MSc Dissertation Zhang 2012

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DIFFERENT APPROACHES TO DERIVE

ANALYTICAL FRAGILITY FUNCTIONS

OF BRIDGES

A Dissertation Submitted in Partial Fulfilment of the Requirements

for the Master of Science Degree in

Earthquake Engineering

By

Xiaoxuan ZHANG

………………………………..

Supervisors: Dr Rui Pinho

Dr Ricardo Monteiro

February, 2013

Istituto Universitario di Studi Superiori di Pavia

Università degli Studi di Pavia



The dissertation entitled “Different Approaches to Derive Analytical Fragility Functions of

Bridges”, by Xiaoxuan Zhang, has been approved in partial fulfilment of the requirements for

the Master of Science Degree in Earthquake Engineering.

Name of Reviewer 1 …… … ………

Name of Reviewer 2………… … ……



Abstract

i

ABSTRACT

Fragility functions play a critically significant role in the assessment of seismic loss, thus the

development of reliable procedures for their calculation has become increasingly popular. The

existence of different analytical methodologies, as well as structure modelling approaches, has an

important influence in the results of fragility functions. In addition it has been recognised that bridges

is one of the most vulnerable structural types during past seismic events. Depending on the seismic

conditions of local site, seismic vulnerability assessment of bridges can be carried out based on

fragility curves. In this study, 3D analytical RC bridge models are generated by OpenSees based on

different material and geometric parameters in order to take structural variability in consideration.

Nonlinear analyses are conducted for each bridge model under different ground motion records.

Curvature ductility is employed as the reference parameter in determining the bridge damage limit

states in terms of structural capacity. Several analytical procedures are used to derive fragility

functions for each simulated bridge. Seven different nonlinear static procedures (Capacity Spectrum

Method, N2, Displacement Coefficient Method, Modified Modal Pushover Analysis, Adaptive

Capacity Spectrum Method, Adaptive Modal Combination Procedures and Modified Adaptive Modal

Combination Procedure), which make use of conventional and adaptive pushover analysis, are

investigated using real ground motion records, as tools to derive fragility functions. A sufficiently

large, thus representative, ensemble of RC bridges is considered in order to duly account for the

uncertainty associated to the use of such methodologies. Furthermore, nonlinear dynamic analyses are

carried out as the baseline method for this parametric study and conclusions are obtained according todirect comparison. Based on the results, N2 has been found as the method providing the best balance

between accuracy and complexity.

Keywords: fragility, bridges, nonlinear static procedures, dynamic analysis



Acknowledgements

ii

ACKNOWLEDGEMENTS

I would like to express my deepest appreciation to my supervisors, Dr. Ricardo Monteiro and Dr. Rui

Pinho, for their guidance, advice and kind help throughout the entire study. I would like to express my

appreciation to Vitor Silva for his support and help during this study. I also want to thank all of my

friends of the ROSE Programme at the UME SCHOOL for their kindness and friendship. Last but not

the least, I want to thank my family for their endless love and encouragement.



Index

iii

TABLE OF CONTENTS

Page

ABSTRACT ............................................................................................................................................. i ACKNOWLEDGEMENTS .................................................................................................................... ii TABLE OF CONTENTS ....................................................................................................................... iii LIST OF FIGURES .............................................................................................................................. vii LIST OF TABLES ................................................................................................................................. xi 1 General Introduction .......................................................................................................................... 1

1.1 Featuring introduction ................................................................................................................ 1 1.2 Object and goal .......................................................................................................................... 4 1.3 Thesis outline ............................................................................................................................. 4

2 State-of-the-art and literature review ................................................................................................. 6 3 Classification and Generation of Bridge Populations ...................................................................... 10

3.1 Bridge characterization ............................................................................................................ 10 3.2 Random generation of bridge models ...................................................................................... 11

4 Bridge Models ................................................................................................................................. 12



Index

iv

4.1 Selection of nonlinear analytical software ............................................................................... 13 4.2 Materials................................................................................................................................... 14

4.2.1 Concrete Modal .............................................................................................................. 14 4.2.2 Steel Model .................................................................................................................... 17

4.3 Abutment .................................................................................................................................. 18 4.3.1 Equivalent linear spring abutments ................................................................................ 18 4.3.2 Diaphragm abutments .................................................................................................... 18 4.3.3 Seat abutments ............................................................................................................... 19

4.4 Fibre Elements ......................................................................................................................... 21 4.5 Deck ......................................................................................................................................... 23 4.6 Mass and Load ......................................................................................................................... 23 4.7 Damping ................................................................................................................................... 23

5 Ground Motion Records .................................................................................................................. 26 5.1 Italian seismicity ...................................................................................................................... 26 5.2 Considered Records ................................................................................................................. 27

5.2.1 Approach 1—Non-scaled records .................................................................................. 28 5.2.2 Approach 2—Scaled Records ........................................................................................ 29

6 Damage Limit States ........................................................................................................................ 34 6.1 Literature Review ..................................................................................................................... 34 6.2 Selection of limit states ............................................................................................................ 40

7 Nonlinear Static Procedures ............................................................................................................ 43

7.1 Pushover Analysis .................................................................................................................... 43



Index

v

7.1.1 Conventional Pushover .................................................................................................. 43 7.1.2 Adaptive Pushover ......................................................................................................... 44

7.2 Nonlinear Static Procedures ..................................................................................................... 46 7.2.1 Capacity Spectrum Method (CSM) ............................................................................... 47 7.2.2 N2 48 7.2.3 Displacement Coefficient Method (DCM) .................................................................... 50 7.2.4 Modified Modal Pushover Analysis (MMPA) .............................................................. 51 7.2.5 Adaptive Capacity Spectrum Method (ACSM) ............................................................. 53 7.2.6 Adaptive Modal Combination Procedures (AMCP) ...................................................... 55 7.2.7 Modified Adaptive Modal Combination Procedures (MAMCP) .................................. 56

7.3 Overall Procedures ................................................................................................................... 57 7.4 Comparison with nonlinear dynamic analysis ......................................................................... 57

8 Results 58 8.1 Introduction .............................................................................................................................. 58 8.2 Bridge Index (BI) based on individual records ........................................................................ 59

8.2.1 Capacity Spectrum Method ............................................................................................ 60 8.2.2 N2 Method ..................................................................................................................... 60 8.2.3 Displacement Coefficient Method ................................................................................. 61 8.2.4 Modified Modal Pushover Analysis .............................................................................. 61 8.2.5 Adaptive Capacity Spectrum Method ............................................................................ 62 8.2.6 Adaptive Modal Combination Procedure ...................................................................... 62

8.2.7 Modified Adaptive Modal Combination Procedure ...................................................... 63



Index

vi

8.2.8 Global Results ................................................................................................................ 63 8.3 Bridge Index (BI) based on spectrum-matching scaled records .............................................. 67 8.4 Fragility Curves ....................................................................................................................... 73

8.4.1 Nonlinear Static Procedures ........................................................................................... 73 8.4.2 Nonlinear Dynamic Analysis ......................................................................................... 76 8.4.3 Comparison of results .................................................................................................... 76

9 CLOSING REMARKS .................................................................................................................... 79 9.1 Conclusions .............................................................................................................................. 79 9.2 Future Recommendation .......................................................................................................... 81

10 REFERENCES ................................................................................................................................ 83



Index

vii

LIST OF FIGURES

Page

Figure 1.1. Damage to bridges during the San Fernando, 1971 and Loma Prieta, 1989

earthquakes .........................................................................................................................1 Figure 1.2. Sample fragility curves for different damage state[Avsar,2009] .............................2

Figure 4.1. Models for seismic bridges analysis[Priestley et al, 1996] .................................... 12

Figure 4.2. Transverse and longitudinal directions in bridges modelling................................. 13 Figure 4.3. Configuration of bridge .......................................................................................... 13 Figure 4.4. Kent-Scott-Park concrete model for unconfined and confined concrete ................ 14 Figure 4.5. Mander concrete model .......................................................................................... 16 Figure 4.6. Conventional pushover using different concrete models ....................................... 16 Figure 4.7. Displacement profile for the two concrete models ................................................. 17 Figure 4.8. Giuffre-Menegotto-Pinto Steel model .................................................................... 17 Figure 4.9. Example of distributed plasticity ............................................................................ 21 Figure 4.10 Example of fibre element’s section ....................................................................... 22

Figure 4. 11 Examples of fibre elements for bridge pier .......................................................... 22



Index

viii

Figure 4.12. Nonlinear Dynamic analysis with zero damping models ..................................... 24

Figure 5.1. Distribution of magnitude of Italian historic earthquake ....................................... 27 Figure 5.2. Distribution of the PGA of those records ............................................................... 29 Figure 5.3. Spectra of the 50 records and NEHRP design spectrum ........................................ 29 Figure 5.4 PGA distributions of the 10 records ........................................................................ 31 Figure 5.5. Spectra of the 10 records and design spectrum ...................................................... 32 Figure 5.6. Median spectra and design spectrum ...................................................................... 32 Figure 5.7. Comparison between two different sets of records ................................................ 33

Figure 6.1 Member limit state and structure limit state ............................................................ 38

Figure 7.1. Pushover curves ...................................................................................................... 45 Figure 7.2 Graphical procedures of CSM ................................................................................. 47 Figure 7.3. Inelastic Spectra by N2 ........................................................................................... 49 Figure 7.4. Bilinearization of the force-displacement curves ................................................... 51 Figure 7.5 Spectral reduction methods ..................................................................................... 54

Figure 8.1. Relationship between the relative error and size of sample ................................... 59 Figure 8.2. CSM median Bridge Index per intensity level ....................................................... 60 Figure 8.3. N2 median Bridge Index per intensity level ........................................................... 60 Figure 8.4. DCM median Bridge Index per intensity level ....................................................... 61 Figure 8.5. MMPA median Bridge Index per intensity level ................................................... 61



Index

ix

Figure 8.6. ACSM median Bridge Index per intensity level .................................................... 62 Figure 8.7. AMCP median Bridge Index per intensity level .................................................... 63 Figure 8.8. MAMCP median Bridge Index per intensity level ................................................. 63 Figure 8. 9. Median Bridge Index per intensity level for all seven methods ............................ 63 Figure 8.10. Median BI for each intensity level of CSM, N2 and DCM .................................. 64 Figure 8.11. Median BI for each intensity level of CSM, N2 and DCM .................................. 64 Figure 8.12. Median Bridge Index for each NSP ...................................................................... 65 Figure 8.13. Median Bridge Index for each PGA ..................................................................... 65 Figure 8.14. Median Brige Index for N2, MMAP and MAMCP ............................................. 66 Figure 8.15. Acceleration spectra ............................................................................................. 67 Figure 8.16. CSM median Bridge Index per intensity level based on scaled spectrum ........... 68 Figure 8.17. N2 median Bridge Index per intensity level based on scaled spectrum ............... 68 Figure 8.18. DCM median Bridge Index per intensity level based on scaled spectrum ........... 68 Figure 8.19. MMPA median Bridge Index per intensity level based on scaled spectrum ........ 69 Figure 8.20. ACSM median Bridge Index per intensity level based on scaled spectrum ......... 69 Figure 8.21. AMCP median Bridge Index per intensity level based on scaled spectrum ......... 69 Figure 8.22. MAMC median Bridge Index per intensity level based on scaled spectrum ....... 70 Figure 8.23. Median Bridge Index per intensity level based on scaled spectrum for each

method ............................................................................................................................... 70 Figure 8.24. Median BI for each intensity levels of CSM, N2 and DCM ................................ 71 Figure 8.25. Median BI for each intensity levels of MMPA, ACSM, AMCP and MAMCP ... 71



Index

x

Figure 8.26. Average values for each intensity......................................................................... 71 Figure 8.27. Median Bridge Index for each NSPs .................................................................... 72 Figure 8.28. Median Bridge Index for each NSPs with two approaches .................................. 72 Figure 8.29. Fragility curves of NSPs ....................................................................................... 75 Figure 8.30. Fragility curves for nonlinear dynamic analysis .................................................. 76 Figure 8.31. General results of each method for each Limit states ........................................... 77



Index

xi

LIST OF TABLES

Page

Table 2.1. Description of three different bridge configurations by Banerjee and Shinozuka

[2008] ..................................................................................................................................7 Table 2.2. Damage states limits for left most columns by Banerjee and Shinozuka [2008] ......8 Table 2.3. Structure attributes for generating bridge sample by Avsar et al [2011] ...................8

Table 3.1. Distribution of material and geometry properties .................................................... 10

Table 4.1 Deck cross section geometric and material properties.............................................. 23 Table 4.2. Different damping factors in OpenSees ................................................................... 25 Table 4.3. Different damping factors in SeismoStruct ............................................................. 25

Table 5.1. Characteristics of scaled of ground motion records................................................. 30

Table 6.1. Damage states given by HAZUS [2003] ................................................................. 34 Table 6.2. Damage states from Hwang et al [2001] ................................................................ 35 Table 6.3 Damage states based on ductility by Hwang et al [2001] ......................................... 35



Index

xii

Table 6. 4. Ductility limits for weak piers and strong bearings ................................................ 36 Table 6. 5. Drift limits from Basoz and Mander [1999] ........................................................... 36 Table 6. 6. Damage states defined by Choi [2004] ................................................................... 37 Table 6. 7. Limit states by J. Kowalsky [2000] ........................................................................ 38 Table 6. 8. Medians and dispersions for bridge component limit states using Baysian updating

........................................................................................................................................... 39 Table 6. 9. Damage states from Karakostas et al [2006] .......................................................... 40 Table 6. 10. Four damage limit states from Neilson et al [2005] ............................................. 41

Table 8.1. Median BI of seven approaches for per intensity level ........................................... 66 Table 8.2. Bridge Index for each intensity ................................................................................ 72 Table 8.3. Statistics of fragility functions by NSPs .................................................................. 74

Table 8.4 Statistics of fragility functions by Nonlinear Dynamic Analysis ............................ 76



Chapter 1 General Introduction

1

1 Introduction

1.1 General scope

Earthquakes have caused severe damage, ever since, to human populations, buildings and

society in general. During notable past earthquake events, which happened in seismic prone

regions all around the world, such as the 1971 San Fernando earthquake in the US, the 1994

Northridge earthquake in the US, the 1995 Kobe earthquake in Japan, the 1999 Marmara

earthquake in Turkey, the 1999 chichi earthquake in Taiwan, the 2008 Wenchuan earthquake

in China, the 2009 L’Aquila earthquake in Italy, the 2011 Christchurch earthquake in New

Zealand, etc. It has been witnessed how bridges are one of the most vulnerable components of

structural network in general, due to the gravity effect of its weight only or to inadequate

lateral forces based design. Figure 1.1 shows the severe damage to bridges in the earthquake

events. Consequently, many countries have revised their seismic hazard levels by increasing

the design peak ground acceleration and reliable seismic vulnerability assessment of bridges

and determination of their performance under seismic actions are more and more seen as

critical.

Figure 1.1. Damage to bridges during the San Fernando, 1971 and Loma Prieta, 1989 earthquakes




2

After characterizing the seismic hazard for a specific region by means of probabilistic or

deterministic approaches, vulnerability assessment can be carried out based on both seismic

information and proper methodologies, which aim at deriving reliable fragility functions.

Besides it is one of the most critical aspects in the seismic loss assessment studies, fragility

functions are also widely applied for global failure probability calculations and other

assessment studies. Fragility functions play a significant role in defining the failure

probability of bridges, however, notwithstanding the extensive available knowledge on the

matter, evidenced by a large number of past and recent studies, there are still many

uncertainties surrounding bridges fragility functions coming whether from the different

structural and material parameters of bridges or the procedures used for their derivation. In

order to minimize such uncertainties, the most reliable methods should be followed. Figure

1.2 shows a sample of fragility curves with different damage states.

Figure 1.2. Sample fragility curves for different damage state[Avsar,2009]

Fragility functions are a fundamental component to determine vulnerability, within risk

assessment procedures, and can be obtained mainly by empirical or analytical methodologies.

