Integrated Multi-Hazard Framework for the Fragility Analysis of Roadway Bridges

12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12
Vancouver, Canada, July 12-15, 2015
Integrated Multi-Hazard Framework for the Fragility Analysis of
Roadway Bridges
Pierre Gehl
Research Associate, EPICentre, Dept. of Civil Engineering, University College London,
London, UK
Dina D’Ayala
Professor, EPICentre, Dept. of Civil Engineering, University College London, London,
UK
ABSTRACT: This paper presents a method for the development of bridge fragility functions that are
able to account for the cumulated impact of different hazard types, namely earthquakes, ground failures
and fluvial floods. After identifying which loading mechanisms are affecting which bridge components,
specific damage-dependent component fragility curves are derived. The definition of the global damage
states at system level through a fault-tree analysis is coupled with a Bayesian Network formulation in
order to account for the correlation structure between failure events. Fragility functions for four system
damage states are finally derived as a function of flow discharge Q (for floods) and peak ground
acceleration PGA (for earthquakes and ground failures): the results are able to represent specific failure
configurations that can be linked to functionality levels or repair durations.
1. INTRODUCTION
Spatially distributed infrastructure networks may
be exposed to a wide range of natural hazards,
whose impacts have to be integrated and homog-
enized in order to assess the reliability of the sys-
tem over the design lifetime of its components (e.g.
roadway bridges). While previous studies have
proposed probabilistic frameworks for multi-hazard
assessment, either for joint independent events or
cascading events (Marzocchi et al., 2012), there re-
mains a lack of fragility models that are able to
account for hazard interactions at the vulnerabil-
ity level. Even though some vulnerability analyses
have addressed the combined effects of earthquakes
and scour (Alipour and Shafei, 2012) or earth-
quakes, scour and truck traffic on bridges (Liang
and Lee, 2013), integrated fragility functions with
respect to the various intensity measures that rep-
resent the hazard types are needed in order to be
coupled to the probabilistic hazard distributions.
To this end, a component-based approach is pro-
posed in this paper: the bridge system is decom-
posed into its various components, so that the
hazard-specific loading mechanisms and their de-
mand can be straightforwardly quantified at the
component level. In the case where some damaged
components influence the response of other compo-
nents with respect to another hazard types, a set of
damage-dependent component fragility curves have
to be derived in order to account for all possible
damage configurations. Finally, the use of proba-
bilistic tools such as Bayesian Networks can facili-
tate the assembly of the component fragility curves
at system level, while accounting for the possible
correlations between the component failure events.
Multi-variate fragility functions are then expected
to be derived, where each input variable repre-
sents a hazard-specific intensity measure, so that
the damage of the bridge system can be assessed
for joint hazard events as well as single events.
1

2. MULTI-HAZARD SCENARIOS
Three main hazard types are considered in the
present study, namely earthquakes (EQ), fluvial
floods (FL) and earthquake-triggered ground fail-
ures (GF). They have been chosen because they
may affect a wide range of bridge components and
they are among the main causes of bridge failures
(Sharma and Mohan, 2011).
Following the classification proposed by Lee and
Sternberg (2008), different types of multi-hazard
scenarios have to be considered:
• Simple events: for instance, EQ, FL or GF
events taken separately (Scenario 1).
• Combined multi-hazard events (i.e. a sin-
gle event triggering multiple loading mecha-
nisms): a GF event triggered by an EQ event
(Scenario 2).
• Subsequent multi-hazard events (i.e. unrelated
single events separated in time): it could be
a FL event followed by an EQ, which in turn
triggers a GF event (Scenario 3).
• Simultaneous multi-hazard events: they repre-
sent the most unlikely scenarios and are not
addressed here.
These different scenarios are then considered
throughout the rest of the paper, as the objective
is to generate multi-hazard fragility curves that are
able to account for different combinations of load-
ing mechanisms.
3. BRIDGE COMPONENTS AND ASSOCI-
ATED FAILURE MECHANISMS
A multi-span concrete bridge with simply-
supported independent decks is considered in the
present study. It is composed of two seat-type
abutments and two piers with three cylindrical
RC columns. Deck displacement is restrained by
elastomeric bearings (i.e. alternation of expan-
sion and fixed devices) in the longitudinal direc-
tion and shear keys in the transversal direction. At
each bridge extremity, an embankment approach is
added in order to simulate the transition between
the plain roadway segment and the bridge (see half
bridge system in Figure 1).
The hazard types that may potentially affect the
various bridge components are detailed in Table 1:
-
-
-
-
-
-
Abutment approach (embankment)
Abutment
Abutment foundation
S hear key
Bearing
Pier
column
Pier foundation
Figure 1: Sketch of the studied bridge system and its
components.
