

2.2 Pushover or capacity curve of the building (Sermin ,2005) 12

2.3 Force displacement characteristics of MDF and equivalent SDF system(Krawinkler, 1998)

14

2.4 Capacity spectrum method (HAZUS MH MR4) 17

2.5 Spectral acceleration coefficients vs. time period (IS 1893-2002) 18

2.6 Traditional response spectrum vs. ADRS spectrum (ATC 40,1996) 19

2.7 Capacity curve vs. capacity spectrum (ATC 40,1996) 20

2.8 Bilinear representation of pushover or capacity curve (FEMA 273, 1997) 23

2.9 Properties of the nth-mode inelastic SDF system from the pushover curve(Chopra , 2002)

24

3.1 Log-normally distributed seismic fragility curves (HAZUS-MHMR1) 28

3.2 Flowchart to develop damage probability matrix 29

3.3 Idealized component load versus deformation curve (FEMA 273,1997) 33

3.4 Damage state thresholds on bilinear capacity spectrum (Barbat , 2008) 35

4.1 Bilinear representation of nonlinear curve 38

4.2 Stress-strain curve for concrete 39

4.3 Stress-strain curve for Fe415 grade steel 39

4.4 Flowchart to develop moment curvature relationship 40

4.5 Moment curvature relationship of with P and without P (C++) 42

4.6 Moment curvature relationship (Matlab) 42

4.7 Mathematical modeling of infill walls (Irtem, 2007) 43

4.8 Load deformation curve of diagonal strut (Irtem, 2007) 43

5.1 Elevation view of 2D frame (Inel, 2006) 46

5.2 Layout of columns of 2D frame (Inel, 2006) 46


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5.3 Comparison of pushover curve of 2D frame (User vs. default propert ies) 47

5.4 Pushover curves of 2D frame (Inel,2006) 47

5.5 Design response spectrum for zone III, MCE, soil type II 48

5.6 Capacity spectrum of 2D frame 49

5.7 Capacity and demand spectra of 2D frame in ADRS format 49

5.8 Bilinear representation of capacity spectra of 2D frame (Trail 1) 50

5.9 Reduced spectra of 2D frame with initial assumed point (Sdi,Sai) (Trail 1) 50

5.10 Bilinear representation of capacity spectra of 2D frame (Trail 2) 51

5.11 Reduced spectra of 2D frame with second assumed point (Sdi,Sai) (Trail 2) 51

5.12 Member level performances of 2D frame for zone III, MCE, soil type II 52

5.13 Performance point of 2D frame for zone III, MCE, soil type I 53

5.14 Member level performances of 2D frame for zone III, MCE, soil type I 53

5.15 Performance point of 2D frame for zone III, MCE, soil type II 54

5.16 Member level performance level for zone III ,MCE, soil type II 54

5.17 Performance point of 2D frame for zone III, MCE, soil type III 55

5.18 Member level performance level for zone III ,MCE, soil type III, 55

5.19 Elevation view of 7 story building (Fajfar,1996) 57

5.20 Plan view and layout of beams and columns of 7 story building (Fajfar,1996) 57

5.21 Reinforcement details of beams and columns of 7 story building(Fajfar,1996) 58

5.22 Comparison of pushover curves for 7 story building 59

5.23 Performance point of 7 storey building for zone III, MCE, soil type II 60

5.24 Member level performance level of 7 storey for zone III, MCE, soil type II 60

5.25 Building configurations with and without infill walls of 3 storey building

(Irtem, 2007)

62

5.26 Pushover curves of 3SBF with different load patterns 63

5.27 Pushover curves of 3SIF-I with different load patterns 63

5.28 Pushover curves of 3SIF-II with different load patterns 63

5.29 Comparison of pushover curves for 3 storey building with and without infill

walls

64

5.30 Pushover curves for 3 storey building (Irtem ,2007) 64


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5.31 Bilinear representation of pushover of 3SBF ( FEMA 273, 1997 ) 65

5.32 Damage state thresholds on capacity spectrum of 3SBF 67

5.33 Seismic fragility curves for 3SBF in terms of spectral displacement 67

5.34 Seismic fragility curves for 3SBF in terms of roof displacement 68

5.35 Probability of different damage states of 3SBF for E4 level of earthquake 68

5.36 Building configuration of example building no1 71

5.37 Another view of building configuration of example building no1 71

5.38 Capacity curves of example building no1 with different load patterns in X-

direction

72

5.39 Pushover curve of the example building no1 pushed in Y-direction 73

5.40 Performance of example building no1 for different zone levels of

earthquake

74

5.41 Capacity spectrum of example building no1 building 74

5.42 Demand imposed by different zones on the example building no1 building 75

5.43 Performance evaluation of example building no1 for zone III, MCE 75

5.44 Performance evaluation of example building no1 for zone IV, MCE 76

5.45 Performance evaluation of example building no1 for zone V, MCE 77

5.46 Building configuration of example building no 2 80

5.47 Pushover curves for original and designed example building no 2 fordifferent load patterns in X-direction

81

5.48 Pushover curves for original and designed example building no 2 in Y-direction

82

5.49 Fragility curves for original example building no 2 in terms of roof

displacement

83

5.50 Fragility curves for original example building no 2 in terms of spectral

displacement

83

5.51 Comparison of fragility curves as per HAZUS and Barbat,2008 method for

original example building no 2

84

5.52 Median damage states in terms of peak ground acceleration for original

example building no 2

84

5.53 Fragility curves of original example building no 2 in terms of peak ground

acceleration

85


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5.54 Fragility curves for designed example building no 2 in terms of roof

displacement

85

5.55 Fragility curves for designed example building no 2 in terms of spectral

displacement

86

5.56 Comparison of fragility curves for original, designed example building no 2 86

5.57 Performance evaluation of original example building no 2 for DBE 87

5.58 Bilinear representation of pushover of original example building no 2 as perFEMA 273, 1997

88

5.59 Member level performances of original example building no 2 for DBE,CSM 89

5.60 Member level performances of original example building no 2 for DBE,DCM 89

5.61 Performance evaluation of original example building no 2 for MCE (Trail1) 90

5.62 Performance evaluation of original example building no 2 for MCE (Trail2) 90

5.63 Member level performances of original example building no 2 for MCE,CSM 92

5.64 Member level performances of original example building no 2 for MCE,DCM

92

5.65 Damage state probabilities of original example building no 2 for DBE 93

5.66 Damage state probabilities of original example building no 2 for MCE 93

5.67 Plan layout of 4, 6, 8 storey buildings with 4m span length 96

5.68 Comparison of capacity curves for varying storey numbers in terms of drift

ratio

97

5.69 Comparison of capacity curves for varying storey height for 6 storey building 97

5.70 Comparison of capacity curves of 6 storey building with varying span length 97

5.71 Fragility curves of collapse resistance of reinforced concrete buildings withvarying storey number

98

5.72 Fragility curves of collapse resistance of reinforced concrete buildings with

varying storey height

98

5.73 Fragility curves of collapse resistance of reinforced concrete buildings with

varying span length

99

5.74 Band of fragility curves of collapse resistance for medium rise reinforced

concrete buildings

99


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List of Tables

Table

No

Description Page

No

2.1 Description of structural performance levels (ATC 40,1996) 5

2.2 Description of nonstructural performance levels (ATC 40,1996) 6

2.3 Building performance levels (ATC 40,1996, FEMA 273,1997) 6

2.4 Earthquake hazard levels (ATC 40,1996) 7

2.5 Earthquake hazard levels (FEMA 273,1997) 7

2.6 Configuration deficiencies in a building (ATC 40,1996) 9

2.7 Global acceptability limits for various performance levels (ATC 40,1996) 26

3.1 Guidance for selection of damage state medians (HAZUS-MH MR1) 32

3.2 Guidance for relating component deformation to the average inter-story

drift ratios of structural damage-state medians (HAZUS-MH MR1)

