ANALYSIS OF BRIDGE PERFORMANCE UNDER THE COMBINED …

The Pennsylvania State University

The Graduate School

College of Engineering

ANALYSIS OF BRIDGE PERFORMANCE UNDER THE

COMBINED EFFECT OF EARTHQUAKE AND FLOOD-

INDUCED SCOUR

A Thesis in

Civil Engineering

by

Gautham Ganesh Prasad

© 2011 Gautham Ganesh Prasad

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

August 2011

ii

The thesis of Gautham Ganesh Prasad was reviewed and approved* by the following:

Swagata Banerjee

Assistant Professor of Civil Engineering

Thesis Adviser

Andrew Scanlon

Professor of Civil Engineering

Jeffrey A. Laman


Peggy A. Johnson


Head of the Department of Civil Engineering

*Signatures are on file in the Graduate School.

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ABSTRACT

Earthquake in the presence of flood-induced scour is a critical multihazard

scenario for bridges located in seismically-active, flood-prone regions of the United

States. Bridge scour causes loss of lateral support at bridge foundations and results in

increased seismic vulnerability of bridges. The present study evaluates the combined

effect of earthquake and flood-induced scour on the performance of four example

reinforced concrete bridges with different number of spans. For the analysis,

Sacramento County in California is considered as the study region where the annual

probabilities of occurrence of earthquakes and floods are reasonably high. The

seismic hazard of the study region is considered through a suite of ground motion

time histories which were generated for this region. Regional flood hazard is

expressed in the form of a flood hazard curve that provides the annual peak

discharges corresponding to flood events with various annual exceedance

probabilities. Bridge scour, an outcome of the flood events, are calculated and used in

the development of finite element models of these bridges. Nonlinear time-history

analyses are preformed to evaluate the seismic performance of the example bridges

having scour at bridge piers. In parallel, analyses are performed to evaluate bridge

seismic performance in the absence of scour. Fragility curves are developed to

represent the performance of example bridges under this multihazard scenario.

Comparison of bridge fragility characteristics obtained in the presence and absence of

scour presents the increased seismic vulnerability of the bridges in the presence of

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flood-induced scour. The study identifies the most sensitive range of bridge scour for

which the rate of degradation of bridge seismic performance is significant.

A sensitivity study is performed to investigate the influence of various input

parameters on the overall performance of the example bridges. For this, the 5-span

example bridge is chosen to analyze under a 100-year flood event. Risk curves of this

example bridge are developed that determine annual probability of exceeding

different levels of societal loss arising from bridge seismic damage in the presence

and absence of flood-induced scour. This societal loss is measured in terms of post-

event bridge repair/restoration cost. Results show that the seismic risk of the example

bridge may increase significantly in the presence of scour.

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TABLE OF CONTENTS

List of Figures ............................................................................................................. vii

List of Tables .................................................................................................................x

Acknowledgement ....................................................................................................... xi

Chapter 1 INTRODUCTION .........................................................................................1

1.1 Scope of present research................................................................................4

1.2 Major objectives and orientation of the thesis ................................................5

Chapter 2 LITERATURE REVIEW ..............................................................................6

2.1 Introduction ...................................................................................................6

2.2 Flood-induced bridge scour ..........................................................................6

2.3 Seismic response of bridges ........................................................................11

2.4 Combined effect of scour and earthquake ..................................................13

Chapter 3 FLOOD AND SEISMIC HAZARD IN THE STUDY REGION ...............16

3.1 The study region ........................................................................................16

3.2 Regional flood hazard ................................................................................16

3.3 Regional seismic hazard ............................................................................20

Chapter 4 ANALYSIS OF EXAMPLE BRIDGES IN THE STUDY REGION.........26

4.1 Example bridges.........................................................................................26

4.2 Foundation of example bridges ..................................................................30

4.3 Calculation of scour depths ........................................................................33

4.4 Modeling of bridges ...................................................................................36

4.5 Time History Analysis and bridge response ..............................................43

Chapter 5 SENSITIVITY ANALYSIS ........................................................................70

5.1 Variability in hazard models .....................................................................71

5.1.1 Regional flood hazard curve with 90% confidence interval .......71

5.1.2 Parameter sensitivity in the calculation of scour depth ..............78

5.1.3 Variability in regional seismic hazard ........................................82

5.2 Variability in bridge response ....................................................................82

5.3: Seismic risk curves of the example bridge ...............................................83

Chapter 6 SUMMARY AND CONCLUSION ............................................................90

6.1 Research significance .................................................................................91

6.2 Assumptions and limitations ......................................................................93

6.3 Major observations.....................................................................................94

6.4 Future study ...............................................................................................95

REFERENCES ............................................................................................................96

Appendix A: p-multiplier design curve .....................................................................102

vi

Appendix B: Post-earthquake restoration cost (CRPm) of the 5-span example bridge

with deq = 0.97 m........................................................................................................103

vii

List of Figures

Figure 1.1: Natural hazard maps of US and Puerto Rico (USGS) ; (a) seismic hazard:

red, orange and pink zones represent high probability of strong shaking and (b) flood

hazard: red zones represent the regions with high flood hazard ...................................2

Figure 2.1: Schematic diagram of local scour ...............................................................8

Figure 3.1 (a) Historic flood data and (b) flood hazard curve for Sacramento County

in CA ............................................................................................................................19

Figure 3.2: Acceleration time history for Los Angeles with a probability of

exceedance (a) 10% in 50 years (b) 2% in 50 years and (c) 50% in 50 years .............25

Figure 4.1: Schematic diagram of 2 span bridge .........................................................27




Figure 4.5: Cross-section of pier ..................................................................................28

Figure 4.6: Typical section of bridge ...........................................................................28

Figure 4.7: Elevation of abutment ...............................................................................28

Figure 4.8: Abutment plan ...........................................................................................29

Figure 4.9: Pile layout ..................................................................................................29

Figure 4.10: Soil profile ...............................................................................................34

Figure 4.11: Expected local scour at foundations of example bridges under flood

events with various frequencies ...................................................................................34

Figure 4.12: Moment-rotation behavior for bridge piers .............................................38

Figure 4.13: Schematic of soil-foundation-structure interaction model;

(a) without flood-induced scour and (b) with flood-induced scour .............................38

Figure 4.14: p-y curves developed for the equivalent pile with 0.97 m diameter .......41

viii

Figure 4.15: Example bridge models ...........................................................................42

Figure 4.16: Acceleration time history data for LA03 ground motion ........................43

Figure 4.17: Direction of earthquake loading ..............................................................44

Figure 4.18: First five mode shapes of 2 span reinforced concrete bridge

with deq = 0.97 m ........................................................................................................47

Figure 4.19: First five mode shapes of 3 span reinforced concrete bridge

with deq = 0.97 m ........................................................................................................48

Figure 4.20(a): First five mode shapes of 4 span reinforced concrete bridge

with deq = 0.97 m (No scour) ......................................................................................49

Figure 4.20 (b): First five mode shapes of 4 span reinforced concrete bridge

with deq = 0.97 m (1.50 m scour) ................................................................................50

Figure 4.21(a): First five mode shapes of 5 span reinforced concrete bridge

with deq = 0.97 m (No scour) ......................................................................................51

Figure 4.21(b): First five mode shapes of 5 span reinforced concrete bridge

with deq = 0.97 m (1.5 m scour) ..................................................................................52

Figure 4.22: Example model for determining the displacement ductility from

rotational ductility ........................................................................................................54

Figure 4.23: Time histories of displacement ductility for 2-span bridge

under a strong motion; The diameter of equivalent pile is taken as

(a) 0.97 m and (b) 4.2 m ..............................................................................................56

Figure 4.24: Change in fragility curve for 2 span bridge with

deq = 1.20 m for different scour depth for (a) minor damage state and

(b) moderate damage state (c) major damage state ......................................................58


deq = 1.20 m for different scour depth for a) minor damage state

b) moderate damage state c) major damage state………………....................... .........60



b) moderate damage state c) major damage state ........................................................61

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b) moderate damage state c) major damage state ........................................................63

Figure 4.28: Three dimensional plots for 2 span bridge ..............................................68




Figure 5.1: Comparison of flood hazard curves developed from empirical and

analytical methods .......................................................................................................75

Figure 5.2: Flood hazard curve with 90% confidence interval ....................................77

Figure 5.3: Tornado diagram developed for the 5-span example bridge .....................81

Figure 5.4: Seismic fragility curves of the 5-span example bridge with

deq = 0.97 m in presence and absence of flood-induced scour; (a) no scour,

(b) 0.56 m scour, (c) 1.22 m scour , (d) 2.85 m scour, (e) 3.08 m scour,

(f) 3.30 m scour and (g) 3.45 m scour ..........................................................................87

Figure 5.5: Risk curve of the 5-span example bridge under the combined effect

of earthquake and flood-induced scour ........................................................................89

Figure 6.1: Flowchart for seismic performance analysis of bridges located

in flood-prone regions ..................................................................................................92

Figure A1: Proposed p-multiplier design curve .........................................................102

x

List of Tables

Table 3.1: Annual peak flood discharge magnitudes for the Sacramento County.......18

Table 3.2: PGA values of LA ground motions having various hazard levels ..............21

Table 4.1: Details of example bridges .........................................................................30

Table 4.2: Calculation of scour for example bridges ...................................................35

Table 4.3: Soil profile ..................................................................................................40

Table 4.4: Fundamental time periods (in sec) ..............................................................46

Table 4.5: Median PGAs for all combinations of Ys and deq for all example bridges .....

......................................................................................................................................65

Table 5.1: Peak discharge flow calculated using analytical method for different

exceeding probability ...................................................................................................73

Table 5.2: Peak flood discharges with 5% and 95% statistical confidence ................76

Table 5.3: Calculation of scour for different flood events ...........................................83

Table A5.1: Post-earthquake restoration cost (CRPm) of the 5-span example bridge

with deq = 0.97 m at no scour condition .....................................................................103


with deq = 0.97 m at 0.56 m scour ..............................................................................105











xi

ACKNOWLEDGEMENT

This research would not have been possible without the support from many

individuals. I would like to express my sincere thanks to people have contributed to

this project in various ways.

First, I would like to thank my advisor, Dr. Swagata Banerjee for all her

comments and suggestions as I worked on this thesis. Without her support, guidance

and advice this project would not be possible. I would like to thank my thesis

committee members, Dr. Andrew Scanlon, Dr. Jeffrey Laman and Dr. Peggy Johnson

for their support and guidance.

I would also like to thank my colleagues and staffs in the Department of Civil &

Environmental Engineering for letting me use the facilities in the CEE dept,

consultations and moral support.

Last but not the least I would like to thank my family for their encouragement

and support.

1

Chapter 1: Introduction

Bridges are important components of highway and railway transportation systems.

Past experience indicates that bridges are extremely vulnerable to natural and

manmade hazards such as earthquakes, floods, high wind, blast and vehicle/vessel

impact at bridge piers. Bridge damage due to such extreme events may cause

significant disruption of the normal functionality of transportation systems, and thus

may result in major economic losses to the society. Therefore, safety and

serviceability of bridges have always been great concerns to the practice and

profession of civil engineering.

A large population of bridges (nearly 70% according to the National Bridge

Inventory, or NBI) in the United States is located in moderate to high seismically

active and flood-prone regions. Figure 1.1a presents the nationwide relative shaking

hazard map provided by the US Geological Survey (USGS), where California falls

under the category of “highly seismically active” region. In last four decades, failure

of a large number of highway bridges is observed during the 1971 San Fernando,

1989 Loma Prieta and 1994 Northridge earthquakes. These extreme natural events

significantly disrupted the normal functionality of the regional highway transportation

networks. Besides seismic events, several damaging flood events are also recorded in

this region (Figure 1.1b represents the flood hazard map of the United States, USGS).

The 1995 California flood (total casualty $1.8 million) and the 1997 Northern

California flood (total casualty $35 million) are two examples of such events that

caused notable damage in the state of California. Present state-of-the-art practice of

bridge engineering considers these extreme events as discrete events to evaluate the

2

performance of bridges. Consequently, loss estimation methodologies and risk

mitigation techniques for bridges are developed based on their failure probabilities

under discrete hazard conditions. However, these two natural hazards (i.e., earthquake

and flood) must be treated as multihazard condition for reliable evaluation of bridge

performance located in regions with high seismic and flood hazards.

(a) Seismic Hazard (b) Flood Hazard

Figure 1.1: Natural hazard maps of US and Puerto Rico (USGS); (a) seismic hazard:

red, orange and pink zones represent high probability of strong shaking and (b) flood

hazard: red zones represent the regions with high flood hazard

The present research evaluates the combined effect of earthquake and flood on

bridge performance considering that these two natural events occur successively.

Flood-induced soil erosion, commonly known as scour, causes loss of lateral support

at bridge foundations (Bennett et al., 2009). This can impose additional flexibility to

bridges which amplify the adverse effect of seismic ground motions on bridge

performance. Hence, an earthquake in the presence of flood-induced scour is a critical

multihazard for bridges located in seismically active, flood-prone regions. Although

3

the joint probability of occurrence of earthquake and flood within the service life of a

bridge is relatively small, past experience indicates that one natural event can occur

just after another (even before the aftermath of the first event is taken care of). For

example, an earthquake of magnitude 4.5 struck the state of Washington on January

30, 2009. This seismic event occurred within three weeks after the occurrence of a

major flood event in that region. Such successive occurrences of extreme events can

significantly increase structural vulnerability from that under discrete hazard events.

The importance of consideration of possible multihazard events for the reliable

performance evaluation of bridges is well understood, however, the availability of

relevant literature is extremely limited. NCHRP Report 489 (Ghosn et al., 2003)

documents reliability indices of bridges subjected to various combinations of extreme

natural hazards. Ghosn et al. (2003) assumed each extreme event to be a sequence of

independent load effects, each lasting for equal duration of time. The service life of a

bridge was also divided into several time intervals with durations equal to that of

load. Occurrence probabilities of independent natural events within each time interval

were calculated and combined to obtain joint load effects. This methodology,

however, cannot be applied for load combinations involving bridge scour. This is

because scour itself does not represent a load; rather it is a consequence of flood

hazard. Therefore, load combination, or load factor design, as proposed in NCHRP

Report 489, may not provide a reliable estimation of bridge performance under a

natural hazard in presence of flood-induced bridge scour. Rigorous numerical study is

required for this purpose.

4

1.1 Scope of the Present Research

The present study evaluates the performance of reinforced concrete bridges under

the combined effect of flood-induced scour and earthquake. Sutter County in

California, a region with high seismic and flood hazards, is chosen as the study

region. Four example reinforced concrete bridges with different lengths are

considered to examine their relative vulnerability under the combined natural hazards.

Flood hazard of the study region is expressed in the form of regional flood hazard

curve. This flood hazard curve is developed using the historic flood discharge data

reported by USGS for this region. Flood hazard with various intensities are

considered and resulting scour depths at bridge foundations are estimated. Seismic

hazard of the same region is modeled through a suite of earthquake ground motions

that were generated for Los Angeles in California. Finite element models of the

example bridges with and without flood-induced scour are developed using SAP2000

Nonlinear (Computer and Structures, Inc. 2000) and analyzed under these ground

motions. Bridge seismic performance in the presence and absence of flood-induced

scour is represented in the form of fragility curves. Change in bridge seismic fragility

characteristics with scour depth demonstrates the change in bridge vulnerability with

the combined demand of these two natural hazards.

Bridge performance under the multihazard condition may vary depending on the

variability involved in various analysis modules. Sensitivity analysis is performed to

identify major uncertain parameters to which bridge performance is greatly sensitive.

Seismic risk curves of bridges are generated as an ultimate outcome of this research.

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1.2 Major Objectives and Orientation of the Thesis

The broader objectives of this research include:

- Development of fragility curves to determine bridge failure probabilities due

to the combined effect of earthquake and flood-induced scour. These fragility

curves are a useful tool for the evaluation of risk and resilience of highway

transportation networks under similar multihazard scenario.

- Investigation of parameters sensitivity in the calculation of flood-induced

scour depth. Four input parameters (discharge rate, bed condition and angle of

attack coefficient, and effective pier width) are considered for this and their

possible variations are taken from existing knowledge-base.

- Development of risk curves to express the annual exceedance probabilities of

various levels of societal loss due to seismic damage of bridges located in

flood-prone regions.

The thesis is organized in the following five chapters: Chapter 2 focuses on the

review of existing literature on the evaluation of bridge performance under flood-

induced scour, earthquakes and the combination of these two natural hazards.

Regional seismic and flood hazard of the study region is discussed in Chapter 3.

Chapter 4 contains the description of example bridges under consideration and the

evaluation of their performance under the combined effect of earthquake and flood-

induced scour. Sensitivity analysis and the expected seismic risk of bridges are

discussed in Chapter 5. Chapter 6 summarizes the present research and presents

conclusions based on the research outcomes. Research significance and

recommendations for further studies on this topic are also presented in this chapter.

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Chapter 2: Literature Review

2.1 Introduction

Literature review for the present study is done by categorizing literatures on

bridge performance evaluation in three groups: (i) due to flood events, (ii) under

earthquake ground motions and (iii) under the combined effect of earthquake and

flood-induced scour. The following sections provide the details.