When following an empirical approach, information about bridge damage is collected from

reconnaissance reports or site surveys. According to the damage level information and bridge

classes, fragility curves can be derived by statistical analysis. Given that the damage states are

identified by means of real structural or non-structural damage information after an

earthquake, the empirical approach can be considered the most realistic method. However,




3

significant drawbacks can be associated to empirical methodologies. Since all the collected

data is based on observation, the subjectivity is another issue that can not be forgotten, so as

the potential lack of accuracy in the determination of the ground motion in a certain region.

Moreover, there are only a few places in the world where post-earthquake damage data has

been collected.

When there is no damage data or expert opinions, an empirical approach is not possible to

follow, and an analytical approach becomes the only method to derive fragility curves for

bridges. As analytical approach requires the definition of a single model, believed to be

representative of a class of bridges, or a set of randomly generated bridge model, usually

making use of finite element techniques. The accurateness of the analytically determined

fragility curves generally depends on three factors: bridge structures sampling and modelling,

ground motion selection and damage state definition.

The methodologies used to derive fragility functions within an analytical approach can be

categorized in two main groups: the ones that make use of Nonlinear Static Procedures (NSP)

or Nonlinear Dynamic Analysis to estimate the structural response. It is well known that the

dynamic based procedures are typically seen as the ones that can provide the most accurate

response estimate by applying a real acceleration time history at the base of the structure.

However, the complexity of the methodology, which includes the definition of the damping

model, the post-yield behaviour model, as well as the expected large amount of computational

work, is often regarded as impractical. On the other hand, NSPs have some merits when

comparing with the complicated nonlinear dynamic analysis, such as low computational

burden, simple definition of post-yield behaviour model, etc. When considering NSPs,

pushover analysis is carried out, which is then incorporated into a nonlinear static procedure

to obtain an estimate the response displacement (or other parameters) experienced by the

structures for a given ground motion record. The critical drawbacks of this methodology lay

under the assumption that the structural performance is estimated from a structure statically

and laterally loaded.




4

1.2 Object and goal

Many tools are currently available for calculating fragility curves, especially based on

analytical approaches, which has become increasingly popular in earthquake engineering

community due to its scientific soundness and because they overcome the lack of data in

empirical approaches. Both static and dynamic nonlinear analysis can be employed for

deriving fragility curves within an analytical approach and both of them feature numerous

different possibilities of application as well as many sources of uncertainty.

In this study, different methodologies (NSPs) are considered to develop analytical fragility

functions of RC bridges. The performance of the different NSPs, when applied to a large

number of randomly generated bridge configurations, is compared with results of an extensivenumber of nonlinear dynamic analyses, considered as reference, in order to identify the most

accurate and efficient method for calculating fragility functions.

1.3 Thesis outline

In order to reach the main goal of the study, each chapter will correspond to every aspect

considered to be relevant within seismic fragility curves of bridges hence this thesis is

composed of seven main chapters, defined as follows:

Chapter 1: General introduction and the outline of the thesis

Chapter 2: State-of-art and literature review

Chapter 3: Classification of bridges based on statistical analysis.

Chapter 4: Selection of analysis tools and generation of 3D nonlinear models based on the

previous studies.

Chapter 5: Selection of earthquake ground motion records. In this part, two different sets of

records were used representing scaling records and unscaling records

respectively.

Chapter 6: Based on the previous studies, choose the proper definition of damage limit

states for calculating bridge fragility functions.




5

Chapter 7: Calculation of seismic performance of bridges using both nonlinear static (NSPs)

and dynamic analysis. Statistical comparison of different engineering demand

parameters and fragility curves obtained through NSPs or nonlinear dynamic

analysis.

Chapter 8: Results of all those analytical approaches.

Chapter 9: Conclusions and future developments.



Chapter 2 State-of-the-art and Literature Review

6

2 State-of-the-art and Literature Review

In the past few decades there has been a significant research effort regarding fragility

functions for buildings whereas, clearly, less thorough investigations have been carried out forthe study of bridges. Fragility functions of bridges is thus a topic that still undergoes

significant development and improvement, particularly in what concerns the classes of

bridges under scrutiny and subsequent loss assessment studies.

Hwang et al [2001] presented an analytical approach for deriving fragility curves for highway

bridges. In this study, a set of earthquake-site-bridge sample were set up to take the

uncertainties from ground motion records, site conditions and bridges into account. The

failure probabilistic characteristics of structural demand were presented as a function of the

correlated ground motion records in terms of spectra acceleration or peak ground acceleration.

In order to calculate the fragility curves, quantitative bridge damage states were defined

according to qualified description given by HAZUS.

Karim and Yamazaki [2003] proposed a simplified method for the definition of fragility

curves for highway RC bridges. Four typical bridge piers and two bridge configurations were

considered and a set of 250 strong ground motion records were selected for nonlinear

dynamic analysis. Fragility curves were obtained using damage indices and ground motion

parameters based on lognormal distribution. Based on the observed relationship between the

fragility curves and the over-strength ratio, a simplified mathematic equation was proposed

for calculating fragility curves by such simplified method.

In a study of Nielson [2005], an expanded methodology for calculating fragility curves of

highway bridges was carried out, which directly estimated the bridge system fragility curves

from component fragilities. OpenSees were used to build the 3D bridge finite element models,

and the bridge classification was based on different bearings and abutments conditions.

Damage limit states for each damage state were defined for different elements according to

different criteria. A Bayesian approach was also applied for dealing with the uncertainties of

the damage states. The conclusion drawn from this study was that the global fragility of

bridge system is higher than any of its individual components.

Kappos et al [2006] introduced an advanced inelastic analysis tools for calculating fragility

curves for bridges. The main steps of the newly developed method included modal pushover




7

analysis based on appropriately selected monitoring points for drawing the pushover curves

and utilizing the capacity spectrum method, and nonlinear dynamic analysis accounting for

the influence of spatial variability of earthquake ground motion on both straight and curved

bridges.

In a study of Karakostas et al [2006], the methodology proposed by Kappos et at [2006],

which took use of pushover analysis and inelastic demand spectrum, for evaluating fragility

curves of bridges is employed for the case of Northern Greece. In this study, a specific bridge

was chosen and the related 3D model was built with finite element program Sap2000. A

pushover analysis was performed for both longitudinal and transverse directions using 1st

mode shape load distribution. Five damage states were defined in terms of damage ratio

y D ! ! /= , where ! is the response displacement calculated from the nonlinear static method,

and ! y is correlated to the yield displacement. The mean expected peak ground acceleration

correspond to each damage limit state was assessed as the point of intersection between the

capacity curve and the demand spectrum. Finally, fragility curves were computed based on

lognormal probability distribution functions.

In a study of Jeong and Elnashai [2007], analytical approaches for deriving fragility functions

were carried out based on the fundamental response quantities of stiffness, strength and

ductility. Single degree of freedom systems were generated to construct a Response Database

of coefficients for defining fragility curves. The uncertainties of modelling simplification

were checked with the more accurate multi-degree of freedom systems.

In a study of Banerjee and Shinozuka [2008], fragility curves for RC bridges were derived

based on mechanistic quantification of damage states. Table 2.1 shows the geometric

parameters of the different three bridges considered in this study.

Table 2.1. Description of three different bridge configurations by Banerjee and Shinozuka [2008]




8

Time history response of the different tested structures was measured in terms of rotational

ductility at the end of columns. Five different damage states were defined by terms of yield

and ultimate rotations obtained from bi-linear moment-rotation plot. Table 2.2 displays the

damage limit states used in this study.

Table 2.2. Damage states limits for left most columns by Banerjee and Shinozuka [2008]

In a study from Avsar et al [2011], analytical fragility curves for ordinary highway bridges

were calculated. Four major bridge classes were considered based on the skew angle, number

of columns per bent and span numbers. Nonlinear dynamic analysis was employed for each

3D bridge model prepared with OpenSees platform. Damage states criterion for structure

critical components was defined for deriving fragility functions. Table 2.3 shows the

geometric and material parameters considered in the 3D bridge modelling.

Table 2.3. Structure attributes for generating bridge sample by Avsar et al [2011]




9

Generally, there are several methodologies for calculating fragility functions and the resulting

curves are conditional on the assumptions and approaches followed in the process, which

includes the parameters used in modelling, definition of the damage states, computational

approaches, etc. These difference caused by different methodologies will lead to critical

discrepancies in the seismic assessment, even when considering the same region. Besides the

limitations of the previous studies are that only unique methodologies was applied in one

study and the lack of comparison among those methodologies. In this study, in order to

improve those drawbacks, several analytical methodologies are employed to calculate fragility

functions for the same bridge type in the same region. Nonlinear dynamic analysis is

calculated as the baseline for the sensitive study, seven different NSPs are applied to make

comparison and to yield the conclusion of the difference among those different methodologies.



Chapter 3 Classification and Generation of Bridge Populations

10

3 Classification and Generation of Bridge Populations

3.1 Bridge characterization

The first step to adopt when calculating fragility curves for a seismic loss assessment study of

a population of bridges can be the classification of the bridges into typological classes, based

on simple parameters that characterize their configuration and influence their seismic

response. In their study, Moschonas et al [2009] categorized all the bridges based on the

characteristics of the piers, bearings and pier-to-bearing connections. In other studies, such as

the one of Avsar [2009], other parameters with significant influence on seismic response of

bridges have been proposed for the classification, which include skew angle, number of spans

and number of bent columns. Afterwards, for each chosen parameter, a statistical probabilistic

distribution can be defined and used for random generation of populations of bridges. The

most typically used distributions are normal, log-normal, gamma, beta or exponential

distributions.

In this study, in order to keep the computational time and effort in a sustainable range, a

single geometrical configuration of bridges with four bays was considered. Nevertheless, the

length, height and configuration of the piers were all submitted to variation, according to pre-defined distributions. In addition, uncertainty was also considered to come from the material

and other geometrical properties of the bridges, according to the distributions presented in

Table 3.1.

Table 3.1. Distribution of material and geometrical properties

The distributions assumed for the different variables intend to represent a scenario of typical

RC bridges and viaducts in Italian. The initial idea behind this study was to use distribution

characterization obtained by statistical analysis of data of real Italian bridges, which for

several bureaucratic/administrative reasons, was not obtained in time for this work.

Parameters Mean ov lower bound upper bound Type of distribution

Steel modulus(Gpa) 200 3% - - Normal

Steel yield strength(MPa) 371.1 11% 250 - Normal

Concrete strength(MPa) 40 20% 20 70 GammaColumn height(m) 15 15% 8 25 Normal

Column diameter(m) 2 14% 1.2 4.79 Lognormal

bay length(m) 50 20% 30 80 Normal

beamcap width(m) 5 4% 2 20 Lognormal

beamcap height(m) 1 2% 1 4 Lognormal



Chapter 3 Classification and Generation of Bridge Populations

11

3.2 Random generation of bridge models

Once the statistical distributions for all the relevant parameters have been defined, the bridge

models can be sampled either following Monte Carlo or Latin Hypercube approaches.

In order to get a reliable sample, which is able to represent the probability distribution as good

as possible, a proper simulation procedure should be applied. As we all know it is the size of

the sample that the appropriateness of the simulation procedure mainly depends on. Among

the available solutions, the most commonly used method is the Monte Carlo approach based

on pure random simulation. However, a more recent and innovative technique is the Latin

Hypercube, which has been initially proposed by Mckay et al [1979], and further developed

by Iman et al [1981]. One of the most important claims of the Latin Hypercube approach isthat it reduces the size of the sampling efficiently.

In this study, an automatic framework was created to randomly simulate the selected

parameters (material and geometrical) for each individual bridge. As a first step, parameters

are randomly generated based on the distributions defined above using Monte Carlo

simulation scheme.



Chapter 4 Bridge Models

12

4 Bridge Models

Proper analytical modelling of bridges is of great importance when calculating seismic

fragility curves and it still encloses a number of challenging issues, among which some needto be further developed and others call for reasonable assumptions. Special attention should

be paid when making simplified assumptions since any simplification will have a direct

impact on the analysis and, consequently, on the reliability of resulting fragility curves. On

the other hand, an excessively refined or complex model will lead to higher computational

demand or even unreachable results. A model should therefore be as simple as possible, while

retaining its soundness.

Priestley et al [1996] suggested various models with different complexity levels for seismic

analysis of bridges, ranging from lumped mass models to finite element models, as shown in

Figure 4.1.

Figure 4.1. Models for seismic bridges analysis [Priestley et al, 1996]

Given that seismic action generally causes much higher damage along the transverse

direction, rather than the longitudinal one, of bridges, 3D models are used for the analysis of

the transverse deformation under different ground motion levels. Figure 4.2 gives the example

of transverse and longitudinal direction of bridges.




13

Figure 4.2. Transverse and longitudinal directions in bridges modelling [Priestley et al, 1996]

Several key aspects, with respect to the modelling task, will be discussed in this chapter, e.g.,

concrete and steel material models, abutment stiffness calculation, fibre elements, lumped

masses and loads, amongst others.

4.1 Selection of nonlinear analytical software

In this study, OpenSees[1] has been employed to carry out all the static and dynamic

nonlinear analysis required by the analytical procedures. OpenSees, that is Open System for

Earthquake Engineering Simulation, which is developed by U.C. Berkeley and funded by

Pacific Earthquake Engineering Research Center (PEER), is widely applied in the structural

or soil seismic response study. At a preliminary stage, its estimates have been compared and

double checked with a different, equally widespread, software program: SeismoStruct. Both

of the software tools enable 3D bridge modelling using force-based fibre elements with

distributed nonlinearity. Figure 4.3 shows the configuration of the bridge model used for the

initial calibration study.

Figure 4.3. Configuration of a bridge model




14

The reason for choosing OpenSees to perform this study is that OpenSees could provide an

automatic computation platform, which allows the calculations to be performed according to a

script automatically without changing parameters manually. That is a clearly fundamental

feature in a random simulation based work.

4.2 Materials

4.2.1 Concrete Modal

Several concrete models are available in literature, with different accuracy levels. Aktan and

Ersoy [1979] pointed out that the concrete model is not the most significant aspect, when

considering the moment-curvature diagram of a reinforced concrete section, thus stating that it

should not be too important to choose a more sophisticated model. It is typically the steel

model, rather than the concrete, that plays a critical role in determining the ductility behaviour

for a given section.

When analysing a reinforced concrete section, there are two different types of concrete to be

considered : confined and unconfined concrete.

The section core concrete, which is confined by transverse reinforcement, is different from the

cover concrete, in terms of stress-strain relationship. OpenSees’ Model “Concrete01”

represents a uniaxial Kent-Scott-Park concrete material object with degraded linear

unloading/reloading stiffness according to the work of Karsan-Jirsa [1969] and no tensile

strength, which can be used for both confined and unconfined concrete. The strain-stress

relationships for both confined and unconfined concrete model are illustrated in Figure 4.4.

Figure 4.4. Kent-Scott-Park concrete model for unconfined and confined concrete




15

According to the plots in the figure above, the parameters needed to characterize the confined

concrete behaviour can be calculated using equations 4.1 to 4.3,

co

yh s f K !

" += 1

(4.1)

K cco

!= 002.0" and K cocco

!=" "

(4.2)

cco

h

s

co

co

s

h

Z

! " #

# $+

$

+

='

75.01000145

29.03

5.0

(4.3)

where K is a parameter related to the limit strain and increasing compression strength due to

confinement. Z is the strain softening slope.

The slope corresponding to the initial stiffness , E c, can be calculated through equation 4.4,

co pcc f E ! /2"=

(4.4)

Assuming the maximum concrete strain as !c0=0.002, E c can be calculated directly based on

the given compression strength.

On the other hand, SeismoStruct make use of Mander [1988] concrete model, for which the

strain-stress relationship is depicted in Figure 4.5. The modulus of elasticity associated to the

initial stiffness branch of the material behaviour is defined by Equation 4.5, with units in MPa.