• Earthquakes affect all structural components
of the bridge (i.e. abutments, piers, bearings,
shear keys).
• Fluvial floods may excavate pier foundations
due to scour, while in extreme cases decks may
be dislodged by hydraulic forces, after shear
keys have been damaged (Padgett et al., 2008).
• Earthquake-triggered ground failures are
likely to affect the approach embankment, as
there is usually a vertical settlement between
the embankment and the bridge, due to the
difference in foundation depth (Puppala et al.,
2009). Deep-seated landslides may also affect
the slope on which the abutment is built.
Table 1: Bridge components and associated hazard
types.
Component EQ FL GF
Pier foundation X
Pier X
Bearing X
Deck X
Abutment foundation X
Abutment X
Abutment approach X
Shear key X X
4. COMPONENT FRAGILITY CURVES
Fragility models are derived to quantify the dam-
age probability of each component to their associ-
ated hazard types.
2

4.1. Earthquake
The layout, dimensions and constitutive mod-
els of the different bridge components are directly
taken from the multi-span simply-supported con-
crete (MSSSC) girder bridge described in the study
by Nielson (2005). However two modifications are
brought to the bridge system:
1. Pier foundations are assumed to be anchored
up to a depth of 8 m: the group of pile foun-
dations, as described by Nielson (2005), is
approximated by an equivalent elastic beam,
which is connected to the ground through a set
of Winkler p-y springs in order to model the
soil resistance. The bending stiffness of the
equivalent pile and the parameters of the p-y
curves are based on the configuration of the
group piles and the soil properties (Prasad and
Banerjee, 2013).
2. External shear keys are added on the pier caps
in order to restrain the displacement of the in-
dependent decks in the transversal direction.
The shear keys are modelled according to a
sliding friction shear mechanism: first, the
deck can slide on the pier cap (i.e. friction
Coulomb law) until the gap with the shear key
is closed. Then, the capacity of the shear key is
engaged until it ruptures through a shear mech-
anism. Once the shear key has failed, it is as-
sumed that the deck can slide again until un-
seating.
The bridge is modelled through a simplified sys-
tem of connectors whose stiffness and hysteretic
models represent the behaviour of the different
bridge components, such as piers, bearings or abut-
ments (Gehl and D’Ayala, 2014). For instance,
each three-column pier is fully modelled with a fi-
nite element program and its cyclic pushover curve
in both longitudinal and transversal directions is
then used to model an hysteretic material spring in
the OpenSees platform.
Following the multi-hazard scenarios defined in
the previous section, the seismic fragility curves
should be derived for various configurations:
• Intact bridge;
• Pier foundations excavated by scour;
• Shear keys damaged by hydraulic forces (i.e.
fluvial flood);
• Both pier foundations and shear keys affected
by a fluvial flood.
Therefore the bridge response has to be re-
assessed for each of the identified configurations,
leading to a series of damage-dependent compo-
nent fragility functions. The effect of scour is in-
troduced by removing the Winkler springs that are
excavated by the scour depth (Alipour et al., 2013).
If damaged, shear keys are simply removed and the
transversal deck movement is only restrained by a
friction law.
Fragility curves for each component, for both
loading directions, and for two damage states (i.e.
yield and collapse for piers and abutments; restraint
failure and unseating for bearings and shear keys)
are then derived using the limit states defined in
Nielson (2005): non-linear dynamic analyses on
the simplified bridge model are carried out with 288
synthetic records for an appropriate range of magni-
tude and epicentral distance (see Gehl and D’Ayala
(2014) for more details). In the longitudinal direc-
tion, 10 components are considered (i.e. piers P1
and P2, abutments Ab1 and Ab2, fixed and expan-
sion bearings B1 to B6), as well as in the transversal
direction, except that the bearings are then replaced
by the shear keys.
Fragility curves have been derived for different
scour depths (i.e. from 0 to 5.1 m): based on the
evolution of the bridge response and the fragility
parameters, three threshold levels of scour depth
have been identified: starting from 1 m, some
changes in the bridge response can be observed (i.e.
scour damage state D1); from 3.6 m, the behaviour
of the pier changes dramatically and the effect of
scour is much more noticeable (D2); finally, it is as-
sumed that, after 5.1 m, the pier has almost a pinned
connection to the soil and the stability of the system
cannot be guaranteed (D3).
