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
Professor of Civil Engineering
Peggy A. Johnson
Professor of Civil Engineering
Head of the Department of Civil Engineering
*Signatures are on file in the Graduate School.
iii
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
iv
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.
v
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 4.2: Schematic diagram of 3 span bridge .........................................................27
Figure 4.3: Schematic diagram of 4 span bridge .........................................................27
Figure 4.4: Schematic diagram of 5 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
Figure 4.25: Change in fragility curve for 3 span bridge with
deq = 1.20 m for different scour depth for a) minor damage state
b) moderate damage state c) major damage state………………....................... .........60
Figure 4.26: Change in fragility curve for 4 span bridge with
deq = 1.20 m for different scour depth for a) minor damage state
b) moderate damage state c) major damage state ........................................................61
ix
Figure 4.27: Change in fragility curve for 5 span bridge with
deq = 1.20 m for different scour depth for a) minor damage state
b) moderate damage state c) major damage state ........................................................63
Figure 4.28: Three dimensional plots for 2 span bridge ..............................................68
Figure 4.29: Three dimensional plots for 3 span bridge ..............................................68
Figure 4.30: Three dimensional plots for 4 span bridge ..............................................69
Figure 4.31: Three dimensional plots for 5 span bridge ..............................................69
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
Table A5.2: Post-earthquake restoration cost (CRPm) of the 5-span example bridge
with deq = 0.97 m at 0.56 m scour ..............................................................................105
Table A5.3: Post-earthquake restoration cost (CRPm) of the 5-span example bridge
with deq = 0.97 m at 1.22 m scour ..............................................................................107
Table A5.4: Post-earthquake restoration cost (CRPm) of the 5-span example bridge
with deq = 0.97 m at 2.85 m scour ..............................................................................109
Table A5.5: Post-earthquake restoration cost (CRPm) of the 5-span example bridge
with deq = 0.97 m at 3.08 m scour ..............................................................................111
Table A5.6: Post-earthquake restoration cost (CRPm) of the 5-span example bridge
with deq = 0.97 m at 3.30 m scour ..............................................................................113
Table A5.7: Post-earthquake restoration cost (CRPm) of the 5-span example bridge
with deq = 0.97 m at 3.45 m scour ..............................................................................115
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.
5
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.
6
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.
8
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).
13
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
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
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 4.2: Schematic diagram of 3 span bridge (not to scale)
Figure 4.3: Schematic diagram of 4 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)
Figure 4.25: Change in fragility curve for 3 span bridge with deq = 1.20 m for
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)
Figure 4.26: Change in fragility curve for 4 span bridge with deq = 1.20 m for
different scour depth for a) minor damage state b) moderate damage state c) major
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)
Figure 4.27: Change in fragility curve for 5 span bridge with deq = 1.20 m for
different scour depth for a) minor damage state b) moderate damage state c) major
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
Figure 4.29: Three dimensional plots for 3 span bridge
69
Figure 4.30: Three dimensional plots for 4 span bridge
Figure 4.31: Three dimensional plots for 5 span bridge
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
50% statistical confidence
95% 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
flood-prone regions.
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,
flood-prone regions.
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
References
Alipour, A., Shafei, B., and Shinozuka, M. (2010). “Evaluation of Uncertainties
Associated with Design of Highway Bridges Considering the Effect of
Scouring and Earthquake.” Proceeding of Structure Congress 2010, Section.
Bridges I, pp.288-297.
Banerjee, S. and Shinozuka, M. (2008a). “Mechanistic quantification of RC bridge
damage states under earthquake through fragility analysis”, Probabilistic
Engineering Mechanics, Vol. 23, No. 1, pp. 12-22.
Banerjee, S. and Shinozuka, M. (2008b). “Experimental verification of bridge seismic
damage states quantified by calibrating analytical models with empirical field
data”, Journal of Earthquake Engineering and Engineering Vibration, Vol. 7, No.
4, pp. 383-393.
Banerjee, S., Shinozuka, M., and Sgaravato, M. (2009). “Uncertainty in Seismic
Perfromance of Highway Network Estimated Using Empirical Fragility
Curves of Bridge.” International Journal of Engineering Under Uncertainty:
Hazards, Assessment and Mitigation, Vol. 1, Nos. 1-2, pp. 1-11.
Basoz, N. and Kiremidjian, A.S. (1997). “Risk assessment of bridges and highway
systems from the Northridge earthquake.” Proceedings of the National
Seismic Conference on Bridges and Highways: “Progress in Research and
Practice.” Sacramento, California: 65–79.