2.2 Flood-Induced Bridge Scour

The extent of flood impact on bridges is commonly measured in terms of scour

depth at bridge foundations. Scour is defined as erosion caused by fast flowing water

which results in removal of sand, earth, or silt from the bottom of the river (Liang et

al., 2009). Bridge scouring has three components (HEC 18, 2001):

1) Local scour: - Local scour occurs at bridge piers, abutments or any other

structural parts that obstruct the normal flow of water.

2) Contraction scour: - Contraction scour occurs when normal stream flow gets

contracted by external objects such as bridge piers. Such contraction reduces the

overall width of the channel and results in accelerated stream flow which causes

scour.

3) Degradation and Aggradations scour: – Degradation and aggradations scour

occurs over time due to continuous flow of water.

Contraction scour and local scour at bridge piers are the expected outcome of

accelerated stream flow due to one-time flood event. In this study, the contraction

7

scour is estimated to be less than 10% of the local scour at bridge piers for all range

of discharge values. Hence, only the pier local scour is considered here as the

immediate outcome of flood events. The guidelines given in HEC -18 (2001) is used

to calculate the pier local scour.

Schematic diagram of the local scour is shown in Figure 2.1. Numerous

researches have been performed to predict scour at bridge piers and a number of

equations have been proposed. Johnson (1995) performed a comparative study with

the scour calculation equations proposed by the Colorado State University (CSU) (as

given in HEC-18 1993), Melville and Sutherland (1988), Breusers et al. (1977), Shen

et al. (1969), Laursen and Toch (1956) and Jain and Fischer (1979). It was observed

that the equations proposed by CSU provided accurate estimates of bridge scour at

very low Froude number (~ 0.1) and hence, suggested to use to evaluate scour depth

at bridge piers. In the present study, Froude numbers calculated for various intensity

flood events fall in a range of 0.11 to 0.16. Thus, scour calculation equations given in

HEC – 18 (Richardson and Davis, 2001) are used here which are the modified version

of that originally proposed by CSU.

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Figure 2.1: Schematic diagram of local scour

(mo.water.usgs.gov/current_studies/Scour/index.htm)

According to HEC-18 (Richardson and Davis, 2001), local scour Ys is expressed

as

43.0

1

65.0

43212 Frh

aKKKhKYs

(2.1)

where h is the flow depth directly upstream to the bridge pier (in m), a is the pier

width (in m), K1, K2, K3, and K4 are correction factors representing pier nose shape,

angle of attack of flow, bed condition, and particle size of soil, respectively. The

Froude number Fr1 is defined as 5.0ghV , V and g being the mean velocity of the

flow directly upstream to the pier (in m/sec) and acceleration of gravity (9.81 m/s2),

respectively. Values of K1, K2, K3 and K4 are determined from HEC-18. V is the

velocity of flow measured at the pier location where scour is calculated and h is the

9

measured flow depth. The flow depth for a given flood discharge rate is calculated as

(Gupta, 2008):

VbhQ (2.2)

where Q is the discharge rate (m3/sec) and b is the passage width. Velocity of the

flood (V) is calculated using the following equation (Gupta, 2008):

2/1

3/2

2

1S

hb

bh

nV

(2.3)

Here n and S represents manning‟s roughness coefficient and slope of bed stream,

respectively. For a given flood event, the annual peak discharge Q is the only known

quantity. To calculate corresponding values of flow velocity V and flow depth h, the

passage width b is assumed to be equal to the total length of the bridge.

Note that the scour depth calculation discussed here provides a rapid estimation of

scour at bridge piers caused by regional flood events. For exact calculation of bridge

pier scour, separate hydraulic analysis at each bridge pier is required.

Johnson and Torrico (1994) suggested another correction factor Kw in Equation

2.1 for h/a < 0.8 in a subcritical flow and uniform noncohesive sediments with a/D50

> 50. In such cases, Kw should be multiplied with the value of scour depth calculated

using in Equation 2.1. Kw is calculated as

c

cw

VVFrah

VVFrahK

for 00.1

for 58.225.0

1

13.0

65.0

1

34.0

(2.4)

where Vc represents critical velocity which is defined as the velocity required for

initiating the motion of bed materials (Richardson and Davis, 2001). This critical

10

velocity can be calculated following the equation below where Ku is taken as 6.19 and

D represent soil partial size.

3/16/1 DhKV uc (2.5)

Bennett et al. (2009) performed analytical study to determine the behavior of a

laterally loaded pile group subjected to scour. The pile group consisted of 8 piles and

each pile in the pile group had a diameter of 0.25 m and length of 10.97 m. The pile

group was converted into group equivalent pile based on the procedure defined in

Mokwa et al. (2000). Effect of scour depth and pile head boundary condition on the

deflection profile of the pile was studied in this paper. Five different scour depths,

measured from the ground level, were considered to evaluate the effect of scour depth

on the pile system. The result showed that the deflection of pile head was

insignificant when the scour depth was less than the depth of the pile head (i.e., scour

did not reach the pile cap). Once the scour depth reached the pile head and proceeded

to further depth, a significant amount of deflection of pile was observed. Deflections

of the pile head under two fixity conditions (fixed and free) were compared. The

study found that the deflection of the pile head under free-head condition was more

than the deflection of the pile head under the fixed-head condition. Increase in scour

depth resulted in reduction of lateral bearing capacity of the pile group. Under the

fixed head case it was observed that the maximum shear force and bending moment

developed at the pile head and these maximum values increased with increase in the

scour depth. From these observations, it can be easily visualized that when the scour

is accompanied by an earthquake, the damage to the structure will be much more

11

intense. This indicates the significance of assessing bridge damageability under scour

and earthquake.

2.3 Seismic Response of Bridges

During last few decades, a number of numerical and experimental studies are

performed to simulate bridge seismic performance. It is well recognized that the

development of fragility curves is one of the most efficient techniques to express

bridge seismic vulnerability. The fragility curves represent the probability of bridge

failure in a particular damage state under certain ground motion intensity (such as

peak ground acceleration or PGA). Such curves are developed either through

empirical method (i.e., using the damage data of the bridges associated with the past

earthquake) or through analytical method (i.e., by simulating damage states based on

dynamic characteristics of bridges).

Basoz and Kiremidjian (1997) developed the fragility curves using the data from

1989 Loma Prieta earthquake and 1994 Northridge earthquake using the regression

analysis. Bridges were categorized into 11 different classes based on substructure and

superstructure type and material. Empirical fragility curves were developed for each

of the bridge class. Shinozuka et al. (2000a) developed empirical fragility curve using

the damage data from 1995 Kobe earthquake and 1994 Northridge earthquake. Along

with the empirical method, studies have been carried to develop the fragility curves

based on analytical method (Hwang et al. 2000, Mander and Basoz 1999, Shinozuka

et al. 2000b, Banerjee and Shinozuka 2008a). This is done through the numerical

simulation of bridge seismic performance using different structural analysis methods

12

such as response spectrum analysis, nonlinear static and dynamic analyses. Hwang

and Huo (1994) developed the fragility curves based on Monte Carlo simulation of

the dynamic characteristics of structures.

In most of these above literatures, fragility curves are generally expressed using a

two-parameter lognormal distribution function. The distribution parameters median

PGAm and log-standard deviation , referred to as fragility parameters, respectively

represent PGA corresponding to 50% probability of exceeding the damage state and

the dispersion of fragility curve. At a damage state k (= minor, moderate, major,

complete collapse), fragility parameters PGAmk and k can be estimated using the

maximum likelihood method. Under the lognormal assumption, the analytical form of

the fragility function F(·) for the state of damage k is given as,

k

mkikmki

PGAPGAPGAPGAF

ln,, (2.6)

PGAi represent PGA of a ground motion i. Care should be taken while developing the

fragility curves in order to make sure that the fragility curves of different damage

state do not intersect each other. This can be achieved by considering a common log-

standard deviation for all damage sates. For the further details on likelihood

method and fragility curve development, readers are referred to Banerjee and

Shinozuka (2008a).

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2.4 Combined Effect of Scour and Earthquake

While a number of research studies have been conducted to evaluate bridge

performance under earthquake and flood-induced scour considering these are discrete

natural disasters, not much attention is given to evaluate bridge response under the

combined effect of these natural hazards. In relation to this, Tsai and Chen (2006)

studied the seismic capacity of a three span reinforced concrete bridge having scoured

group piles. The length and diameter of bridge columns were 10 m and 2.2 m,

respectively. The bridge was supported on a group pile foundation consisting of nine

piles with diameter of 0.7 m and having length equal to 30 m. In numerical analysis,

soil springs were provided in lateral direction of piles to incorporate the pile soil

interaction. For scour condition, these springs were removed up to the scour depth.

Results from this numerical study showed that the scouring of pile group resulted in

lower seismic capacity of the bridge. Exposure of piles due to scour resulted in

shifting of plastic hinges from bottom of the pier to the top of the pile which resulted

in lesser lateral force resistance.

Chen (2008) performed a small-scale experiment to demonstrate the effect of

scour and earthquake on a bridge pier. The experimental set-up consisted of a flume

of 10 cm wide, 2 m long and 20 cm deep that was filled with sand. The pier model of

length 14 cm and diameter 1.27 cm made out of aluminum plate was placed in the

flume box. Two linear motors were installed in the flume for the purpose of shaking

the entire system. The test was performed for different flow rates and different

shaking frequencies. Scour depth around the piers were evaluated for concurrent

14

application of scour due to flow and earthquake and then compared with that obtained

from sequential application of these two independent events. The result showed that

the scour depth for the concurrent event case and sequential event case is different. In

general it was observed that sequential event case resulted in shallower scour depth as

compared to scour depth due to concurrent event case. Thus, this study confirmed the

higher structural vulnerability under this combined hazard scenario.

Another related study on the combined effect of earthquake and scour on bridge

performance is done by Alipour et al. (2010). In this study, three bridges, one short,

one medium and one long span, were modeled in OpenSees (2009). These bridge

models were supported by pile shaft foundations. Soil-structure interaction was

modeled using American Petroleum Institute (API) guidelines. These were

incorporated in the model using bi-linear springs along the length of the shaft.

Seismic performance of bridges under 16%, 50% and 84% probability of occurrence

of scour is evaluated. The effect of scour was modeled by removing the bilinear

springs from the pile shaft up to the scour depth range. Both pushover and time-

history analyses were performed to investigate bridge performance. For the push over

analysis, the strength of the bridge was evaluated in terms of base shear capacity.

From the study it was concluded that bridges subjected to higher scour have a lesser

base shear capacity. The response of the bridge for time history loading was evaluated

in terms of deck drift ratio and the bridge was subjected to 60 ground motions. For

the evaluation of seismic performance of bridges, fragility curves were plotted.

Ductility limits were developed for each damage state (i.e., slight, moderate,

extensive and complete collapse damage state) and was compared with bridge

15

ductility to evaluate the performance of the bridge. Fragility curves for the medium

span bridge at minor and moderate damage states were plotted for different scour

levels. In the result, change in the fragility curves was observed with the change in

scour levels. The probability of exceeding a particular damage state increased with

higher scour depth.

The above three studies show the importance of considering earthquake and

flood-induced scour for bridge performance evaluation. However, none of these

studies have strategically modeled the seismic and flood hazards for any region and

investigated the combined impact of these hazards on performance of bridges

populated in that region. However, for multihazard analysis, it is very important to

identify the region specific seismic and flood hazard levels as bridge safety depends

on maximum demands of these two natural disasters. In addition, many of the

analysis parameters and modules may be associated with uncertainties. A sensitivity

analysis is needed to identify the critical input parameters and the influence of their

variability on the seismic performance of bridges in the presence of flood-induced

scour.

16

Chapter 3: Flood and Seismic Hazard in the Study Region

3.1 The Study Region

Overlapping of seismic and flood hazard maps (shown in Figure 1.1) indicates

that California, Washington and partly Oregon in Western US and the New Madrid

Seismic Zone in Eastern US are regions with high seismic and flood hazards. In this

study, Sacramento County in California is chosen as the bridge site. Nevertheless, the

method presented in this study is transportable and can easily be applied to any other

region of interest.

3.2 Regional Flood Hazard

Regional flood hazard is generally expressed in the form of flood hazard curve

that provides probability of exceedance of annual peak discharges in the region. Such

curve can be developed through flood-frequency analysis (Gupta, 2008) performed

using (1) empirical method, (2) analytical method and (3) graphical method. In this

section, empirical method is used to develop the flood hazard curve. For the study

region, 104 data (from year 1907 to 2010) of annual peak discharge is collected from

USGS National Water Information System (USGS 2011). Table 3.1 and Figure 3.1a

display the magnitudes of these annual peak flood discharges. Plotting of these data in

a log-normal probability paper provides the probability of exceedance of different

annual peak discharges (Figure 3.1b). This represents the flood hazard curve of the

study region. Thus the relation between flood hazard level and physical measure of

flood characteristics (i.e., peak discharge rate) is established. Federal Highway

17

Administration or FHWA requires all bridges over water must be able to withstand

the scour associated with 100-year floods (i.e., flood events having probability of

exceedance once in 100 years) (Richardson and Davis, 2001). Hence, 100-year flood

events are considered in this study as extreme flood scenario. From the hazard curve

(Figure 3.1b), annual peak discharge corresponding to 100-year flood (exceedance

probability 0.01) is estimated to be equal to 2200 m3/s. Five other more-frequent

flood events with exceedance probabilities of 0.90, 0.50, 0.10, 0.05, and 0.02

(respectively for 1.1-year, 2-year, 10-year, 20-year, and 50-year flood) are also

considered. Corresponding annual peak discharges are estimated to be 60 m3/s, 305

m3/s, 900 m

3/s, 1300 m

3/s, and 1900 m

3/s. Amount of bridge scour resulting from the

above various frequency flood events are estimated and presented in Chapter 4.

18

Table 3.1: Annual peak flood discharge magnitudes for the Sacramento County

Year

Annual

peak

discharge

(m3/sec)

Year

Annual

peak

discharge

(m3/sec)

Year

Annual peak

discharge

(m3/sec)

Year

Annual

peak

discharge

(m3/sec)

1907 2009.3 1933 25.2 1959 122.8 1985 178.0

1908 62.3 1934 202.9 1960 317.0 1986 1276.3

1909 803.7 1935 568.8 1961 13.8 1987 55.2

1910 272.8 1936 515.1 1962 210.6 1988 34.0

1911 803.7 1937 433.0 1963 1115.0 1989 195.3

1912 48.1 1938 546.2 1964 113.5 1990 34.5

1913 48.1 1939 54.6 1965 1061.3 1991 188.8

1914 515.1 1940 741.5 1966 81.5 1992 151.1

1915 232.1 1941 262.6 1967 450.0 1993 270.8

1916 294.3 1942 693.4 1968 119.4 1994 30.6

1917 648.1 1943 648.1 1969 636.8 1995 690.5

1918 336.8 1944 240.3 1970 475.4 1996 300.0

1919 622.6 1945 597.1 1971 243.1 1997 2631.9

1920 104.7 1946 356.6 1972 108.7 1998 840.5

1921 583.0 1947 111.2 1973 424.5 1999 633.9

1922 300.0 1948 176.6 1974 254.1 2000 317.0

1923 328.3 1949 382.1 1975 311.3 2001 33.4

1924 31.7 1950 236.6 1976 12.3 2002 95.9

1925 673.5 1951 781.1 1977 5.7 2003 107.5

1926 109.0 1952 353.8 1978 233.5 2004 139.0

1927 322.6 1953 115.5 1979 197.8 2005 362.2

1928 648.1 1954 109.2 1980 967.9 2006 993.3

1929 89.4 1955 114.9 1981 166.7 2007 140.9

1930 172.3 1956 1188.6 1982 1047.1 2008 49.0

1931 45.8 1957 196.1 1983 738.6 2009 209.1

1932 300.0 1958 829.2 1984 560.3 2010 143.5

19

190

0

191

0

192

0

193

0

194

0

195

0

196

0

197

0

198

0

199

0

200

0

201

0

Year

1

10

100

1000

10000

Annu

al p

eak d

isch

arge

(m3/s

)

(a)

0.9

90.9

8

0.9

5

0.9

0

0.8

0

0.7

00.6

00.5

00.4

00.3

0

0.2

0

0.1

0

0.0

5

0.0

20.0

1

Probability of exceedance(Probability of annual discharge being

equal or exceeded)

1

10

100

1000

10000

Ann

ual

pea

k d

isch

arge

(m3/s

)

(b)

Figure 3.1 (a) Historic flood data and (b) flood hazard curve for Sacramento County

in CA; Dots represent 104 data and the solid curve indicates mean hazard curve

20

3.3 Regional Seismic Hazard

Seismic hazard of this region is modeled by considering sixty ground motions

with various hazard levels. These motions were originally generated by FEMA for the

area of Los Angeles in California

(http://nisee.berkeley.edu/data/strong_motion/sacsteel/ground_motions.html). These

include both recorded and synthetic motions and are categorized into three sets

having annual exceedance probabilities of 2%, 10% and 50% in 50 years. Each set

has 20 ground motions; LA01 to LA20 represent moderate motions with annual

exceedance probability of 10% in 50 years, LA21 to LA40 represent strong motions

with annual exceedance probability of 2% in 50 years, and LA41 to LA60 represent

weak motions with annual exceedance probability of 50% in 50 years. Table 3.2

represents the range of PGA values of the ground motions under each of these three

sets. Figure 3.2 shows the acceleration time history of one ground motion from each

set of strong, moderate and weak motions.

