Hazard-Consistent Intensity Measure Conversion of Fragility Curves

Abstract

In seismic risk assessment of structures, fragility functions are the typical representation of seismic vulnerability, expressing the probability of exceedance of a given performance level as a function of a ground motion intensity measure (IM). Fragility curves, in general, are structure-and site-specific, thus a comparison of fragility curves is not straightforward across multiple structures and/or sites. The study presented in this paper discusses possible strategies to convert a fragility curve from an original IM to a target IM for a given site. In particular, three conversion cases, under different assumptions on the explanatory power with respect to structural failure of the involved IMs, are considered: (i) a vector-valued IM consisting of two different IMs (to say, original and target), magnitude, and source-to-site distance, (ii) a vector-valued IM consisting of the original and target IMs, and (iii) the original IM only, supposed to be a sufficient one; i.e., the structural response given IM statistically-independent of the other ground motion characteristics. The original fragility functions are supposed to be obtained through the state-of-the-art methods, then the fragility functions in terms of the target IM are obtained via applications of the probability calculus rules, which ensure consistency with the seismic hazard at the site of interest. The considered cases are illustrated via an example referring to an Italian code-conforming RC building designed for a site in L'Aquila. As far as the case-study is concerned, all conversion cases show agreement, likely because of the hazard-consistent record selection and to the explanatory power of the original IM with respect to structural failure.

Full text

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
Hazard-Consistent Intensity Measure Conversion of Fragility
Curves
Akiko Suzuki
Graduate Student, Dipartimento di Strutture per l’Ingegneria e l’Architettura, Università degli Studi di
Napoli Federico II, Naples, Italy
Iunio Iervolino
Professor, Dipartimento di Strutture per l’Ingegneria e l’Architettura, Università degli Studi di Napoli
Federico II, Naples, Italy
ABSTRACT: In seismic risk assessment of structures, fragility functions are the typical representation
of seismic vulnerability, expressing the probability of exceedance of a given performance level as a
function of a ground motion intensity measure (IM). Fragility curves, in general, are structure- and site-
specific, thus a comparison of fragility curves is not straightforward across multiple structures and/or
sites. The study presented in this paper discusses possible strategies to convert a fragility curve from an
original IM to a target IM for a given site. In particular, three conversion cases, under different
assumptions on the explanatory power with respect to structural failure of the involved IMs, are
considered: (i) a vector-valued IM consisting of two different IMs (to say, original and target), magnitude,
and source-to-site distance, (ii) a vector-valued IM consisting of the original and target IMs, and (iii) the
original IM only, supposed to be a sufficient one; i.e., the structural response given IM statistically-
independent of the other ground motion characteristics. The original fragility functions are supposed to
be obtained through the state-of-the-art methods, then the fragility functions in terms of the target IM are
obtained via applications of the probability calculus rules, which ensure consistency with the seismic
hazard at the site of interest. The considered cases are illustrated via an example referring to an Italian
code-conforming RC building designed for a site in L’Aquila. As far as the case-study is concerned, all
conversion cases show agreement, likely because of the hazard-consistent record selection and to the
explanatory power of the original IM with respect to structural failure.
1. INTRODUCTION
Probabilistic seismic risk assessment of
structures evaluates the mean annual frequency of
exceeding a given performance level (i.e., failure
rate) by integrating seismic fragility and seismic
hazard, both expressed in terms of the same
ground motion (GM) intensity measure (IM)
serving as a link between the two probabilistic
models. The choice of the IM to be employed in
the risk analysis is structure-specific. In principle,
it is mainly determined by the desired properties
of the selected IM, e.g., sufficiency and efficiency,
and also considering issues such as robustness to
GM scaling (Tothong and Luco, 2007).
In earthquake engineering practice, the peak
ground acceleration (PGA) and the spectral
acceleration at the fundamental vibration period
of the structure,


Sa T
, are common IMs. PGA is
convenient because hazard models are typically
developed in terms of PGA.