34

3.3 Damage state thresholds (Barbat ,2008) 34

3.4 Average inter-story drift ratio for structural damage states

(HAZUS-MH MR1)

35

4.1 Beam details and validation of program with reported results 41

5.1 Comparison of modal analysis results of 2D frame with reported results 45

5.2 Parameters used in estimating seismic inelastic displacement of 2D frame

by CSM

50

5.3 Summary of member level performances of 2Dframe for zone III, MCE,soil type II (Manually estimated)

52

5.4 Summary of member level performances of 2D frame for given level of

earthquake under different soil conditions

56

5.5 Comparison of modal analysis results of 7 storey building with reported

results

58

5.6 Summary of member level performances of 7 storey building for zone III,

MCE, soil type II

59

5.7 Details of diagonal strut used in mathematical modeling of 3 storey 61


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building

5.8 Comparison of modal analysis results of 3 storey buildings with reported

results

61

5.9 Comparison of parameters used in DCM for 3SBF with reported results 66

5.10 Damage state thresholds and variability for 3SBF 69

5.11 Results of modal analysis of example building no1 70

5.12 Design base shear values and member level performances of example

building no1 in various zones using SCM

73

5.13 Member level performances of example building no1 in different zones

using CSM

77

5.14 Comparison of estimated inelastic displacement of example building no1

by SCM and CSM

78

5.15 Comparison of modal analysis results of original and designed example

building no 2

80

5.16 Damage state thresholds and variability of original and designed example

building no 2

82

5.17 Parameter values used in evaluation of original example building no 2using CSM for DBE,MCE

91

5.18 Element performance levels of original example building no 2 using

CSM,DCM for DBE,MCE

91

5.19 Damage state probabilities of original example building no 2 for DBE and

MCE

94


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Declaration sheet

I declare that this written submission represents my ideas in my own words and where

others ideas or words have been included; I have adequately cited and referenced the

original sources. I also declare that I have adhered to all principles of academic honesty and

integrity and have not misrepresented or fabricated or falsified any idea/data/fact/source

in my submission. I understand that any violation of the above will be cause for disciplinary

action by the Institute and can also evoke penal action from the sources which have thus not

been properly cited or from whom proper permission has not been taken when needed.

A.Pavan Kumar

(Roll No. 08304032)

Date: 2nd July, 2010

Place: IIT Bombay


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Chapter 1

Introduction

1.1 Background

Earthquakes can create serious damage to structures. The structures already built are

vulnerable to future earthquakes. The damage to structures causes deaths, injuries, economic

loss, and loss of functions. Earthquake risk is associated with seismic hazard, vulnerability of

buildings, exposure. Seismic hazard quantifies the probable ground motion that can occur at

site. Vulnerability of building is important in causing risk to life. The seismic vulnerability ofa structure can be described as its susceptibility to damage by ground shaking of a given

intensity. The aim of a vulnerability assessment is to obtain the probability of a given level of

damage to a given building type due to a scenario earthquake. The level of damage is directly

associated with deaths, injuries, economic losses. Damage functions are to be developed to

assess the damage level for given level of earthquake. The outcome of vulnerability

assessment can be used in loss estimation. Loss estimation is essential in disaster mitigation,

emergency preparedness.

Although force-based procedures are well known by engineering profession and easy to

apply, they have certain drawbacks. Structural components are evaluated for serviceability in

the elastic range of strength and deformation. Post-elastic behavior of structures could not be

identified by an elastic analysis. However, post-elastic behavior should be considered as

almost all structures are expected to deform in inelastic range during a strong earthquake. The

seismic force reduction factor "R" is utilized to account for inelastic behavior indirectly by

reducing elastic forces to inelastic.

Elastic methods can predict elastic capacity of structure and indicate where the first yielding

will occur, however they dont predict failure mechanisms and account for the redistribution

of forces that will take place as the yielding progresses. Real deficiencies present in the

structure could be missed. Moreover, force-based methods primarily provide life safety but

they cant provide damage limitation and easy repair. The drawbacks of force-based

procedures and the dependence of damage on deformation have led the researches to develop

displacement-based procedures for seismic performance evaluation. Displacement-based

procedures are mainly based on inelastic deformations rather than elastic forces and use


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nonlinear analysis procedures considering seismic demands and available capacities

explicitly.

1.2 Scope of the Dissertation

1. Seismic performance evaluation of reinforced concrete buildings using capacityspectrum method (ATC 40, 1996) and displacement coefficient method (FEMA273,

and FEMA 356, 2000) is studied.

2. Basics of pushover analysis, advantages and limitations are discussed in detail. 3. Estimation of seismic inelastic displacement by capacity spectrum method,

displacement coefficient method, seismic coefficient method and modal pushover

analysis is explained in detail.

4. Modeling of member nonlinearity in flexural members, axial members, and shearwalls has been studied in detail.

5. Developed coding to generate moment-curvature relationship for reinforced concretesections in Matlab and C++.

6. Comparison of capacity curves developed using SAP 2000 default hinge propertiesand user defined hinge properties developed based on IS 456-2000 is studied andimportance of using user defined hinge properties is discussed.

7. Effect of infill walls on lateral resistance and capacity of building is studied andcompared with bare frame.

8. Influence of soil conditions on seismic inelastic displacement of building for givenlevel of earthquake is studied.

9. Different methods of developing fragility curves and damage probability matrix arestudied and comparisons were made.

10.Applicability of HAZUS based drift ratios and fragility curves for buildings designedas per IS 456-2000 have been studied.

11.Influence of age of construction of buildings on fragility curves of buildings is studied.12.Influence of structural parameters on vulnerability of commercial building class is

studied.


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1.3 Organization of the Report

Second chapter discuss seismic performance evaluation of RCC buildings as per ATC 40,

1996 and FEMA 273, 1997 guidelines. The methods for evaluation of performance of

structure are discussed in brief. Basics of pushover analysis, advantages and limitations have

been discussed in detail. It also explains methods for estimating seismic inelastic

displacement using capacity spectrum method (CSM), displacement coefficient method

(DCM), and modal pushover analysis (MPA).

Thirdchapter explains different procedures in development of fragility curves and damage

probability matrices. The different ways of defining median damage states for developing

fragility curves have been discussed along with their limitations.

Fourth chapter discusses modeling of nonlinear component properties in flexural members,

axial members, and shear walls. The validation of developed moment curvature relationships

with already published results has been done. The modeling of infill walls as diagonal struts

and its load deformation characteristics has been studied.

Fifth chapter deals with modal analysis, performance evaluation and fragility analysis of

buildings. It starts with models taken from published literature and results were compared.

Fragility curves developed with different methods were compared and conclusions were

drawn. Influence of structural parameters on vulnerability of commercial building class has

been studied.

Sixth chapter deals with discussions, conclusions and scope for further work.