'

5000cc f E = (4.5)




16

Figure 4.5. Mander concrete model

Again, a brief comparison between the two selected software tools, considering the nonlinear

behaviour of the concrete material, has been carried out, in terms of pushover curves of a

bridge with a simple geometrical configuration. The results obtained with OpenSees and

SeismoStruct, employing their corresponding concrete models, are shown in Figure 4.6,

which denotes how the stiffness associated to Mander’s concrete model degrades slower than

when applying Kent-Scott-Park model.

Figure 4.6. Conventional pushover using different concrete models

In Figure 4.6, the uniform distributed load pushover analyses give much higher base shear

than the parabolic distributed load pushover analyses in both SeismoStruct and OpenSees

results. Moreover the pushover curves from the two software are followed the same changing




17

trend under two different load patterns. However, as the two software make use of two

different concrete types, there are discrepancies between the pushover curves derived from the

two software even under the same distributed load pattern. In order to check whether the

different concrete material will lead huge difference in displacement profile, Figure 4.7 is

generated to show the bridge displacement profile at different performance stage.

Figure 4.7. Displacement profile for the two concrete models

From the observation of figure 4.7, one can easily realize how the displacement profiles

obtained with different concrete models show much more acceptable differences between

them, when compared to the force-displacement curves in Figure 4.6. Given that OpenSees

provides a sophisticated and stable analysis algorithm for Kent-Scott-Park model, the

“Concrete01” model has been chosen for all the models and analyses to be carried out with

OpenSees.

4.2.2 Steel Model

Reinforcement bars are modelled with “Steel01” in OpenSees, which makes use of the well-

known Giuffre-Menegotto-Pinto model with isotropic strain hardening. The hysteretic

behaviour of the steel model is depicted in Figure 4.8.

Figure 4.8. Giuffre-Menegotto-Pinto Steel model




18

Loading and unloading paths are involved in the monotonic loading bi-linear behaviour law

and the stress-strain relationship expression is defined by Equation 4.6.

*

/1*

**

)1()1( !

!

! " bb

R R +

+

#=

(4.6)

4.3 Abutments

Abutments are usually detailed and designed for service loads and then checked for seismic

performance. There are different kinds of abutments applied to the bridges structures and a

short description of the most commonly employed types is carried out in what follows.

4.3.1 Equivalent linear spring abutments

Commonly, equivalent linear springs are used in structural models to simulate the deck

restrains provided by the abutments. The characterization of the equivalent springs should

represent the dynamic behaviour of the soil behind the abutments, the structure components of

the abutments and the interaction between them.

In design applications, stiffness values of these springs are usually determined based either on

simplified rules and iterative process or from abutment capacity and expected deformationduring the earthquake.

4.3.2 Diaphragm abutments

Diaphragm abutment is one of the most popular bridge abutments types, which is good at

absorbing energy during an earthquake. The longitudinal resistance for the seismic analysis of

a diaphragm abutment should be based on mobilizing the backfill equal to the depth of the

superstructure plus the shear capacity of the abutment diaphragm. The ultimate passive

resistance can be assumed by CALTRANS (California Department of Transportation) as

370KPa. The transverse resistance for seismic analysis of a diaphragm abutment should be

based on the ultimate shear capacity of one wingwall and all piles. When Class 400, Class 625

or standard 400mm CIDH piles are used an ultimate shear capacity of 180KN per pile may be

used. The ultimate shear capacity of the piles is the limiting force for transverse keys for

diaphragm abutment. To reduce the possible damage to the piles, transverse keys should be

designed for 75% of the limiting force.




19

4.3.3 Seat abutments

Another commonly used type of abutments is seat abutment, which gives the designers more

control over the amount of forces the abutments could resist, but introduces the potential for

the superstructure becoming unseated leading to collapse of the end span. The longitudinal

stiffness assumed for the seismic analysis should be based on mobilizing only the soil equal to

the depth of the superstructure. The limiting force for the transverse shear keys may be

approximated by adding the ultimate shear capacity of one wingwall plus the ultimate shear

capacity of the piles.

Priestley et al [1996] proposed that all the aforementioned three types of abutments should

follow the same properties:

1. They are massive structures;

2. Mobilize and interact with large soil masses;

3. Based on their geometry, exhibit significantly higher stiffness values than do other bridge

bents and thus attract proportionally higher seismic forces;

4. Feature some or all of the following highly nonlinear elements and behaviour

characteristics: breakaway shear keys, expansion joint restrainers, sacrificial wing and

back walls and a potential for inelastic pile action.

In the CALTRANS procedure, the design value for abutment capacity in longitudinal

direction is computed as the sum of the resistance values provided by the foundation and the

soil behind the backwall, which utilize an abutment capacity based on a maximum effective

soil pressure of 239kPa, amplified by about 50% to 368kPa for dynamic seismic loads,

capacity levels recently verified by large-scale abutment tests. For the projected abutment area

Aeff in the loading direction, the nominal dynamic abutment capacity, F abutment , can be

determined using Equation 4.7,

eff abutment AkPa F •= 368

(4.7)

The design value of abutment stiffness is calculated as a ratio of the abutment capacity and

acceptable deformation. The backwall in longitudinal and shear key in transverse direction are




20

generally designed as sacrificial element to protect the whole structures against the inelastic

action.

CALTRANS guidelines set the initial abutment stiffness estimate according to Equation 4.8,

p p sa n Bk k k 7115 +=+=

(4.8)

where B represents the effective abutment width and n p is the number of piles. In the

transverse direction the effective width B is taken as the length of the wing walls multiplied

by a factor of 8/9 to account for differences in participation of both wing walls. The

displacement at the abutments may be underestimated by this approach unless the secant

stiffness or substitute structure approach is extended to the entire system including equivalent

bent characteristics.

Goel and Chopra [1997] concluded that the values proposed by CALTRANS procedures are

too high for both transverse and longitudinal directions. Reliable values to represent the

interaction of abutment and soil were instead estimated from the force-deformation relations

of the system soil-abutment by dynamic equilibrium of the road deck. Hysteretic loops were

applied for the analysis and provided the results of 7500 and 12000 kip/ft, which are equal to

109,454kN/m and 175,127kN/m for positive and negative deformation respectively. The same

study also proposed the stiffness of abutments sitting directly on piles to be 709,150kN/m and

229,970kN/m in longitudinal and transverse direction respectively.

In the study of Alvarez [2004] and J.C. Oritiz Restrepo [2007], 75000kN/m was suggested for

the abutment stiffness with limited displacement of 100mm.

Casarotti et al [2005] considered, in their study, two kinds of abutments. In one of them the

abutment is directly sitting on top of the piles, for which a bi-linear behaviour was assumed

for modelling purposes. The pre-yielding stiffness was defined as 229,970kN/m for transverse

direction and 0.5% post-yielding stiffness ratio, when the displacement exceeded the design

value of 25mm. The other considered type is when the abutment sits on top of pot bearings,

for which elastic behaviour was assumed as 26,329kN/m.

The work presented herein did not have the scope of going deep into abutment design but

rather choosing a reasonable model to define the abutments of the bridges used in theextensive fragility curve parametric study. Indeed, no uncertainty related to the abutments has




21

been considered within the random simulation phase (see Table 2.1), although it should be

seen as one of the future developments (see chapter 9). Accordingly, for this study, the type of

abutments considered has been the one sitting on top of pot bearings, allowing a maximum

elastic deformation of 500mm. The stiffness of the abutments was set as 26,329kN/m,

according to what proposed in the study of Casarotti et al [2005].

4.4 Fibre Elements

According to the types of the plasticity formulation, nonlinear elements can be divided into

two major groups: lumped plasticity elements and distributed plasticity elements. The latter

one, shown in Figure 4.9, can be derived through displacement-based formulation, force-

based formulation and mathematic integration.

Figure 4.9. Example of distributed nonlinearity

OpenSees provides a powerful library of possible fibre elements shown in Figure 4.10, which

will have a significant importance in modelling properly the distributed material nonlinearity.




22

Figure 4.10 Example of fibre element’s section

The displacement and flexibility matrix of the force-based elements can be calculated as per

Equations 4.9 and 4.10.

! ="

L sT

Qe dx xe N

0

(4.9)

! =

L

Q sT

Qe xdx xN xf N f

0

(4.10)

where s stands for section and e stands for element, N Q is the force shape function.

Those equations are the exact solution for beam theory independent of the material properties.

In this study, force-based elements (nonlinear beam-column element) have been adopted to

accurately represent the inelastic deformation of the analysed bridges. Figure 4.11 shows the

definition of the fibre elements section.

Figure 4. 11 Examples of fibre elements for bridge pier




23

4.5 Deck

Decks are typically modelled as linear elastic elements, which represent the expected

behaviour of this structural component under seismic actions. Given that it is often made up

of pre-stressed, rather than regular reinforced concrete, no damage or nonlinear deformation is

expected to occur.

In this study, the deck was modelled with a 3D elastic element, whose characteristics are

described in Table 4.1, capable of reproducing local geometric properties.

Table 4.1 Deck cross section geometric and material properties

4.6 Mass and Load

Masses and gravity loads have been defined considering the reinforcement and concrete parts

of the bridges as well as the relative density of asphalt, thus self-weight only. Indeed,

according to common design codes and guidelines, e.g. AASHTO LRFD [2007] and EC8

[2005], as far as bridge analysis is concerned, no traffic live loads are to be considered when

carrying out seismic response calculations.

4.7 Damping

Damping is known to be as controversial as important within nonlinear dynamic analysis

hence a damping model should be chosen and employed carefully.

Generally Rayleigh damping is widely used when defining system damping, which can be

calculated as Equation 4.11,

K

i

M

i

i !

" !

" #

22

1+=

(4.11)

Where "i is the frequency of the correlated mode, # M and # K are the damping factors for mass

matrix and stiffness matrix respectively.

EI2 KNm2 EI3 KNm2 GJ KNm21.32E+08 1.32E+08 4.75E+09




24

However, as Joan F. Hall [2005] pointed out that Rayleigh damping typically using the initial

linear stiffness matrix for the stiffness-proportional part damping consistently overestimates

the damping ratio for the system, especially when the structure has softening nonlinearity, the

damping forces generated by such matrix can become unrealistically large compared to the

restoring forces. In order to overcome the drawbacks of Rayleigh damping, Priestley and

Grant [2005] suggested that the tangent-stiffness proportional damping could be applied as

Equation 4.12,

!

" #

T

K =

(4.12)

A first calibration of OpenSees and SeismoStruct, in terms of dynamic analysis, is carried out

using models with no damping and the same parameters, except for the concrete model,

although both concrete models have the same compression strength. Figure 4.12 displays the

nonlinear dynamic analysis results obtained for those two models. The peak displacement

occurred at the same time and the difference between the two curves can be disregarded.

Figure 4.12. Nonlinear dynamic analysis with zero damping models

In OpenSees, although the command for defining damping is called Rayleigh, damping

factors are defined separately for mass matrix and stiffness matrix, while stiffness matrix

affected damping plays a more important role.

Table 4.2 presents different maximum displacement observed using the same geometrically

simple bridge model as previous examples when using different damping factors, applied to

mass and stiffness matrices. The column named “Ratio” shows the ratio between the




25

maximum displacement deriving from each damping combination and the maximum

displacement measured when no damping is considered.

Table 4.2. Different damping factors in OpenSees

From Table 4.3, one can say that the mass matrix gives less critical influence in the structural

damping analysis, when compared to the zero damping model. The maximum displacement

with stiffness damping of 2% decreases to 95% while by increasing the mass damping of 1%,

the maximum displacement only drops by 1%. Moreover 5% stiffness damping will lead to

more than 10% decrease in maximum displacement.

By carrying out the same brief parametric study on damping with SeismoStruct, Table 4.3 is

obtained when choosing Rayleigh damping. The column Mode1 and Mode2 values are the

damping coefficients applied in Rayleigh damping when considering the same mass and

stiffness proportional damping listed in the second and third columns. The results show that

the displacement is smaller with respect to OpenSees, which means that the amount of

estimated damping is slightly larger but still comparable.

Table 4.3. Different damping factors in SeismoStruct

For this study, stiffness proportional damping was employed and the damping coefficient

used in Opensees was 2% for stiffness matrix in order to minimize the effect of damping and

keep it in a reasonable range.

Rayleigh Damping Mass Matrix Stiffness Matrix Max displ(m) Ratio

ok(%) 1 5 0.282 87%

ok(%) 0 5 0.2844 88%

ok(%) 1 2 0.3053 94%

ok(%) 0 2 0.3081 95%

ok(%) 0 0 0.3244 100%

pen ees

Rayleigh Damping Mass Matrix Stiffness Matrix Max displ(m) Ratio Mode1 Mode2ok(%) 1 5 0.2699 82% 2.6 3.1

ok(%) 0 5 0.2683 81% 2.6 0

ok(%) 1 2 0.3034 92% 1.05 1.26

ok(%) 0 2 0.3034 92% 1.05 0

ok(%) 0 0 0.3307 100% 0 0

eismo tru t



Chapter 5 Ground Motion Records

26

5 Ground Motion Records

Ground motion records are generally recognized as the essential input for any seismic

assessment based on nonlinear dynamic analysis. Generally, there are two kinds of recordscould be applied in the dynamic analysis: the artificial records and the real records. Given that

the former one makes use of computational techniques to amplify the original records, which

will lead to changes of frequency content and energy release. On the contrary, the real records

represent better the local seismic action. Thus, real records are more commonly employed in

dynamic analysis. If the real records are considered, it is recommended that a sufficiently

large number of different records is used, in order to duly represent an intended seismic

hazard scenario, characterized by predefined ranges of parameters such as magnitude and

peak ground acceleration (PGA), most common fault failure mechanism, frequency content,

duration and epicentre distance.

Moreover, they play an important role when calculating fragility curves, given that each

ground motion record is necessarily related to specific seismic hazard and geotechnical

conditions of the region where it took place, which will induce important uncertainties in

fragility functions calculation. Consequently, in order to minimize such uncertainties, proper

care should be paid when records are chosen for analysis.

5.1 Italian seismicity

The ground motion records usually are collected in terms of the level of peak ground motion

acceleration (PGA), as well as the distance and magnitude. The majority of the strongest

earthquakes that occurred in Italy in past featured dip-slip mechanism and some of those

events were associated to strike faults. The average depth of an Italian damaging earthquake is

10km.

Based on the data from Catalogue of Strong Earthquakes in Italy [2000] from 461 B.C. to

1997, the chart in Figure 5.1 shows the distribution of historical earthquakes, in terms of

magnitude, that occurred in Italy.




27

Figure 5.1. Distribution of magnitude of Italian historic earthquake

5.2 Considered records

The parametric study presented herein considered two approaches, with respect to the

selection of ground motion records. Such approaches take the uncertainty associated to the

seismic ground motion into account in two different ways.

The first approach considers non-scaled records, directly chosen from one of the available

ground motion databases, restricted to the predefined ranges of magnitude and epicenter

distance. A sufficiently large number of records (e.g. 50 or 100) are considered, so as to duly

incorporate the epistemic/aleatory uncertainties. In addition, a uniform distribution of the peak

ground acceleration, within a representative range (e.g. 0 – 1g), is sought. When following

this sort of approach, each record is used only once, given that the entire range of ground

motion intensity (measured in terms of peak ground acceleration) is covered by the different

time-histories. Given that there is no need for scaling, the fact that the records keep their

original form can be seen as a major advantage. On the other hand, each ground motion

intensity level is somewhat represented by a single accelerograms hence not taking into

account record-to-record variability.

The second approach considers a smaller number of real records, chosen under spectral

matching criteria. The selected records are chosen assuring that their median (or mean)

response spectrum matches (within a given tolerance) the PSHA-based response spectrum for

the region of interest, associated to a certain probability of exceedance for a given return

period. The median response spectrum is associated to a level of ground motion intensity, i.e.,




28

peak ground acceleration thus the selected records can be scaled until the entire range of

interest, of PGAs is covered. Under such conditions, for each ground motion intensity level,

the record-to-record variability is contemplated by use of the entire set of selected records.