97
Bennett, C.R., Lin, C., Parsons, R., and Han, J. (2009). “Evaluation of behavior of a
laterally loaded bridge pile group under scour conditions.” Proceedings of SEI
2009 Structures Congress, Texas, pp. 290-299.
Breusers, H. N. C., Nicollet, G., and Shen, H.W. (1977). “ Local Scour Around
Cylindrical Piers.” Journal of Hydraulic Research, Vol. 15, No. 3, pp. 211-
252.
Brown, D.A., O‟Neill, M.W., Hoit, M., McVay, M., El Naggar, M.H., and
Chakraborty, S. (2001). “Static and Dynamic Lateral Loading of Pile Groups.”
NCHRP Report 461, Transportation Research Board, Washington, D.C.
California Department of Transportation (CALTRANS), Bridge Design Aids, 1988.
California Department of Transportation (CALTRANS), Seismic design criteria,
Version 1.4, 2006.
Chen, H -C. (2008). “Multi-Hazard of a bridge pier due to earthquake and scour.”
report presented to University of Maryland, at College Park, Md., in partial
fulfillment of the requirement for the degree of Master of Science.
Computer and Structures, Inc., (1995), SAP2000 (Structural Analysis Program),
Berkley, CA, Version 14.1.0.
Federal Emergency Management Agency (FEMA), 1999. Earthquake loss estimation
methodology: HAZUS 99 (SR2). Washington DC: FEMA. Technical manual.
Flood Insurance Study for Sutter County, California, Federal Emergency
Management Agency (FEMA), Flood Insurance Study Number
060394V000A, 2008.
98
Ghosn, M., Moses, F., and Wang, J. (2003). “Design of Highway Bridge for extreme
events.” NCHRP Report 489, Transportation Research Board, Washington,
D.C., 2003.
Gupta, R.S. (2008). Hydrology and hydraulic systems. 3rd ed. Prientice-Hall, Inc., A
division of Simon & Schuster, New Jersey.
Hwang, H.H.M., and Huo, J-R. (1994). “Generation of hazard-consistent fragility
curves for seismic loss estimation studies.” Technical Report NCEER-94-
0015, National Center for Earthquake Engineering Research, State University
of New York at Buffalo, Buffalo, New York.
Hwang, H., Jernigan, J. B., and Lin, Y.W. (2000). “Evaluation of seismic damage to
Memphis bridges and highway systems.” Journal of Bridge Engineering
2000, Vol. 5, No. 4, pp. 322–330.
Interagency Advisory Committee on Water Data (IACWD). (1982). “Guidelines for
Determining Flood Flow Frequency.” Bulletin 17B, U.S. Dept. of the Interior,
Office of Water Data Coordination, Reston, VA, 1982.
Jain, S.C. and Fischer, E. E. (1979). “Scour around Circular Bridge Piers at High
Froude Numbers.” Rep. No.FHwa-RD-79-104, Federal Hwy. Administration
(FHWA), Washington, D.C.
Johnson, P. A. (1995). “Comparison of Pier-Scour Equations Using Field Data.”
Journal of Hydraulic Engineering, Vol. 121, No. 8, pp. 626-629.
Johnson, P.A. and Dock, D.A. (1998). “Probabilistic Bridge Scour Estimates.”
Journal of Hydraulic Engineering, Vol. 124, No. 7, pp. 750-754.
99
Johnson, P.A. and Torrico, E.F. (1994). "Scour Around Wide Piers in Shallow
Water." Transportation Research Board Record 1471, Transportation
Research Board, Washington, D.C.
Kapur, J. (2011). “Bridge Design Manual.” Washington State Department of
Transportation, M 23-50.04.
Laursen, E. M., and Toch, A. (1956). “ Scour around Bridge Piers and Abutments.”
Bulletin No. 4, Iowa Highway Research Board , Ames, Iowa
Liang, F., Bennett, C. R., Parsons, R. L., Han, J., and Lin, C. (2009). “A Literature
Review on Behavior of Scoured Piles under Bridges.” 2009 International
Foundation Congress and Equipment Expo, ASCE.
Mander, J. B., and Basoz, N. (1999). “Seismic fragility curve theory for highway
bridges.” Fifth U.S. Conference on Lifeline Earthquake Engineering, Seattle,
WA, U.S.A. ASCE.
Melville, B. W., and Sutherland, A.J. (1988). “Design Method for Local Scour at
Bridge Piers.” Journal of Hydraulic Engineering, ASCE, Vol. 114, No. 10, pp.