21

Table 3.2: PGA values of LA ground motions having various hazard levels

Los Angeles Ground Motions Having a Probability of Exceedance of 10% in 50 Years

SAC

Name Record

Earthquake

Magnitude

Distance

(km)

Scale

Factor

Number

of

Points

DT

(sec)

Duration

(sec)

PGA

(cm/sec2)

LA01

Imperial Valley,

1940, El Centro 6.9 10 2.01 2674 0.02 39.38 452.03

LA02

Imperial Valley,

1940, El Centro 6.9 10 2.01 2674 0.02 39.38 662.88

LA03

Imperial Valley,

1979, Array #05 6.5 4.1 1.01 3939 0.01 39.38 386.04

LA04

Imperial Valley,

1979, Array #05 6.5 4.1 1.01 3939 0.01 39.38 478.65

LA05

Imperial Valley,

1979, Array #06 6.5 1.2 0.84 3909 0.01 39.08 295.69

LA06

Imperial Valley,

1979, Array #06 6.5 1.2 0.84 3909 0.01 39.08 230.08

LA07

Landers, 1992,

Barstow 7.3 36 3.2 4000 0.02 79.98 412.98

LA08

Landers, 1992,

Barstow 7.3 36 3.2 4000 0.02 79.98 417.49

LA09

Landers, 1992,

Yermo 7.3 25 2.17 4000 0.02 79.98 509.70

LA10

Landers, 1992,

Yermo 7.3 25 2.17 4000 0.02 79.98 353.35

LA11

Loma Prieta,

1989, Gilroy 7 12 1.79 2000 0.02 39.98 652.49

LA12

Loma Prieta,

1989, Gilroy 7 12 1.79 2000 0.02 39.98 950.93

LA13

Northridge,

1994, Newhall 6.7 6.7 1.03 3000 0.02 59.98 664.93

LA14

Northridge,

1994, Newhall 6.7 6.7 1.03 3000 0.02 59.98 644.49

LA15

Northridge,

1994, Rinaldi

RS

6.7 7.5 0.79 2990 0.005 14.945 523.30

LA16

Northridge,

1994, Rinaldi

RS

6.7 7.5 0.79 2990 0.005 14.945 568.58

LA17

Northridge,

1994, Sylmar 6.7 6.4 0.99 3000 0.02 59.98 558.43

http://nisee.berkeley.edu/data/strong_motion/sacsteel/motions/images/la01.gif

















22

LA18

Northridge,

1994, Sylmar 6.7 6.4 0.99 3000 0.02 59.98 801.44

LA19

North Palm

Springs, 1986 6 6.7 2.97 3000 0.02 59.98 999.43

LA20 North Palm

Springs, 1986 6 6.7 2.97 3000 0.02 59.98 967.61

Los Angeles Ground Motions Having a Probability of Exceedence of 2% in 50 Years

SAC

Name Record

Earthquake

Magnitude

Distance

(km)

Scale

Factor

Number

of

Points

DT

(sec)

Duration

(sec)

PGA

(cm/sec2)

LA21 1995 Kobe 6.9 3.4 1.15 3000 0.02 59.98 1258.00

LA22 1995 Kobe 6.9 3.4 1.15 3000 0.02 59.98 902.75

LA23

1989 Loma

Prieta 7 3.5 0.82 2500 0.01 24.99 409.95

LA24

1989 Loma

Prieta 7 3.5 0.82 2500 0.01 24.99 463.76

LA25 1994 Northridge 6.7 7.5 1.29 2990 0.005 14.945 851.62

LA26 1994 Northridge 6.7 7.5 1.29 2990 0.005 14.945 925.29

LA27 1994 Northridge 6.7 6.4 1.61 3000 0.02 59.98 908.70

LA28 1994 Northridge 6.7 6.4 1.61 3000 0.02 59.98 1304.10

LA29 1974 Tabas 7.4 1.2 1.08 2500 0.02 49.98 793.45

LA30 1974 Tabas 7.4 1.2 1.08 2500 0.02 49.98 972.58

LA31

Elysian Park

(simulated) 7.1 17.5 1.43 3000 0.01 29.99 1271.20

LA32

Elysian Park

(simulated) 7.1 17.5 1.43 3000 0.01 29.99 1163.50

LA33

Elysian Park

(simulated) 7.1 10.7 0.97 3000 0.01 29.99 767.26

LA34

Elysian Park

(simulated) 7.1 10.7 0.97 3000 0.01 29.99 667.59

LA35

Elysian Park

(simulated) 7.1 11.2 1.1 3000 0.01 29.99 973.16

LA36

Elysian Park

(simulated) 7.1 11.2 1.1 3000 0.01 29.99 1079.30

LA37

Palos Verdes

(simulated) 7.1 1.5 0.9 3000 0.02 59.98 697.84

LA38

Palos Verdes

(simulated) 7.1 1.5 0.9 3000 0.02 59.98 761.31

LA39 Palos Verdes 7.1 1.5 0.88 3000 0.02 59.98 490.58






















23

(simulated)

LA40

Palos Verdes

(simulated) 7.1 1.5 0.88 3000 0.02 59.98 613.28

Los Angeles Ground Motions Having a Probability of Exceedence of 50% in 50 Years

SAC

Name Record

Earthquake

Magnitude

Distance

(km)

Scale

Factor

Number

of

Points

DT

(sec)

Duration

(sec)

PGA

(cm/sec2)

LA41

Coyote Lake,

1979 5.7 8.8 2.28 2686 0.01 39.38 578.34

LA42

Coyote Lake,

1979 5.7 8.8 2.28 2686 0.01 39.38 326.81

LA43

Imperial Valley,

1979 6.5 1.2 0.4 3909 0.01 39.08 140.67

LA44

Imperial Valley,

1979 6.5 1.2 0.4 3909 0.01 39.08 109.45

LA45 Kern, 1952 7.7 107 2.92 3931 0.02 78.6 141.49

LA46 Kern, 1952 7.7 107 2.92 3931 0.02 78.6 156.02

LA47 Landers, 1992 7.3 64 2.63 4000 0.02 79.98 331.22

LA48 Landers, 1992 7.3 64 2.63 4000 0.02 79.98 301.74

LA49

Morgan Hill,

1984 6.2 15 2.35 3000 0.02 59.98 312.41

LA50

Morgan Hill,

1984 6.2 15 2.35 3000 0.02 59.98 535.88

LA51

Parkfield, 1966,

Cholame 5W 6.1 3.7 1.81 2197 0.02 43.92 765.65

LA52

Parkfield, 1966,

Cholame 5W 6.1 3.7 1.81 2197 0.02 43.92 619.36

LA53

Parkfield, 1966,

Cholame 8W 6.1 8 2.92 1308 0.02 26.14 680.01

LA54

Parkfield, 1966,

Cholame 8W 6.1 8 2.92 1308 0.02 26.14 775.05

LA55

North Palm

Springs, 1986 6 9.6 2.75 3000 0.02 59.98 507.58

LA56

North Palm

Springs, 1986 6 9.6 2.75 3000 0.02 59.98 371.66

LA57

San Fernando,

1971 6.5 1 1.3 3974 0.02 79.46 248.14

LA58

San Fernando,

1971 6.5 1 1.3 3974 0.02 79.46 226.54

LA59 Whittier, 1987 6 17 3.62 2000 0.02 39.98 753.70

LA60 Whittier, 1987 6 17 3.62 2000 0.02 39.98 469.07




















http://nisee.berkeley.edu/data/strong_motion/sacsteel/motions/images/a59.gif


24

0 10 20 30 40

Time (sec)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Acc

eler

atio

n (

g)

(a) probability of exceedance: 10% in 50 years

0 10 20 30 40 50 60

Time (sec)

-1.5

-1

-0.5

0

0.5

1

1.5

Acc

eler

atio

n (

g)

(b) probability of exceedance: 2% in 50 years

25

0 10 20 30 40

Time (sec)

-0.2

-0.1

0

0.1

0.2

Acc

eler

atio

n (

g)

(c) probability of exceedance: 50% in 50 years

Figure 3.2: Acceleration time history for Los Angeles with a probability of

exceedance (a) 10% in 50 years (b) 2% in 50 years and (c) 50% in 50 years

(http://nisee.berkeley.edu/data/strong_motion/sacsteel/ground_motions.html)

26

Chapter 4: Analysis of Example Bridges in the Study Region

4.1 Example Bridges

The example bridges used in this study are adopted from the five-span (two

exterior spans @ 39.6 m and three interior spans @ 53.3 m) reinforced concrete

bridge model presented by Sultan and Kawashima (1993). The bridge was designed

following bridge design aids (Caltrans, 1988). The bridge deck is composed of 2.1 m

deep and 12.9 m wide prestressed concrete hollow box-girders. This literature does

not provide the reinforcement in the bridge girder. In fact, this is not a crucial factor

for the seismic performance analysis of bridges as bridge girders generally remains

elastic during earthquakes. Bridge piers are 19.8 m long and have circular cross

sections (2.4 m diameter). Keeping all structural conditions the same, the present

study considers three additional example bridges with 2, 3 and 4 spans by omitting

one or more interior spans from the original five-span bridge model. Figures 4.1 to

4.4 present these bridge models. The pier and girder cross-sections of these bridges

are shown in Figures 4.5 and 4.6, respectively. The elevation and the plan of

abutment are presented in the Figure 4.7 and Figure 4.8 respectively. Details of the

exterior and interior (as applicable) span lengths, cross-sectional and material

properties, and bridge foundations of the example bridges are given in Table 4.1.

Figure 4.9 represents the pile layout for the bridge. Cross section of bridge girder

generally changes with change in span numbers. However, to avoid complexity in

developing the numerical models of these bridges, the same girder cross-section is

used here for all example bridges.

27

Figure 4.1: Schematic diagram of 2 span bridge (not to scale)



Figure 4.4: Schematic diagram of 5 span bridge (After Sultan and Kawashima 1993)

(not to scale)

28

# 14,

total 44

# 6 spiral @

0.11mColumn

2.4m Ø

CL

Figure 4.5: Cross-section of pier Figure 4.6: Typical section of bridge

Figure 4.7: Elevation of abutment

Center Line of Bridge

0.53 m

6.5 m

0.53 m

6.5 m

2.13 m

0.91 m

4.58 m

5.5 m

1.14 m

1.22 m

0.84 m 3.05 m

0.61 m 0.61 m

1.83

m

29

Figure 4.8: Abutment plan

Figure 4.9: Pile layout

12.80 m

13.72 m

0.3 m

3.05 m

6.4 m 0.38 m

Ø pile

6.4 m

5.5 m

5.5 m

0.45 m

0.45 m

30

Table 4.1: Details of example bridges

Span

#

Total

span

Interior

span

Exterior

span

Deck

type

Type, height and

diameter of piers

Foundation at

pier bottom

2 79.2 m N/A 39.6 m

Hollow

box-

girder

Circular, 19.8 m

long and 2.44 m

diameter

Forty (40)

0.38 m

diameter and

18.3 m long

concrete piles

3 132.5 m 53.3 m 39.6 m

4 185.8 m 53.3 m 39.6 m

5 239.1 m 53.3 m 39.6 m

4.2 Foundation of Example Bridges

As shown in Figure 4.9, Sultan and Kawashima (1993) assumed a group of forty

0.38 m diameter, 18.3 m long piles as the foundation below each pier of the five-span

model bridge. In this study, the same foundation is used for all example bridges for

the purpose of modeling simplicity. The movement of ground due to seismic shaking

imposes lateral load to the pile foundation. For pile groups, Brown et al. (2001)

suggests the use of reduction factors (p-multipliers) that should be applied to the p-y

curve for each single pile to obtain a set of p-y curves for piles acting as a group. The

“p” value in the p-y curve for the group equivalent pile is adjusted by considering the

reduction factors which are defined in the Mokwa et al. (2000). Values of these

reduction factors depend on the location of a pile in the pile group with respect to the

point of load application. The equivalent pile “p” value is obtained by adding all the

adjusted “p” values of the each pile in the pile group. GEP “p” value is obtained using

the Equation 4.1.

31

(4.1)

where pi is the p-value for the single pile, fmi is the p-multiplier obtained from the

Figure A1 (Mokwa et al., 2000) and N is the number of piles in the group.

Randolph (2003) outlined an approach to calculate the stiffness of an axially

loaded pile group using an equivalent pile that will represent the functional behavior

of the pile group. Yin and Konagai (2001) adopted the same approach for laterally

loaded pile groups. The pile group is replaced by an equivalent pile with bending

stiffness EIeq (where E is modulus of elasticity for the pile material and Ieq is the

moment of inertial for the equivalent pile cross section) equal the bending stiffness

EIGroup (where IGroup is the moment of inertia for the entire group) of the pile group.

Additionally, pile foundation may have sway and/or rocking motions during seismic

excitations. The load-deflection characteristic and the bending stiffness of a pile

group vary under these two types of seismic motion. Accordingly, the dimension of

the equivalent pile will be different for sway and rocking motions. For sway motion,

the bending stiffness of the equivalent pile is calculated as the summation of stiffness

of all piles in the group (Yin and Konagai, 2001)

ppGroupeq EInEIEI (4.2)

where np is the number of piles in a pile group and Ip is the moment of inertia of a

single pile cross section. To calculate EIeq under a rocking motion, the relative

location of each individual pile (with respect to the center of gravity of the group) is

considered in addition to the bending stiffness EIp of an individual pile. According to

32

Yin and Konagai (2001), EIeq for a pile group subjected to rocking ground motion can

be calculated as

pn

i

pippppGroupeq xxAEIEEIEI1

2

0 (4.3)

where Ap is the cross-sectional area of a single pile, xpi and x0 are, respectively, the

coordinates of pile i and the centroid of the pile group with respect to a fixed origin.

The present study uses p-y curve of an equivalent pile which is functionally (with

respect to bending stiffness) identical to the pile group that it represents. Using

equations (4.2) and (4.3), the diameters of equivalent piles deq under sway and

rocking motions are calculated to be equal to 0.97 m and 4.2 m, respectively. These

values of deq indicate that the pile foundation has much higher rotational rigidity

against lateral rocking motion compared to that under lateral sway motion. As

identical foundation is considered for all example bridges, the above calculations of

deq remain the same for all four example bridges. The length of the equivalent pile is

considered to be the same as of the individual piles.

The value of deq will change according to the geometry and dimensions of a

bridge foundation. Therefore, to facilitate the identification of the effect of deq on

bridge dynamic characteristics in presence and absence of flood-induced scour, a

parametric study is performed as part of the present study. Five different deq values

(4.2 m, 2.4 m, 1.6 m, 1.2 m, and 0.97 m) are considered with an upper bound

calculated for lateral rocking motion and a lower bound calculated for lateral sway

motion.

33

4.3 Calculation of Scour Depths

The subsurface soil profile assumed by Sultan and Kawashima (1993) at the

model bridge site is considered for this study. Bridge scour is not a prominent

phenomenon in fine-grained soils (e.g., clays, plastic silts); however, for coarse-

grained soils (e.g., sand or silty sand) scouring is very common. Therefore, to account

for the worst possible scenario of flood-induced scouring, the present study considers

an example bridge site that consists of sand and silty sand down to infinite depth. The

soil profile considered for this study is shown in Figure 4.10. Using the Equations 2.1,

2.2 and 2.3, and the flood data provide in Figure 3.1, the scour depths for all example

bridges are calculated. As all bridge piers considered here have circular shape, the K1

is taken as 1.0. K2 is taken as 1.0 considering zero angle of attack of flow. K3 is 1.1

for clear-water scour. K4 is taken as 1.0 for the size of soil particles D50 and D95 being

less than 2 mm and 20 mm. For coarser particles (i.e., D50 > 2 mm and D95 > 20 mm),

K4 is less than 1.0 and this results in less scour. This is apparent because coarser

particles require higher flow velocity to be eroded. For this study, n = 0.08 (FEMA

2008) and S = 0.1% which represents Mannings roughness coefficient and slope of

bed stream are used in Equation 2.3 to calculate the velocity of flow and this

calculated velocity is used in Equation 2.1 to calculate scour. The calculated scour

depth values are given in Table 4.2 and plotted in Figure 4.11.

34

Figure 4.10: Soil profile

0 500 1000 1500 2000 2500

Annual peak discharge (m3/s)

0

1

2

3

4

5

Loca

l sc

our

Ys (m

)

2-span bridge

3-span bridge

4-span bridge

5-span bridge

Figure 4.11: Expected local scour at foundations of example bridges under flood

events with various frequencies

35

Table 4.2: Calculation of scour for example bridges

Flood event 1.1-year 2-year 10-year 20-year 50-year 100-

year

Exceedance

probability

0.90 0.50 0.10 0.05 0.02 0.01

Discharge rates

(m3/s)

60 305 900 1300 1900 2200

Local

scour at

bridge

piers Ys

(m)

2-span 1.07 2.83 3.69 4.02 4.40 4.55

3-span 0.89 2.51 3.28 3.59 3.94 4.08

4-span 0.79 2.30 3.02 3.31 3.64 3.77

5-span 0.73 2.16 2.84 3.11 3.42 3.55

Figure 4.11 shows that the scour depths calculated for the four example bridges

for the flood events considered in this study spread over a wide range from 0.73 m to

4.55 m. To investigate the relative impact of different combinations of earthquake and

flood-induced scour depths on the performance of all example bridges, it is important

to consider the same ground motions and scour depths for the numerical analyses.