From Figure 2 it can be observed that scour has
mainly a detrimental effect on the bearings, which
tend to fail earlier as scour depth increases. Since
the pier connection is relaxed at their base, they ex-
perience lower bending moments and their failure
probability decreases for higher scour levels (how-
ever shear failure is not currently modelled). Fi-
nally, the response of abutments seems to remain
3

0 5 10 15
P1
P2
Ab1
B1
B2
B3
B4
B5
B6
Ab2
D1 (EQ) : Yield / Restraint failure
α [m/s
2
]
0 5 10 15
P1
P2
Ab1
B1
B2
B3
B4
B5
B6
Ab2
D2 (EQ): Collapse / Unseating
α [m/s
2
]
scour D0
scour D1
scour D2
Figure 2: Mean fragility parameter α in the longitu-
dinal direction, for the different components and their
two damage states, for different initial states of scour
damage.
stable across the different scour depth. Similar ob-
servations have been made when fragility curves
are computed in the transversal direction.
The effect of previous shear key failure is only
observed for fragility curves in the transversal di-
rection: the influence of shear keys is not signif-
icant for lower damage states (i.e. yield), however
the absence of shear keys leads to a much earlier oc-
currence of heavier damage states, especially deck
unseating.
4.2. Ground failure
As discussed in the previous section, earthquake-
triggered ground failures are more likely to affect
the components that are located at the bridge ex-
tremities, namely the approach embankment and
the abutment foundations. First, the fragility of ap-
proach embankments with respect to lateral spread-
ing of the supporting soil is taken from Kaynia et al.
(2012). Using the first two damage states (i.e. slight
damage D1 with permanent ground displacement
of 3 cm and moderate D2 with permanent ground
displacement of 15 cm), fragility parameters for
D1 (respectively D2) are the mean α1 = 1.96 m/s2
(resp. α2 = 4.12 m/s2) and the standard deviation
β1 = 0.70 (resp. β2 = 0.70), for a European soil
type D and an embankment height of 2 m.
Regarding the abutments, their deep foundations
prevent the occurrence of damage due to superfi-
cial landslides with a planar sliding surface: how-
ever the damage due to a deep-seated circular land-
slide that occurs below the foundations may have
to be considered. In this case, the factor of safety
FS is estimated with the limit equilibrium method
(i.e. Bishop’s simplified method), assuming a cir-
cular slip surface (see Figure 3). The surface is
subdivided into a number n of vertical slices and
the factor of safety FS is then expressed as the ratio
of resisting versus destabilizing moments of all the
slices.
0 5 10 15 20 25
−2
0
2
4
6
8
10
12
X [m]
Z[m] R
Moment arm L
i
θ
iPore pressure u1
= 0 kPa
Unit weight ρ1
= 18 kN/m
3
Friction angle φ1
~ N(30°,6)
Cohesion c1
~ N(50kPa,20)
Cohesion c
2
~ N(75kPa,30)
Friction angle φ2
~ N(34°,6.8)
Weight W
i
Width bi
Pore pressure u2
= 0 kPa
Unit weight ρ
2
= 18 kN/m3
Figure 3: Slice equilibrium method for the estimation
of the factor of safety. The black shape is a simplified
view of the studied bridge and its foundations.
Fragility functions for slope instability (D1) are
then derived following the method proposed by
Wu (2014): for each increasing value of PGA,
the reliability index of lnFS > 0 is estimated
using a Mean-Value First-Order Second Moment
(MFOSM) method. The input random variables
are the cohesion and friction angle of each soil
layer, while a correlation factor of −0.4 is assumed
between the cohesion and the friction angle (Wu,
2013). The search algorithm for the probabilis-
tic critical surfaces proposed by Hassan and Wolff
(1999) is used in order to ensure that the minimum
reliability index is found for each combination of
the soil parameters, which is different that finding
the surface with the smallest factor of safety. An ad-
ditional constraint is introduced by the location of
the abutment and pied foundations, since the crit-
4

ical surface is unlikely to generate any failures if
it intersects with the bridge foundations. Finally,
the reliability index is converted into the probabil-
ity of failure Pf and the points PGA − Pf are fitted
into a fragility curve with a lognormal cumulative
distribution function: a mean of 12.62 m/s2 and a
standard deviation of 1.03 have been found.
4.3. Scour
For the purpose of the demonstration, only lo-
cal scour at piers is considered, since it could be
assumed that there is no contraction of the river
bed cross-section. Also, general scour due to river
bed degradation is usually neglected with respect
to local scour. Empirical equations from HEC-18
(Richardson and Davis, 1995) are used to quantify
the excavated depth ys due to local scour at piers:
ys = 2·K1 ·K2 ·K3 ·K4 ·y·
D
y
0.65
·F0.43
(1)
Where y represents the flow height, D is the pier
width, F is the Froude number and the Ki param-
eters are corrective coefficients (see Table 2). Fi-
nally, the flow discharge Q can be expressed as a
function of velocity v and height y (Alipour et al.,
2013), with river section width b:
Q = b·y·v =
b·y
n
·
b·y
b+2y
2/3
·S
1/2
0 (2)
The Manning’s roughness coefficient n and slope
grade S0 are specified in Table 2.