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Chapter 2

Seismic Performance Evaluation of Buildings

2.1 Seismic performance evaluation

The seismic performance evaluation and retrofit of existing buildings pose a great challenge

for the owners, architects, and engineers. The risk, measured in both lives and dollars, are

high. Equally high is the uncertainty of where, when, and how large future earthquakes will

be. The inherent complexity of concrete buildings and of their performance during

earthquakes compounds uncertainty. Traditional procedures developed primarily for newconstruction are not wholly adequate tools for meeting this challenge. Filiatrault et al, 1997

studied seismic behavior of two half scale reinforced concrete structures experimental and

analytically. Performance based evaluation procedure provides insight about the actual

performance of buildings during earthquake. The steps to be followed in seismic performance

evaluation of structures and rehabilitation of structures are given below

1. Select the performance objective of the building as required by owner to achieve forgiven seismic hazard.

2. Review the existing building conditions by visual inspections, existing drawings, andtests on structure and perform preliminary evaluation of the building.

3. Formulate a strategy for achieving the desired performance objective for given level ofseismic hazard.

4. Assess the performance of the retrofitted structure with any analysis procedures.5. Check the performance of the structure with desired performance objective. 6. If performance objective is not achieved, formulate new strategy and assess the

performance of the structure again. Do the above process till desired performance

objective is achieved.

The steps for performance based design remain same as above. In design we have flexibility

in changing the configuration of building, and sections. Once the design is fixed the above

process remains same. The explanation of performance levels, preliminary evaluation of

building and formulation of the retrofit strategy is discussed in brief in the following sections.


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2.1.1 Performance objectives

Performance objective specifies the desired seismic performance of building. Seismic

performance is described by designating the maximum allowable damage state for an

identified seismic hazard.

2.1.2 Performance levels

Performance level describes a limiting damage condition which may be considered

satisfactory for a given building and a given ground motion. Performance levels are

qualitative statements of damage the structure going to experience in future prescribed

earthquakes. Performance levels are described for structural components and nonstructuralcomponents.ATC 40, 1996 defines 6 levels of structural damage or performance levels and 5

levels of nonstructural damage. The brief details of structural and non-structural performance

levels are given in table 2.1 and table 2.2.

Table 2.1 Description of structural performance levels (ATC 40, 1996)

Structural

performance level

Damage description

Immediate

occupancy(IO)

Very limited structural damage and risk to life is negligible. Vertical

and lateral resisting system retains all pre-earthquakes characteristics.

Damage control Range with more damage than IO and less than LS

Life safety (LS) Significant damage to structural elements with some residual strength.

Risk to life from structural damage is very low.

Limited safety Range with more damage than LS and less than SS

Structural

stability(SS)

Building is on verge of partial or total collapse. Significant degradation

in stiffness and strength of lateral resisting system. Gravity load

resisting remains to carry gravity demand.

There is not considered (NC) option in performance level. This is option for owner weather to

consider structural or nonstructural performance level. FEMA 273, 1997 defines same

definitions of performance levels as described in ATC 40, 1996 but instead of structural

stability (SS) FEMA 273, 1997 describes as collapse prevention (CP).


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Table 2.2 Description of nonstructural performance levels (ATC 40, 1996)

Nonstructural

performance level

Damage description

Operational Nonstructural systems are in place and functional. All equipment

and machinery will be in working condition

Immediate occupancy Minor disruption of nonstructural elements and functionality is not

considered. Seismic safety status should not be affected

Life safety Considerable damage to nonstructural elements. Risk to life from

nonstructural damage is very low.

Hazards reduced Extensive damage to nonstructural damage. Risk to life because of

collapse or falling of large and heavy items should be considered

Building performance level is combination of structural and nonstructural performance levels.

There so many combinations of performance levels for owner to choose based on

requirement. Building performance levels that commonly used are given in table 2.3. The

building performance levels represented on pushover curve and load deformation curve are

shown in figure 2.1

Table 2.3 Building performance levels (ATC 40, 1996 and FEMA 273, 1997)

Building performance

levels

Combination of structural and nonstructural performance

level

Operational Immediate occupancy(S)+ Operational (NS)

Immediate occupancy Immediate occupancy(S)+Immediate occupancy(NS)

Life safety Life safety (S)+ Life safety(NS)

Structural stability (or)

Collapse prevention

Structural stability (or) Collapse prevention (S)+Not

considered

2.2 Earthquake g round motion

Earthquake ground motion is combined with a desired building performance level to perform

a performance objective. The earthquake ground motion can be specified as level of shaking


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associate with a given probability of occurrence or in terms of maximum shaking from single

event. ATC 40, 1996 defines three levels of earthquake ground motions as given in table 2.4

Table 2.4 Earthquake hazard levels (ATC 40, 1996)

Level of earthquake Definition

Serviceability earthquake

(SE)

Ground motion with a 50 percent chance of being exceeded in

50 year period

Design earthquake (DE) Ground motion with a 10 percent chance of being exceeded in

50 year period

Maximum earthquake

(ME)

Ground motion with a 5 percent chance of being exceeded in 50

year period

FEMA 273, 1997 defines two levels of earthquake ground motions as given in table 2.5

Table 2.5 Earthquake hazard levels (FEMA 273, 1997)

Level of earthquake Definition

Basic safety earthquake 1

(BSE~1)


50 year period

Basic safety earthquake 2

(BSE~2)


50 year period

Figure 2.1 Performance levels on pushover curve and load deformation curve (Irtem, 2007)


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2.3 Basic safety objective

As per ATC 40, 1996 Basic performance objective is defined as achieving life safety

performance level for design earthquake (DE) and structural stability performance level for

maximum earthquake (ME). As per FEMA 273,1997 guidelines basic safety objective is

defined as achieving life safety performance level for basic safety earthquake~1 (BSE~1) and

collapse prevention performance level for basic safety earthquake~2(BSE~2). The wide

variety of building performance level can be combined with various levels of ground motion

to form many possible performance objectives. Performance objectives for any building may

be assigned using functional, policy, preservation or cost considerations.

2.4 Preliminary evaluation of structure

A general sense of expected building performance should be developed before performing a

detailed analysis. Preliminary evaluation involves acquisition of building data, review of the

seismic hazard, limited analysis, and characterization of potential deficiencies. With

preliminary analysis we can judge whether to go for higher level of analysis or to use

simplified analysis. If sufficient data of the building is not there then material testing has to be

done. Sometimes with preliminary analysis it is observed that the building meets the desired

performance objective. The typical deficiencies that could be observed in a building during

preliminary evaluation are described in the table 2.6

2.5 Retrofit strategy and retrofit system

Retrofit strategy is a basic approach adopted to improve the seismic performance of the

building or otherwise reduce the existing seismic risk to an acceptable level. Both technical

strategies and management strategies can be employed to obtain seismic risk reduction.

Technical strategies include such approaches as increasing building strength, correcting

critical deficiencies, altering stiffness, and reducing demand. Management strategies include

such approaches as change of occupancy, incremental improvement, and phased construction.

Retrofit system is the specific method used to achieve the selected strategy. If the basic

strategy is to increase building strength, then the alternative systems that may used to

accomplish this strategy could include addition of shear walls, thickening of existing shear


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walls, and addition of braced frames. It is necessary to select a specific system in order to

complete a design.

Table 2.6 Configuration deficiencies in a building (ATC 40, 1996)

Configuration

deficiencies

Explanation of deficiencies

Incomplete load

path

Complete load path is required to transfer lateral load to foundation.

Missing links in the load path must be identified.

Vertical

irregularities

Vertical irregularities typically occur in a story which is significantly

weaker, more flexible or heavier than the stories above or below.