Thus, three representative values of Ys (= 0.6 m, 1.5 m, and 3.0 m) are used for

seismic analysis of example bridges under the 60 ground motions described in

Chapter 3. Seismic performance of example bridges does not change for pier scour

more than 3.0 m (will be demonstrated further in this chapter). Thus, the scour depth

beyond 3.0 m is not considered here. In addition, analyses are performed for a „no

scour‟ condition (i.e., Ys = 0). The analysis results will demonstrate the relationship

between bridge seismic performance and increasing scour depth and will identify the

36

critical range of scour depth within which bridge performance degradation is

significant.

4.4 Modeling of Bridges

Finite element (FE) analyses are performed using SAP2000 Nonlinear (Computer

and Structures, Inc. 2000). During earthquakes, bridge girders are expected to

respond in the elastic range. These are modeled in SAP2000 by using linear beam-

column elements. These elements are aligned along the center line of bridge decks. At

the two extreme ends, bridge girders are generally supported on abutments which

provide full restraint against the vertical movement (translation) of girders at these

locations. The horizontal (longitudinal) movement of girders at these locations is

allowed up to an initially provided gap (in couple of centimeters) between girder and

abutment. To numerically simulate bridge failure, the present study assumed no

constraint at abutment locations for the longitudinal movement of girders. This

allowed the bridge models to translate freely in this direction. Hence, at abutments,

bridge girders are modeled to have unconstrained degrees of freedom along the

longitudinal translation (along the axis of the bridge). This type of modeling is also

adopted by the Washington State Department of Transportation (Kapur, 2011). The

degrees of freedom for vertical movement at these locations are taken as fully

constrained. The girders are also allowed to have in-plane rotations at abutments.

Probability of bridge failure under seismic shaking is then calculated by measuring

the horizontal displacement of the bridge at superstructure level. Failure (i.e.,

37

complete collapse) is said to occur when the maximum displacement reaches to the

ultimate limit state which is defined in the following section of this chapter.

Bridge piers are modeled as single column bents. During seismic excitation,

maximum bending moment generates at pier ends which can lead to the formation of

plastic hinges at these locations if the generated moment exceeds the plastic moment

capacity of these sections. To model such nonlinear behavior of bridge piers,

nonlinear rotational springs (Plastic Wen Nlink) are introduced in FE model at the top

and bottom of each pier where plastic hinges are likely to form. The properties of

these nonlinear links are decided based on the bi-linear moment-rotation behavior of

bridge piers (Figure 4.12; Priestley et al., 1996). This bi-linear behavior is

approximated from the original nonlinear moment-rotation relationship (also shown

in Figure 4.12) in order to input this in SAP2000. Rigid elements are assigned at

girder-pier connections to ensure full connectivity at these intersections of monolithic

concrete bridges.

The equivalent pile is assumed to be drilled in sand as the conhesionless soil is

considered in the present study. Nonlinear p-y springs are used along the length of the

equivalent pile to model the soil-foundation-structure interaction (Figure 4.13a). In

presence of scour, a part of the bridge foundation system loses lateral support from

soil. To model this loss of lateral support, springs within a length Ys from the top of

the equivalent pile are removed (Figure 4.13b). For example, if flood results in bridge

scour of 1.5 m depth, p-y links upto a depth of 1.5 m from the top of pile are removed

to model scouring. Hinge condition is assumed at the tip of the equivalent piles

(Priestley et al., 1996).

38

Figure 4.12: Moment-rotation behavior for bridge piers (1 kip-ft = 1.36 KN-m)

(a) (b)

Figure 4.13: Schematic of soil-foundation-structure interaction model; (a) without

flood-induced scour and (b) with flood-induced scour

p-y springs

p-y springs

39

Stiffness of p-y springs is calculated following API recommendations (API 2000).

The nonlinear lateral soil resistance deflection (p-y curve) behavior of sand is

expressed in the equation below (API 2000).

y

pA

HkpAp

u

u tanh (4.4)

where p and y are the lateral soil resistance and corresponding lateral deflection at a

depth H, respectively. A represents the factor to account for static or cyclic loading

condition which is given as 0.9 for cyclic loading. pu is the ultimate bearing capacity

of soil at a depth H and k is the initial modulus of subgrade reaction (1220 ton/cubic

meter). pu is depth dependent; API (2000) proposes two equations (Equations 4.5 and

4.6) to calculate pu for shallow and deep depths. The smaller of these two values is

used as the ultimate lateral resistance in the calculation of p-y curve (Equation 4.4).

HdCHCp 21u :depth shallowFor (4.5)

HdCp 3u :depth deepFor (4.6)

where coefficients C1, C2 and C3 are determined according to the friction angle of

sand (Figure 6.8.6-1 of API 2000), d is the average pile diameter measured from top

of the pile to depth H (in m) and γ is the effective soil weight (0.92 ton/cubic meter).

For all equivalent piles (with various deq), p-y curves are developed for an interval of

0.3 m along the length of the piles. The pile is driven into cohesionless soil with

properties shown in the Table 4.3. Using the properties of piles and the properties

shown in Table 4.3, load deflection curve for pile soil interaction is calculated using

the Equation 4.4.

40

Table 4.3: Soil profile

Soil type Frictional angle

(deg)

Soil layer depth

(m) Cohesion (KN/m

2)

Silty Sand 31 0 - 11.6 0

Sand 29 11.6 – 17.0 0

Sand 32 17.0 – 18.3 0

Figure 4.14 shows two example curves developed at depths of 7.3 m and 11.3 m

for 0.97 m diameter equivalent pile. As expected, increasing lateral resistance of soil

is achieved at greater depths. Such curves are assigned to the pile nodes (at 0.3 m

interval) of the FE bridge model to characterize the nonlinear behavior of soil during

seismic events. These p-y curves are incorporated in to the SAP2000 bridge model by

assigning multi-linear elastic links (having nonlinear properties). Figure 4.15 shows

the line diagram of all four example bridges modeled in SAP2000.

41

-0.10 -0.05 0.00 0.05 0.10

Deflection (m)

-800

-400

0

400

800

Lat

eral

Res

ista

nce

(K

N)

p-y curve at 11.3 m depth

p-y curve at 7.3 m depth

Figure 4.14: p-y curves developed for the equivalent pile with 0.97 m

diameter

42

(a) Two span Reinforced Concrete Bridge

(b) Three span Reinforced Concrete Bridge

(c) Four span Reinforced Concrete Bridge

(d) Five span Reinforced Concrete Bridge

Figure 4.15: Example bridge models (from SAP2000)

43

4.5 Time History Analysis and Bridge Response

Time history analysis results in dynamic response of a structure subjected to

arbitrary loading which varies with time (like earthquake). Nonlinear modal time

history analysis is performed on the bridge model and only material nonlinearity in

terms of Nlinks is considered. Modal analysis is performed using Ritz vector as it is

recommended for nonlinear time history analysis. Constant damping of 5% is

considered for all the modes. The acceleration versus time history data which is

developed by FEMA is imported into SAP2000 using the time history function

command. Figure 4.16 below shows the acceleration time history data for one of the

earthquake ground motion (LA03) that is used in the study.

Figure 4.16: Acceleration time history data for LA03 ground motion (FEMA)

Common loads such as thermal load, wind load (regular wind) and live load act

on the bridges simultaneously with extreme loads due to earthquakes and floods. This

research primarily focuses on the bridge performance under extreme loads, and

44

hence, simultaneous occurrences of other common loads and their effects on bridge

performance are not studied here.

Earthquake loading is considered to act along the longitudinal direction of bridge

as shown in the Figure 4.17. As the bridge is modeled in 2D, the earthquake loading

is considered only in longitudinal direction of bridge. No phase lag of earthquake at

various bridge supports is considered.

Figure 4.17: Direction of earthquake loading

Nonlinear time history analyses of the example bridges are performed in the

presence and absence of flood-induced scour (for Ys = 0, 0.6 m, 1.5 m, 3.0 m; deq =

0.97 m, 1.2 m, 1.6 m, 2.4 m, 4.2 m). For each combination of Ys and deq, 60 ground

motions are considered that represent the seismic hazard of the bridge site. Thus 4800

numerical simulations are performed in total for all bridges considered herein. The

fundamental time periods of these bridges are given in Table 4.4. The fundamental

time period increases with increasing bridge scour indicating higher flexibility of

bridges obtained due to the release of lateral resistance at foundations. Time-histories

of bridge response are recorded at various bridge components. To characterize the

global nature of bridge performance under the combined action of earthquake and

45

scour, the study considered bridge responses measured in terms of longitudinal

displacement (along the bridge axis) at the bridge girder level for the fragility

analysis. It is reasonable to consider that flexible bridges (due to scour at foundation)

have higher tendency to deflect laterally. During earthquake, the abutment and the

girder may deflect in the longitudinal and transverse directions. In the present study,

bridge deflection only in the longitudinal direction (i.e., along the bridge axis) is

considered as the example bridges are modeled in 2D. Axial force develops at the

interface of girder and abutment when the girder and abutment strike, which may

eventually lead to the failure of abutment backwalls. In case of out-of-phase

movement of abutment and girder, excessive separation among these two components

may occur resulting in the unseating of bridge girders. The present study assumed an

unrestrained longitudinal movement of bridge girders at abutment locations. Thus,

with this assumption, bridge failure is defined here in terms of released degree of

freedom (i.e., longitudinal deflection of girder).

The first five mode shapes for reinforced concrete bridge with deq = 0.97 m is

shown in Figure 4.18, 4.19, 4.20 and 4.21 for 2 span, 3 span, 4 span and 5 span

bridges respectively for no scour and 1.5 m scour.

46

Table 4.4: Fundamental time periods (in sec)

deq (m) Scour depths Ys (m) Scour depths Ys (m)

0 0.6 1.5 3.0 0 0.6 1.5 3.0

2-span bridge 3-span bridge

0.97 2.39 2.97 3.40 3.84 2.19 2.72 3.12 3.53

1.2 2.23 2.60 2.97 3.42 2.05 2.38 2.73 3.14

1.6 2.15 2.30 2.51 2.83 1.97 2.11 2.31 2.60

2.4 2.10 2.13 2.19 2.29 1.93 1.96 2.01 2.11

4.2 2.09 2.09 2.10 2.12 1.92 1.92 1.93 1.94

4-span bridge 5-span bridge

0.97 2.12 2.64 3.02 3.42 2.09 2.59 2.97 3.36

1.2 1.99 2.31 2.64 3.04 1.95 2.27 2.60 2.99

1.6 1.90 2.01 2.16 2.39 1.87 1.98 2.14 2.38

2.4 1.87 1.90 1.95 2.04 1.84 1.87 1.92 2.01

4.2 1.86 1.86 1.87 1.89 1.83 1.84 1.87 1.90

Following figures (Figure 4.18 to 4.21) show the first five modes of the example

bridges for 0 m scour and 1.5 m scour along with their modal periods.

47

0 m scour 1.5 m scour

Mode 1: 2.39 sec

Mode 1: 3.40 sec

Mode 2: 0.25 sec

Mode 2: 0.25 sec

Mode 3: 0.20 sec

Mode 3: 0.20 sec

Mode 4: 0.092 sec

Mode 4: 0.10 sec

Mode 5: 0.091 sec

Mode 5: 0.094 sec

Figure 4.18: First five mode shapes of 2 span reinforced concrete bridge with deq =

0.97 m

48

0 m scour 1.5 m scour

Mode 1: 2.19 sec

Mode 1: 3.12 sec

Mode 2: 0.46 sec

Mode 2: 0.46 sec

Mode 3: 0.23 sec

Mode 3: 0.23 sec

Mode 4: 0.22 sec

Mode 4: 0.22 sec

Mode 5: 0.16 sec

Mode 5: 0.16 sec

Figure 4.19: First five mode shapes of 3 span reinforced concrete bridge with deq =

0.97 m

49

Mode 1: 2.12 sec

Mode 2: 0.53 sec

Mode 3: 0.39 sec

Mode 4: 0.23 sec

Mode 5: 0.22 sec

Figure 4.20(a): First five mode shapes of 4 span reinforced concrete bridge with deq =

0.97 m (No scour)

50

Mode 1: 3.02 sec

Mode 2: 0.53 sec

Mode 3: 0.40 sec

Mode 4: 0.23 sec

Mode 5: 0.22 sec

Figure 4.20(b): First five mode shapes of 4 span reinforced concrete bridge with deq =

0.97 m (1.50 m scour)

51

Mode 1: 2.09 sec

Mode 2: 0.56 sec

Mode 3: 0.45 sec

Mode 4: 0.34 sec

Mode 5: 0.18 sec

Figure 4.21(a): First five mode shapes of 5 span reinforced concrete bridge with deq =

0.97 m (No scour)

52

Mode 1: 2.97 sec

Mode 2: 0.57 sec

Mode 3: 0.45 sec

Mode 4: 0.37 sec

Mode 5: 0.22 sec

Figure 4.21(b): First five mode shapes of 5 span reinforced concrete bridge with deq =

0.97 m (1.5 m scour)

53

Measured displacements are converted to displacement ductility (µΔ) which is

used here as signature representing bridge seismic performance. By definition,

displacement ductility is the ratio of displacement of the bridge pier to the yield

displacement (Caltrans, 2006). Displacements at top of bridge piers are found to be

the same as the displacements of their immediate nodes in girder due to the

monolithic pier-girder connections of example bridges. µΔ corresponding to the

ultimate state (i.e., complete collapse) is calculated to be equal to 5.0. This value is in

accordance with Caltrans recommendation for target displacement ductility (Caltrans,

2006). Beyond yielding and before complete collapse, three intermediate states of

bridge damage namely minor (or slight), moderate and major (or extensive) damage

(definition follows HAZUS 1999 physical descriptions of bridge seismic damage) are

considered. Banerjee and Shinozuka (2008a, b) quantified these intermediate bridge

damage states in terms of rotational ductility. In this study in order to convert the

rotational ductility values at different damage states to their corresponding

displacement ductility values a simple beam was modeled in SAP2000 (Figure 4.22).

The beam was subjected to nonlinear push over analysis at the top. The rotation of the

plastic wen link and the corresponding displacement at the top of the beam was

measured. Based on the rotational ductility in the intermediate damage states as

defined in Banerjee and Shinozuka (2008a, b), the corresponding displacement

ductility was calculated. The values of µΔ corresponding to minor, moderate and

major damage states are thus calculated to be equal to 2.25, 2.90 and 4.60,

respectively. These values represent the threshold limits (lower bound) of their

corresponding damage states. The state of bridge damage under a specific ground

54

motion is decided by comparing the calculated displacement ductility µΔ with the

threshold limits of all damage states (i.e., minor, moderate and major). Thus, bridge

damage state under a ground motion can be decided as

damageMajor ;60.4

damage Moderate ;60.490.2

damageMinor ;90.225.2

damage No ;25.2

The ultimate displacement using the above procedure is calculated to be 0.81 m.

The ultimate displacement is also calculated using the procedure provided in Priestley

et al. (1996). The ultimate displacement from Priestley et al. (1996) is calculated to

be 0.83 m. Hence, the procedure used above to calculate the displacement ductility

can be declared to be accurate.

Figure 4.22: Example model for determining the displacement ductility from

rotational ductility

Plastic wen link

Push over load Roller support

55

Figures 4.23a and 4.23b show the time histories of displacement ductility for 2-

span bridge with 0.97 m (purely sway motion) and 4.2 m diameter (purely rocking

motion) equivalent piles under one of the strong motions. As observed the bridge has

increased displacement ductility with increasing scour depth if the foundation has

purely sway motion during earthquake (Figure 4.23a), however, no such change is

observed in case of pure rocking motion (Figure 4.23b).

56

0 10 20 30 40 50

Time (sec)

-10

-5

0

5

10

Dis

pla

cem

ent

duct

ilit

y

0m scour

0.6m scour

1.5m scour

3.0m scour

(a)

0 10 20 30 40 50

Time (sec)

-10

-5

0

5

10

Dis

pla

cem

ent

duct

ilit

y

0m scour

0.6m scour

1.5m scour

3.0m scour

(b)

Figure 4.23: Time histories of displacement ductility for 2-span bridge under a strong

motion; The diameter of equivalent pile is taken as (a) 0.97 m and (b) 4.2 m

57

Figures 4.24 to 4.27 represent the fragility curves of all four example bridges at

three damage states. Fragility curves are developed based on the median PGA‟s value

presented in Table 4.5 and the log-standard deviation of 0.50. As shown in all of these

figures, fragility curves shift from right to left with increasing depth of scour. Hence

for a specific PGA value, the probability of exceeding any damage state increases

with increasing scour. This indicates higher seismic damageability of bridges in

presence of flood-induced scour.