The main drawback is the need for record scaling.

Scaled records (Approach 2) will be used to assess the individual performance of the different

nonlinear static procedures (NSPs), i.e. CSM, N2, MMPA, ACSM, AMCP and MAMCP,

through comparison with nonlinear dynamic analysis, whereas non-scaled records (Approach

1), which lead to a lower computational onus, will be used for the extension of the parametric

study to the computation of fragility curves.

5.2.1 Approach 1 Non-scaled records

The selection of real earthquake records, under the first approach, was carried out using the

PEER database [2], which is short for Pacific Earthquake Engineering Research Center. In

agreement with the historical characteristic of the Italian territory seismicity, briefly described

in Section 4.1 and Figure 4.1, the following intervals for the different parameters were used

for the selection of the records. As those records were selected based on historical magnitude

distribution, 50 real records were chosen in order to avoid changes of frequency content or

event duration, which are inherent to the magnitude of the event.

1) Magnitude Range: 5.0—7.5 Mw.

2) Distance Range: 10— 150 km.

3)

Rupture mechanism: Strike-slip and Dip-slip.

The distribution of peak ground acceleration and the median response spectrum of the 50

selected records are respectively shown in Figure 5.2 and Figure 5.3.




29

Figure 5.2. Distribution of the PGA of the selected records

Figure 5.3. Spectra of the 50 records and the median spectrum

5.2.2 Approach 2 Scaled Records

Within the second approach, for the sake of agreement with the first one, a response spectrum

associated to a given probability of exceedance, for the Italian territory, should be considered

for the matching and selection of records. Given that this is not a clear endeavour, if one

considers the Italian territory (as considered in the ranges defined in the first approach) and

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P G A ( g )

No. of ground motion records

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

S a ( g )

Period(s)




30

because the study to be carried out is essentially parametric, an already existing set of 10

records, which was used in the aforementioned study by R. Monteiro [2011], chosen from Los

Angeles Strong Motion Database, was considered instead. The employed set of records are

selected from a suite of historical ground motion data [SAC，1997] scaled to match 10%

exceeding probability in 50 years (475 years return period) uniform hazard spectrum for Los

Angeles, which is known as NEHRP.

Another reason for the consideration of a second approach is the fact that many guidelines

prescribing the use of Nonlinear Static Procedures, such as Capacity Spectrum Method or N2,

recommended, in agreement with the original formulation of the methods, the employment of

smoothed design spectra.

Table 5.1 shows the detailed information of each of the 10 scaled records, including

magnitude, duration and PGA. Keeping the SAC Project designation, the 10 records are

labelled as: LA02, LA04, LA06, LA08, LA10, LA12, LA14, LA16, LA18, LA20 and those

ten records contain very large energy and frequency complexity.

Table 5.1. Characteristics of scaled of ground motion records [SAC,1997]




31

Based on Table 5.1, the main characteristics of the records can be summarized as follows:

1) Magnitude range: 6.0 – 7.3 Mw.

2) Distance to fault range: 1.2 – 36 km.

3) Peak ground acceleration: 0.2 – 1.0g.

Figure 5.4, in tandem with Figure 5.2 for Approach 1, plots the distribution of the peak

ground acceleration of the 10 records.

Figure 5.4 PGA distributions of the 10 records

Figure 5.5 depicts the spectra of the 10 real ground motion records and the NEHRP design

spectrum to which they have been scaled to match.

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P G A ( g )

No. of ground motion records




32

Figure 5.5. Spectra of the 10 records and design spectrum

Finally, Figure 5.6 illustrates the comparison between the median of the 10 ground motion

records and, again, the “matched” NEHRP design spectrum. Figure 5.7 displays the difference

of those two different approaches.

Figure 5.6. Median spectra and design spectrum

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

S a ( g )

Period(s)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

S a ( g )

Period(s)

NEHRP

Median




33

Figure 5.7. Comparison between two different sets of records

From Figure 5.7, we can see that the non-scaled records present lower spectral ordinates, with

respect to the 10 scaled records thus the different sets of records will be useful to check

whether the nonlinear static procedures could provide good estimations or not under both less

and more demanding conditions.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

S a ( g )

Period(s)

NEHRP

non-scaled Median

scaled Median



Chapter 6 Damage Limit States

34

6 Damage Limit States

The definition of the damage states is a fundamental step when calculating fragility curves. In

addition to quantitative information, each damage state lies also on functional and operationalinterpretation of the status of the structure it refers to. A large number of studies have

addressed the definition of damage states for the specific seismic assessment of bridges. A

brief overview is presented next.

6.1 Literature review

In HAZUS [FEMA, 2003], five different limit states are defined, based on damage observed

in the bridges components. Those are: None, Slight/Minor, Moderate, Extensive and

Complete, for which a detailed qualitative description is provided but, unfortunately, no

corresponding parameter quantification is not carried out. The lack of quantitative

characterization of limit states is a significant limitation to the employment of the HAZUS

approach within analytical computation of fragility curves. Table 6.1 shows the quantitive

description of the five damage states given by HAZUS.

Table 6.1. Damage states given by HAZUS [2003]




35

Hwang et al [2001] proposed two different approaches for seismic damage limits based on the

description of five damage states by HAZUS. In the first approach, illustrated in Table 6.2,

four damage states were defined according to the flexural capacity of the columns.

Table 6.2. Damage states from Hwang et al [2001]

In Table 6.2, M1 is the column moment at the first yielding of the longitudinal bar and My is

the yield moment at the idealized moment curvature curve of the given column section. ! p is

the plastic hinge rotation with !c equal to 0.002 and 0.004 respectively for the columns with or

without lap splices at the bottom of the column.

In the second approach, damage states were defined using the displacement ductility ratio of

the columns to derive the overall fragility functions, which is defined by Equation 6.1,

1cy

d !

!=µ

(6.1)

In which " is the relative displacement at the top of a column obtained from seismic response

analysis, and "cy1 is the relative displacement of a column when the longitudinal reinforcing

bars at the bottom of the column reach the first yield. Five damage states, described in Table

6.3 were defined using the parameter of displacement ductility ratio of columns, µd .

Table 6.3 Damage states based on ductility by Hwang et al [2001]




36

As "cy1 is defined from the first yield of the longitudinal bars at the bottom of a column, µcy1

is equal to 1. µcy is the yield displacement ductility ratio of a column and µc2 is the

displacement ductility ratio when !c=0.002. µmax is the displacement ductility ratio defined

as µc2+3.0.

Liao and Loh [2004] proposed 4 damage states using displacement and ductility illustrated in

Table 6.4. For each damage state, ductility limits were specified for weak piers and strong

bearings by considering the design type of bridges, which was either seismic or conventional

design.

Table 6. 4. Ductility limits for weak piers and strong bearings

In the work by Basoz and Mander [1999], five damage states were defined, as represented in

Table 6.5. Drift limits were used for defining damage states of seismically or non-seismically

design bridges, which are applicable for bridges with weak pier and strong bearings.

Displacement limits increase as the drift limits grow bigger.

Table 6. 5. Drift limits from Basoz and Mander [1999]




37

In the study of Choi et al [2004], damage states were defined based on column ductility

demand, steel fixed and expansion bearing deformations, and elastomeric bearing

deformations, which are summarized in Table 6.6

Table 6. 6. Damage states defined by Choi [2004]

Choi et al [2004] pointed out that the quantified data for the damage states were based on the

previous studies and tests results. The displacement at the complete damage limit state was

assumed to be #=255mm, which accounts for the unseating PSC-girders.

In a study of Kibboua et al [2011], the damage assessment of bridge piers is carried out using

the quantified Park-Ang damage index DI expressed asu

hd DI

µ

! µ µ += ,where

d µ is the

displacement ductility,u

µ is the ultimate ductility of the bridge piers, $ is the cyclicloading

factor taken as 0.15 andh

µ is the cumulative energy ductility.

Priestley et al [1996] proposed two kinds of limit states: section limit state and structure limit

state based on strain properties and ductility respectively, which is illustrated in Figure 6.1.




38

Figure 6.1. Member limit state and structure limit state

The pushover curves are converted into bilinear representation by the following procedures.

The yielding point of the longitudinal reinforcement is specified to determine linear elastic

portion and initial slope of the bilinear curve. The ultimate curve point is specified when

reinforced steel or confined concrete extreme fibre has reach its ultimate strain value or when

the moment capacity has decreased.

Priestley et al. proposed three limit states for the assessment of bridges: serviceability,

damage control and survival.

For the serviceability limit state, the typical value for displacement ductility factors is 2.

For the damage control limit state, generally the displacement ductility factors is from 3 to

6, while it is appropriate in Europe that it is equal to 4.

Although survival limit state is of critical significance, there is no specific proposed

corresponding ductility value.

J. Kowalsky [2000] proposed a method to determine damage limits for circular bridge

columns, which considered two damage states: serviceability and damage control. It is

assumed that those damage states are related to concrete compression and steel tension strain

limits, as detailed in Table 6.7.

Table 6. 7. Limit states by J. Kowalsky [2000]




39

After establishing the two damage states based on dimensionless curvature relationships,

expression for drift ratio can be derived easily, which is shown in equation 6.2,6.3,

D

L K L

D

K K Y p

Y S S

3+

!

=" (6.2)

D

L K L

D

K K Y p

Y DC DC

3

+

!

=" (6.3)

Where L p represents the member plastic hinge length according to Priestley et al [1996].

Curvature ductility factors for serviceability and damage control limit states can be obtained

by dividing the design limit states curvature by the yield curvature.

In a study of Nielson et al [2005], four damage limit states were defined based on HAZUS

guidelines, using a component level approach, as described in Table 6.8. The proposed

framework includes both physics-based and descriptive features, integrated by means of a

Bayesian approach.

Table 6. 8. Medians and dispersions for bridge component limit states using Baysian updating

In a study of Karakostas et al [2006], five damage state criteria were used to characterize the

damage level the structure was under, for each ground motion record. Those five damage

states were defined by , as described in Table 5.9, which refers to ratio between the

performance displacement and yield displacement thus ductility in displacements.

y

! ! /




40

Table 6. 9. Damage states from Karakostas et al [2006]

Generally, damage states described by HAZUS are widely accepted in the development of

fragility curves. However, as different studies may focus on different regions with different

configuration of bridges, the detailed limit states and criteria may have big difference among

one another. On the other hand, most of those studies make use of global damage states when

carrying out the seismic assessment analysis. In this study, four limit states based on the

description of HAZUS are considered, which also concentrated on the global damage states of

the whole bridges.

6.2 Selection of limit states

According to Hwang et al [2001], element-level limit state criteria are more suitable when

detailed seismic damage assessment is required, such as the assessment of the seismic retrofit

scheme of a bridge. For other purposes, such as seismic fragility analysis of bridge

populations, other approaches, based on the assessment of global damage of a bridge, should

be applied.

Accordingly, in this particular study, given that a large number of bridges, defined with

randomly generated properties, it is more appropriate to use structure-level, rather than

element-level, limit state criteria to define damage states. The chosen engineering demand

parameter, used for quantification of damage of each pier, was ductility in displacements. On

a simplified fashion, the maximum value among the different bridge piers was taken as

representative of the behaviour of the entire bridge.




41

In this study, the definitions of each damage limit state is following the description of damage

states proposed by HAZUS [2003], thus four damage states were considered: slight damage,

moderate damage and extensive damage and collapse.

The use of strain measurements from nonlinear numerical analysis can be seen as of

questionable accurateness. Consequently, curvature, a parameter far more stable, has been

chosen to check damage limit states occurrence. Therefore, as Neilson et al [2005] used the

curvature ductility to check each limit states, four damage limit states were also defined based

on the HAZUS description and Bayesian approach. Thus, Neilson’s study was chosen as the

definition for each damage states.

The ductility in curvatures corresponding to each of the four damage limit states is

presented in Table 6.10.

Table 6. 10. Four damage limit states from Neilson et al [2005]

Accordingly, displacement ductility can be calculated based on the corresponding observed

curvature ductility using equations 6.4 and 6.5, as proposed by Priestley et al [1996].

)]/(5.01[)/()1(31 l l l l p p !"!!"+=# $ µ µ (6.4)

b p d l l 908.0 += (6.5)

Accordingly, incorporating the data from Table 5.10 in Equation 5.4, the limit states in terms

of displacement ductility for a pier of, e.g., height 15m, using 30mm reinforcement bars, will

result as follows.

md l l b p 47.103.091508.0908.0 =!+!=+=

For LS1, curvature ductility is equal to 1.29, thus the related displacement ductility is:

Slight Moderate Extensive collapse

Curvature

ductiltiy 1.29 2.10 3.52 5.24




42

081.1)]15/47.1(5.01[)15/47.1()129.1(31)]/(5.01[)/()1(31 =!"!!"+=#"##"+=$

l l l l p p% µ µ


308.1)]15/47.1(5.01[)15/47.1()101.2(31)]/(5.01[)/()1(31 =!"!!"+=#"##"+=$

l l l l p p% µ µ


705.1)]15/47.1(5.01[)15/47.1()152.3(31)]/(5.01[)/()1(31 =!"!!"+=#"##"+=$

l l l l p p% µ µ


185.2)]15/47.1(5.01[)15/47.1()124.5(31)]/(5.01[)/()1(31 =!"!!"+=#"##"+=$

l l l l p p% µ µ

When the different damage states are very distant from each other, in terms of the

corresponding values for ductility, the allowance range for each damage state will be larger,

and so will be the tolerance for differences between each method, which means that the

comparison between methods, in terms of fragility functions, could hide important differences.

However, from the 15m tall pier example above, one can see that the gap between two

different damage states is small, thus the comparison of corresponding fragility functions can

become sensitive enough, which will render the comparison between different NSPs an

dynamic analysis more clear.



Chapter 7 Nonlinear Static Procedures

43

7 Nonlinear Static Procedures

7.1 Pushover Analysis

The earthquake engineering community has been witnessing a significant development of the

performance-based seismic engineering concepts and methodologies in the last two decades.

New ideas and specific procedures, such as pushover analysis, have become increasingly

popular.

It is widely accepted that the most accurate method of analysis for evaluating the seismic

response of structures is nonlinear dynamic analysis. This sort of analysis is however

associated to critical issues, such as the selection of appropriate ground motion records or

intrinsic complexities, such as definition of damping model and post-elastic behaviour, and

typically requires considerable computational effort with time-step integration. As a

consequence, alternative nonlinear static procedures, based on pushover analysis, are

recognized as simplified, yet reliable, techniques for the assessment of the seismic response of

existing structures.

In this study, three different types of pushover analysis are considered for each bridge: 1)

adaptive displacement-based pushover analysis; 2) conventional pushover analysis with

uniformly distributed lateral load; and 3) conventional pushover analysis with 1st mode

proportional distributed lateral load.

7.1.1 Conventional Pushover

A pushover curve shows the relationship between the total base shear and the displacement of

a reference node of a MDOF system. The advantages of using pushover analysis, with respect

to nonlinear dynamic analysis, are mostly related to time and computational onus saving. On

the other hand, typical disadvantages pointed out to the employment of pushover analysis are

the eventually questionable level of accuracy of seismic performance estimates, especially in

irregular structures, including bridges. Such drawback has been particularly associated to

conventional pushover, which makes use of constant load patterns, such as uniform or 1 st

mode proportional shapes, which can hardly take higher modes and inelastic behaviour into

proper consideration.




44

A first attempt to overcome such inherent limitation was proposed in a study of Chopra and

Goel [2002], which presented a modal pushover analysis (MPA) procedure. In this procedure,

multiple pushover analyses are carried out using the load pattern of each relevant vibration

mode and the final response of a structure is calculated as the combination of the different

modal response estimates, which enables the higher modes contribution to be considered.