Horizontal

irregularities

Horizontal irregularities are typically due to odd plan shapes, re-entrant

corners, diaphragm openings and d iscontinuities.

Weak

column/Strong

beam

Optimum seismic performance is gained when frame members have shear

strengths greater than bending strengths of column are greater than beams

to have controlled failure mode. Column hinging can lead to story

mechanism creating large deflections and inelastic rotations.

Detailing concern Non-ductile frame exhibit poor seismic performance. Quantity, spacing,

splicing, location, size, anchorages of bars are to be checked.

Beam column

joint

The lateral stability of the frame is dependent upon beam column joint

capacity. Adequate stiffness and strength must be provided to sustain

repeated cyclic stress reversals. Adequate reinforcement should be

provided in joint.

2.6 Methods of analysis for evaluation of seismic performance

evaluation of buildings

Basically two methods of analysis are available to predict the seismic performance of

structures. Each method has its own advantages and limitations. The details of the two

methods are given below.


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2.6.1 Elastic method of analysis

It is assumed that the structure will remain elastic under probable loads. So the strains and

stress are linear along the depth of section. But to design a building to remain elastic for

earthquake forces is uneconomical.

2.6.1.1 Seismic coefficient method

In seismic coefficient method the maximum base shear is calculated based on the fundamental

time period, importance factor, reduction coefficient. Lateral forces are distributed

proportional to square of height. R factor is used to allow structure to go into inelastic to

dissipate energy through yielding.

2.6.1.2 Linear elastic dynamic analysis

This analysis required for Irregular buildings and Tall buildings. Dynamic Analysis can be

time history analysis or response spectrum analysis. Sufficient number of modes must be

considered in analysis such that total mass participation is at least 90%.Elastic Methods can

predict elastic capacity of structure and indicate where the first yielding will occur, however

they dont predict failure mechanism and account for the redistribution of forces that will take

place as the yielding progresses. Moreover, force-based methods primarily provide life safety

but they cant provide damage limitation and easy repair.

2.6.2 Inelastic method of analysis

Inelastic method of analysis incorporates material nonlinear behavior and geometric

nonlinearity. Material nonlinearity is modeled using nonlinear stress-strain curve. Geometric

nonlinearity is incorporated in structure by calculating secondary moment for each time step.

2.6.2.1 Inelastic time history analysis or nonlinear response history analysis

In NRH analysis the reduced stiffness in nonlinear range is considered and the force

deformation is not single valued function. It depends on direction of motion as well. The

inelastic time history analysis is the most accurate method to predict the force and

deformation demands at various components of the structure. However, the use of inelastic

time history analysis is limited because dynamic response is very sensitive to modeling and

ground motion characteristics. It requires proper modeling of cyclic load deformation

characteristics considering deterioration properties of all important components. Also, it


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requires availability of a set of representative ground motion records that accounts for

uncertainties and differences in severity, frequency and duration characteristics. Moreover,

computation time, time required for input preparation and interpreting voluminous output

make the use of inelastic time history analysis impractical for seismic performance

evaluation.

2.6.2.2 Inelastic static analysis or pushover analysis

In pushover analysis the structure is subjected to monotonically increasing lateral loads until

target displacement is reached. A predefined load pattern is applied and increased till yielding

in one member occurs then the structure is modified and lateral loads are increased further.

Sermin et al, 2005 studied application of pushover of procedure for frame structures. He

studied the effect of different lateral load patterns on capacity of structure. The pushover or

capacity curve of the building is shown figure 2.2. Lateral loads are increased till structure

reaches its ultimate capacity. The pushover is expected to provide information on many

response characteristics that cannot be obtained from an elastic static or dynamic analysis.

The following are examples of such response characteristics are taken from Krawinkler et al,

1998.

1. The realistic force demands on potentially brittle elements, such as axial forcedemands on columns, force demands on brace connections, moment demands on

beam-to-column connections, shear force demands in deep reinforced concrete

spandrel beams, shear force demands in un reinforced masonry wall piers, etc.

2. Estimates of the deformation demands for elements that have to deform in elasticallyin order to dissipate the energy imparted to the structure by ground motions.

3.

Consequences of the strength deterioration of individual elements on the behavior ofthe structural system.

4. Identification of the critical regions in which the deformation demands are expectedto be high and that have to become the focus of thorough detailing.

5. Identification of the strength discontinuities in plan or elevation that will lead tochanges in the dynamic characteristics in the inelastic range.

6. Estimates of the inter story drifts that account for strength or stiffness discontinuitiesand that may be used to control damage and to evaluate P- effects.


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7. Verification of the completeness and adequacy of load path, considering all theelements of the structural system, all the connections, the stiff nonstructural elements

of significant strength, and the foundation system.

Figure 2.2 Pushover or capacity curve of the building (Sermin, 2005)

2.6.2.2.1 Background to pushover analysis

Pushover analysis is based on the assumption that the response of the structure can be related

to the response of an equivalentsingle degree-of- freedom (SDOF) system. The formulation of

the equivalent SDOF system is not unique, but the basic underlying assumption common to

all approaches is that the deflected shape of the MDOF system can be represented by a shape

vector that remains constant throughout the time history, regardless of the level of

deformation. Accepting this assumption and defining the relative displacement vector of

an MDOF system as .The governing differential equation of an MDOF system

can be written as

(1)

Where M and C are mass and damping matrices, Q denotes storey force vector and is

ground acce leration. If we define the reference SDOF displacement as

(2)

Pre-multiply differential equation by and substitute for we obtain the followingdifferential equation for the response of the equivalent SDOF system as


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(3)

Where are the properties of the equivalent SDOF system and are given by

(4)

(5)

(6)

Presuming that the shape vector is known, the force deformation characteristics of the

equivalent SDOF system can be determined from the results of a nonlinear

incremental static analysis of the MDOF structure, which usually produces a base shear

versus roof displacement diagram of the type shown in figure 2.3 below. In order to identify

nominal global strength and displacement quantities, the multi linear diagram needs to

be represented by a bilinear relationship that defines a yield strength and effective elastic

stiffness and post elastic stiffness for the structure.

The simplified pushover is shown in the figure 2.3 is needed to define the properties of the

equivalent SDOF. The yield value of base shear and corresponding roof displacement

are used to compute force displacement relationship for equivalent SDF system as follows.

(7)

(8)

Where is story force vector at yield i.e.,

The initial period of equivalent SDOF is given by following equation

(9)


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Figure 2.3 Force displacement characteristics of MDF and equivalent SDF system

(Krawinkler, 1998)

The strain hardening ratio of of the MDF system defines the strain hardening ratio

of equivalent SDF system. The fundamental question in the execution of the pushover

analysis is the magnitude of the target displacement at which seismic performance evaluation

of the structure is to be performed. The target displacement serves as an estimate of the global

displacement the structure is expected to experience in a design earthquake. A convenientdefinition of target displacement is the roof displacement at the center of mass of the struct ure.

The properties of the equivalent SDOF system, together with spectral information for inelastic

SDOF system provide the information necessary to estimate the target displacement. The

target displacement for inelastic SDOF system is calculated from inelastic spectra using the

period calculated above. Then this target displacement is converted into global roof

displacement. To use inelastic response spectrum we need ductility factor which we can get

from R-factor (i.e. ratio of elastic strength to yield strength). The R-factor can be calculated

from the above data. The displacement obtained from inelastic spectrum has to be modified to

account for effect, stiffness degradation, strength deterioration.For performance

evaluation of a structure, the structure is pushed to calculated target displacement, and then

desired responses at target displacement are found out from pushover data base.