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

3.0 m

1.5 m

0.6 m

0.0 m

(a)

58

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

3.0 m

1.5 m

0.6 m

0.0 m

(b)

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

3.0 m

1.5 m

0.6 m

0.0 m

(c)

Figure 4.24: Change in fragility curve for 2 span bridge with deq = 1.20 m for

different scour depth for (a) minor damage state and (b) moderate damage state (c)

major damage state

59

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

3.0 m

1.5 m

0.6 m

0.0 m

(a)

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

3.0 m

1.5 m

0.6 m

0.0 m

(b)

60

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

3.0 m

1.5 m

0.6 m

0.0 m

(c)


different scour depth for a) minor damage state b) moderate damage state c) major

damage state

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

3.0 m

1.5 m

0.6 m

0.0 m

(a)

61

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

3.0 m

1.5 m

0.6 m

0.0 m

(b)

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

3.0 m

1.5 m

0.6 m

0.0 m

(c)



damage state

62

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

3.0 m

1.5 m

0.6 m

0.0 m

(a)

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

3.0 m

1.5 m

0.6 m

0.0 m

(b)

63

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

3.0 m

1.5 m

0.6 m

0.0 m

(c)



damage state

Comparison of bridge fragility characteristics obtained for different combination

of deq and Ys is made in terms of median PGA (PGAm) in Figures 4.28 to 4.31.

Higher median PGA represents stronger curves (i.e., less vulnerable). This is because,

fragility curves move towards right with increase in median value and thus, result in

less probability of exceeding a specific damage state. For example, seismic fragility

curves developed for 2-span bridge at minor damage state have median PGAs equal

to 0.65g and 0.48g for no scour (solid curve with solid dots) and 3.0 m scour (solid

curve without dots), respectively (Figure 4.24a). Among these, the former is stronger

as it requires higher PGA to reach to a certain level of failure probability.

64

The horizontal axes of the three dimensional (3D) plots shown in Figures 4.28 to

4.31 represent scour depths (Ys) and diameter of equivalent pile (deq). Obtained

median PGAs for all combinations of Ys and deq are plotted along the vertical axis of

these 3D plots. These median PGAs values are represented in Table 4.5. In each of

these plots, three surfaces are shown representing three states of bridge damage. The

lowermost surface represents minor damage and the uppermost represents major

damage. These surfaces show the change in bridge fragility characteristics with scour

depth and diameter of equivalent pile. In general, seismic fragility characteristics of

example bridges degrade with increasing scour depth. The rate of degradation (i.e.,

rate of change of median PGA) is significant for lower values of deq. As observed

from Figures 4.28 to 4.31, very high rate of degradation is observed for deq less than

1.6 m when scour condition at bridge foundation changes from zero to 0.6 m. During

this phase, more than 20% reductions of median PGAs are estimated at minor damage

for all example bridges with deq = 0.97 m. For any further change in scour depth

beyond 0.6 m, degradation of bridge seismic performance is not significant and the

rate of change gradually becomes zero (corresponding to the flat regions of surfaces

in Figures 4.28 to 4.31 for sour 1.0 m). For higher deq (more than 1.6 m), bridge

seismic performance is almost independent of scour depth. This is quite expected due

to higher foundation stiffness contributed by bigger diameter piles.

Observed tends are nearly identical for all example bridges. In general, weaker

fragility curves are obtained for the 2-span bridge than that for other three bridges.

However, the difference is not significant for any damage state in particular.

65

Table 4.5: Median PGAs for all combinations of Ys and deq for all example bridges

Bridge

Type

Equivalent

pile

diameter

Damage

state

Median PGAs value

0 m

scour

0.6 m

scour

1.5 m

scour

3.0 m

scour

2 span

bridge

deq =

0.97m

Minor 0.65 0.51 0.48 0.48

Moderate 0.88 0.73 0.68 0.69

Major 1.27 1.13 1.08 1.03

deq =

1.20m

Minor 0.71 0.56 0.51 0.50

Moderate 0.88 0.78 0.73 0.69

Major 1.28 1.28 1.13 1.09

deq =

1.60m

Minor 0.73 0.67 0.61 0.51

Moderate 0.93 0.88 0.82 0.73

Major 1.36 1.28 1.28 1.22

deq =

2.40m

Minor 0.75 0.73 0.73 0.65

Moderate 0.97 0.92 0.88 0.85

Major 1.36 1.36 1.28 1.28

deq =

4.20m

Minor 0.75 0.75 0.75 0.75

Moderate 0.97 0.97 0.97 0.97

Major 1.36 1.36 1.36 1.36

3 span

bridge

deq =

0.97m

Minor 0.71 0.53 0.49 0.50

Moderate 0.91 0.75 0.72 0.69

Major 1.28 1.28 1.09 1.09

deq =

1.20m

Minor 0.73 0.65 0.52 0.50

Moderate 0.93 0.85 0.75 0.73

66

Major 1.35 1.28 1.28 1.13

deq =

1.60m

Minor 0.74 0.73 0.67 0.58

Moderate 0.93 0.90 0.89 0.78

Major 1.43 1.34 1.34 1.28

deq =

2.40m

Minor 0.76 0.74 0.73 0.71

Moderate 0.97 0.93 0.93 0.89

Major 1.43 1.43 1.43 1.35

deq =

4.20m

Minor 0.76 0.76 0.76 0.76

Moderate 0.97 0.97 0.97 0.97

Major 1.44 1.44 1.44 1.44

4 span

bridge

deq =

0.97m

Minor 0.73 0.55 0.50 0.51

Moderate 0.90 0.78 0.73 0.72

Major 1.27 1.28 1.13 1.09

deq =

1.20m

Minor 0.74 0.67 0.56 0.51

Moderate 0.97 0.85 0.78 0.75

Major 1.34 1.27 1.28 1.18

deq =

1.60m

Minor 0.77 0.73 0.71 0.67

Moderate 0.97 0.93 0.90 0.85

Major 1.34 1.27 1.21 1.21

deq =

2.40m

Minor 0.77 0.74 0.74 0.71

Moderate 0.97 0.97 0.97 0.94

Major 1.34 1.34 1.34 1.27

deq = Minor 0.77 0.77 0.77 0.74

67

4.20m Moderate 0.97 0.97 0.97 0.97

Major 1.34 1.34 1.34 1.34

5 span

bridge

deq =

0.97m

Minor 0.73 0.56 0.50 0.51

Moderate 0.90 0.78 0.71 0.72

Major 1.27 1.28 1.13 1.09

deq =

1.20m

Minor 0.74 0.71 0.56 0.50

Moderate 0.97 0.90 0.78 0.73

Major 1.34 1.27 1.22 1.18

deq =

1.60m

Minor 0.77 0.76 0.75 0.67

Moderate 0.97 0.97 0.93 0.83

Major 1.34 1.34 1.21 1.21

deq =

2.40m

Minor 0.77 0.77 0.74 0.73

Moderate 0.97 0.97 0.97 0.93

Major 1.34 1.34 1.34 1.27

deq =

4.20m

Minor 0.77 0.77 0.77 0.74

Moderate 0.97 0.97 0.97 0.97

Major 1.34 1.34 1.34 1.34

68

Figure 4.28: Three dimensional plots for 2 span bridge


69



70

Chapter 5: Sensitivity Analysis

The analysis modules and parameters may be associated with uncertainties. For

reliable performance evaluation of bridges under the combined action of earthquake

and flood-induced scour, it is therefore important to quantify these uncertainties and

estimate their influences on bridge seismic performance. Parametric uncertainty may

get introduced in the analyses from various sources such as the calculation of

discharge rate, flow depth and velocity of flood flow, occurrence rate and peak

intensity of seismic motions, geometry and material properties of bridges (Johnson

and Dock 1998, Ghosn et al. 2003, Perkins 2002). Approximate characterization of

underlying mechanism for different physical processes (e.g., scouring, ground

shaking) induces model uncertainty. Statistical uncertainty, which is due to the

limited number of available observation data, also has a high potential to introduce

significant uncertainty in the final outcome (Banerjee et al., 2009). For example,

annual peak discharge rates corresponding to certain flood hazard are estimated from

regional flood hazard curve. This curve is generated based on the measured data.

Hence, the values of annual peak discharge obtained for given flood hazard levels

may be associated with significant statistical uncertainty. For reliable estimation of

peak discharges, it is thus important to develop the confidence bounds of flood hazard

curves (Gupta, 2008).

This chapter presents a sensitivity study to identify major uncertain parameters.

For this, variability of different analysis parameters is determined based on available

knowledge.

71

5.1 Variability in Hazard Models

5.1.1 Regional Flood Hazard Curve with 90% Confidence Interval

As mentioned in Chapter 3, regional flood hazard curve can be developed through

(1) empirical method, (2) analytical method and (3) graphical method. The flood

hazard curve of the study region shown in Figure 3b is developed using empirical

method. Estimation of confidence intervals of this curve, however, requires the curve

to be developed using analytical procedure (Gupta, 2008). Therefore the same 104

data of annual peak discharges collected for the study region is used here to perform

analytical method in order to generate the mean flood hazard curve and its 90%

confidence interval.

In the analytical method, flood hazard curve is developed using a probability

distribution function. The present study used Gamma-Type distribution as this

distribution has been adopted by US Federal agencies for flood analysis (Gupta,

2008). Other distributions such as Gaussian, Lognormal and Gumbel can also be used

for this purpose. The annual peak discharge (Q) of a specified flood event is

expressed in terms of the mean ( X ) and standard deviation () of logarithmic values

of sample annual peak discharges ( Q̂ ) collected over years for a specific watershed.

The relations are given as:

KXQ ln (5.1)

QEX ˆln (5.2)

72

22

ˆln XQE

(5.3)

where K, known as the frequency factor, represents the property of the probability

distribution under consideration at specified annual occurrence probability (or return

period T) of flood events. For Gamma-Type distribution, values of K can be obtained

using the coefficient of skewness (g) of sample annual peak discharges ( Q̂ ). For a

sample of size n, g is obtained as

3

1

3

21

ˆln

nn

XQn

g

n

i

i

(5.4)

For the present set of 104 data of annual peak flood discharge Q̂ collected for the

study region (Table 3.1), gX and, , are calculated to be equal to 3.16 m3/sec, 0.50

and –0.65 respectively. Using these statistical parameters, values of K for various

flood occurrence probabilities are estimated from Table 8.7 (IACWD, 1982) of Gupta

(2008) and presented in Table 5.1. The table also provides the values of Q obtained

for these flood events (Equation 5.1). Developed flood hazard curve (Q vs. annual

exceedance probability of flood events) is presented in Figure 5.1. The same obtained

from empirical method (Figure 3b) is also plotted in the same figure to make a

comparison between these two curves. The observed difference is purely due to

different procedures undertaken in empirical and analytical methods to establish the

flood discharge-frequency relationship.

73

Table 5.1: Peak discharge flow calculated using Analytical method for different

exceeding probability

Probability

of

exceedance

Frequency

factor (K)

Logarithimic

discharge

(ft3/sec)

Probability

of

exceedance

Frequency

factor (K)

Logarithimic

discharge

(ft3/sec)

0.010 1.843 4.80 0.505 0.090 3.96

0.019 1.707 4.76 0.514 0.065 3.95

0.029 1.609 4.72 0.524 0.035 3.93

0.038 1.526 4.68 0.533 0.008 3.92

0.048 1.466 4.65 0.543 -0.022 3.91

0.057 1.418 4.63 0.552 -0.049 3.89

0.067 1.366 4.60 0.562 -0.079 3.88

0.076 1.318 4.58 0.571 -0.106 3.86

0.086 1.265 4.55 0.581 -0.136 3.85

0.095 1.218 4.53 0.590 -0.163 3.83

0.105 1.175 4.50 0.600 -0.194 3.82

0.114 1.144 4.49 0.610 -0.224 3.80

0.124 1.111 4.47 0.619 -0.251 3.79

0.133 1.081 4.46 0.629 -0.281 3.78

0.143 1.048 4.44 0.638 -0.308 3.76

0.152 1.017 4.42 0.648 -0.338 3.75

0.162 0.984 4.41 0.657 -0.365 3.73

0.171 0.954 4.39 0.667 -0.395 3.72

0.181 0.920 4.38 0.676 -0.422 3.71

0.190 0.890 4.36 0.686 -0.452 3.69

0.200 0.857 4.34 0.695 -0.479 3.68

0.210 0.832 4.33 0.705 -0.509 3.66

0.219 0.809 4.32 0.714 -0.536 3.65

0.229 0.784 4.31 0.724 -0.566 3.63

0.238 0.762 4.30 0.733 -0.594 3.62

0.248 0.737 4.28 0.743 -0.624 3.60

0.257 0.714 4.27 0.752 -0.651 3.59

0.267 0.689 4.26 0.762 -0.681 3.58

0.276 0.667 4.25 0.771 -0.708 3.56

0.286 0.642 4.24 0.781 -0.738 3.55

0.295 0.619 4.23 0.790 -0.765 3.53

0.305 0.594 4.21 0.800 -0.795 3.52

0.314 0.572 4.20 0.810 -0.900 3.47

74

Probability

of

exceedance

Frequency

factor (K)

Logarithimic

discharge

(ft3/sec)

Probability

of

exceedance

Frequency

factor (K)

Logarithimic

discharge

(ft3/sec)

0.324 0.547 4.19 0.819 -0.994 3.42

0.343 0.499 4.17 0.838 -1.190 3.32

0.352 0.477 4.15 0.848 -1.299 3.27

0.362 0.452 4.14 0.857 -1.393 3.22

0.371 0.430 4.13 0.867 -1.498 3.17

0.381 0.405 4.12 0.876 -1.593 3.12

0.390 0.382 4.11 0.886 -1.698 3.07

0.400 0.357 4.09 0.895 -1.792 3.02

0.410 0.332 4.08 0.905 -1.897 2.97

0.419 0.310 4.07 0.914 -1.991 2.92

0.429 0.285 4.06 0.924 -2.100 2.87

0.438 0.262 4.05 0.933 -2.191 2.82

0.448 0.237 4.03 0.943 -2.300 2.77

0.457 0.215 4.02 0.952 -2.390 2.72

0.467 0.190 4.01 0.962 -2.495 2.67

0.476 0.167 4.00 0.971 -2.590 2.62

0.486 0.142 3.99 0.981 -2.695 2.57

0.495 0.120 3.98 0.990 -2.789 2.52

75

0.9

9

0.9

8

0.9

5

0.9

0

0.8

0

0.7

0

0.6

0

0.5

0

0.4

0

0.3

0

0.2

0

0.1

0

0.0

5

0.0

2

0.0

1

Probability of exceedance(Probability of annual discharge being equal or exceeded)

1

10

100

1000

10000

An

nu

al p

eak d

isch

arge

(m3/s

)

Analytical curve

Empirical curve

Figure 5.1: Comparison of flood hazard curves developed from empirical and

analytical methods

76

Developed flood hazard curve through analytical method (Figure 5.1) is

associated with 50% statistical confidence. It provides the most expected (or mean)

estimates of annual peak discharges corresponding to various flood hazard levels. To

develop the 90% confidence interval of this mean hazard curve, error limits on both

sides of the mean estimate for various probability of exceedance are estimated

(Gupta, 2008). Table 5.2 represents these error limits for 5% and 95% statistical

confidence and the corresponding values of peak discharge are calculated. Figure 5.2

represents flood hazard curves of the study region with 5%, 50% and 95% statistical

confidence. Lower statistical confidence indicates higher exceedance probability.

Thus, for a particular flood hazard, the annual peak discharges obtained for 5%

statistical confidence is always higher than that obtained for other two confidence

levels.

Table 5.2: Peak flood discharges with 5% and 95% statistical confidence

Probability

of

exceedance

Error limit Error limit × Standard

deviation ()

Qln = Discharge

(ft3/sec)

5%

confidence

95%

confidence

5%

confidence

95%

confidence

5%

confidence

95%

confidence

A B1 B2 C1 C2

D1

(Table 5.1 +

C1)

D2

(Table 5.1

– C2)

0.01 0.36 0.29 0.18 0.145 4.98 4.65

0.10 0.25 0.21 0.125 0.105 4.64 4.41

0.50 0.17 0.17 0.085 0.085 4.05 3.88

0.90 0.21 0.25 0.105 0.125 3.09 2.86

0.99 0.29 0.36 0.145 0.18 2.66 2.34

77

0.9

9

0.9

8

0.9

5

0.9

0

0.8

0

0.7

0

0.6

0

0.5

0

0.4

0

0.3

0

0.2

0

0.1

0

0.0

5

0.0

2

0.0

1

Probability of exceedance(Probability of annual discharge being equal or exceeded)

1

10

100

1000

10000

An

nu

al p

eak d

isch

arge

(m3/s

)

5% statistical confidence



Figure 5.2: Flood hazard curve with 90% confidence interval

78

5.1.2 Parameter Sensitivity in the Calculation of Scour Depth

Depending on expected scour depth from flood events, bridge seismic response

may change to a great extent (Figures 4.28 to 4.31). Parameters involved in the scour

depth calculation have high variability which introduces variability in calculated

scour depth. To estimate the level of variability associated with calculated scour

depths and to investigate the sensitivity of different input parameters, a sensitivity

study is performed here. This is done by considering the 5-span example bridge and a

100-year flood event. This particular bridge is chosen as it is directly adopted from

the literature without any alteration. Based on the bridge response observed in

Chapter 4, it is expected that other example bridges will have the similar trend as of

the 5-span bridge. Sensitivity of four input parameters (discharge rate, coefficient for

angle of attack of flow, and bed condition coefficient and effective pier width) are

studied. Variability of these parameters is discussed in the following section.