7.1.2 Adaptive Pushover

A different proposal to overcome the drawbacks of traditional pushover consists of using

adaptive or fully adaptive pushover analysis, developed Antoniou and Pinho [2004]. In this

approach, instead of applying an invariant load vector, the structural properties of the model

are evaluated at each step of the analysis and the loading pattern is updated accordingly thus

taking into account the influence of higher mode effects, degradation characters and spectral

amplifications due to ground motion frequency content.

1. The adaptive pushover logarithm, applied to th analysis of 3D bridges, can be described

as follows.Define nominal load vector P 0 and inertial mass. This step is only carried out

at the beginning of the analysis. In adaptive pushover, the load vectors are definedautomatically at each step therefore the nominal vector of load should follow a uniform

distribution. Moreover, the inertial masses of the structure should be modelled so that

eigenvalue analysis could be carried out at each step.

2. Compute load factors. This and the next two steps are repeated at each equilibrium

stage of the pushover analysis. The magnitude of the loading vector increment ! P at

any step is given, in general terms, by the product of its nominal counterpart P 0, defined

previously, and the load factor "# at that step. The latter is automatically increased till a

predetermined target is reached.

(6.1)

3. Calculate normalised scaling vector. The normalised modal scaling vector ,

computed at the start of each load increment, reflects the actual stiffness state of the

structure, the contribution of the different modes and the influence of the frequency of a

0 P P !"=" #

D




45

particular input. In order to determine the scaling vector at each step, eigenvalue

analysis is carried out at the end of the previous load increment.

(6.2)

The lateral load profiles of each vibration mode are then combined using SRSS rule.

The displacement obtained at each location after their modal combination are

normalised so that the maximum displacement remains proportional to the load vector.

(6.3)

4. Update the loading vector. Once the normalised scaling vector and load factor have

been determined, as well as the value of initial nominal load vector, the loading force

vector at the given analysis step is updated by incremental updating technique.

The calculation of the updated loading vector is performed in two successive steps: the

updating of the loading vector per each structural mode of relevance and the

computation of the current normalised modal scaling vector.

(6.4)

A brief comparison between the different types of pushover curves, for a simple bridge

model, using OpenSees, is carried out in Figure 7.1

Figure 7.1. Pushover curves

jd ij jij s D ,! "=

)max(i

i

i

D

D D =

01 P D P P

k k k k ! "+=#




46

The observation of the different curves indicates that the adaptive pushover exhibits the

higher stiffness degradation after yielding, when compared to 1st proportional pushover

analysis.

7.2 Nonlinear Static Procedures

It is commonly accepted that deformations are of higher importance, with respect to seismic

performance thus it is believed that seismic design and/or assessment should rely on

displacements. Nonlinear static procedures (NSP) represent a simplified approach for the

assessment of the seismic behaviour of structures based on pushover analysis. Generally they

consist of the following steps: 1) define a SDOF model from pushover analysis of a MDOF

structure; 2) bi-linearize the capacity curve of the SDOF system; 3) “intersect” the capacity

with the response spectrum to find out the performance point of the structure.

NSPs can currently be found in codes and guidelines, such as the ATC-40 [ATC,1996],

FEMA-273 [ATC,1997] or the European Code [CEN, 2005a, 2006a]. The reason why NSP

have become such popular within the engineering community is that, when compared to

nonlinear dynamic analysis, NSPs have proved to provide reasonable estimates, requiring less

time and computational effort.

There is a group of pioneering methods, corresponding to the first proposals of nonlinear

static analysis based procedures, which have led to reasonably accurate results. Capacity

Spectrum Method (CSM), introduced by Freeman et al. [1975] and implemented in ATC-40

guidelines, is one of those. Similarly, the N2 method, has been proposed by Fajfar and

Fischinger [1988] and included afterwards in the recommended simplified procedures in

European Code [CEN,2005a]. These first methodologies are mainly focused on simplicity and

consider either the first mode or uniform load distribution for the pushover computation.

Recently, an improved version of CSM has been presented in FEMA-440 guidelines

[ATC,2005], including updated empirical equations to estimate equivalent viscous damping

and spectral reduction factor. Displacement Coefficient Method (DCM) was initially

introduced in ATC-40 and provides a considerable simple empirical equation to calculate the

seismic response of structures using different specific coefficients. Given that all those first

methods fail to take higher modes into consideration, Modal Pushover Analysis was proposed




47

by Chopra and Goel [2002] and has been modified recently according to the seismic

behaviour of higher modes. On the other hand, with the development of the adaptive pushover

analysis, the methodologies based on adaptive or fully adaptive scope enjoy an increasing

development. Several methods were proposed by different scholars and some of them have

already been put into utilization, such as Adaptive Capacity Spectrum Method (ACSM) or

Adaptive Modal Combination Procedures (AMCP).

7.2.1 Capacity Spectrum Method (CSM)

The Capacity Spectrum Method (CSM) was initially proposed by Freeman et al. [1975],

which, by means of a graphical procedure, illustrated in Figure 7.2, iteratively compares the

structure capacity and the seismic demand, represented by a capacity curve and a response

spectrum, respectively. In order to account for the nonlinear behaviour of the structural

system, the ground motion spectrum is computed for a level of equivalent viscous damping at

each iteration.

Figure 7.2 Graphical procedures of CSM

According to ATC-40, the procedure for employing CSM is the following:

1. Carry out a conventional pushover analysis of the multi degree of freedom structural




48

model;

2. Define the capacity curve for the MDOF structure;

3. Transform the MDOF capacity curve into a SDOF capacity curve based on the

participation factor of the fundamental mode;

4. Approximate a bi-linear curve from the SDOF capacity curve. Based on ATC-40, the

stiffness before yielding is equal to the initial stiffness of the structure. In order to derive

the yielding point, the principle of equal areas is applied;

5. Intersect the capacity curve with the demand spectrum, initially damped of 5% and

determine the performance point. Iteration is carried out by reducing the demand spectrum

based on the updated equivalent viscous damping until the convergence in performance

point estimate is reached.

Recently, FEMA-440 proposed an update to this method, introducing new equations for

calculating equivalent viscous damping and periods. In this study, the procedure was applied

following the updated version.

In this study, CSM were carried out using two different lateral load patters for the

conventional pushover analysis: uniform distributed lateral load and 1st mode proportional

lateral load.

7.2.2 N2

The N2 method was formally proposed by Fajfar [2000], as a simplified procedure for

estimating the seismic response of structures. N stands for nonlinear analysis whereas 2

represents two individual mathematical models applied, which are pushover analysis and

spectrum approach separately.

N2 method is similar to CSM in some aspects, given that both of them carry out an analysis

based on capacity curves and response spectra. On the other hand, the major difference

between the methods is that N2 employs inelastic spectra, given in Figure 7.3, rather than

elastic spectra with equivalent viscous damping, which avoids the need for assumptions and

empirical expressions. Moreover, according to FEMA356 [ATC,2000], the post yielding




49

stiffness is equal to zero for the bi-linearization step. In addition, the performance point could

be derived without iteration.

Figure 7.3. Inelastic Spectra by N2

The most essential concept of N2 method is the time-independent lateral displacement profile,

which has an important influence over the pushover curves and the final performance point of

structures. In order to get reasonable pushover curves, at least two load patterns should be

applied, which are uniform distributed load and 1st mode shape proportional distributed load.

The procedure for application of N2 is defined by the steps below:

1. Carry our a conventional pushover analysis for the multi degree of freedom model;

2. Derive the capacity curve for the MDOF structure, based on the total base shear and

reference node displacement;


participation factor of the fundamental mode;

4. Calculate the bi-linear curve from the SDOF capacity curve. Build the elasto-perfectly

plastic capacity curves and calculate the corresponding period of the equivalent SDOF

systems;




50

5. Determine the target displacement by comparing the equivalent period of the system and

the given spectral corner periods, placing the structure in the proper period range;

Check if the target displacement equal to the displacement assumed from the bi-linearcurve; if not, iteration can be carried out to define a new bilinear curve, based on the target

displacement, until convergence is reached.

7.2.3 Displacement Coefficient Method (DCM)

The Displacement Coefficient Method (DCM) was initially implemented in ATC-40 [1996],

then revised and modified by FEMA440 recently. DCM consists of modifying the elastic

spectral displacement for the effective fundamental period (extracted from the capacity

curve), using four coefficients related to different key factors in determining the seismic

response of a structure, shown in equation 7.5.

g T

S C C C C eat 2

2

3210

4!

" =

(7.5)

C0 is the modification factors to spectral displacement of the equivalent SDOF system to the

top displacement of the MDOF system. In this study, it is equal to the first modal participation

factor at the level of the control node.

C1 is the modification factor to expected maximum inelastic displacement, applied to the

displacement calculated for linear elastic response.

C2 is the modification factor related to the effect of hysteric shape, stiffness degrading and

strength degrading on maximum displacement.

C3 is the modification factor to P-delta effect.

The definition of the coefficients is mostly empirical, essentially developed by means of

statistical analysis of the dynamic behaviour of SDOF models.

The DCM procedure, according to FEMA256, and further improved by FEMA440, is

described next:




51

1. Carry out a conventional pushover analysis for a multi degree of freedom model;

2. Derive the capacity curve for the MDOF structure based on the total base shear and

reference node displacement;


participation factor of the fundamental mode.

4. Calculate the bi-linear curve from the SDOF capacity curve, illustrated in Figure 7.4.

According to FEMA356, the stiffness of the pre-yielding branch should be equal to the

secant stiffness calculated based on the base shear equal to 60% of the yield strength.

Figure 7.4. Bilinearization of the force-displacement curves

5. Calculate target displacement by a set of four coefficients. In this step, iterations may need

for re-bilinearization till the target displacement equal to the displacement assumed from

the bi-linear curve.

DCM has been developed and optimized with a view to application to buildings thus the

results for bridges rely on some simplifications and specific assumptions.

7.2.4 Modified Modal Pushover Analysis (MMPA)

The Modal Pushover Analysis was initially introduced by Chopra and Goel [2002],

suggesting repeated nonlinear static procedures for every significant mode of the structure.




52

This method has been widely used due to the the difficulty of the traditional procedurs in

taking higher modes and the redistribution of the inertia forces into consideration, which play

important roles in bridge analysis.

Recently, Chopra and Goel formulated that the response of higher modes during earthquakes

remains in the elastic range hence elastic response spectra should be used for higher modes

effects estimates. They proposed the Modified Modal Pushover Analysis (MMPA), based on

MPA to calculate the performance point of a specific structure. The novelty is that in MMPA,

inelastic response is only being considered for the fundamental mode while for higher modes,

the whole structure is assumed to exhibit elastic behaviour. Chopra and Goel [2004] have

addressed the comparison between the two methods showing that the MPA results

underestimate the seismic response of structures whilst the MMPA yields better estimates,

through the use of elastic SDOF systems.

The MMPA procedure can be summarized in the following steps:

1. Compute the n natural frequencies and modes for the linearly elastic vibration of the

structure;

2. For the first mode, develop the base shear-roof displacement pushover curve for force

distribution proportional to the mass and mode shape;

3. Idealize the pushover curve as a bilinear curve and convert it, computing the first mode

inelastic SDOF system quantities;

4. Compute the peak deformation of the first mode inelastic SDOF system defined

previously using nonlinear response history analysis, inelastic design spectrum or

empirical equations for the ratio of deformations of inelastic and elastic systems;

5. Compute the dynamic response due to the first mode combining the effects of lateral and

gravity loads;

6. Compute the dynamic response due to higher modes under the assumption that the system

remains elastic, performing a classical modal analysis of a linear MDOF system, skipping

the need for additional pushover analysis;

Determine the total response combining the peak modal responses using SRSS rule.




53

7.2.5 Adaptive Capacity Spectrum Method (ACSM)

The Adaptive Capacity Spectrum Method (ACSM) was proposed by Casarotti and Pinho

[2007] and allows the estimation of the seismic response of structures using a fully adaptive

perspective.

Rather than using any elastic or inelastic mode of vibration to convert the MDOF structure

into the equivalent SDOF system, the equivalent SDOF adaptive capacity curve is step by

step derived by calculating the equivalent system displacement and acceleration based on the

actual deformed shape at each analysis step. The developed adaptive capacity curve is

intersected with the demand spectrum, providing an estimate of the inelastic acceleration and

displacement demand (performance point) of the structure. An iterative procedure is then

required until convergence is reached in terms of the equivalent viscous damping to be used in

the reduction of the demand spectrum.

One of the key issues within ACSM is the definition of spectral Reduction Factors, which can

be estimated according to several approaches.

Reduction factors can be roughly categorized into two groups: damping based, computed

according to Equations (7.6), (7.7) and (7.8), and ductility based, as defined by Equations (7.9)

and (7.10). The former can be described as deriving reduction factors associated with

equivalent viscous damping and applying it to both acceleration and displacement spectrum

ordinates. The latter use 5%-damping spectra and then reduce the acceleration ordinate by a

factor defined as a function of ductility. The spectral reduction within ductility-based methods

is not exactly vertical, given that displacements are modified as well.

%5,, !"= el adampa S BS (7.6)

%5,, !"= el d dampd S BS (7.7)

2

,

,!

dampa

dampd

S S = (7.8)

R

S S el a

duct a%5,

,!= (7.9)




54

%5,%5,, !!"==

el d el aduct d S C S

RS

µ (7.10)

Figure 7.5 Spectral reduction methods

A recent study, by Casarotti et al. [2009], has compared the employment of a number of

approaches to estimate spectral reductions factors and concluded that, among the different

approaches, the damping-based model proposed by Priestley et al. [2007] was the most

accurate.

The ACSM procedure is summarized as follows:

1. Carry out an adaptive pushover analysis for the multi degree of freedom model of the

structure;

2. Derive the equivalent SDOF adaptive capacity curve;

3. Intersect the SDOF adaptive capacity curve with the demand spectrum, calculate the

performance point for an assumed damping;

4. Bi-linearize the capacity curve at the performance point and calculate the corresponding

damping;

Check if the calculated damping equal to the assumed one, if not iterate step 3 to 5 till

they are equal and get convergence.




55

7.2.6 Adaptive Modal Combination Procedures (AMCP)

With the main purpose of taking higher modes into consideration, Adaptive Modal

Combination (AMCP) was proposed by Kalkan and Kunnath [2006], based on adaptive

pushover analysis.

In this procedure, higher modes are taken into account by combining individual modal

pushover response with effects of varying dynamic properties, defining parameters for

calculating inelastic spectrum. Moreover, the procedure also makes use of Capacity Spectrum

Method and Modal Pushover Analysis concepts, and, at the same time, it eliminates the need

to pre-estimate the target displacement. When calculating the increment step for capacity

curve of the equivalent SDOF system, an energy-based method is applied. A key aspect of themethod is that a set of capacity spectra based on a series of predetermined ductility levels are

used for each mode to approximate displacement demand.

The steps to be followed when employing AMC are:

1. Compute modal properties of the multi degree of freedom model, such as frequencies,

mode shapes and modal participation factors;

2. For each considered mode, apply an adaptive pushover analysis using the corresponding

mode proportional lateral force, which, similarly to adaptive pushover analysis, could be

calculated after every load step;

3. Calculate the equivalent SDOF adaptive capacity curve for each mode;

4. Calculate the bi-linear curve from the SDOF capacity curve based on elasto-perfectly

plastic post-yielding behaviour.

5. Intersect the SDOF adaptive capacity curve with the demand spectrum and calculate the


6. Check if the calculated damping is equal to the assumed one; if not, iterate steps 4 to 6

until convergence is reached.





56

This method was proposed having building structures, instead of bridge ones, in mind thus its

application within this study required some simplification.