2.6.2.2.2 Implementation of pushover analysis

The process is to represent the structure in a two- or three dimensional analytical model that

accounts for all important linear and nonlinear response characteristics, apply gravity loads

followed by lateral loads in predetermined or adaptive patterns that represent approximately


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the relative inertia forces generated at locations of substantial masses, and push the structure

under these load patterns to target displacements that are associated with specific performance

levels. The internal forces and deformations computed at these target displacements are used

as estimates of the strength and deformation demands, which need to be compared to

available capacities. The emphasis in performance evaluation needs to be on the following

points.

1. Verification that an adequate load path exists.2. Verification that the load path remains sound at the deformations associated with the

target displacement level.

3. Verification that critical connections remain capable of transferring loads between theelements that form part of the load path.

4. Verification that individual elements that may fail in a brittle mode and are importantparts of the load path are not overloaded.

5. Verification that localized failures (should they occur) do not pose a collapse or lifesafety hazard, i.e. that the loads tributary to the failed element(s) can be transferred

safely to other elements and that the failed element itself does not pose a falling

hazard.

2.6.2.2.3 Limitations of pushover analysis

A carefully performed pushover analysis will provide insight into structural aspects that

control performance during severe earthquakes. For structures that vibrate primarily in the

fundamental mode, such an analysis will very likely provide good estimates of global as well

as local inelastic deformation demands. It will also expose design weaknesses that may

remain hidden in an elastic analysis. Such weaknesses include story mechanisms, excessive

deformation demands, strength irregularities, and overloads on potentially brittle elements,

such as columns and connections. Although pushover analysis posses a lot of advantages, it

has several limitations also.

1. Pushover analysis is approximate in nature and based on static loading, so it cannotrepresent dynamic phenomena in large accuracy. It may not detect some important

deformation modes that may occur in a structure subjected to severe earthquakes, and

it may exaggerate others.

2. Limitations are imposed also by the load pattern choices. Whatever load pattern ischosen, it is likely to favor certain deformation modes that are triggered by the load


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pattern and miss others that are initiated and propagated by the ground motion and

inelastic dynamic response characteristics of the structure. Thus, good judgment needs

to be employed in selecting load patterns and in interpreting the results obtained from

selected load patterns.

3. Pushover analysis will give reasonable results when the structure is vibrating infundamental mode. But its accuracy decreases when the higher modes become

important in particular structure.

2.7 Estimation of inelastic displacement demand

The structure undergoes inelastic displacement for severe earthquake. Linear analysis

methods cannot predict the inelastic displacement. Nonlinear response history analysis gives

exact behavior of the buildings under severe earthquakes. Nonlinear response history analysis

is very sensitive to ground motions and building characteristics. The other method which uses

inelastic static analysis (pushover analysis) is effective way of estimating inelastic

displacement. Three methods for estimating inelastic displacement are discussed in detail.

2.7.1 Capacity spectrum method

ATC 40, 1996 has developed a simple iterative procedure to estimate seismic inelastic

displacement for given level of earthquake. For seismic evaluation of existing structures the

procedure can be easily implemented. This procedure requires pushover curve which is

obtained from nonlinear static analysis of structure. Demand spectrum has to be developed for

the given site considering level of earthquake (Serviceability earthquake (SE), Design

earthquake (DE), and Maximum earthquake (ME)). This are defined based on percentage

chances of probability of exceeding particular ground motion during 50 year time period.

IS1893 defines two levels of earthquakes (Design basis earthquake (DBE), Maximum

considered earthquake (MCE)). The procedure to estimate seismic inelastic displacement as

per ATC 40, 1996 procedure is given below.

1. Develop design demand spectrum ( vs T) for the given site considering soil effects,level of earthquake.

2. Convert demand spectrum ( vs T) into acceleration displacement responsespectrum (ADRS) format.

3. Develop the capacity curve i.e., pushover curve obtained with incremental invariantlateral load pattern applied to structure until structure reaches ultimate capacity.


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4. Convert capacity curve into capacity spectrum which is representation of capacitycurve in acceleration-displacement response spectra (ADRS) format.

5. Bilinear representation of capacity spectrum is needed to estimate the effectivedamping and appropriate reduction of spectral demand associated with

displacement .

6. Calculate the effective viscous damping associated with maximum displacementi.e. hysteretic damping represented as equivalent viscous damping plus inherent

viscous damping.

7. Calculate spectral reduction factors which are required to reduce 5%damped elastic design response spectrum to account for yielding.

8. Draw demand spectrum in ADRS format on the same plot as the capacity spectrum asshown in the figure 2.4

9. If reduced demand spectrum intersects the capacity spectrum at initially assumeddisplacement then it is the performance point. Performance point is the inelastic

displacement of the structure for the given level of earthquake.

10.If reduced demand spectrum does not intersects the capacity spectrum at initiallyassumed displacement then assume next displacement based on judgment. Repeat

steps 5 to 8 until convergence is achieved. The plot showing capacity spectrum

method is given in figure 2.4

Figure 2.4 Capacity spectrum method (HAZUS MH MR 4)


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Conversion of capacity curve and demand spectrum into acceleration-displacement response

spectrum is explained in the following sections. Calculation of effective damping and spectral

reduction factors are also explained in the following sections.

2.7.1.1 Design demand spectrum ( vs T)

The design demand spectrum has to be developed for given site considering range of

earthquakes or IS 1893-2002 code gives design response spectrum for different zones. IS

1893-2002codes gives design response spectrum for three sites i.e., rocky or hard soil,

medium soil, soft soil sites is represented in figure 2.5. The classification of site into above

mentioned categories is based on IS 1893-2002. The design response spectrum is for 5%

damped structure. IS1893 gives modification factors for other damping values. For special

structure site design spectrum has to be developed. The reduction factors given in IS1893 for

other damping can be used as reduction factors to get reduced design demand response

spectrum.

Figure 2.5 Spectral acceleration coefficients vs. time period (IS 1893-2002)

The mathematical expressions to calculate spectral acceleration coefficient for different sites

as per IS1893 are given below.

For rocky or hard soil sites

(10)

For medium soil sites


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(11)

For soft soil sites

(12)

2.7.1.2 Conversion of design response spectrum into ADRS format

Every point on response spectrum curve has associated with it a unique spectral

acceleration , spectral velocity , spectral displacement and time period T. To convert

a spectrum from standard to ADRS format it is necessary to determine the value

for each point on the curve . The line radiating from origin to point on the curve

represents the time period . The representation of response spectrum in traditional and in

ADRS format is shown in figure 2.6. It is observed that the period lengthens as the structure

undergoes inelastic displacement. This spectral displacement is related to spectral acceleration

and time period as given in equation no 13.

(13)

Figure 2.6 Traditional response spectrum vs. ADRS spectrum (ATC 40, 1996)

2.7.1.3 Conversion of capacity curve into capacity spectrum

To use capacity spectrum method it is necessary to convert the capacity curve into capacity

spectrum. Capacity spectrum is plot in terms spectral acceleration and spectral

displacement . The demand imposed on the structure i.e., design response spectrum ( vs


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T) is also converted into ADRS format. Both capacity spectrum and design response spectrum

in ADRS format are p lotted on graph. The performance point i.e., inelastic seismic demand is

intersection of capacity spectrum and reduced demand spectrum such that capacity of

structure and demand imposed on the structure are equal. The required equations to make

transformation are given in equation no 14 to equation no 17. Conversion of capacity curve

into capacity spectrum is shown in figure 2.7.