Based on the probabilistic distribution of the in-
put parameters, a Monte Carlo scheme is used with
sampled values of flow height y in order to gener-
ate around 10,000 couples of points Q − ys, which
are related by the combination of Equations 1 and
2. This data set is then used to generate fragility
curves with respect to flow discharge Q, using the
three scour depth thresholds that have been previ-
ously defined. A comparative analysis of differ-
ent statistical models has shown that the lognormal
cumulative distribution function is the best fit for
these scour fragility curves. The derived fragility
parameters for damage states D1, D2 and D3 are
Table 2: Values used in the scour equations. Some of
the parameter distributions are taken from Alipour and
Shafei (2012).
Variable Description Distribution
K1 Factor for pier K1 = 1
nose shape
K2 Factor for flow U (1,1.5)
angle of attack
K3 Factor for bed N 1.1,0.0552
condition
K4 Factor for bed K4 = 1
material size
n Manning’s roughness lnN 0.025,0.2752
coefficient
S0 Slope grade lnN 0.02,0.52
means α1 = 2.74 m3/s, α2 = 285.77 m3/s, α3 =
847.62 m3/s and standard deviations β1 = 0.55,
β2 = 0.57 and β3 = 0.53.
4.4. Submersion
While several studies have dealt with the vul-
nerability of bridges due to hurricanes and related
storm surges (Kameshwar and Padgett, 2014; Pad-
gett et al., 2008), the impact of fluvial floods on
bridge superstructures remains difficult to quantify.
Therefore, as a very raw approximation and for the
sake of the demonstration, the fragility curve from
Kameshwar and Padgett (2014) for bridge failure
due to storm surge is taken in order to represent the
probability of deck unseating (D2), while keeping
the coefficient related to wave height equal to 0.
Regarding damage to shear keys (D1), a conser-
vative assumption could consist in considering fail-
ure as soon as the flow height reaches the top of the
pier cap. These threshold values can then be con-
verted in terms of flow discharge using Equation 2.
5. GLOBAL DAMAGE STATES
Global damage states should be defined so that
they are consistent in terms of gravity of dam-
age and consequences on the bridge functional-
ity. Therefore a rationale is proposed here in or-
der to identify homogeneous system damage states
(SDSs) depending on all possible combinations of
component damage states, based on the function-
ality level of the bridge and the repair operations
5

required:
• SDS1 (slight repairs and no closing time):
slight damage to approach embankments (D1)
without a significant impact on the bridge
functionality, even though repair operations
are eventually necessary;
• SDS2 (moderate repairs and short closing
time): structural damage to bridge components
(i.e. piers, abutments, bearings and shear keys
in damage state D1 due to earthquake, ap-
proach embankments in damage state D2 due
to ground failure, damaged shear keys D1 due
to flood);
• SDS3 (extensive repairs and closing time):
deck unseating that induces long term closure
of the bridge, even though temporary deck
spans could be installed if the substructure
components have not collapsed (i.e. deck un-
seating D2 due to flood, bearings and shear
keys in damage state D2 due to earthquake);
• SDS4 (irreparable with full collapse): sub-
structure components have collapsed and in-
duce the total failure of the bridge system (i.e.
piers and abutments in damage state D2 due
to earthquake, scour damage state D3 at pier
foundations, slope failure D1 beneath abut-
ment foundations).
These system damage states can be represented
through a fault-tree analysis, as shown by the exam-
ple in Figure 4, where the cascading failures lead-
ing to the global failure event can be detailed: the
seismic response of all bridge components is de-
pendent on the foundations conditions, therefore
the failure events of bearings and shear keys (i.e.
BEQ,2 and ShEQ,2) should also be expressed as a
function of the damaged pier foundations due to
floor/scour (PfFL,i representing the different scour
damage states). For each scour damage state, the
bearing failure events are differentiated into specific
events B◦
EQ,2, B′
EQ,2 and B′′
EQ,2, in order to repre-
sent the different failure probabilities for different
scour levels (see Figure 2). The same development
has to be applied to the ShEQ,2 event, which also
has to account for the likelihood of having the shear
keys previously damaged by flood, thus doubling
the number of elementary failure events to assess.
Figure 4: Fault-tree analysis for deck unseating, with
the bearing failure event expanded in order to show the
joint effects of earthquake and scour events.