Annual peak discharge (Q): Uncertainties associated with Q for any particular

flood hazard level can be estimated from the flood hazard curve with 90% confidence

interval (Figure 5.2). This may have significant impact on the expected bridge scour

depth. To demonstrate the impact, present study considered 100-year flood event.

Values of Q corresponding to 5%, 50% and 95% confidence levels are estimated to

be equal to 2720 m3/sec, 1960 m

3/sec and 1280 m

3/sec, respectively. Respective

values of flow velocity and flow depth are calculated using Eqs. 2.2 and 2.3. These

two variables (i.e., flow velocity and flow depth) are considered here as independent

variables although, in reality, some correlation may exist among them. The

79

randomness of these two variables is assumed to be the sole result of the randomness

observed in the annual peak discharge.

Correction factors for HEC equation (K): This study considered K1 and K4 as

deterministic variables and K2 and K3 as random variables. The correction factor K1

corresponds to the shape of the bridge piers which are considered to be circular.

Therefore K1 has no variation. K4 corresponds to the size of subsurface soil particles

and remains constant (equal to 1.0) for D50 < 2mm and D95 < 20mm. Any variation in

soil particle size distribution assumed not to alter this criterion. K2 and K3 correspond

to the angle of attack and bed condition, respectively. During flooding, the angle of

attack may vary depending on the direction of flood water to the bridge site.

Therefore, some variability may be present in K2. This variation is taken here

following the recommendation of Johnson and Dock (1998). K3 depends on the bed

condition. Many a times it is very difficult to accurately predict the type of bed

condition, particularly during flood events. This prediction uncertainty is accounted

here by considering a variation in K3 (Johnson and Dock, 1998).

As presented in Johnson and Dock (1998), K2 is assumed to follow a normal

distribution with mean and coefficient of variation equal to 1.00 and 0.05. Similarly,

K3 is considered as a random variable having a mean equal to 1.10 with 5% variation

(Johnson and Dock, 1998). Therefore the values of mean ± 2(standard deviation) are

calculated to be 0.9 and 1.10 for K2 and 1.21 and 0.99 for K3. These values are

considered in the sensitivity analysis.

80

Effective pier width (a): For analysis purpose, it is assumed that diameter of

bridge piers has 10% variation. Hence the values of mean ± (standard deviation) of

„a‟ becomes 2.16 m and 2.64 m, where the mean value is 2.4 m.

Other input parameters for the calculation of scour depth (Chapter 2) are

considered to be deterministic assuming these parameters can be measured with

relatively high certainty. Values of these deterministic parameters are kept unaltered

from the previous analysis presented in Chapter 4. In the sensitivity study, all

uncertain parameters are first kept at their mean values and the scour of the 5-span

example bridge is calculated. This scour depth is referred to as the most expected

scour depth and calculated to be 3.44 m for a 100-year flood event. Following this,

value of one uncertain parameter is changed (with the assigned variation) at a time

keeping all other parameters at their respective mean values. Scour depths are

calculated for each case and are plotted in Figure 5.3. This figure, known as Tornado

diagram, shows the parameter sensitivity in the bridge scour calculation. The result

indicates that the calculated scour depth is most sensitive to angle of attack of flow

(K2) and bed condition coefficient (K3).

81

Figure 5.3: Tornado diagram developed for the 5-span example bridge

The scour depth is directly proportional (has linear relation) to coefficient of angle

of attack of flow (K2) and bed condition coefficient (K3) (Equation 3.1). Thus, scour

depth increases with increase in K2 and K3.

Due to the variability in annual peak discharge (Q) obtained for 5% and 95%

confidence levels, almost the same amount of change in scour is observed on both

sides of the most expected scour depth.

Increase in effective width of bridge piers (a) results in increase in scour depth

due to the fact that more flow is obstructed by a bigger pier. Note that this analysis

considered all bridge piers have the same diameter irrespective of the variability

considered in its value.

82

5.1.3 Variability in Regional Seismic Hazard

Three different seismic hazard levels in the study region are considered (Chapter

3). These include seismic events with annual exceedance probabilities of 2%, 10%

and 50% in 50 years. Accordingly example bridges have a wide variability in seismic

response under these motions which have demonstrated through fragility analysis in

Chapter 4. It should be noted that the uncertainties associated with primary seismic

parameters (such as magnitude, epicenter, and occurrence rate) and ground motion

distribution (i.e., attenuation function) are not considered here. This is because not

only these parameters are uncertain, but also many other source mechanism

parameters are not explicitly included in the analysis. This requires extensive further

study.

5.2 Variability in Bridge Response

To analyze the 5-span example bridge for 1.1, 2, 10, 20, 50 and 100 years flood

events, corresponding annual peak discharges at 5%, 50% and 95% statistical

confidence levels are estimated from Figure 5.2. Scour calculated for each of these

flood events are presented in Table 5.3. Result depicts that for any particular level of

flood hazard, calculated scour depths for 5% and 95% confidence levels vary within

10% of that obtained for 50% confidence level. Such a small variation of scour depth

will not produce any notable variation of bridge seismic response. Hence, scour depth

corresponding to the annual peak discharge with 50% statistical confidence is used in

the following part of this study. The same has been done in Chapter 4 as well.

83

Table 5.3: Calculation of scour for different flood events

Statistical

confidence

of annual

peak

discharge

1.1 year

flood

event

2 years

flood

event

10 years

flood

event

20 years

flood

event

50 years

flood

event

100 years

flood

event

Scour depth (m)

5 % 0.6 1.32 3.05 3.29 3.61 3.74

50 % 0.56 1.22 2.85 3.08 3.30 3.45

95 % 0.50 1.14 2.68 2.79 3.03 3.11

5.3 Seismic Risk Curves of the Example Bridge

Risk curves of the 5-span example bridge are developed due to the regional

seismic hazard in the presence of flood-induced scour. These curves represent the

annual probability of exceeding different levels of societal loss arising from system

degradation due to regional hazard events (Banerjee et al., 2009). In the present

study, the societal loss is measured in terms of post-event bridge repair or restoration

cost (Zhou et al., 2010). Losses due to network downtime during bridge restoration

are not considered as a part of the loss estimation.

To estimate the risk of example bridges, the 5-span example bridge is analyzed

under 60 LA ground motions considering that it was pre-exposed to 1.1, 2, 10, 20, 50

and 100-year flood events. The mean expected scour depths of this bridge for these

scenario floods are calculated to be 0.56 m, 1.22 m, 2.85 m, 3.08 m, 3.30 m and 3.45

m, respectively. In addition, no scour condition is also considered. Figures 5.4 show

the fragility curves of the 5-span bridge with deq = 0.97 m at minor, moderate, major

84

damage and complete collapse states. As expected, fragility curves become weak with

increase in pier scour up to a certain depth. These curves are used further in the

development of risk curves of the example bridge.

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0P

robab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

teMinor

Moderate

Major

Collapse

(a)

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

Minor

Moderate

Major

Collapse

(b)

85

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

Minor

Moderate

Major

Collapse

(c)

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

Minor

Moderate

Major

Collapse

(d)

86

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

Minor

Moderate

Major

Collapse

(e)

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

Minor

Moderate

Major

Collapse

(f)

87

0.0 0.2 0.4 0.6 0.8 1.0

PGA (g)

0.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y o

f E

xce

edin

g a

Dam

age

Sta

te

Minor

Moderate

Major

Collapse

(g)

Figure 5.4: Seismic fragility curves of the 5-span example bridge with deq = 0.97m

in presence and absence of flood-induced scour; (a) no scour, (b) 0.56m scour, (c)

1.22m scour , (d) 2.85m scour, (e) 3.08m scour, (f) 3.30m scour and (g) 3.45m scour

Loss due to bridge damage under the combined effect of earthquake and flood-

induced scour is calculated according to the extent of damage that the bridge suffers

from these natural hazards. Post-event repair cost of a bridge is taken as proportional

to its replacement value (Zhou et al., 2010). This proportionality factor, known as the

damage ratio rk (HAZUS; FEMA 1999), depends on the state of bridge damage k.

Values of rk for minor, moderate, major damage and complete collapse state are 0.03,

0.08, 0.25 and 0.40 respectively, as given in HAZUS (FEMA 1999). The total

replacement cost (C) is estimated by multiplying unit deck area replacement cost with

the total area of a bridge deck. Caltrans recommended that the unit area replacement

88

cost can be taken as $1292/m2. Therefore, the total bridge restoration cost (CRm) after

an earthquake event m can be written as (Zhou et al., 2010)

4

1

k

kmRm rCkDSpC (5.5)

where k represents bridge damage states (= 1, 2, 3 and 4 respectively for minor,

moderate, major damage and collapse state). pm (DS = k) is the probability that the

example bridge can sustain damage state k during the seismic event m. This value

under different ground motions is obtained from fragility curves shown in Figures

5.4. Table A5.1 in appendix displays the calculation of bridge restoration cost (CRm)

for the example 5-span bridge with deq = 0.97 m under 60 LA motions. The first two

columns of the table represent the earthquakes and their annual probabilities of

occurrence. Following four columns represents the probability of bridge damage in

different damage states. For example, at no scour condition the bridge has 0.1993,

0.1033, 0.0255 and 0.0138 probability of exceeding minor, moderate, major damage

and complete collapse, respectively under LA60. The replacement cost C is

calculated by multiplying the deck area 3099 m2 (which is the product of width and

length of the deck) with unit area replacement cost $1292/m2. Thus, CRm for LA60 is

computed to be [(0.1993 × 0.03 × 4003500) + (0.1033 × 0.08 × 4003500) + (0.0255 ×

0.25 × 4003500) + (0.0138 × 0.40 × 4003500)] = $104643.

Figures 5.5 show the risk curve of the bridge under the combined effect of

regional seismicity and flood-induced scour resulted from various frequency flood

events. The seismic risk of the example bridge increases with increase in scour depth.

89

However, the rate of increase in bridge seismic risk decreases for higher values of

scour depths.

0.0 0.5 1.0 1.5 2.0 2.5

Restoration Cost (Million Dollars)

0

0.1

0.2

0.3

0.4

Ann

ual

Pro

bab

ilit

y o

f E

xce

edan

ce 3.45 m Scour

3.30 m Scour

3.08 m Scour

2.85 m Scour

1.22 m Scour

0.56 m Scour

0.0 m Scour

Figure 5.5: Risk curve of the 5-span example bridge under the combined effect of

earthquake and flood-induced scour

90

Chapter 6: Summary and Conclusions

Seismic performance of four example reinforced concrete bridges in the presence

and absence of flood-induced scour is studied here. Bridge scour causes the loss of

lateral support at bridge foundations, and thus, the adverse effects of earthquakes on

bridges get enhanced. In this study, Sacramento County in California is considered as

the bridge site which is a seismically-active, flood-prone region. Seismic and flood

hazards of the study region are evaluated and bridge scour resulting from various

frequency flood events are calculated. Finite element models of the example bridges

with a varying depth of scour (zero to 3 m) are developed. The pile foundations below

bridge piers are modeled as equivalent single piles which represent the same bending

stiffness values as of the entire pile foundations under lateral movement. A

parametric study is performed by considering different diameters of equivalent pile.

The pile-soil interaction was included in the model using multi-linear elastic links,

known as p-y springs. These p-y springs are assigned at an interval of 0.3 m along the

entire length of the equivalent pile. In the presence of flood-induced scour, these p-y

springs are removed up to a depth of bridge scour to model the loss of lateral support

provided by subsurface soil.

Modal analysis of these bridges shows the change in fundamental dynamic

properties (i.e., fundamental time period) due to the presence and absence of flood-

induced scour at bridge foundations. Nonlinear time history analyses are performed

under 60 LA motions that are considered to represent seismicity of the study region.

The bridge response is measured in terms of displacement at the top of pier, which is

91

further converted to displacement ductility. Damage states of the example bridges

under seismic ground motions are identified by comparing the calculated values of

displacement ductility with its threshold values for these damage states. From the

bridge damage information, fragility curves are generated which represent the

probability of bridge failure at a particular damage state under certain intensity of

ground motion. These curves are the useful tool for the seismic risk evaluation of

bridges under similar multihazard scenarios. This study is further extended to

measure the variability of different uncertain input parameters and their sensitivity on

the calculation of bridge scour.

6.1 Research Significance

From the observed result, it can be stated that in the presence of flood-induced

scour, an earthquake is capable of producing more damaging effect than that is when

there is no scour. Bridge response under a low intensity earthquake may go to the

nonlinear range, which could otherwise be linear elastic, due to pre-existing flood

consequence. On the other hand, a low intensity (i.e., more frequent) flood event that

could result in very small amount of bridge scour may cause significant degradation

of seismic performance of bridges. Hence in seismic design of bridges located in

moderate to high seismic zones, it is important to consider the effect of flood-induced

bridge scour.

The procedure discussed here is transportable and can easily be applied to any

other bridge type and region of interest. A general flowchart (Figure 6.1) is presented

92

here that can be followed to evaluate the seismic performance of bridges located in


Figure 6.1 Flowchart for seismic performance analysis of bridges located in flood-

prone regions

Bridge model

Flood hazard Bridge scour

Bridge model

with scour Seismic hazard

Bridge model

without scour

Bridge seismic performance

in absence of scour

Bridge seismic performance

in presence of scour

Fragility curves

93

6.2 Assumptions and Limitations

Key assumptions and limitations that are applied to the present study include:

1) In the calculation scour depths at bridge piers, width of the flood plain is

assumed to be equal to the length of the example bridges. This assumption is

made to estimate the flow velocity and the flow depth from the peak

discharge, which is the only known quantity for a given flood event.

2) For a given flood discharge, the same amount of local scour around all piers

of a bridge is considered. In reality, various scour depths are generally

observed at different bridge piers based on the local hydraulic characteristics

around piers.

3) Spatial variation of earthquake ground motion is not considered here.

Therefore, ground motions do not have any phase lag between any two bridge

supports.

4) The pile is assumed to be drilled in cohesionless type of soil. Pile tips are

assumed to be drilled into hard rock.

5) Abutment is not modeled and the end of the bridge girder is allowed to

translate without any constraint in the longitudinal direction (i.e., along bridge

axis).

94

6.3 Major Observations

Based on the results obtained in this study following conclusions can be made.

1) Seismic damageability of bridges increases with the increase in bridge scour.

2) Even a low intensity (which is more frequent) flood event that causes very

small bridge scour may significantly degrade the seismic performance of

bridges.

3) Diameter of equivalent pile (deq) has significant influence on the performance

of example bridges under the combined effect of earthquake and flood-

induced scour.

4) The observed difference in the fragility characteristics of four example bridges

is not significant for any damage state and for any combination of scour depth

and deq. This suggests that the number of bridge span does not have any major

influence on their seismic performance in the presence of flood-induced scour.

5) The evaluation of seismic risk of bridges located in flood-prone regions must

consider the combined effect of regional seismic and flood hazards on bridge

performance.

6) Bridge supported on caisson type of foundation (deq > pier diameter)

performed better under combined action of scour and earthquake.

Note that bridge performance observed in this paper may vary depending on

bridge type, subsurface conditions and hazard characteristics of a region. For bridges

having statistically identical structural attributes and configurations as of the example

bridges located in similar sites, developed fragility curves can be used to describe

their damageability under the same regional multihazard condition.

95

6.4 Future Study

Future study in this topic should be focused on

1) Development of three dimensional (3D) bridge models and detailed modeling

of abutment to identify all possible failure modes that have potential to govern

global nature of bridge failure under seismic events in the presence of flood-

induced scour.

2) Investigation of the effectiveness of currently existing seismic retrofit

techniques for bridges under the similar multihazard.

3) Development of design guidelines for bridges located in seismically-active,


4) Evaluation of bridge performance located in other seismically-active flood-

prone regions. This will attribute to the development of a comprehensive

knowledge-base on multihazard bridge performance evaluation.