1210-1216
Mokwa, R. L., Duncan, J. M., and Charles, E. V. (2000). "Investigation of the
resistance of pile caps and integral abutments to lateral loading." Virginia
Transportation Research Council, Charlottesville, Virginia, 2000.
National Bridge Inventory, Federal Highway Administration (FHWA), U.S.
Department of Transportation,
http://www.fhwa.dot.gov/bridge/nbi.htm>(Mar .11, 2010)
100
OpenSees Development Team. (2009). OpenSees: Open System for Earthquake
Engineering Simulation, Berkeley, CA.
Perkins, D.M. (2002). Uncertainty in Probabilistic Seismic Hazard Analysis, in
Acceptable Risk Processes: Lifelines and Natural Hazards, Monograph No.
21, Edited by C. Taylor and E. VanMarcke, Technical Council on Lifeline
Earthquake Engineering, ASCE, Reston VAUSA.
Priestley, M.J.N., Seible, F., and Calvi, G.M. (1996). Seismic design and retrofit of
bridges, John Wiley and Sons, Inc., NY; 1996.
Randolph, M. F. (2003). “Science and Empiricism in Pile Foundation Design.”
Geotechnique 53, No. 10, pp. 847-875.
“Recommended practice for planning, designing and constructing fixed offshore
platforms.” (2000). 21st Ed., API Recommended Pract. 2A-WSD (RP 2A),
American Petroleum Institute.
Richardson, E.V. and Davis, S. R. (2001). “Evaluating Scour at Bridges.” Publication
No. FHWA NHI 01-001, Hydraulic Engineering Circular No – 18, Federal
Highway Administration, U.S. Department of Transportation, Washington,
D.C., 2001.
Shen, H .W., Schneider, V. R., and Karaki, S. (1969). “Local Scour around Bridge
Piers.” Asce Proc, Journal of Hydraulic Division, Vol. 95, No. 6, pp. 1919-
1940.
Shinozuka, M., Feng, M.Q., Lee, J., and Naganuma, T. (2000a). “Statistical analysis
of fragility curves.” Journal of Engineering Mechanics, ASCE, Vol. 126, No.
12, pp. 1224-1231.
101
Shinozuka, M., Feng, M.Q., Kim, H.-K., and Kim, S.-H. (2000b). “Nonlinear static
procedure for fragility curve development.” Journal of Engineering
Mechanics, ASCE, Vol. 126, No. 12, pp. 1287-1295.
Sultan, M., and Kawashima, K. (1993). “Comparison of the Seismic Design of
Highway Bridges in California and in Japan.” 9th
U.S.-Japan Bridge
Workshop, Tsukuba, Japan, May, 1993.
Tsai, C. and Chen, Y. (2006). “Seismic Capacity Evaluation of Bridges with Scoured
Group Pile Foundations.” 4th
International Conference on Earthquake
Engineering, Taipei, Taiwan, Paper No. 053.
U.S. Department of Transportation. (1993). “Evaluating Scour at Bridges.” Hydraulic
Engineering Circular No – 18, FHWA-IP-90-017, Federal Highway
Administration, Washington, D.C.
Yin ,Y. and Konagai, K. (2001). “A simplified method for expression of the dynamic
stiffness of large-scaled grouped piles in sway and rocking motions.” Journal
of Applied Mechanics, JSCE, Vol. 4: 415-422.
Zhou, Y., Banerjee, S., and Shinozuka, M. (2010). “Socio-Economic Effect of
Seismic Retrofit of Bridges for Highway Transportation Networks: A Pilot
Study.” Structure and Infrastructure Engineering, Vol. 6, Nos. 1-2, pp. 145-
157.
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
Table A5.1: Post-earthquake restoration cost (CRPm) of the 5-span example bridge
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
Table A5.2: Post-earthquake restoration cost (CRPm) of the 5-span example bridge
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
Table A5.3: Post-earthquake restoration cost (CRPm) of the 5-span example bridge
with deq = 0.97m at 1.22 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 0.002 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
Table A5.4: Post-earthquake restoration cost (CRPm) of the 5-span example bridge
with deq = 0.97m at 2.85 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 0.3906 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
Table A5.5: Post-earthquake restoration cost (CRPm) of the 5-span example bridge
with deq = 0.97m at 3.08 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 0.4053 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
Table A5.6: Post-earthquake restoration cost (CRPm) of the 5-span example bridge
with deq = 0.97m at 3.30 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 0.1942 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
Table A5.7: Post-earthquake restoration cost (CRPm) of the 5-span example bridge
with deq = 0.97m at 3.45 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 0.4053 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