6. MULTI-HAZARD FRAGILITY FUNC-
TIONS
Once the component fragility curves and the sys-
tem damage states have been fully described, it
is possible to derive system-level fragility func-
tions that are able to account for the joint effect
of the different loading mechanisms: since hazard-
specific fragility curves from previous sections are
expressed with either flow discharge Q (i.e. fluvial
flood) or PGA (i.e. earthquake and ground failure),
then it is possible to derive a fragility surface that
expresses the damage failure with respect to two
statistically independent intensity measures.
The fault-tree analysis in Figure 4 for the deck
unseating event may be translated into a Bayesian
Network formulation (see Figure 5): this enables
to clearly identify the potential common cause fail-
ures induced by the hazard types. While the source
events (e.g. EQ or FL) are present at multiple lo-
cations in the fault-tree, they are only represented
once in the Bayesian Network, thus fully emphasiz-
ing the statistical dependency between the different
failure events.
Correlations between failure events can then be
approximated with a Dunnett-Sobel class of Gaus-
sian random variables, as proposed by Song and
Kang (2009): for each component i, the standard-
6

Figure 5: Bayesian Network for deck unseating (SDS3).
Dunnett-Sobel variables are represented in red.
ized safety factor Zi (i.e. ratio of demand over ca-
pacity) can be expressed as a combination of ran-
dom variables such as Zi =
√
1−t2 · Vi + ti · U,
where the tis have to be optimized by fitting the cor-
relation matrix of Zi.
Numerical seismic analysis of the bridge system
enables to obtain a straightforward correlation ma-
trix of the component responses, however this is not
the case for floods and ground failures. It can how-
ever be assumed that flood- and earthquake-related
failures are statistically independent, therefore a
correlation factor of 0 is used between the different
hazard types. The random variable representing the
correlation between events has to be sampled over
the whole support of the standard Gaussian distri-
bution, which requires numerous assessments of the
Bayesian Network: finally, the fragility surfaces for
SDS1 to 4 are displayed in Figure 6. A cumulative
representation of the damage probabilities has been
chosen, however it can be seen that the definition of
the four SDSs does not follow a strictly hierarchi-
cal model (e.g. the probability of reaching SDS4
is higher than the probability of reaching SDS1 for
some combinations of Q and PGA).
It can be seen that the effect of fluvial flood
is mainly significant for heavier system damage
states, such as deck unseating or full collapse: for
these damage states, it has been observed that the
effect of scour or shear key removal has the most in-
fluence on the seismic response of the other bridge
0
10
0
500
1000
0
0.5
1
PGA [m/s
2
]
SDS4
Q [m3
/s] 0
10
0
500
1000
0
0.5
1
PGA [m/s2
]
SDS3
Q [m3
/s]
0
10
0
500
1000
0
0.5
1
PGA [m/s2
]
SDS2
Q [m3
/s] 0
10
0
500
1000
0
0.5
1
PGA [m/s
2
]
SDS1
Q [m3
/s]
Figure 6: Multi-hazard fragility functions for the four
system damage states, expressed as a function of Q and
PGA.
components. Finally, it can checked that the ob-
tained fragility functions are consistent with the
three multi-hazard scenarios that have been previ-
ously defined: if the loading history on the system
is respected, the cumulative damages can be quan-
tified for the various configurations.
7. CONCLUSIONS
This paper has shown through an elaborate ex-
ample that multi-hazard fragility functions can be
assembled from hazard-specific fragility curves and
a corresponding Bayesian Network formulation. It
should be noted that this framework allows to treat
component fragility curves that have been derived
through various techniques (e.g. empirical, analyt-
ical, judgement-based, with plain Monte Carlo or
MFOSM methods), which are usually inherent to
each hazard type. Limited knowledge on a given
quantitative fragility curve could also be addressed
by assigning upper and lower bound to the dam-
age probabilities and analysing their effects on the
system-level fragility functions.
Finally, this study has assumed that all pier foun-
dations or all shear keys are in the same damage
state (i.e. correlation factor of 1) due to fluvial
flood, while there could actually be many config-
urations where only a few components have failed
at the same time. Accounting for this level of de-
7

tail would lead to an almost intractable number of
damage configurations, while the fragility functions
presented here are already based on a large amount
of damage-dependent fragility curves (e.g. assem-
bly of 65 component fragility curves for SDS2).
Such difficulties are mainly due to the nature of
the seismic response of a structural system, where
any component may have an effect on the behaviour
of the other components, and reciprocally. The
use of meta-models or surface responses could be
a reasonable compromise to adjust the intact-state
fragility curves, depending on which components
have been previously damaged.
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