7.2.7 Modified Adaptive Modal Combination Procedures (MAMCP)

As Chopra pointed out, with respect to higher modes, the seismic response generally remain

in elastic range thus the Modified Adaptive Modal Combination Procedure is proposed by the

author. The procedure is mostly based on AMCP, with the only difference being that, in

MACMP, inelastic response is only considered for the fundamental mode whilst, for the

higher modes, the structural response is assumed to be elastic. An immediate consequence is

that MACMP requires less computational effort.

The procedure can be applied according to the steps below:

1. Carry out a 1st mode-based adaptive pushover analysis for the multi degree of freedom

model of the structure;

2. Derive the equivalent SDOF adaptive capacity curve;

3. Intersect the SDOF adaptive capacity curve with the demand spectrum and calculate the


4. Bi-linearize the capacity curve at the performance point and calculate the corresponding

damping;

5. Check if the calculated damping equals the assumed one; if not, iterate steps 3 to 5 until

convergence occurs;

6. Compute the dynamic response due to higher modes under the assumption that the system

remains elastic, performing a classical modal analysis of a linear MDOF system, skipping

the need for additional pushover analysis.





57

7.3 Overall Procedures

The overall procedure that has been followed to assess the performance of the different NSPs,

when calculating fragility curves, can be summarized in the following steps.

1. Random generation of a population of 3D bridge FE models using Monte Carlo simulation;

2. Pushover analysis of each bridge and conversion to the equivalent SDOF system curve;

3. Estimate nonlinear target displacement for each bridge, for each of the selected Nonlinear

Static Procedures, using a large set of ground motion records;

4. Identification of the global damage state based on the nonlinear response;

5. Representation of the cumulative percentage of bridges in each damage state versus the

value of the intensity measure corresponding to each record;

6. Regression analysis to calculate the parameters defining the fragility functions.

7.4 Comparison with nonlinear dynamic analysis

Nonlinear dynamic analysis is widely recognized as the most accurate and reliable tool to

estimate the seismic response of a structure. However, the requirements of this approach are

relatively more complicated, if compared to the previously described static procedures. Such

high complexity is frequently related to the level of the detail the model, the definition of

initial masses, the determination of system damping, etc., all leading to a significant increase

in the computational onus.

In this study, nonlinear dynamic analysis is carried out as the baseline for the sensitivity study

to yield the conclusion that which nonlinear static procedures gives the best compromising

between the accuracy and complexity. The closer the results derived from nonlinear static

analysis to the nonlinear dynamic analysis, the more accurate the methodology provides.

In this study, OpenSees was used to carry out the dynamic analysis of the randomly generated

portfolio bridges, with a specific set of ground motion records. Only the significant duration

of the records was considered based on 5% maximum PGA according to Bommer and Pereira[1999].



Chapter 8 Results

58

8 Results

8.1 Introduction

The presentation of results has been divided in two main parts: one corresponds to the

validation of the different NSPs, through direct comparison by means of the parameter Bridge

Index, BI calculated as equation 8.1, which is defined as median of the ratios between the

response parameter quantities, at different bridge locations, estimated for the performance

point of the structure obtained through a specific NSP and the maximum response parameter

quantities estimated with nonlinear dynamic analysis, whereas the other regards the relative

performance of the different NSPs in terms of Fragility Curves.

Dynamic

NSP BI !

!= (8.1)

The response of the bridge for a given intensity level can be measured at different locations

thus the parameter Bridge Index (BI) was chosen to directly compare the different methods,

based on the performance point (PP) response parameters, that is the maximum response

structural displacement of the deck.

In tandem with the two approaches described in Section 4.2, the comparison of the different

NSPs with dynamic analysis was carried out by calculating BI ratios in two different ways.

One was to derive performance points of each NSP using intersection1 with individual real

spectra whereas the other was to compute performance points from median spectra matching a

predefined design spectrum, as prescribed by the majority of the tested NSPs and applied in

Monteiro [2011].

With respect to the computation of fragility functions using nonlinear static procedures,

although almost all available guidelines suggest the use of smoothed design spectra to

calculate performance points for NSPs this study used the ground motion records selected

from the PEER database, which are real, non-scaled records (in line with the approach

described in Section 5.2.1).

1 To be precise, not all of the NSPs use the intersection of capacity curve with response spectrum to find the performance

point.



Chapter 8 Results

59

In order to make sure that the size of sample would be high enough to assure soundness

results, a brief preliminary parametric study was carried out, testing statistical convergence. In

this study, 10’000 RC bridges were generated to estimate the maximum displacement of the

middle pier and the mean value of those 10’000 bridges was assumed as the exact solution.

Afterwards samples of different sizes were generated (from 5 to 300), and the corresponding

mean values were used to calculate the relative error, which is defined as the difference

between the mean value derived from each sample and the exact solution divided by the latter

one. In Figure 8.1, which illustrates such “calibration” procedure, one can see that when the

size of sample is larger than 170, the relative error remains below 4% thus, conservatively, the

size of the samples, to use as case study, has been chosen as 200.

Figure 8.1. Relationship between the relative error and size of sample

It is not always straightforward to employ NSPs that rely on capacity-demand interaction in

an automatic fashion, when using real accelerograms, given that the intersections of the

structure capacity curve with the response spectrum can occur more than one time. In such

case, according to Casarotti et al [2005], the first intersection point was chosen for the

analysis.

8.2 Bridge Index (BI) based on individual records

In this part of study, 200 bridges were generated randomly and the set of ground motion

records was the 10 LA records.

0 50 100 150 200 250 3000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

R e l a t i v e E r r o r

Size of Sample



Chapter 8 Results

60

8.2.1 Capacity Spectrum Method

Figure 8.2 illustrated the median BIs obtained with CSM-based estimation.

Figure 8.2. CSM median Bridge Index per intensity level

Generally, using pushover analysis that makes use of 1st mode shape proportional or uniform

load patterns yields similar results, within the employment of CSM. Furthermore, with

increasing intensity level, CSM produces more accurate seismic performance estimates, when

compared with nonlinear dynamic analysis. One of the possible reasons for the

underestimating trend may be the fact that CSM uses equivalent viscous damping, which may

lead to overestimation of damping.

8.2.2 N2 Method

Figure 8.3 illustrates the median BIs obtained from N2-based estimates.

Figure 8.3. N2 median Bridge Index per intensity level

The observation of the N2 results leads to similar conclusions to what has been found for

CSM: the 1st mode proportional load pattern yields slightly more accurate predictions. In

addition, generally speaking, the larger the intensity level, the closer the results get to

dynamic analysis ones.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.6

0.8

1

1.2

1.4

1.6

M e d i a n B I

Intensity Level:PGA(g)

BI CSM

CSMuni

CSM1st

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.6

0.8

1

1.2

1.4

1.6

M e d i a n B I


BI N2

N2uni

N21st



Chapter 8 Results

61

8.2.3 Displacement Coefficient Method

Figure 8.4 illustrates the median BIs obtained from DCM-based estimates.

Figure 8.4. DCM median Bridge Index per intensity level

It seems quite evident that the 1st mode proportional load pattern leads to better predictions,

which clearly states for the significant influence of the choice for the lateral load patterns of

pushover analysis on the prediction of the bridge performance point.

On the other hand, as DCM was initially proposed for buildings and some of the coefficients

are not important or even relevant for bridges, the results derived from this procedure might

be compromised, and were indeed not as good as the ones obtained with other procedures.

8.2.4 Modified Modal Pushover Analysis

Figure 8.5 illustrates the median BIs obtained from MMPA-based estimates.

Figure 8.5. MMPA median Bridge Index per intensity level

Modified modal pushover analysis takes higher modes contribution into consideration, which,

according to the indices in Figure 8.5, do seems to have led to better predictions, especially

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

M e d i a n B I


BI DCM

DCMuni

DCM1st

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.6

0.8

1

1.2

1.4

1.6

M e d

i a n B I


BI MMPA



Chapter 8 Results

62

when compared to other methods that make use of 1st mode based conventional pushover

analysis.

8.2.5 Adaptive Capacity Spectrum Method

Figure 8.6 shows the Bridge Index obtained with Adaptive Capacity Spectrum Method.

Figure 8.6. ACSM median Bridge Index per intensity level

Similarly to CSM, but contrarily to N2 and MMPA, ACSM is a method that relies

considerably on damping, estimating it to then again estimate the spectral reduction factor. In

spite of making use of an adaptive pushover analysis, ACSM did not yield particularly

accurate predictions, exhibiting a general underestimating trend. As has been observed for

other procedures, the accuracy of the ACSM predictions increases with the increasing of the

intensity level.

8.2.6 Adaptive Modal Combination Procedure

Figure 8.7 illustrates the comparison with nonlinear dynamic analysis, obtained when AMCP

is employed.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.6

0.8

1

1.2

1.4

1.6

M e d i a n B I


BI ACSM

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.6

0.8

1

1.2

1.4

1.6

M e d i a n B I


BI AMCP



Chapter 8 Results

63

Figure 8.7. AMCP median Bridge Index per intensity level

Considering both higher modes effects and post-yield stiffness degradation, AMCP provides

fairly good estimates for lower intensity levels whilst for higher intensity levels, consideringthe higher modes contribution through SRSS combination for the final response parameter

prediction, AMCP tends to slightly overestimate nonlinear dynamic analysis.

8.2.7 Modified Adaptive Modal Combination Procedure

MAMCP results in terms of Bridge Index are shown in Figure 8.8.

Figure 8.8. MAMCP median Bridge Index per intensity level

Similarly to the previous methods, MAMCP provides slightly overestimating indexes for high

levels of intensity. Nevertheless, the average Bridge Index for this procedure is around 0.9,

which somehow denotes global underestimation of the performance.

8.2.8 Global Results

Figure 8. 9. Median Bridge Index per intensity level for all seven methods

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

M e d i a n B I


BI MAMCP

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

M e d i a n B I


BICSM

N2

DCM

MMPA

ACSM

AMCP

MAMCP



Chapter 8 Results

64

Figure 8.9 plots the Bridge Index for all of the nonlinear static procedures applied in this

study. Generally, NSP-based response estimates of bridges get closer to the dynamic-based

ones, thus more accurate, with increasing intensity level.

In order to render the comparison more clear, the set of NSPs was divided in two groups, one

containing the pioneering and/or conventional ones and the other including the improved

and/or adaptive ones. The corresponding results are presented in Figures 8.10 and 8.11.

Figure 8.10. Median BI for each intensity level of CSM, N2 and DCM

From the observation of Figure 8.10, one can conclude that, generally, the “classic”

procedures tend to underestimate the response. In addition, it can be seen that DCM has a

very variable behaviour within the considered intensity range, whereas CSM can be seen as

the method with the steadiest behaviour, although it sometimes heavily underestimates the

response displacement estimates.

Figure 8.11. Median BI for each intensity level of CSM, N2 and DCM

Figure 8.11, on the other hand, shows that the so-called innovative NSPs, do bring in some

improvement, given that BIs are definitely closer to unity. Despite that MAMCP considers the

higher modes as elastic, ACSM and MAMCP yield somewhat similar results, which indicate

that within the use of adaptive modal pushover analysis, the 1st mode contribution is more

important than the one from higher modes.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

M e d i a n B I


BI

CSM

N2

DCM

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

1.5

2

M e d i a n B I


BI

MMPA

ACSM

AMCP

MAMCP



Chapter 8 Results

65

Figure 8.12. Median Bridge Index for each NSP

Finally, Figure 8.12 shows overall comparison of the tested NSPs, by calculating the median

Bridge Index, across all the intensity levels. MMPA, ACSM and, particularly, N2 and

MAMCP stand as the best perfomring procedures, yielding global BI ratios pratically equal to

one., i.e., staitic-based estimates that globally match the dynamic ones.

Figure 8.13. Median Bridge Index for each PGA

Figure 8.13 shows the global average Bridge Index results from a different perspective, i.e.,

for each peak ground motion acceleration, the median BI across all the tested NSPs is plotted.

The results shows, again, that with increasing intensity level, the accuracy of the predictions

of the deifferent NSPs becomes higher. As a general conclusion one can say that the NSP

predictions are mostly within a -/+20% range, with respect to the dynamic analysis ones.

As a final output, Figure 8.14 represents the median BI values for N2, MMPA, ACSM and

MAMCP, which have been recognised as the best performing procedures among all the seven

NSPs. When eliminating the less accurate procedures from the representation, the range in

which the majority of the estimates fall within becomes -/+10%.

CSM N2 DCM MMPA ACSM AMCP MAMCP0.5

1

1.5

M

e d i a n B I

Global BI

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

1

1.5

M e d i a n B I

Intensity levels:PGA(g)

Displacement BI



Chapter 8 Results

66

Figure 8.14. Median Brige Index for N2, MMAP and MAMCP

Table 8.1 represents the quantitative version of the results, given by the exact BI values for

every considered NSP, per intensity level.

Table 8.1. Median BI of seven approaches for per intensity level

The observation of the numbers reinforces the main trend identified so far for nearly all of thetested Nonlinear Static Procedures: with the increase of the intensity measure (peak ground

acceleration), the static-based displacement estimates become closer to dynamic analysis

based ones. Nevertheless, for some of the low intensity ground motion records, some NSPs

can still provide fair predicitons.

Based on the results presented so far, even though adaptive pushover analysis considers the

actual stiffness of the structure at each step, it still yielded underestimated predictions, which

might be due to a further needed calibration of the equivalent viscous damping model. On the

other hand, the N2 method, which makes use of a constant load pattern and inelastic spectrum,

provided fairly good estimates. A similar scenario was found for the application of MMPA,

which also make use of inelastic spectrum but takes higher modes into consideration. With

respect to other characteristics of the procedures, such as computional and time efforts, N2

would be preferable.

It is worth mentioning that the proposed MAMCP also provides comparetively good estimates,

coupled with considerable time saving by accounting for higher modes in elastic regime.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

1

1.5

M

e d i a n B I

Intensity levels:PGA(g)

Displacement BI

PGA(g) 0.676 0.479 0.235 0.426 0.360 0.969 0.657 0.580 0.817 0.986 Median

BI_CSM1st 0.728 0.664 0.542 0.786 0.742 0.873 0.972 0.618 0.941 0.740 0.761

BI_N21st 1.186 0.525 0.772 1.249 1.066 1.375 1.031 0.988 0.889 0.900 0.998

BI_DCM1st 1.118 0.407 0.358 1.141 0.207 1.895 0.919 0.671 0.562 0.659 0.794

BI_MPA 1.162 0.540 0.744 1.220 1.061 1.335 0.950 0.951 0.867 0.875 0.970

BI_ACSM 0.746 0.744 0.544 1.164 0.715 0.882 0.900 0.796 1.359 1.000 0.885

BI_AMC 1.084 0.989 0.709 1.543 1.119 1.173 1.209 1.224 2.102 1.566 1.272

BI_MAMC 0.850 0.843 0.622 1.323 0.739 1.184 0.980 0.839 1.451 1.044 0.987

Median 0.982 0.673 0.613 1.204 0.807 1.245 0.994 0.870 1.167 0.969 0.952



Chapter 8 Results

67

8.3 Bridge Index (BI) based on spectrum-matching scaled records

In agreement with what was mentioned in Section 5.2.2, the records applied within this

approach are known to feature high energy and frequency content, which renders them

particularly demanding for some structures. Furthermore, some Nonlinear Static Procedures,

such as CSM, N2, DCM and MMPA, are originally formulated with a view to the use of

design smoothed, rather than real, response spectra. A second approach was thus followed, in

which design spectrum was used within the four mentioned NSP methodologies, whereas the

median real spectrum was used within ACSM, AMCP and MAMCP.

These two spectra, applied to the different NSPs, are compared in Figure 8.15. The period of

the population of bridges generally ranges between 0.4s and 1.2s. Within this range, accordingto Figure 8.15, there is a relatively good matching between the design spectrum of NEHRP

and the median spectra.

Figure 8.15. Acceleration spectra

In order to minimize the influence of the uncertainties related to the different records, the

mean nonlinear dynamic result is calculated using the maximum displacements deriving from

the ten time-history analyses.