96

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102

Appendix A: p-multiplier design curve

Figure A1: Proposed p-multiplier design curve (Mokwa et al., 2000)

103

Appendix B: Post-earthquake restoration cost (CRPm) of the 5-span example

bridge with deq = 0.97m


with deq = 0.97m at no scour condition

EQ

(PGA)

Annual

Prob of

Occur

pm

(DS=1)

pm

(DS=2)

pm

(DS=3)

pm

(DS=4)

C

(in $)

CRm

Sum

(in $)

1 0.0021 0.1793 0.0906 0.0214 0.0114 4003500 90227

2 0.0021 0.4394 0.2839 0.1038 0.0654 4003500 352324

3 0.0021 0.1086 0.0492 0.0096 0.0048 4003500 46096

4 0.0021 0.2108 0.1108 0.028 0.0153 4003500 113331

5 0.0021 0.0386 0.0144 0.002 0.0009 4003500 12691

6 0.0021 0.0116 0.0036 0.0004 0.0001 4003500 3107

7 0.0021 0.1359 0.0646 0.0137 0.007 4003500 61934

8 0.0021 0.1407 0.0673 0.0145 0.0074 4003500 64817

9 0.0021 0.2489 0.1364 0.0371 0.0208 4003500 144022

10 0.0021 0.0792 0.0337 0.0059 0.0028 4003500 30695

11 0.0021 0.427 0.2733 0.0983 0.0614 4003500 335529

12 0.0021 0.7154 0.5598 0.2952 0.2149 4003500 904815

13 0.0021 0.4418 0.286 0.105 0.0662 4003500 355767

14 0.0021 0.4173 0.2652 0.0941 0.0585 4003500 322922

15 0.0021 0.2659 0.1482 0.0416 0.0236 4003500 158831

16 0.0021 0.323 0.1899 0.0586 0.0345 4003500 213515

17 0.0021 0.3102 0.1803 0.0545 0.0319 4003500 200635

18 0.0021 0.5898 0.424 0.1893 0.1289 4003500 602523

19 0.0021 0.7482 0.5987 0.3304 0.2451 4003500 1004806

20 0.0021 0.7271 0.5735 0.3073 0.2252 4003500 939213

21 0.000404 0.8705 0.7612 0.5085 0.4091 4003500 1512426

22 0.000404 0.6791 0.5185 0.2604 0.1858 4003500 805796

23 0.000404 0.1327 0.0627 0.0132 0.0067 4003500 59960

24 0.000404 0.193 0.0993 0.0242 0.013 4003500 100023

25 0.000404 0.6363 0.4721 0.224 0.1562 4003500 701962

26 0.000404 0.6966 0.5382 0.2766 0.1993 4003500 852041

27 0.000404 0.6838 0.5238 0.2646 0.1893 4003500 817867

28 0.000404 0.8851 0.7829 0.5372 0.4373 4003500 1595014

29 0.000404 0.582 0.4162 0.184 0.1247 4003500 587057

104

30 0.000404 0.7305 0.5775 0.3109 0.2283 4003500 949470

31 0.000404 0.8749 0.7676 0.5169 0.4172 4003500 1536383

32 0.000404 0.8346 0.7102 0.4464 0.3497 4003500 1334503

33 0.000404 0.5557 0.3902 0.1666 0.1115 4003500 537017

34 0.000404 0.445 0.2888 0.1064 0.0672 4003500 360051

35 0.000404 0.7309 0.578 0.3113 0.2287 4003500 950719

36 0.000404 0.7946 0.6568 0.3878 0.2959 4003500 1167789

37 0.000404 0.4802 0.3197 0.1236 0.0795 4003500 411087

38 0.000404 0.5495 0.3843 0.1628 0.1085 4003500 525776

39 0.000404 0.2253 0.1204 0.0313 0.0173 4003500 124653

40 0.000404 0.379 0.2337 0.0785 0.0478 4003500 275485

41 0.0139 0.3353 0.1993 0.0627 0.0372 4003500 226430

42 0.0139 0.0586 0.0235 0.0037 0.0017 4003500 20990

43 0.0139 0.0006 0.0001 0 0 4003500 104

44 0.0139 0.0001 0 0 0 4003500 12

45 0.0139 0.0006 0.0001 0 0 4003500 104

46 0.0139 0.0012 0.0003 0 0 4003500 240

47 0.0139 0.0618 0.0251 0.0041 0.0019 4003500 22608

48 0.0139 0.0421 0.016 0.0023 0.001 4003500 14084

49 0.0139 0.0488 0.019 0.0029 0.0013 4003500 16931

50 0.0139 0.2817 0.1595 0.046 0.0264 4003500 173235

51 0.0139 0.554 0.3886 0.1656 0.1107 4003500 534019

52 0.0139 0.3866 0.2398 0.0814 0.0498 4003500 284457

53 0.0139 0.4596 0.3015 0.1133 0.0721 4003500 380625

54 0.0139 0.5636 0.398 0.1717 0.1154 4003500 551814

55 0.0139 0.2462 0.1346 0.0364 0.0204 4003500 141780

56 0.0139 0.0951 0.042 0.0078 0.0038 4003500 38766

57 0.0139 0.0171 0.0056 0.0006 0.0003 4003500 4928

58 0.0139 0.0107 0.0033 0.0003 0.0001 4003500 2802

59 0.0139 0.5415 0.3766 0.1579 0.1048 4003500 511519

60 0.0139 0.1993 0.1033 0.0255 0.0138 4003500 104643

105


with deq = 0.97m at 0.56 m scour

EQ

(PGA)

Annual

Prob of

Occur

pm

(DS=1)

pm

(DS=2)

pm

(DS=3)

pm

(DS=4)

C

(in $)

CRm

Sum

(in $)

1 0.0021 0.349 0.1467 0.0206 0.0165 4003500 135943

2 0.0021 0.6472 0.3878 0.101 0.0858 4003500 440425

3 0.0021 0.2408 0.0859 0.0092 0.0072 4003500 77171

4 0.0021 0.3922 0.1746 0.027 0.0218 4003500 164960

5 0.0021 0.1081 0.0287 0.0019 0.0014 4003500 26319

6 0.0021 0.0411 0.0082 0.0003 0.0002 4003500 8183

7 0.0021 0.2848 0.1091 0.0131 0.0103 4003500 98754

8 0.0021 0.2922 0.1132 0.0139 0.011 4003500 102878

9 0.0021 0.4412 0.2088 0.0358 0.0292 4003500 202457

10 0.0021 0.1893 0.0614 0.0056 0.0043 4003500 54892

11 0.0021 0.6354 0.3758 0.0956 0.0809 4003500 421913

12 0.0021 0.8642 0.6688 0.2898 0.2593 4003500 1023295

13 0.0021 0.6495 0.3902 0.1021 0.0867 4003500 444012

14 0.0021 0.6261 0.3665 0.0914 0.0773 4003500 407849

15 0.0021 0.4621 0.2243 0.0402 0.0329 4003500 220261

16 0.0021 0.5282 0.277 0.0568 0.047 4003500 284273

17 0.0021 0.5139 0.265 0.0528 0.0436 4003500 269263

18 0.0021 0.7756 0.5377 0.1851 0.1617 4003500 709576

19 0.0021 0.8847 0.7041 0.3248 0.2925 4003500 1125260

20 0.0021 0.8716 0.6813 0.3018 0.2707 4003500 1058453

21 0.000404 0.9515 0.8405 0.5023 0.4658 4003500 1632147

22 0.000404 0.8402 0.6303 0.2553 0.2268 4003500 921506

23 0.000404 0.2798 0.1064 0.0127 0.0099 4003500 96248

24 0.000404 0.3682 0.1588 0.0233 0.0187 4003500 148350

25 0.000404 0.8102 0.5855 0.2193 0.1932 4003500 813715

26 0.000404 0.8519 0.6488 0.2714 0.2419 4003500 969131

27 0.000404 0.8434 0.6353 0.2595 0.2307 4003500 933940

28 0.000404 0.9583 0.8573 0.531 0.4945 4003500 1713030

29 0.000404 0.7695 0.5297 0.1798 0.1568 4003500 693130

30 0.000404 0.8738 0.685 0.3054 0.2741 4003500 1068951

31 0.000404 0.9535 0.8455 0.5106 0.4741 4003500 1655587

32 0.000404 0.9336 0.7996 0.4402 0.4044 4003500 1456417

106

33 0.000404 0.7486 0.503 0.1628 0.1412 4003500 640072

34 0.000404 0.6524 0.3933 0.1036 0.088 4003500 448936

35 0.000404 0.874 0.6854 0.3058 0.2745 4003500 1070144

36 0.000404 0.9119 0.7549 0.3818 0.3474 4003500 1289764

37 0.000404 0.6846 0.4277 0.1204 0.103 4003500 504657

38 0.000404 0.7436 0.4968 0.159 0.1378 4003500 628237

39 0.000404 0.4113 0.1875 0.0302 0.0245 4003500 178912

40 0.000404 0.5879 0.3298 0.0762 0.0639 4003500 354834

41 0.0139 0.5418 0.2885 0.0608 0.0505 4003500 299198

42 0.0139 0.1499 0.0446 0.0036 0.0027 4003500 40215

43 0.0139 0.0032 0.0004 0 0 4003500 512

44 0.0139 0.0006 0.0001 0 0 4003500 104

45 0.0139 0.0034 0.0004 0 0 4003500 536

46 0.0139 0.0059 0.0007 0 0 4003500 933

47 0.0139 0.1563 0.0472 0.0039 0.0029 4003500 42437

48 0.0139 0.1158 0.0315 0.0022 0.0016 4003500 28761

49 0.0139 0.1299 0.0368 0.0027 0.002 4003500 33293

50 0.0139 0.481 0.2387 0.0445 0.0365 4003500 237211

51 0.0139 0.7473 0.5013 0.1617 0.1403 4003500 636829

52 0.0139 0.5956 0.3369 0.0791 0.0664 4003500 364939

53 0.0139 0.6659 0.4075 0.1104 0.094 4003500 471520

54 0.0139 0.755 0.511 0.1678 0.1458 4003500 655773

55 0.0139 0.438 0.2064 0.0352 0.0286 4003500 199743

56 0.0139 0.2178 0.0746 0.0075 0.0058 4003500 66846

57 0.0139 0.0562 0.0122 0.0006 0.0004 4003500 11898

58 0.0139 0.0384 0.0075 0.0003 0.0002 4003500 7635

59 0.0139 0.7371 0.4888 0.1541 0.1334 4003500 612944

60 0.0139 0.3768 0.1644 0.0246 0.0198 4003500 154239

107



EQ

(PGA)

Annual

Prob of

Occur

pm

(DS=1)

pm

(DS=2)

pm

(DS=3)

pm

(DS=4)

C

(in $)

CRm

Sum

(in $)

1 0.002 0.4053 0.1942 0.0301 0.0206 4003500 173992

2 0.002 0.7005 0.4614 0.1329 0.101 4003500 526668

3 0.002 0.2893 0.1193 0.0141 0.0092 4003500 101801

4 0.002 0.4501 0.2272 0.0388 0.027 4003500 208899

5 0.002 0.1382 0.0435 0.0032 0.0019 4003500 36776

6 0.002 0.0559 0.0134 0.0006 0.0003 4003500 12087

7 0.002 0.3371 0.1484 0.0197 0.0131 4003500 128713

8 0.002 0.345 0.1535 0.0208 0.0139 4003500 133677

9 0.002 0.5002 0.2668 0.0507 0.0358 4003500 253602

10 0.002 0.232 0.0877 0.0089 0.0056 4003500 73829

11 0.002 0.6895 0.4489 0.1262 0.0956 4003500 505990

12 0.002 0.8939 0.7339 0.3478 0.2898 4003500 1154605

13 0.002 0.7027 0.4638 0.1342 0.1021 4003500 530764

14 0.002 0.6807 0.4391 0.1212 0.0914 4003500 490064

15 0.002 0.5212 0.2844 0.0564 0.0402 4003500 274512

16 0.002 0.5867 0.3432 0.0778 0.0568 4003500 349213

17 0.002 0.5726 0.33 0.0727 0.0528 4003500 331782

18 0.002 0.8174 0.6113 0.2317 0.1851 4003500 822283

19 0.002 0.911 0.7655 0.3852 0.3248 4003500 1260262

20 0.002 0.9001 0.7452 0.3607 0.3018 4003500 1191097

21 0.0004 0.9646 0.8819 0.5669 0.5023 4003500 1770087

22 0.0004 0.8736 0.6987 0.3102 0.2553 4003500 1048012

23 0.0004 0.3317 0.145 0.019 0.0127 4003500 125634

24 0.0004 0.4252 0.2086 0.0338 0.0233 4003500 189021

25 0.0004 0.8478 0.657 0.2703 0.2193 4003500 933973

26 0.0004 0.8836 0.7157 0.3278 0.2714 4003500 1098056

27 0.0004 0.8763 0.7033 0.3149 0.2595 4003500 1061240

28 0.0004 0.9699 0.8955 0.595 0.531 4003500 1849165

29 0.0004 0.8121 0.6036 0.2256 0.1798 4003500 804587

30 0.0004 0.9019 0.7485 0.3646 0.3054 4003500 1202039

31 0.0004 0.9662 0.886 0.5751 0.5106 4003500 1793092

108

32 0.0004 0.9506 0.8481 0.5049 0.4402 4003500 1596079

33 0.0004 0.7934 0.5775 0.206 0.1628 4003500 747141

34 0.0004 0.7054 0.467 0.1359 0.1036 4003500 536217

35 0.0004 0.9021 0.7489 0.365 0.3058 4003500 1203232

36 0.0004 0.9333 0.81 0.4451 0.3818 4003500 1428425

37 0.0004 0.7352 0.5023 0.1562 0.1204 4003500 598323

38 0.0004 0.789 0.5714 0.2016 0.159 4003500 734170

39 0.0004 0.4697 0.2423 0.0432 0.0302 4003500 225617

40 0.0004 0.6445 0.4003 0.1023 0.0762 4003500 430032

41 0.01 0.5999 0.3558 0.0829 0.0608 4003500 366344

42 0.01 0.1871 0.0653 0.0057 0.0036 4003500 54856

43 0.01 0.005 0.0007 0 0 4003500 825

44 0.01 0.001 0.0001 0 0 4003500 152

45 0.01 0.0052 0.0007 0 0 4003500 849

46 0.01 0.009 0.0014 0 0 4003500 1529

47 0.01 0.1944 0.0688 0.0062 0.0039 4003500 57835

48 0.01 0.1473 0.0474 0.0036 0.0022 4003500 39999

49 0.01 0.1639 0.0546 0.0044 0.0027 4003500 45900

50 0.01 0.5401 0.3007 0.062 0.0445 4003500 294493

51 0.01 0.7922 0.5759 0.2048 0.1617 4003500 743522

52 0.01 0.6518 0.408 0.1059 0.0791 4003500 441622

53 0.01 0.718 0.4817 0.1441 0.1104 4003500 561535

54 0.01 0.7992 0.5854 0.2118 0.1678 4003500 764180

55 0.01 0.4968 0.2641 0.0498 0.0352 4003500 250467

56 0.01 0.2639 0.1049 0.0116 0.0075 4003500 88914

57 0.01 0.075 0.0196 0.001 0.0006 4003500 17247

58 0.01 0.0525 0.0124 0.0006 0.0003 4003500 11358

59 0.01 0.7831 0.5635 0.196 0.1541 4003500 717479

60 0.01 0.4342 0.2152 0.0356 0.0246 4003500 196099

109



EQ

(PGA)

Annual

Prob of

Occur

pm

(DS=1)

pm

(DS=2)

pm

(DS=3)

pm

(DS=4)

C

(in $)

CRm

Sum

(in $)

1 0.0021 0.3906 0.1942 0.0427 0.0301 4003500 200051

2 0.0021 0.6872 0.4614 0.17 0.1329 4003500 613288

3 0.0021 0.2764 0.1193 0.0209 0.0141 4003500 114904

4 0.0021 0.4351 0.2272 0.0542 0.0388 4003500 241407

5 0.0021 0.1299 0.0435 0.0051 0.0032 4003500 39763

6 0.0021 0.0517 0.0134 0.0011 0.0006 4003500 12563

7 0.0021 0.3233 0.1484 0.0287 0.0197 4003500 146632

8 0.0021 0.3311 0.1535 0.0301 0.0208 4003500 152365

9 0.0021 0.485 0.2668 0.0695 0.0507 4003500 294453

10 0.0021 0.2205 0.0877 0.0135 0.0089 4003500 82336

11 0.0021 0.6759 0.4489 0.1621 0.1262 4003500 589291

12 0.0021 0.8868 0.7339 0.408 0.3478 4003500 1306887

13 0.0021 0.6894 0.4638 0.1715 0.1342 4003500 617904

14 0.0021 0.667 0.4391 0.1561 0.1212 4003500 571071

15 0.0021 0.506 0.2844 0.0768 0.0564 4003500 319047

16 0.0021 0.5718 0.3432 0.1036 0.0778 4003500 406876

17 0.0021 0.5576 0.33 0.0973 0.0727 4003500 386470

18 0.0021 0.8072 0.6113 0.2828 0.2317 4003500 946828

19 0.0021 0.9047 0.7655 0.4471 0.3852 4003500 1418184

20 0.0021 0.8933 0.7452 0.4216 0.3607 4003500 1345556

21 0.000404 0.9616 0.8819 0.6282 0.5669 4003500 1934531

22 0.000404 0.8655 0.6987 0.3682 0.3102 4003500 1193007

23 0.000404 0.318 0.145 0.0277 0.019 4003500 142785

24 0.000404 0.4104 0.2086 0.0476 0.0338 4003500 217870

25 0.000404 0.8387 0.657 0.3252 0.2703 4003500 1069499

26 0.000404 0.8759 0.7157 0.387 0.3278 4003500 1246702

27 0.000404 0.8684 0.7033 0.3732 0.3149 4003500 1207360

28 0.000404 0.9672 0.8955 0.6551 0.595 4003500 2011483

29 0.000404 0.8016 0.6036 0.276 0.2256 4003500 927115

30 0.000404 0.8952 0.7485 0.4256 0.3646 4003500 1357090

31 0.000404 0.9633 0.886 0.6361 0.5751 4003500 1957087

110

32 0.000404 0.9466 0.8481 0.5679 0.5049 4003500 1762265

33 0.000404 0.7824 0.5775 0.254 0.206 4003500 863043

34 0.000404 0.6922 0.467 0.1736 0.1359 4003500 624090

35 0.000404 0.8954 0.7489 0.4261 0.365 4003500 1358384

36 0.000404 0.9282 0.81 0.5082 0.4451 4003500 1592336

37 0.000404 0.7226 0.5023 0.1973 0.1562 4003500 695276

38 0.000404 0.7778 0.5714 0.2491 0.2016 4003500 848586

39 0.000404 0.4546 0.2423 0.0598 0.0432 4003500 261236

40 0.000404 0.6302 0.4003 0.1335 0.1023 4003500 501338

41 0.01379 0.5851 0.3558 0.1099 0.0829 4003500 426981

42 0.01379 0.1771 0.0653 0.0089 0.0057 4003500 60221

43 0.01379 0.0045 0.0007 0 0 4003500 765

44 0.01379 0.0009 0.0001 0 0 4003500 140

45 0.01379 0.0046 0.0007 0 0 4003500 777

46 0.01379 0.0081 0.0014 0.0001 0 4003500 1521

47 0.01379 0.1841 0.0688 0.0096 0.0062 4003500 63684

48 0.01379 0.1387 0.0474 0.0057 0.0036 4003500 43310

49 0.01379 0.1546 0.0546 0.007 0.0044 4003500 50108

50 0.01379 0.5249 0.3007 0.0838 0.062 4003500 342511

51 0.01379 0.7812 0.5759 0.2527 0.2048 4003500 859163

52 0.01379 0.6376 0.408 0.1378 0.1059 4003500 514762

53 0.01379 0.705 0.4817 0.1832 0.1441 4003500 653075

54 0.01379 0.7883 0.5854 0.2606 0.2118 4003500 882175

55 0.01379 0.4816 0.2641 0.0684 0.0498 4003500 290638

56 0.01379 0.2516 0.1049 0.0174 0.0116 4003500 99807

57 0.01379 0.0698 0.0196 0.0018 0.001 4003500 18064

58 0.01379 0.0485 0.0124 0.001 0.0006 4003500 11758

59 0.01379 0.7718 0.5635 0.2428 0.196 4003500 830062

60 0.01379 0.4192 0.2152 0.0499 0.0356 4003500 226226

111



EQ

(PGA)