Sa T
is generally
considered more efficient than PGA, and
sufficient in several situations. Hence, it is often
used as the IM for the development of fragility
functions; however, fragility expressed in terms of
spectral acceleration at different vibration periods
cannot be directly compared.
Several studies addressed approaches for
converting IMs of fragility curves in the last
decades. For example, Ohtori and Hirata (2007)

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
2
presented an approach based on the first-order
second moment approximation although the
relationship between original and target IMs (to
convert from and to, denoted as
1
IM
and
2
IM
,
respectively) is not completely characterized.
Michel et al. (2018) recently proposed a
probabilistic approach to convert fragility curves
to a common IM considering the conditional
probability of
2
IM
given
1
IM
, yet the converted
fragility functions in terms of
2
IM
are conditional
to a specific earthquake scenario, beyond
2
IM
.
Furthermore, these past studies seem to assume
1
IM
of a specific structure to be sufficient, which
is not the general case.
Aiming at discussing fragility conversion
between spectral accelerations in a rigorous
probabilistic framework, the study presented
herein examines three conversion cases, with
different assumptions on the IMs involved. In
particular, the fragility curve of a structure in
terms of the target intensity (i.e.,
2
IM
) is derived
through hazard-consistent conversion of a
fragility function varying the number of the
original IMs; i.e.,
1
IM
,
2
IM
and the GM
characteristics for hazard assessment of the site.
The IMs considered in each case are: (i) a vector-
valued intensity measure (
ΙΜ
) consisting of
1
IM
,
2
IM
, magnitude


M
and source-to-site
distance


R
, hereafter denoted as
4v
ΙΜ
; (ii) a
vector-valued
ΙΜ
consisting of
1
IM
and
2
IM
,
hereafter denoted as
2
v
ΙΜ
; and (iii) the original
1
IM
which is supposed to be a sufficient IM.
The converted fragility functions are
obtained with the state-of-the-art methods for
structural response analysis within the
Performance-Based Earthquake Engineering
framework (PBEE; Cornell and Krawinkler,
2000), which ensures consistency with the
earthquake scenarios at the site of interest; i.e., the
multiple stripe analysis method (MSA; Jalayer
and Cornell, 2003), and hazard-consistent record
selection based on the conditional spectra (CS;
Lin et al., 2013).
The considered cases are investigated for an
Italian code-conforming RC building for which
the fragility conversion is performed from the
spectral acceleration at a period close to the
fundamental vibration period of the structure (i.e.,
1
IM
) to PGA (i.e.,
2
IM
). For comparison, a
fragility curve expressed in terms of
2
IM
(i.e.,
PGA) is also evaluated by performing nonlinear
dynamic analyses (NLDAs) carried out with
reference to
2
IM
.
2. METHODOLOGY
The considered framework assumes that
structural response data are obtained through
NLDA, aiming to assess the fragility in terms of
1
IM
, and intends to convert to
2
IM
without
carrying out further structural analyses.
2.1. Conversion equations
When structural failure (denoted as F) is
defined as the exceedance of a given performance
level, the probability of failure given a certain


y
value of the target
2
IM
, that is


2
|
P F IM y

,
can be computed via Eq. (1) based on the total
probability theorem. In the equation: the first term
of the integrands
1 2
P F IM x IM y M w R z
 
      
 
is the
failure probability conditional to


1 2
, , ,
IM IM M R
;


1 2
, , |
, , |
IM M R IM
f x w z y
is a
probability density function (PDF), given by the
product of the following two PDFs:


1 2 , ,
, ,
IM IM M R
f x y w z
and


2
,,
M R IM
f w z y
. The
former can be obtained from a ground motion
prediction equation (GMPE), considering the
statistical dependency between
1
IM
and
2
IM
conditional to
M
and
R
, while the latter is
computed through seismic hazard disaggregation
(e.g., Bazzurro and Cornell, 1999) that provides
the probability (density) of a certain
M
and
R
scenario given the occurrence of
2
IM
.


1 2
1
2 1 2 , , | , , |
IM M R IM
IM M R
P F IM y P F IM x IM y M w R z f x w z y dz dw dx
   
            
   
  
(1)

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
3
If the vector


1 2
,
IM IM
is a sufficient
ΙΜ
,
the structural response given
ΙΜ
is, by definition,
statistically-independent of
M
and
R
, and Eq.
(1) can be reduced to Eq. (2).
If the original
1
IM
is a sufficient one (also
with respect to
2
IM
), then Eq. (2) can be further
simplified as per Eq. (3).
2.2. Fragility modelling
The fragility functions within the integrals at
the right-hand sides of Eqs. (1)-(3) can be
obtained via NLDA, which provides the seismic
demand in terms of an Engineering Demand
Parameter (EDP). The modelling approaches
discussed herein assume structural assessment
procedures such as the cloud method (Cornell et
al., 2002) and MSA method. The former typically
employs a set of unscaled GM records, while the
latter approach employs, for different values of
1
IM
, different record sets each of which is
selected consistent with the seismic hazard at the
site of interest in terms of
1
IM
.
Log-linear regression models are often
employed to calibrate the relationship between the
EDP and IMs. For example, in the case of
4
v
ΙΜ
,
the logarithm of EDP, in its simplest format, is
given by Eq. (4), where
ln
EDP
is the conditional
mean,