Four different scale factors are used for the consideration of five different intensity levels

(0.5,1.0,1.5,2.0 and 2.5), which means amplifying the median spectrum and design spectrum

using those scale factors.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

S a ( g )

Period(s)

NEHRP

Median



Chapter 8 Results

68

Figure 8.16 provides the results obtained when employing Capacity Spectrum Method.

Figure 8.16. CSM median Bridge Index per intensity level based on scaled spectrum

It can be seen that with increasing of intensity level, the method provides more accurate

estimates, which is a similar trend to one observed when using individual record derived

spectra.

Furthermore, similar results are found when using N2, DCM and MMPA, as evidenced by

Figures 8.17 to 8.19.

Figure 8.17. N2 median Bridge Index per intensity level based on scaled spectrum

Figure 8.18. DCM median Bridge Index per intensity level based on scaled spectrum

0 0.5 1 1.5 2 2.5 3

0.4

0.6

0.8

1

1.2

1.4

1.6

M e d i a n B I

Intensity Levels

BI CSM

CSMuni

CSM1st

0 0.5 1 1.5 2 2.5 30.5

1

1.5

M e d i a n B I

Intensity Levels

BI N2

N2uni

N21st

0 0.5 1 1.5 2 2.5 30.5

1

1.5

M e d i a n B I

Intensity Levels

BI DCM

DCMuni

DCM1st



Chapter 8 Results

69

Figure 8.19. MMPA median Bridge Index per intensity level based on scaled spectrum

Figure 8.17 to 8.19 present the performance of the procedures that make use of smoothed

design spectra. Among the four approaches, N2, closely followed by MMPA, provides the

most accurate estimates, when compared to dynamic analysis.

The latter presents a slight drop down branch for high intensity levels. A possible reason for

this might be the fact that when calculating the performance point, elastic behaviour is

assumed for the higher modes, which might not necessarily happen with considerably high

intensity levels.

Figures 8.20 to 8.22 represent the results obtained with the procedures that make use of

median response spectrum of the ten selected records.

Figure 8.20. ACSM median Bridge Index per intensity level based on scaled spectrum

Figure 8.21. AMCP median Bridge Index per intensity level based on scaled spectrum

0 0.5 1 1.5 2 2.5 30.5

1

1.5

M

e d i a n B I

Intensity Levels

BI MMPA

0 0.5 1 1.5 2 2.5 30.5

1

1.5

M e d i a n B I

Intensity Levels

BI ACSM

0 0.5 1 1.5 2 2.5 30.5

1

1.5

M e d i a n B I

Intensity Levels

BI AMCP



Chapter 8 Results

70

Figure 8.22. MAMC median Bridge Index per intensity level based on scaled spectrum

Results show that AMCP generally provides overestimated results, when compared to

dynamic analysis, whereas, on the other hand, by considering elastic behaviour for higher

modes, MAMCP yields fairly good estimates. On the other hand, ACSM yields similarestimates to the ones obtained when using real ground motion spectrum, which utilizing the

equivalent viscous damping model and spectrum reduction factor, rather than inelastic

spectrum, have led to underestimated predictions.

Figure 8.23 plots the median BIs for the tested methods all-together.

Figure 8.23. Median Bridge Index per intensity level based on scaled spectrum for each method

When compared to Figure 8.11, Figure 8.23 shows that when using a median design response

spectrum, the dispersion among the different NSPs is considerably lower than the one

associated to the real records spectrum approach. Moreover, ACSM and MAMCP generally

provide the best estimates, whilst AMCP tends to overestimate the nonlinear dynamic

response predictions.

In order to render thr presentation of the results of the seven NSPs more clear, two different

sets of NSPs, traditional and improved, were again plotted in Figures 8.24 and 8.25.

0 0.5 1 1.5 2 2.5 30.5

1

1.5

M

e d i a n B I

Intensity Levels

BI MAMCP

0 0.5 1 1.5 2 2.5 30.5

1

1.5

M e d i a n B I

Intensity Levels

BI

CSM

N2

DCMMMPA

ACSM

AMCP

MAMCP



Chapter 8 Results

71

Figure 8.24. Median BI for each intensity levels of CSM, N2 and DCM

Figure 8.25. Median BI for each intensity levels of MMPA, ACSM, AMCP and MAMCP

It can be seen from Figure 8.24 and 8.25 that the previous conclusion can be confirmed that

the higher the seismic intensity, the closer the NSPs’ results get to the dynamic ones.

Moreover, the second set of the approaches provide better performance than the first one.

The global median results (i.e. across all the NSPs) for each intensity level, and for each NSP

are presented in Figures 8.26 and 8.27, respectively, which demonstrate that the conclusion

that the accuracy of the predicitons increases with intensity level is again confirmed.

Figure 8.26. Average values for each intensity

0 0.5 1 1.5 2 2.5 30.5

1

1.5

M

e d i a n B I

Intensity Levels

BI

CSM

N2

DCM

0 0.5 1 1.5 2 2.5 30.5

1

1.5

M e d i a n B I

Intensity Levels

BI

MMPA

ACSM

AMCP

MAMCP

0 0.5 1 1.5 2 2.5 30.5

1

1.5

M e d i a n

B I

Intensity Levels

Displacement BI



Chapter 8 Results

72

Figure 8.27. Median Bridge Index for each NSPs

Finally, Figure 8.28 represents the comparison between the two different approaches, in terms

of global median Bridge Index for each NSP.

Figure 8.28. Median Bridge Index for each NSPs with two approaches

Figure 8.28 shows that N2, MMPA and ACSM yield similar and the most accurate estimates,

CSM and DCM underestimate and AMCP overestimates the nonlinear dynamic response in

terms of deck displacements. In addition, the use of two approaches allows the verification of

the relative performance of each NSP, together with the fact that the first approach, which

makes use of non-scaled records, provides predictions closer to the dynamic analysis.

Table 8.2 shown illustrates the numbers behind the different plots, for median Bridge Index

(BI) for each intensity.

Table 8.2. Bridge Index for each intensity

CSM N2 DCM MMPA ACSM AMCP MAMCP0.5

1

1.5

M

e d i a n B I

Gloabal BI

CSM N2 DCM MMPA ACSM AMCP MAMCP0

0.5

1

1.5

2

M e d i a n B I

Displacement BI

real spectra

median spectra

Intensity Level 0.5 1 1.5 2 2.5 edi nBI_CSM1st 0.992 0.744 0.836 0.878 0.901 0.878

BI_N21st 0.743 0.815 0.877 0.926 0.955 0.877

BI_DCM1st 0.603 0.812 0.732 0.730 0.719 0.730

BI_MPA 0.859 0.774 0.893 0.948 0.867 0.867

BI_ACSM 0.934 0.818 0.817 0.936 0.901 0.901

BI_AMC 1.129 1.032 1.096 1.206 1.161 1.129BI_MAMC 0.833 0.841 0.847 0.890 0.893 0.847



Chapter 8 Results

73

As one can see from the table, accurate estimates are obtained not only the higher intensity

levels but also, for some methods, at the intensity level of 0.5, to provide comparatively good

results. The better quality for some intensity levels, with respect to others, may have to do

with the regular shape of the used spectra around the intersection region, which thus features

lower uncertainty.

The next section presents the results obtained with the different NSPs, in terms of fragility

functions, which have been calculated using of non-scaled real spectra, given that, according

to Figure 6.33, such approach leads to more accurate predictions.

8.4 Fragility Curves

Response estimation for fragility curves calculations was based on statistic analysis and

lognormal distribution has been assumed for each fragility curve as a reasonable choice. The

procedure is described as follows:

1. Derive the linear regression function of the relationship between natural logarithm

distribution of Probability of exceedance and ln(PGA);

2. Calculate according to the linear function;

3. Calculate the lognormal distribution for each PGA and get the curve.

As mentioned in Chapter 6, the global displacement ductility for each bridge has been

assumed as the highest among the different piers.

The same set of 50 ground motion records described in Section 4.2 was used for calculating

fragility functions.

8.4.1 Nonlinear Static Procedures

The mean value and dispersion, lamda and kesi, for the lognormal distribution calculated

using the predictions of the different NSPs are presented in Table 8.3 to 8.9 and the

corresponding fragility functions, calculated for the different limit states, are plotted in Figure

8.29. Each NSP was applied in its optimized fashion, according to the results of Section 8.2.

! µ ! •"== bm

,1



Chapter 8 Results

74

Table 8.3. Statistics of fragility functions by NSPs

NSP LS Lamda Kesi

CSM

LS1 -1.167 0.260

LS2 -0.941 0.257

LS3 -0.501 0.293

LS4 -0.098 0.334

N2

LS1 -0.859 0.212

LS2 -0.731 0.222

LS3 -0.360 0.280

LS4 0.201 0.375

DCM

LS1 -1.389 0.333

LS2 -1.202 0.334

LS3 -0.972 0.293

LS4 -0.795 0.289

MMPA

LS1 -0.780 0.322LS2 -0.594 0.329

LS3 -0.285 0.303

LS4 0.267 0.387

ACSM

LS1 -0.691 0.229

LS2 -0.342 0.273

LS3 0.010 0.359

LS4 0.987 0.490

AMCP

LS1 -1.090 0.210

LS2 -0.915 0.213

LS3 -0.800 0.214

LS4 -0.499 0.265

MAMCP

LS1 -1.132 0.179

LS2 -0.999 0.204

LS3 -0.790 0.221

LS4 -0.412 0.271



Chapter 8 Results

75

Figure 8.29. Fragility curves of NSPs

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E x c e e d e n c e

P r o

b a b i l i t y

PGA(g)

Fragility CSM

LS1

LS2

LS3

LS4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E x c e e d e n c e

P r o

b a b i l i t y

PGA(g)

Fragility N2

LS1

LS2

LS3

LS4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E x c e e d e n c e

P r o

b a b i l i t y

PGA(g)

Fragility DCM

LS1

LS2

LS3

LS4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E x c e e d e n c e

P r o

b a b i l i t y

PGA(g)

Fragility MMPA

LS1

LS2

LS3

LS4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E x c e e d e n c e

P r o

b a b i l i t y

PGA(g)

Fragility ACSM

LS1

LS2

LS3

LS4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E x c e e d e n c e

P r o b a b i l i t y

PGA(g)

Fragility AMCP

LS1

LS2

LS3

LS4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E x c e e d e n c e


PGA(g)

Fragility MAMCP

LS1

LS2

LS3

LS4



Chapter 8 Results

76

8.4.2 Nonlinear Dynamic Analysis

The mean value and dispersion, lamda and kesi, for the lognormal distribution calculated

using nonlinear dynamic analysis predictions are presented in Table 8.10 and the

corresponding fragility functions, calculated for the different limit states, are plotted in Figure

8.30.

Table 8.4 Statistics of fragility functions by Nonlinear Dynamic Analysis

Figure 8.30. Fragility curves for nonlinear dynamic analysis

8.4.3 Comparison of results

The comparison of static- and dynamic-based fragility curves, for each of the limit states, is

illustrated in Figure 8.31.

l m kesiLS1 -0.870 0.242

LS2 -0.626 0.271

LS3 -0.309 0.309

LS4 0.146 0.382

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E x c e e d e n c e


PGA(g)

Fragility Dynamic

LS1

LS2

LS3

LS4



Chapter 8 Results

77

Figure 8.31. General results of each method for each Limit states

Among the seven selected nonlinear static procedures, N2 provides the closest to dynamic

analysis estimations, featuring also with the lower computational onus efforts, followed by

MMPA. Generally, the methods requiring the definition of an equivalent viscous damping

model, such as ACSM, AMCP and MAMCP, yield fragility curves farther from dynamic

estimates. On the contrary, the methods making use of inelastic spectra, i.e. N2 and MMPA,

did get closer to dynamic analysis.

Adaptive pushover based methods, such as ACSM, did not necessarily provide equally

improved predictions, as observed in previous studies [Pinho et al., 2009]. A possible reason

could be related to the equivalent viscous damping model used in the procedure, which may

be still leading to underestimation of the displacements when using ground motion records

with low peak ground accelerations, such as the ones applied in this study.

Given that many records feature low peak ground acceleration, the chosen intensity measure,

higher modes did not play a predominant role, since the structures did not go far beyond the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E x c e e d e n c e

P r o b a b

i l i t y

PGA(g)

Fragility LS1

CSM

N2

DCM

MMPA

ACSM

AMCP

MAMCPDynamic

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E x c e e d e n c e


PGA(g)

Fragility LS2

CSM

N2

DCM

MMPA

ACSM

AMCP

MAMCPDynamic

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E x c e e d e n c e


PGA(g)

Fragility LS3

CSM

N2

DCM

MMPA

ACSM

AMCP

MAMCP

Dynamic

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E x c e e d e n c e


PGA(g)

Fragility LS4

CSM

N2

DCM

MMPA

ACSM

AMCP

MAMCP

Dynamic



Chapter 8 Results

78

elastic range thus AMCP and MAMCP provided similar results. Adding this to the

computational effort required for carrying out adaptive pushover analysis for each relevant

mode, such procedures can become rather elaborated while still not providing more accurate

estimates of fragility curves. The author would thus not recommended AMCP or MAMCP for

the prediction of seismic performance of bridges, by means of fragility curves, within loss

assessment studies.

CSM and DCM did not succeed as well to yield accurate estimates, when compared to

nonlinear dynamic analysis, in tandem with what had been verified before, in terms of Bridge

Index results.



Chapter 9 Closing Remarks

79

9 Closing Remarks

9.1 Conclusions

A number of past studies have addressed the ability of nonlinear static procedures to,

individually or comparatively to alternative counterparts, assess the structural performance of

single bridges. Much less attention has been paid, on the other hand, to such evaluation within

a given region context, comprising a population of bridges, assessing the performance of the

different approaches in terms of vulnerability models, loss estimations or seismic risk

calculations. In this study, seven different nonlinear static procedures were applied to

investigate the most suitable analytical approach(es) for deriving fragility functions of bridges

for use in loss assessment studies.

In order to fulfil such goal, a sufficient large number (200) of 3D bridge models, using a

representative bridge configuration of three piers and four spans, were randomly generated

using a fibre model based structural analysis software (OpenSees). The generation of different

bridges was based on the variation of a number of parameters, which included concrete

strength, steel strength, pier heights, column diameter and bay lengths. For all those

considered parameters a statistical distribution was defined.

After generating a detailed 3D model, two different pushover analyses were carried out,

conventional and adaptive. The former was tested with two different load distribution patterns:

uniform and first mode proportional. Based on the pushover curves derived from the

nonlinear static analysis, seven nonlinear static procedures, which include Capacity Spectrum

Method (CSM), Displacement Coefficient Method (DCM), N2, Modified Modal Pushover

Analysis (MMPA), Adaptive Capacity Spectrum Method (ACSM), Adaptive Modal

Combination Procedure (AMCP) and Modified Adaptive Modal Combination Procedure

(MAMCP), were employed for estimating response displacement for a number of ground

motion records.

As a tool to measure accuracy of the nonlinear static procedures, nonlinear dynamic analyses

were carried out, using two different sets of ground motion records. One consisted of non-

scaled real records selected from PEER database, which presented similar fault mechanisms

and potential seismic intensity to the Italy territory characteristics. The second consisted of




80

scaled records in order to match a smoothed design spectrum, associated to a predefined

exceedance probability.

Damage states were defined for different levels in terms of displacement ductility, which wasconsidered as the highest value amongst the different piers. Probability of exceeding a

particular damage limit state was determined for each bridge by checking whether the

maximum response displacement calculated through NSPs or Nonlinear dynamic analysis

went beyond the limit state threshold. Lognormal distribution functions were used to

characterize the fragility functions for each method.

The comparison of the response estimates derived from NSPs and nonlinear dynamic analysis

was carried out in terms of Bridge Index, a ratio that directly divides the response

displacement from each NSP by the maximum response displacement from nonlinear

dynamic analysis, and in terms of corresponding fragility functions, using different NSP

estimates.