Figure 2.7 Capacity curve vs. capacity spectrum (ATC 40, 1996)

(14)

(15)

(16)

(17)

The meaning of notations are given below

=Mode shape vector at level i

=Modal mass participation factor for the first natural mode

V= Base shear

W= Seismic weight of the building i.e., dead load and likely live load

Mass assigned to level i

=Spectral acceleration

=Spectral d isplacement


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2.7.1.4 Bilinear representation of capacity spectrum

A bilinear representation of the capacity spectrum is needed to estimate the effective damping

and appropriate spectral reduction factors . Construction of the bilinear

representation requires definition of the point . This point is the trial performance

point which is estimated to develop a reduced demand response spectrum. The first estimate

of the point is designated as the second , and so on. Equal

displacement approximation is used as an estimate of ap1, dp1. Equal displacement

approximation is based on the assumption that the inelastic spectral displacement is the same

as that which would occur if the structure remained perfectly elastic. The procedure to

construct bilinear curve is explained in section 4.3. Once bilinear capacity spectrum

developed we can calculate the effective damping and spectral reduction factors

using the expressions given in equation no 18.

(18)

Where are initial assumed values based on equal displacement approximation

method. The yield values are obtained from bilinear curve of capacity spectrum. -

factor is measure of the extent to which actual building hysteresis is well represented by

idealized parallelogram. The - factor depends on structural behavior of the building and

duration of ground shaking. The values for factor can be obta ined from ATC 40, 1996.

2.7.1.5 Calculation of spectral reduction factors

Spectral reduction factors are required to reduce the elastic 5% damped design response

spectrum to account for yielding i.e. hysteric effect. The spectral reduction factors

are given in equation no 19, and 20.

(19)

(20)

2.7.2 Displacement coefficient method

FEMA273, 1997 has developed displacement coefficient method for calculating the

displacement demand. The displacement coefficient method provides a numerical process for

calculating the displacement demand. There are some differences in terminology used


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compared to ATC 40, 1996. Bilinear representation of pushover curve is different from the

procedure used in ATC 40, 1996.

The step by step procedure to calculate the inelastic displacement of structure is given below.

1. Develop a capacity curve (base shear versus roof displacement) of the overall structure bypushover analysis.

2. Bilinear representation of pushover is developed to know initial stiffness, secant stiffnessand post elastic stiffness. These parameters are required to modify time period of the

structure. The method for bilinear representation of pushover curve as per FEMA 273,

1997 guidelines is explained in section 2.8.1

3. Calculate the effective fundamental time period using expression given below(21)

Where

=Initial elastic lateral stiffness of building in the direction under consideration.

=Effective lateral stiffness of building in the direction under consideration.

=Effective fundamental time period

=Elastic fundamental time period calculated by elastic dynamic analysis

Calculate the target displacement as

(22)

Where

=Modification factor to relate spectral displacement and likely building roof displacement

=Modification factor to relate expected maximum inelastic displacement to displacement

calculated for linear elastic response

=Modification factor to represent the effect of hysteresis shape on the maximum

displacement response

=Modification factor to represent increased displacements due to second order effects.

The values and expressions to calculate above factors are given in FEMA 273, 1997.

In this approach, a line representing the average post-elastic stiffness, of capacity curve is

first drawn by judgement. Then, a secant line representing effective elastic stiffness is

drawn such that it intersects the capacity curve at 60% of the yield base shear. The yield base

shear is defined at the intersection of and lines. The process is iterative because the


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value of yield base shear is not known at the beginning. An illustrative capacity curve and its

bilinear representation are shown in figure 2.8

Figure 2.8 Bilinear representation of pushover or capacity curve (FEMA 356, 2000)

2.7.3 Modal pushover analysis

Chopra et al, 2007 developed an improved pushover analysis procedure named Modal

Pushover Analysis (MPA) to account for the effects of higher modes on structural response

and for the redistribution of inertial forces during progressive yielding. This procedure is

shown to be equivalent to response spectrum analysis (RSA) when applied to linear elastic

systems. In the modal pushover analysis, the seismic demand due to the individual terms in

the modal expansion of effective earthquake forces is determined by pushover analysis using

the inertia force distribution at each mode. Combining these modal demands due to first two

or three terms of the expansion provide an estimate of the total seismic demand on the

inelastic systems. The differential equation governing the response o f a multistory building to

horizontal earthquake ground motion is

(23)

The standard approach is to directly solve these coupled equations leading to exact nonlinear

response history analysis. The spatial distribution s of effective earthquake forces is

expanded into modal coordinates with assumption that coupling of modal coordinates

arising from yielding of the system is neglected. The governing equation for the nth mode

inelastic SDF system is

(24)

The resisting force is given as


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The vibration properties natural frequency and damping ratio are properties ofnth mode

of the corresponding linear MDF system. The relationship is obtained from

pushover curve with load distribution with structure pushed till structure reaches

its ultimate capacity.

The pushover curve ( is converted to as explained in the following

equations. The two sets of forces and displacements are related as follows

, (25)

The above equation is used to convert pushover curve to desired relation, the yield

values of and are

, (26)

Where = is the effective modal mass

The vibration period of the nth mode inelastic SDF system is computed as

(27)

Figure 2.9 Properties of the nth-mode inelastic SDF system from the pushover curve

(Chopra, 2002)


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The procedure to estimate the seismic demands of inelastic systems consists of following

steps.

1. Determine the natural frequencies and mode shapes for linearly elasticvibration of the structure.

2. For the nth mode, develop the 'modal' capacity curve (base shear versus roofdisplacement) of the overall structure for the lateral load distribution

where is the mass matrix.

3. Obtain the force-displacement relationship of the nth mode inelastic SDOF from thecorresponding 'modal' capacity curve as described in preceding paragraphs.

4. Perform a nonlinear dynamic analysis for the ground motion excitation by utilizing theforce-d isplacement relationship ofnth mode inelastic SDOF system to obtain the peak

deformation of n-th mode inelastic SDOF system.

5. Calculate the peak roof displacement of MDF system associated with the n-thmode inelastic SDOF system as

(28)

Where

Modal participation factor for the nth mode

Mode shape value ofnth mode MDF system at roof level

: Peak spectral displacement ofnth mode inelastic SDOF system

6. Extract any peak response parameter (shear force, bending moment, drift ratios,plastic rotation) from the pushover results at roof displacement .

7. Repeat above steps for as many modes required. The condition for number modes tobe considered is the total mass participation should be at least 90%.

8. Determine the peak value of total response by combining the peak modal responsesusing any appropriate modal combination (square root of sum of squares (SRSS),

complete quadratic combination (CQC)).

,


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2.8 Acceptance criteria for different performance levels

To determine whether a building meets a specified performance objective, response quantities

obtained from above mentioned analysis procedures are compared with limits for appropriateperformance levels. The response limits fall into two categories.