Annual

Prob of

Occur

pm

(DS=1)

pm

(DS=2)

pm

(DS=3)

pm

(DS=4)

C

(in $)

CRm

Sum

(in $)

1 0.0021 0.4053 0.1866 0.0427 0.0366 4003500 209791

2 0.0021 0.7005 0.4503 0.17 0.1524 4003500 642558

3 0.0021 0.2893 0.1139 0.0209 0.0175 4003500 120169

4 0.0021 0.4501 0.2188 0.0542 0.0467 4003500 253169

5 0.0021 0.1382 0.041 0.0051 0.0041 4003500 41400

6 0.0021 0.0559 0.0125 0.0011 0.0008 4003500 13099

7 0.0021 0.3371 0.142 0.0287 0.0243 4003500 153606

8 0.0021 0.345 0.147 0.0301 0.0255 4003500 159479

9 0.0021 0.5002 0.2577 0.0695 0.0603 4003500 308738

10 0.0021 0.232 0.0833 0.0135 0.0112 4003500 85991

11 0.0021 0.6895 0.4378 0.1621 0.145 4003500 617476

12 0.0021 0.8939 0.7247 0.408 0.3803 4003500 1356838

13 0.0021 0.7027 0.4527 0.1715 0.1538 4003500 647334

14 0.0021 0.6807 0.4281 0.1561 0.1395 4003500 598499

15 0.0021 0.5212 0.275 0.0768 0.0669 4003500 334677

16 0.0021 0.5867 0.3329 0.1036 0.0912 4003500 426825

17 0.0021 0.5726 0.32 0.0973 0.0855 4003500 405567

18 0.0021 0.8174 0.6005 0.2828 0.2589 4003500 988152

19 0.0021 0.911 0.7569 0.4471 0.4187 4003500 1469833

20 0.0021 0.9001 0.7362 0.4216 0.3936 4003500 1396177

21 0.000404 0.9646 0.8763 0.6282 0.6006 4003500 1987065

22 0.000404 0.8736 0.6889 0.3682 0.3414 4003500 1240805

23 0.000404 0.3317 0.1387 0.0277 0.0234 4003500 149459

24 0.000404 0.4252 0.2006 0.0476 0.0409 4003500 228456

25 0.000404 0.8478 0.6466 0.3252 0.2997 4003500 1114342

26 0.000404 0.8836 0.7061 0.387 0.3597 4003500 1295637

27 0.000404 0.8763 0.6935 0.3732 0.3462 4003500 1255293

28 0.000404 0.9699 0.8904 0.6551 0.6282 4003500 2063340

29 0.000404 0.8121 0.5928 0.276 0.2525 4003500 967994

30 0.000404 0.9019 0.7395 0.4256 0.3976 4003500 1407859

31 0.000404 0.9662 0.8805 0.6361 0.6087 4003500 2009481

112

32 0.000404 0.9506 0.8414 0.5679 0.5394 4003500 1815847

33 0.000404 0.7934 0.5665 0.254 0.2315 4003500 901676

34 0.000404 0.7054 0.4559 0.1736 0.1557 4003500 653828

35 0.000404 0.9021 0.7399 0.4261 0.398 4003500 1409152

36 0.000404 0.9333 0.8023 0.5082 0.4795 4003500 1645571

37 0.000404 0.7352 0.4912 0.1973 0.1779 4003500 727984

38 0.000404 0.789 0.5604 0.2491 0.2268 4003500 886763

39 0.000404 0.4697 0.2336 0.0598 0.0517 4003500 273875

40 0.000404 0.6445 0.3896 0.1335 0.1186 4003500 525732

41 0.01379 0.5999 0.3454 0.1099 0.0969 4003500 447848

42 0.01379 0.1871 0.0619 0.0089 0.0073 4003500 62895

43 0.01379 0.005 0.0006 0 0 4003500 793

44 0.01379 0.001 0.0001 0 0 4003500 152

45 0.01379 0.0052 0.0007 0 0 4003500 849

46 0.01379 0.009 0.0013 0.0001 0 4003500 1597

47 0.01379 0.1944 0.0652 0.0096 0.0079 4003500 66490

48 0.01379 0.1473 0.0447 0.0057 0.0047 4003500 45240

49 0.01379 0.1639 0.0516 0.007 0.0057 4003500 52346

50 0.01379 0.5401 0.2911 0.0838 0.0733 4003500 359358

51 0.01379 0.7922 0.5649 0.2527 0.2302 4003500 897637

52 0.01379 0.6518 0.3972 0.1378 0.1226 4003500 539752

53 0.01379 0.718 0.4706 0.1832 0.1647 4003500 684070

54 0.01379 0.7992 0.5745 0.2606 0.2377 4003500 921470

55 0.01379 0.4968 0.255 0.0684 0.0594 4003500 304923

56 0.01379 0.2639 0.0999 0.0174 0.0145 4003500 104327

57 0.01379 0.075 0.0183 0.0018 0.0014 4003500 18913

58 0.01379 0.0525 0.0115 0.001 0.0008 4003500 12271

59 0.01379 0.7831 0.5525 0.2428 0.2208 4003500 867610

60 0.01379 0.4342 0.2071 0.0499 0.0429 4003500 237123

113



EQ

(PGA)

Annual

Prob of

Occur

pm

(DS=1)

pm

(DS=2)

pm

(DS=3)

pm

(DS=4)

C

(in $)

CRm

Sum

(in $)

1 0.0021 0.1942 0.0871 0.0206 0.0118 4003500 90735

2 0.0021 0.4614 0.2765 0.101 0.0672 4003500 352676

3 0.0021 0.1193 0.047 0.0092 0.005 4003500 46597

4 0.0021 0.2272 0.1066 0.027 0.0158 4003500 113755

5 0.0021 0.0435 0.0136 0.0019 0.0009 4003500 12923

6 0.0021 0.0134 0.0034 0.0003 0.0002 4003500 3319

7 0.0021 0.1484 0.0618 0.0131 0.0073 4003500 62419

8 0.0021 0.1535 0.0645 0.0139 0.0077 4003500 65337

9 0.0021 0.2668 0.1316 0.0358 0.0215 4003500 144454

10 0.0021 0.0877 0.032 0.0056 0.0029 4003500 31031

11 0.0021 0.4489 0.266 0.0956 0.0631 4003500 335842

12 0.0021 0.7339 0.5511 0.2898 0.219 4003500 905412

13 0.0021 0.4638 0.2786 0.1021 0.068 4003500 356019

14 0.0021 0.4391 0.258 0.0914 0.0601 4003500 323094

15 0.0021 0.2844 0.1432 0.0402 0.0244 4003500 159331

16 0.0021 0.3432 0.184 0.0568 0.0356 4003500 214011

17 0.0021 0.33 0.1746 0.0528 0.0329 4003500 201088

18 0.0021 0.6113 0.4154 0.1851 0.1319 4003500 602951

19 0.0021 0.7655 0.5901 0.3248 0.2495 4003500 1005571

20 0.0021 0.7452 0.5648 0.3018 0.2294 4003500 939822

21 0.000404 0.8819 0.7543 0.5023 0.4145 4003500 1514028

22 0.000404 0.6987 0.5097 0.2553 0.1895 4003500 806153

23 0.000404 0.145 0.06 0.0127 0.007 4003500 60553

24 0.000404 0.2086 0.0955 0.0233 0.0135 4003500 100580

25 0.000404 0.657 0.4633 0.2193 0.1596 4003500 702370

26 0.000404 0.7157 0.5294 0.2714 0.2032 4003500 852557

27 0.000404 0.7033 0.515 0.2595 0.1931 4003500 818371

28 0.000404 0.8955 0.7764 0.531 0.4428 4003500 1596784

29 0.000404 0.6036 0.4076 0.1798 0.1276 4003500 587337

30 0.000404 0.7485 0.5688 0.3054 0.2325 4003500 950067

31 0.000404 0.886 0.7608 0.5106 0.4227 4003500 1538041

32 0.000404 0.8481 0.7026 0.4402 0.3549 4003500 1335812

114

33 0.000404 0.5775 0.3818 0.1628 0.1141 4003500 537306

34 0.000404 0.467 0.2812 0.1036 0.069 4003500 360339

35 0.000404 0.7489 0.5693 0.3058 0.2329 4003500 951316

36 0.000404 0.81 0.6486 0.3818 0.3007 4003500 1168694

37 0.000404 0.5023 0.3119 0.1204 0.0816 4003500 411404

38 0.000404 0.5714 0.3758 0.159 0.1112 4003500 526204

39 0.000404 0.2423 0.116 0.0302 0.0179 4003500 125145

40 0.000404 0.4003 0.227 0.0762 0.0492 4003500 275837

41 0.01379 0.3558 0.1932 0.0608 0.0384 4003500 226958

42 0.01379 0.0653 0.0223 0.0036 0.0018 4003500 21471

43 0.01379 0.0007 0.0001 0 0 4003500 116

44 0.01379 0.0001 0 0 0 4003500 12

45 0.01379 0.0007 0.0001 0 0 4003500 116

46 0.01379 0.0014 0.0002 0 0 4003500 232

47 0.01379 0.0688 0.0238 0.0039 0.002 4003500 22992

48 0.01379 0.0474 0.0151 0.0022 0.0011 4003500 14493

49 0.01379 0.0546 0.018 0.0027 0.0013 4003500 17107

50 0.01379 0.3007 0.1542 0.0445 0.0273 4003500 173760

51 0.01379 0.5759 0.3802 0.1617 0.1133 4003500 534219

52 0.01379 0.408 0.233 0.0791 0.0512 4003500 284789

53 0.01379 0.4817 0.2938 0.1104 0.0741 4003500 381113

54 0.01379 0.5854 0.3895 0.1678 0.1181 4003500 552131

55 0.01379 0.2641 0.1298 0.0352 0.0211 4003500 142312

56 0.01379 0.1049 0.04 0.0075 0.004 4003500 39322

57 0.01379 0.0196 0.0053 0.0006 0.0003 4003500 5132

58 0.01379 0.0124 0.0031 0.0003 0.0001 4003500 2943

59 0.01379 0.5635 0.3682 0.1541 0.1074 4003500 511831

60 0.01379 0.2152 0.0994 0.0246 0.0143 4003500 105204

115



EQ

(PGA)

Annual

Prob of

Occur

pm

(DS=1)

pm

(DS=2)

pm

(DS=3)

pm

(DS=4)

C

(in $)

CRm

Sum

(in $)

1 0.0021 0.4053 0.1866 0.0427 0.0427 4003500 219560

2 0.0021 0.7005 0.4503 0.17 0.17 4003500 670742

3 0.0021 0.2893 0.1139 0.0209 0.0209 4003500 125614

4 0.0021 0.4501 0.2188 0.0542 0.0542 4003500 265180

5 0.0021 0.1382 0.041 0.0051 0.0051 4003500 43002

6 0.0021 0.0559 0.0125 0.0011 0.0011 4003500 13580

7 0.0021 0.3371 0.142 0.0287 0.0287 4003500 160652

8 0.0021 0.345 0.147 0.0301 0.0301 4003500 166846

9 0.0021 0.5002 0.2577 0.0695 0.0695 4003500 323471

10 0.0021 0.232 0.0833 0.0135 0.0135 4003500 89674

11 0.0021 0.6895 0.4378 0.1621 0.1621 4003500 644860

12 0.0021 0.8939 0.7247 0.408 0.408 4003500 1401197

13 0.0021 0.7027 0.4527 0.1715 0.1715 4003500 675679

14 0.0021 0.6807 0.4281 0.1561 0.1561 4003500 625082

15 0.0021 0.5212 0.275 0.0768 0.0768 4003500 350530

16 0.0021 0.5867 0.3329 0.1036 0.1036 4003500 446683

17 0.0021 0.5726 0.32 0.0973 0.0973 4003500 424463

18 0.0021 0.8174 0.6005 0.2828 0.2828 4003500 1026425

19 0.0021 0.911 0.7569 0.4471 0.4471 4003500 1515313

20 0.0021 0.9001 0.7362 0.4216 0.4216 4003500 1441016

21 0.000404 0.9646 0.8763 0.6282 0.6282 4003500 2031264

22 0.000404 0.8736 0.6889 0.3682 0.3682 4003500 1283722

23 0.000404 0.3317 0.1387 0.0277 0.0277 4003500 156345

24 0.000404 0.4252 0.2006 0.0476 0.0476 4003500 239185

25 0.000404 0.8478 0.6466 0.3252 0.3252 4003500 1155178

26 0.000404 0.8836 0.7061 0.387 0.387 4003500 1339355

27 0.000404 0.8763 0.6935 0.3732 0.3732 4003500 1298531

28 0.000404 0.9699 0.8904 0.6551 0.6551 4003500 2106418

29 0.000404 0.8121 0.5928 0.276 0.276 4003500 1005627

30 0.000404 0.9019 0.7395 0.4256 0.4256 4003500 1452698

31 0.000404 0.9662 0.8805 0.6361 0.6361 4003500 2053359

32 0.000404 0.9506 0.8414 0.5679 0.5679 4003500 1861487

116

33 0.000404 0.7934 0.5665 0.254 0.254 4003500 937708

34 0.000404 0.7054 0.4559 0.1736 0.1736 4003500 682493

35 0.000404 0.9021 0.7399 0.4261 0.4261 4003500 1454151

36 0.000404 0.9333 0.8023 0.5082 0.5082 4003500 1691531

37 0.000404 0.7352 0.4912 0.1973 0.1973 4003500 759052

38 0.000404 0.789 0.5604 0.2491 0.2491 4003500 922474

39 0.000404 0.4697 0.2336 0.0598 0.0598 4003500 286847

40 0.000404 0.6445 0.3896 0.1335 0.1335 4003500 549592

41 0.01379 0.5999 0.3454 0.1099 0.1099 4003500 468666

42 0.01379 0.1871 0.0619 0.0089 0.0089 4003500 65457

43 0.01379 0.005 0.0006 0 0 4003500 793

44 0.01379 0.001 0.0001 0 0 4003500 152

45 0.01379 0.0052 0.0007 0 0 4003500 849

46 0.01379 0.009 0.0013 0.0001 0.0001 4003500 1758

47 0.01379 0.1944 0.0652 0.0096 0.0096 4003500 69213

48 0.01379 0.1473 0.0447 0.0057 0.0057 4003500 46841

49 0.01379 0.1639 0.0516 0.007 0.007 4003500 54428

50 0.01379 0.5401 0.2911 0.0838 0.0838 4003500 376173

51 0.01379 0.7922 0.5649 0.2527 0.2527 4003500 933668

52 0.01379 0.6518 0.3972 0.1378 0.1378 4003500 564093

53 0.01379 0.718 0.4706 0.1832 0.1832 4003500 713696

54 0.01379 0.7992 0.5745 0.2606 0.2606 4003500 958142

55 0.01379 0.4968 0.255 0.0684 0.0684 4003500 319335

56 0.01379 0.2639 0.0999 0.0174 0.0174 4003500 108971

57 0.01379 0.075 0.0183 0.0018 0.0018 4003500 19553

58 0.01379 0.0525 0.0115 0.001 0.001 4003500 12591

59 0.01379 0.7831 0.5525 0.2428 0.2428 4003500 902841

60 0.01379 0.4342 0.2071 0.0499 0.0499 4003500 248333

Documents

ANALYSIS OF BRIDGE PERFORMANCE UNDER THE COMBINED …