The main conclusions from the results of this study can be summarized in the following

points:

• Generally, Nonlinear Static Procedures were able to provide reasonable predictions in

terms of response displacements, when compared to the nonlinear dynamic estimates.

Notable exceptions include the Capacity Spectrum Method (CSM) and Displacement

Coefficient Method (DCM), which yielded poor estimates, mostly due to the fact that

they constitute pioneering approaches (CSM) or feature a strong empirical basis (DCM)

which can be hard to apply in bridge analysis. The Adaptive Modal Combination

Procedure (AMCP) has considerably overestimated the dynamic results, for high ground

motion intensity levels, whereas for low intensity levels it provided similar results to

Modified Adaptive Modal Combination Procedure (MAMCP). The Adaptive Capacity

Spectrum Method (ACSM) provided fairly good estimates for large peak ground

acceleration records, while, for low intensity ground motion records, it tends to

underestimate the response displacement. N2, followed by Modified Modal Pushover

Analysis (MMPA), provided quite accurate estimates, with respect to nonlinear dynamic

results, for both high and low intensity levels.




81

Globally, nonlinear static procedures provided better estimates with increasing ground

motion intensity. The improved methods, which make use of adaptive pushover analysis

and take higher modes contribution into account did not perform significantly better than

the traditional ones. Indeed, if a single procedure should be selected, by taking

computational effort and time in consideration, N2 would rank first among the seven

NSPs.

• When comparing the results of fragility curves derived from different nonlinear static

approaches, one can see that the conclusion summarized by BI could be confirmed. i.e.,

N2 provides the closest results to the nonlinear dynamic ones, followed by MMPA. CSM

and DCM, initially proposed for building structures, failed to provide good estimates offragility functions of bridges. As the set of ground motion records considered in fragility

analysis did not send the structures highly into the nonlinear range, AMCP and MAMCP

yielded similar results due to the elastic behaviour of higher modes. ACSM, making use

of equivalent viscous damping, turned out to underestimate the results when applied with

low intensity ground motion records. Moreover, there is still significant dispersion among

the different methodologies, which, apart from the conceptual differences, might be

related to the definition of limit states or the lognormal distribution chosen.

9.2 Future Recommendations

The main goal of this study was to provide an overall perspective of the different available

analytical approaches for the derivation of fragility functions of bridges. Some details and

simplified assumptions could be further addressed more precisely thus the author would like

to refer to some possible future developments.

• With respect to 3D modelling, the stiffness of the abutments plays an important role,

which has influence in the results of pushover analysis. However, in this study only one

kind of abutment was taken into consideration thus no variability was considered within

the randomly generated bridges. More attention should be paid to the stiffness of the

abutments without necessarily going through a full abutment design procedure, by, e.g.

establishing a simplified relationship between the stiffness of the abutments and the

bay/deck lengths;




82

• In this study, only one bridge configuration was considered. However, different bridge

configurations could have a critical influence on the seismic assessment. Thus the

consideration of additional bridge configurations, possibly based on real case study will

be developed in the future;

• Other intensity measures could be considered, when calculating fragility functions,

instead of PGA. Possible alternatives, such as spectral acceleration (Sa) or Arias Intensity

(ASI), could be considered in a short parametric study, to list the best performing

intensity measure;

• Further refinement could be considered with respect to the type of distribution used to

characterize the fragility functions through nonlinear regression analysis;

• The comparison of different NSPs should be extended to the calculation of vulnerability

functions and loss estimates.



References

83

10 REFERENCES

ATC [1996] Seismic Evaluation and Retrofit of Concrete Buildings, Volumes 1 and 2. Report No.

ATC-40, Applied Technology Council, Redwood City, CA.

ATC [1997] NEHRP Guidelines for the Seismic Rehabilitation of Buildings. Report No. FEMA-273,

Federal Emergency Management Agency, Washington, DC.

ATC [2000] Prestandard and Commentary for the Seismic Rehabilitation of Buildings. Report No.

FEMA-365, Federal Emergency Management Agency, Washington, DC.

ATC [2005] Improvement of Nonlinear Static Seismic Analysis Procedures. Report No. FEMA-440,

Federal Emergency Management Agency, Washington, DC.

Antoniou, S. and Pinho, R. [2004] “Development and verification of a displacement-based adaptive

pushover procedure,” Journal of Earthquake Engineering , Vol.8, No.5, pp. 643-661.

Aktan, A. E. and Ersoy, U. [1979] “Analytical Study of R/C Material Hysteresis,” Proceedings of the

AICAP-CEB Symposium on Structural Concrete Under Seismic Actions, CEB Bulletin

d'Information No. 132, pp. 97-104.

Avsar, O. [2009] “Fragility based seismic vulnerability assessment of ordinary highway bridges inTurkey,” Thesis, Middle East Technical University, Turkey.

Bal, I.E., Crowley, H., Pinho, R., Gulay, F.G. [2008] “Detailed assessment of structural characteristics

of Turkish RC building stock for loss assessment models,” Soil Dynamics and Earthquake

Engineering , Vol. 28, pp. 914-932.

Bal, I.E., Crowley, H., Pinho, R. [2010] Displacement-based Earthquake Loss Assessment: Method

Development and Application to Turkish Building Stock , ROSE Research Report, IUSS Press,

Pavia, Italy.

Banerjee, S., Shinozuka, M. [2008] “Mechanistic quantification of RC bridge damage states under

earthquake through fragility analysis,” Probabilistic Engineering Mechanics, Vol.23, No.1, pp. 12-

22.



References

84

Bommer,J.J.,Martinez-Pereira,A. [1999] “The effective duration of earthquake strong motion,”

Journal of Earthquake Engineering , Vol.3, No. 2, pp. 127-172.

CALTRANS,(various dates), Bridge Memo To Designers,California Department of Transportation,

Sacramento, California.

Cardone, D., Perrone, G., Sofia, S. [2011] “A performance based adaptive methodology for the

seismic evaluation of multi-span simply supported deck bridges,” Bulletin of Earthquake

Engineering , Vol. 9, No. 5, pp. 1463-1498.

Casarotti, C., Pinho, R. and Calvi, G. M. [2005] Adaptive pushover-based methods for seismic

assessment and design of bridge structures. ROSE Research Report No. 2005/06, IUSS Press,

Pavia, Italy

Casarotti, C. and Pinho, R. [2006] “Seismic response of continuous span bridges through fiber-based

finite element analysis,” Earthquake Engineering and Engineering Vibration, Vol.5 No. 1, pp. 119-

131.

Casarroti, C., Pinho, R. [2007] “An adaptive capacity spectrum method for assessment of bridges

subjected to earthquake action,” Bulletin of Earthquake Engineering , Vol. 5, No. 3, pp. 377-390.

Casarotti, C., Monteiro, R. and Pinho, R. [2009] “Verification of spectral reduction factors for seismic

assessment of bridges,” Bulletin of the New Zealand Society for Earthquake Engineering , Vol. 42,

No. 2, pp. 111-121.

CEN [2005a] Eurocode 8: Design of Structures for Earthquake Resistance - Part 1: General rules,

seismic actions and rules for buildings. EN 1998-2, Comité Européen de Normalisation, Brussels,

Belgium.

CEN [2005b] Eurocode 8: Design of Structures for Earthquake Resistance - Part 2: Bridges. EN

1998-2, Comité Européen de Normalisation, Brussels, Belgium

Choi, E., DesRoches, R., Nielson, B. [2004] “Seismic fragility of typical bridges in moderate seismic

zones,” Engineering Structures, Vol. 26, No. 2, pp. 187-199.

Chopra, A. K. [1995] Dynamics of Structures: Theory and Applications to Earthquake Engineering .

Prentice-Hall, Englewood Cliffs, NJ.

Chopra, A. K. and Goel, R. K. [2001] A modal pushover analysis procedure to estimating seismic

demands for buildings: Theory and preliminary evaluation. PERR Report 2001/03, Pacific

Earthquake Engineering Research Center, University of California, Berkeley, CA.

Chopra, A. K. and Goel, R. K. [2002] “A modal pushover analysis procedure for estimating seismic

demands for buildings,” Earthquake Engineering & Structural Dynamics, Vol.31,No.3, pp. 561-

582.

Chopra, A.K., Geol, R.K., Chintanapakdee, C. [2004] “Evaluation of a modified MPA procedure

assuming higher modes as elastic to estimate seismic demands,” Earthquake Spectra, Vol. 20, No.

3, pp. 757-778.

Elnashai, A.S., Borzi, B. [2004] “Deformation-based vulnerability functions for RC bridges,”

Structural Engineering and Mechanics, Vol. 17, No. 2, pp. 215-244.



References

85

Fajfar, P. and Fischinger, M. [1988] “N2 - A method for non-linear seismic analysis of regular

buildings,” Proceedings of the 9th World Conference in Earthquake Engineering , August 2-9,

Tokyo-Kyoto, Japan.

Fajfar, P. [1999] “Capacity spectrum method based on inelastic demand spectra,” Earthquake Engineering & Structural Dynamics, Vol. 28, No. 9, pp. 979-993.

Fajfar, P. [2000] “A nonlinear analysis method for performance based seismic design”, Earthquake

Spectra, Vol. 16, 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

1st U.S. National Conference on Earthquake Engineering , Berkley, USA.

Geol, R.K., Chopra, A.K. [1997] “Evaluation of bridge abutment capacity and stiffness during

earthquakes,” Earthquake Spectra, Vol. 13, No. 1.

Hall, J.F. [2005] “Problems encountered from the use (or misuse) of Rayleigh damping”, Earthquake

Engineering and Structural Dynamics, Vol. 35, No.5, pp. 525-545.

HAZUS [2003], Multi-hazard Loss Estimation Methodology Earthquake Model , Federal Emergency

Management Agency, Washington, D.C., USA.

Hwang, H., Liu,J.B., Chiu, Y. [2001] “Seimic fragility analysis for highway bridges,” Individual study,

Center for Earthquake Research and Information, University of Memphis, USA.

Iman, R. L., Helson, J. C. and Campbell, J. E. [1981] An Approach to Sensitivity Analysis of

Computer Models: Part I - Introduction, Input Variable Selection and Preliminary Variable

Assessment. Journal of Quality Technology, Vol.13, No.3, pp. 174-183.

Jeong, S., Elnashai, A.S. [2007] “Probabilistic fragility analysis parameterized by fundamental

response quantities,” Engineering Structures, Vol.29, No. 6, pp. 1238-1251.

Kalkan, E., Kunnath, S.K. [2006] “Adaptive modal combination procedure for nonlinear static

analysis of building structures,” Journal of Structural Engineering , Vol. 132, No. 11, pp. 1721-

1731.

Kappos, A.J., Chryssanthopoulos, M.K., Dymiotis, C. [1999] “Uncertainties analysis of strength and

ductility of confined reinforced concrete members,” Engineering Structures, Vol. 21, pp. 195-208.

Kappos, A.J., Moschonas, I., Paraskeva, T., Sextos, A. [2006] “A methodology for derivation of

seismic fragility curves for bridges with the aid of advanced analysis tools,” First European

Conference on Earthquake Engineering and Seismology, Geneva, Switzerland.

Karakostas, C., Makarios, T., Lekidis, V., Kappos, A. [2006] “Evaluation of vulnerability curves for

bridges-a case study,” First European Conference on Earthquake Engineering and Seismology,

Geneva, Switzerland.

Karim, K.R., Yamazaki, F. [2003] “a simplified method of constructing fragility curves for highway

bridges,” Earthquake Engineering and Structural Dynamics, Vol. 32, No.10, pp. 1603-1626.

Karsan, I.D, Jirsa, J.O. [1969] “Behaviour of concrete under compressive loading”, Structural

Division, ASCE, USA, 95(2).



References

86

Kibboua, A., Naili, M., Benouar, D., Kehila, F. [2011] “Analytical fragility curves for typical Algerian

reinforced concrete bridge piers,” Structural Engineering and Mechanics, Vol.39, No. 3, pp. 411-

425.

Kowaslsky, M.J. [2000] “Deformation limit states for circular reinforced concrete bridge columns,” Journal of Structural Engineering , Vol. 126, No.8, pp. 869-878.

Liao, W., Loh, C.H. [2004] “Preliminary study on the fragility curves for highway bridges in Taiwan,”

Journal of the Chinese Institute of Engineers, Vol. 27, No. 3, pp. 367-375.

Lupoi, G., Franchin, P., Lupoi, A., Pinto, P.E. [2006] “Seismic fragility analysis of structural systems,”

Journal of Engineering Mechanics, Vol. 132, No. 4, pp. 385-395.

Mander, J.B. [1999] “Fragility curves development for assessing the seismic vulnerability of highway

bridges,” Individual Study, University at Buffalo, State University of New York, USA.

McKay, M. D., Beckman, R. J. and Conover, W. J. [1979] A Comparison of Three Methods for

Selecting Values of Input Variables in the Analysis of Output from a Computer Code.

Technometrics, Vol.21, No.2, pp. 239-245.

Monteiro, R. [2011] “Probabilistic Seismic Assessment of Bridges”, Thesis, Faculty of Engineering,

University of Porto, Portugal.

Moschonas, I.F., Kappos, A.J., Panetsos, P., Papadopoulos,V., Makarios, T., Thanopoulos, P. [2009]

“Seismic fragility curves for greek bridges: methodology and case studies,” Bulletin Earthquake

Engineering , Vol. 7, pp. 439-468.

Nielson, B.G. [2005] “Analytical fragility curves for highway bridges in moderate seismic zones,”

Thesis, School of Civil and Environment Engineering, Georgia Institute of Technology, USA.

Nielson, B.G., DesRoches, R. [2007] “Seismic fragility methodology for highway bridges using a

component level approach,” Earthquake Engineering and Structural Dynamics, Vol. 26, No. 6 pp.

823-839.

Papanikolaou, V.K., Elnashai, A.S. [2005] “Evaluation of conventional and adaptive pushover

analysis I: Methodology,” Journal of Earthquake Engineering , Vol. 9, No. 6, pp. 923-941.

Pinho, R. and Antoniou, S. [2005] “A displacement-based adaptive pushover algorithm for assessment

of vertically irregular frames,” Proceedings of the 4th European Workshop on the Seismic

Behaviour of Irregular and Complex Structures, August 26-27, Thessaloniki, Greece.

Pinho, R., Casarotti, C. and Monteiro, R. [2007] “An Adaptive Capacity Spectrum Method and other

Nonlinear Static Procedures Applied to the Seismic Assessment of Bridges,” Proceedings of the 1 st

US-Italy Seismic Bridge Workshop, April 19-20, Pavia, Italy.

Pinto, P.E. [2006] “Seismic assessment of concrete bridges,” Proceeding of the 2nd

International

Congress, Naples, Italy.

Priestley, M.J.N., Seible, F., Calvi, G.M. [1996] Seismic Design and Retrofit of Bridges, Wiley-

Interscience, New York.

Priestley, M.J.N., Grant, D.N. [2005] “Viscous damping in seismic design and analysis,” Journal of

Earthquake Engineering, Vol. 9, No. 2, pp. 229-255.



References

Priestley, M. J. N., Calvi, G. M. and Kowalsky, M. J. [2007] Displacement-based Seismic Design of

Structures. IUSS Press, Pavia, Italy.

Rossetto, T., Elnashai, A. [2005] “A new analytical procedure for derivation of displacement-based

vulnerability curves for populations of RC structures,” Engineering Structures, Vol. 27, No. 3, pp.397-409.

Shinozuka, M., Feng, M.Q., Kim, H.K., Kim, S.H. [2000] “Nonlinear static procedure for fragility

curve development,” Journal of Engineering Mechanics, Vol. 126, No. 12, pp. 1287-1295.

Web Reference:

[1]OpenSees: http://opensees.berkeley.edu/

[2]PEER strong motion database: http://peer.berkeley.edu/smcat/

MSc Dissertation Zhang 2012

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