These response limits include requirements for the vertical load capacity, lateral load

resistance, and lateral drift. The gravity load capacity of the building structure should remain

intact for acceptable performance at any level. The lateral load resistance of the building

should not degrade by more than 20 percent of the maximum resistance of the structure. The

deformation limits for various performance levels are given in table 2.7

Table 2.7 Global acceptability limits for various performance levels (ATC 40, 1996)

Inter-story drift limit Immediate

occupancy

Damage

control

Life

safety

Structural

stability

Maximum total drift 0.01 0.01-0.02 0.02

Maximum inelastic

drift

0.005 0.005-0.015 No limit No limit

Where =Total calculated lateral shear force in story i

= Total gravity load at story i

Each component must be checked to determine if its components respond within acceptable

limits. The acceptable limits for various performance levels for beams and columns are given

in ATC 40, 1996 and FEMA 273, 1997 guidelines.

2.9 Discussions

Seismic performance evaluation of buildings is discussed. The various steps to be followed in

evaluation procedure are explained in detail. Various seismic methods of analysis of structure

are discussed in brief. Basics of pushover or nonlinear static analysis, advantages and

limitations are explained clearly. Methods of estimating seismic inelastic displacement as per

ATC 40, 1996 and FEMA356, 2000 guidelines are explained in detail. Modal pushover

analysis procedure proposed by Chopra et al, 2007is also explained.


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Chapter 3

Seismic Vulnerability and Fragility Analysis of Buildings

3.1 Seismic vulnerability of building

Earthquake risk assessment is needed to estimate the casualties, losses (direct losses,

economic losses, social impact) and to mitigate the risk associated. Earthquake risk is depends

on hazard, vulnerability, and exposure. A significant component of a loss model is a

methodology to assess the vulnerability of the built environment. The seismic vulnerability of

a structure can be described as its susceptibility to damage by ground shaking of a givenintensity. The aim of a vulnerability assessment is to obtain the probability of a given level of

damage to a given building type due to a scenario earthquake. There are two methods of

assessing vulnerability of given building type. Empirical methods developed based on

observed damage in past earthquakes. Analytical methods developed by simulation done on

computer model. Lang et al, 2002 studied seismic vulnerability of existing buildings in

Switzerland. He developed analytical capacity curves for masonry building reinforced

buildings. Damage grades were defined on capacity curves. Calvi et al, 2006 has discussed

the development of vulnerability assessment methods in 30 years.

3.2 Fragility curves of building

Fragility curves describe the probability of damage to building. Building fragility curves are

lognormal functions that describe the probability of reaching, or exceeding, structural and

nonstructural damage states, given median estimates of spectral response, for example

spectral displacement. These curves take into account the variability and uncertaintyassociated with capacity curve properties, damage states and ground shaking.

The fragility curves distribute damage among slight, moderate, extensive and complete

damage states. For any given value of spectral response, discrete damage-state probabilities

are calculated as the difference of the cumulative probabilities of reaching, or exceeding,

successive damage states. The probabilities of a building reaching or exceeding the various

damage levels at a given response level sum to 100%. Discrete damage-state probabilities are

used as inputs to the calculation of various types of building-related loss. Each fragility curve

is defined by a median value of the demand parameter (e.g., spectral displacement) that


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corresponds to the threshold of that damage state and by the variability associated with that

damage state. The typical fragility curve is shown in figure 3.1.

Figure 3.1 Log-normally distributed seismic fragility curves (HAZUS-MHMR1)

3.2.1 Procedure to develop damage probability matrix (DPM) of building

for given level of earthquake

The steps involved in development of fragility curve as per HAZUS-MH MR1 are explained

in the flowchart shown in figure 3.2.

3.2.1.1 Building type and classification

Buildings are classified both in terms of their use, or occupancy class, and in terms of their

structural system, or model building type. Damage is predicted based on model building type,

since the structural system is considered the key factor in assessing overall buildingperformance, loss of function and casualties. Occupancy class is impo rtant in determining

economic loss, since building value is primarily a function of building use

Buildings are classified based on structural characteristics like number of stories as

1. Low-rise (1-3 stories),2. Mid-rise (4-7 stories)3. High-rise (8+ stories)

Building classification is done based on the material used for construction: steel frame,

concrete frame, brick masonry burned and unburned, stone masonry and mud wall.


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Building Model Type

Structural Building Characteristics

Seismic Design Level

Develop Response Spectra Generate Capacity Curve

Calculate Building Peak Response (Sd)

Generate Fragility Curve for defined Damage States

Calculate Discrete Damage Probabilities for Damage States

Ground Motion and Seismic Data

Damage Probability Matrix for

particular Building Model Type

Figure 3.2 Flowchart to develop damage probability matrix

3.2.1.2 Seismic design levels and quality of construction

The building damage functions distinguish among buildings that are designed to different

seismic standards, have different construction quality, or are otherwise expected to perform

differently during an earthquake. These differences in expected building performance are

determined primarily on the basis of seismic zone location, design vintage and use (i.e.,

special seismic design of essential facilities).Damage functions are provided for three Code

seismic design levels, labeled as High-code, Moderate-code and Low-code, and an additional

design level for Pre-code buildings.

3.2.1.3 Damage states

Damage states are defined separately for structural and nonstructural systems of a building.

Damage is described by one of four discrete damage states: slight, moderate, extensive, and

complete. Loss functions relate the physical condition of the building to various loss

parameters (i.e., direct economic loss, casualties, and loss of function).


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Slight structural damage: Flexural or shear type hairline cracks in some beams and columns

near joints or within joints.

Moderate structural damage: Most beams and columns exhibit hairline cracks. In ductileframes some of the frame elements have reached yield capacity indicated by large flexural

cracks and some concrete spalling.

Extensive structural damage: Some of the frame elements have reached their ultimate

capacity indicated in ductile frames by large flexural cracks, spalled concrete and buckled

main reinforcement; non-ductile frame elements may have suffered shear failures or bond

failures at reinforcement splices or broken ties or bucked main reinforcement in columns

which may result in partial collapse.

Complete structural damage: Structure is collapsed or in imminent danger of collapse due

to brittle failure of non-ductile frames or loss of frame stability.

3.2.1.4 Calculation of cumulative damage probabilities of particular damage state

The damage function is assumed to be lognormal function. To define a probability

distribution median and standard deviation values are required. For a given median spectral

displacement and standard deviation for a particular damage state , design level

the conditional probability of being in or exceeding is defined by

(29)

Where

= Median value of spectral displacement at which the building reaches the threshold of

damage state,

=Standard deviation of the natural logarithm of spectral displacement for damage state,

= Standard normal cumulative distribution function.

= Given peak spectral displacement

Probability of being in or exceeding slight damage state,

Probability of being in or exceeding slight moderate state,

Probability of being in or exceeding slight extensive state,

Probability of being in or exceeding collapse damage state,


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3.2.1.5 Calculation of discrete damage probabilities of damage states

The probability of discrete damage state is given below

Probability of complete damage =

Probability of extensive damage =

Probability of moderate damage =

Probability of slight damage =

Probability of no damage =

3.2.1.6 Median spectral displacements for different damage s tates

There are certain key aspects to the damage functions of which users must be aware when

developing fragility parameters. First, the damage functions should predict damage without

bias such as that inherent to the conservatism of seismic design codes and guidelines. In

general, limit states of the NEHRP guidelines (or ATC 40, 1996) will under-predict the

capability of the structure, particularly for the more critical performance objectives, such as

collapse prevention (CP). The NEHRP guidelines criteria for judging CP certainly do not

intend that 50 out of 100 buildings that just meet CP limits would collapse. Most engineers

would likely consider an acceptable fraction of CP failures (given that buildings just meet CP

criteria) to be between 1 and 10 in every 100 buildings. In contrast, the median drift value of

the Complete structural damage state ofHAZUS is the amount of building displacement that

would cause, on the average, 50 out of 100 buildings of the building type of interest to have

Complete damage (e.g., full financial loss). In general, users should not derive median values

ofHAZUS damage states directly from the performance limits of the NEHRP guidelinesand

ATC 40, 1996.

Fragility parameters of the more extreme damage states are particularly difficult to estimate

since these levels of damage are rarely observed even in the strongest ground shaking. In the

1995 Kobe earthquake, the worst earthquake disaster to occur in a modern urban region, only

about 10 in every 100 mid-rise commercial buildings located close to fault rupture had severe

damage or collapse. Typically, the fraction of modern buildings with such damage (e.g.,

complete structural damage) is much less than 10 in 100. In selecting median values ofdamage states, users should be mindful that median values represent the 50 percentile (e.g., 50


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in every 100 buildings have reached the state of damage of interest). Median values of

spectral displacement (or spectral acceleration) for the more extensive states of damage may

appear large relative to seismic code or guideline design criteria.

Development of damage-state medians involves three basic steps

1. Develop a detailed understanding of damage to elements and components as acontinuous function of building response (e.g., average inter-story drift or floor

acceleration)

2. Select specific values of building response that best represent the threshold of eachdiscrete damage state

3. Convert damage-state threshold values (e.g., average inter-story drift) to spectralresponse coordinates (i.e., same coordinates as those of the capacity curve).

The general guidelines for selection of damage state medians are given table 3.1

Table 3.1 Guidance for selection of damage state medians (HAZUS-MH MR1)

Damage

state

Likely amount of damage, direct economic loss, building condition

Range of

possible lossratios

Probability of

long-termbuilding closure

Probability of

partial or fullcollapse

Slight 0% - 5% P = 0 P = 0

Moderate 5% - 25% P = 0 P = 0

Extensive 25% - 100% P 0.5 P 0

Collapse 100% P 1.0 P

In using the acceptance criteria of the NEHRP guidelinesusers must be aware and account for

each of the following four issues

Conservative deformation limits: The deformation limits of the NEHRP guidelines are, in

general, conservative estimates of true component or element capacity. The Collapse

Prevention deformation limits of primary components or elements are defined as 75% of that

permitted for secondary elements, reflecting added conservatism for design of primary

components or elements. While appropriate for design, conservatism should be removed from

deformation limits used to estimate actual damage and loss.


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Deformation limits vs. Damage states :The NEHRP guidelinesprovide limits on component

or element deformation rather than explicitly defining damage in terms of degree of concrete

cracking, nail pull-out, etc., or whether component of element damage is likely to repairable

(or not). For estimating direct economic loss it is important to understand the type of damage,

not just the degree of yielding, to establish if repair would be required and what the nature

(and cost) of such repairs would be.

Global vs. Local damage: Local damage (as inferred from the deformation limits of the

NEHRP guidelines) of individual components and elements must be accumulated over the

entire structure to represent a global damage state.

Collapse failure: Reaching the collapse prevention deformation limit of components or

elements does not necessarily imply structural collapse. Typically, structural systems can

deform significantly beyond Collapse Prevention deformation limits before actually

sustaining a local or global instability.

The load deformation curve used as per NEHRP guidelines is given in figure 3.3.

Figure 3.3 Idealized component load versus deformation curve (FEMA 273, 1997)

The median of Slight damage is defined by the first structural component to reach control

point C on its load deformation curve. Moderate damage is defined by a median value for

which a sufficient number of components have each reached control point C (on their

respective load deformation curves) such that it will cost at least 5% of the replacement value

of the structural system to repair (or replace) these components. Extensive damage is defined

by a median value similar to moderate damage, except that damage repair now costs at least

25% of the value of the structural system. Complete damage is defined by a median value for

which at least 50% (in terms of repair/replacement cost) of structural components have each

lost full lateral capacity, as defined by control point E on their respective load deformation


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curves. The general guidance for relating component deformation to the average inter-story

drifts ratios of structural damage-state medians are given in table 3.2

Table 3.2 Guidance for relating component deformation to the average inter-story drift ratios

of structural damage-state medians (HAZUS-MH MR1)

Damage state Component (Criteria Set No. 1) Component (Criteria Set No. 2)

Fraction2 Limit3 Factor4 Fraction2 Limit3 Factor4

Slight > 0% C 1.0 50% B 1.0

Moderate 5% C 1.0 50% B 1.5

Extensive 25% C 1.0 50% B 4.5

Collapse 50% E 1.0-1.55 50% B 12

1. The average inter-story drift ratio of structural damage state is lesser of the two drift ratios

defined by criteria sets no. 1 and no.2, respectively.

2. Fraction defined as the repair or replacement cost of components at limit divided by the

total replacement value of the structural system.

3. Limit defined by the control points of figure 3.3 and the acceptance criteria of NEHRP

guidelines.

4. Factor applied to average inter-story drift of structure at deformation (or deformation ratio)

limit to calculate average inter-story drift ratio of structural damage-state median.

5. Complete factor is largest value in range for which the structural system is stable.

Barbat et al, 2008 has defined median spectral displacement for each damage state. This

method uses capacity spectrum to define threshold for each damage state. Push the structure

to its ultimate capacity. Convert pushover in to capacity spectrum (ADRS format). Bilinear

the capacity spectrum with ultimate and yield points. Bilinear representatio n of capacity

spectrum with damage thresholds is shown figure 3.4.

Table 3.3 Damage state thersholds (Barbat ,2008)

Median spectral displacement Damage state

Slight

Moderate

Extensive

Complete


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Figure 3.4 Damage state thresholds on bilinear capacity spectrum (Barbat, 2008)

The damage state medians for different code level designs and heights of the generic building

type are given in table 3.4 (HAZUS-MH MR1). These values doesnt represent for a specific

building.

Table 3.4 Average inter-story drift ratio for structural damage states (HAZUS-MH MR1)

Model building

type

Structural damage states

Concrete

moment frame

(C1)

Slight Moderate Extensive Collapse

Low rise buildingHigh- code design level

0.005 0.010 0.030 0.080

Low rise buildingModerate- code design level

0.005 0.009 0.023 0.060

Low rise buildingLow- code design level

0.005 0.008 0.020 0.050

Low rise buildingPre- code design level

0.004 0.006 0.016 0.040

Mid-rise buildings

2/3*LR 2/3*LR 2/3*LR 2/3*LR

High rise buildings

1/2*LR 1/2*LR 1/2*LR 1/2*LR


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Mid-rise and high-rise buildings have damage-state drift values based on low-rise (LR) drift

criteria reduced by factors of 2/3 and 1/2, respectively, to account for higher-mode effects and

differences between average inter-story drift and individual inter-story drift.

The median spectral displacement for each damage state is given by equation

(30)

Average inter-story drift ratio at the threshold of damage state,

=Height of building at the roof level

=Modal mass participation factor for the first natural mode

3.2.1.7 Development of damage state variability

Lognormal standard deviation values describe the total variability of fragility-curve

damage states. Three primary sources contribute to the total variability of any given state

namely, the variability associated with the capacity curve, the variability associated with

the demand spectrum , and the variability associated with the discrete threshold of each

damage state

(31)

HAZUS gives standard deviation values based on the following criteria

1. Building height group - Low-rise buildings, Mid-rise buildings, High-rise buildings2. Post-yield degradation of the structural system Minor, Major and Extreme

degradation

3. Damage-state threshold variabilitySmall, Moderate

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