0 1 2 3 4
, , , ,
    
are regression
coefficients, and

(i.e., the regression residual)
is a zero-mean Gaussian random variable, with
standard deviation


. At this point, if
f
EDP
is a
threshold identifying failure, the lognormal
fragility is given by Eq. (5) (Baker, 2007).
It should be noted that the numerical model
of the structure does not yield meaningful EDP
values in cases numerical instability, according to
the definition in Shome and Cornell (2000),
occurs. Even in such cases, one can derive a
fragility model that accounts for the contribution
from these data (e.g., Elefante et al., 2010).
Furthermore, in case of MSA, the fragility
function can be also modelled estimating different
regression coefficients for each
1
IM
stripe (an
option not considered in the following
application).
2.3. Hazard terms
The calculations to obtain 1 2
, ,
IM IM M R
f have
been discussed in previous research (e.g., Baker
and Cornell, 2005; Iervolino et al., 2010)
Provided that
1
T
and
2
T
denote the vibration
periods corresponding to


1 1
IM Sa T
 and


2 2
IM Sa T
, it is often assumed that 1 2
, ,
IM IM M R
f
is a lognormal distribution. The mean value of
1
ln
IM
given


2
ln , ,
IM M R
and the standard
deviation of
1
ln
IM
given
2
ln
IM
, denoted as
1 2
ln ln , ,
IM IM M R

and
1 2
ln ln
IM IM

, respectively, can
be calculated as:
1 1 2 2 1
1 2
1 2 1
1 2
ln | , , ln
ln ln , ,
2
, ln
ln ln 1
IM M R T T T IM
IM IM M R
T T IM
IM IM
    
  
   

  


(6)
where 1
ln | ,
IM M R

and
1
ln
IM

are the mean and
standard deviation of
1
ln
IM
conditional to
M
and
R
(i.e., from a GMPE),
1 2
,
T T

is the
correlation coefficient between the logarithms of


1 2
1
2 1 2 , , | , , |
IM M R IM
IM M R
P F IM y P F IM x IM y f x w z y dz dw dx
   
        
   
   (2)


1 2
1
2 1 , , | , , |
IM M R IM
IM M R
P F IM y P F IM x f x w z y dz dw dx
   
      
   
   (3)
0 1 2 3 4
ln ln ln ln lnEDP EDP x y w z
      
           
(4)
ln ln
1f
f
EDP EDP
P F P EDP EDP


 

   
    
 
   
 
 
IM IM (5)

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
4
the two IMs, and
2
T

is the number of the standard
deviations by which
2
ln
IM
is away from the
mean conditional to
M
and
R
(provided by the
GMPE). Finally,
2
,
M R IM
f can be computed via
hazard disaggregation.
3. ILLUSTRATIVE EXAMPLE
3.1. Structure and site
This study considered, as an example, one of
the code-conforming RC residential buildings
examined in the RINTC project; see Ricci et al.
(2018) for details. In particular, the selected case
study is the equivalent single-degree-of-freedom
(ESDoF) system of the six-story
5 3

bay RC
pilotis-frame building in Figure 1a. The building
is supposed to be located in L’Aquila, central Italy
(42.35° N, 13.40° E, on soil C-type, according to
Eurocode 8 classification; see CEN, 2004). The
ESDoF system was calibrated according to
Baltzopoulos et al. (2017); i.e., leading to the
ESDoF mass
*
m
, vibration period
*
T
, the critical
viscous damping ratio
*

, and the
characterizations of the hysteretic behavior for the
static pushover (SPO) backbone. Figure 1b shows
the tri-linear backbones of the ESDoF systems for
the two horizontal directions (X and Y in Figure
1a). Each backbone of the ESDoF system is
compared with that of the original three
dimensional (3D) structural model scaled by the
modal participation factor of the first-mode
vibration,

. In the figure, the definitions of
backbone parameters, that is, yield strength and
displacement


,
y y
F

 
, post-yielding hardening
and softening ratio


,
h c
a a
, capping ductility


c c y
  
 
, and the residual strength ratio


p
r
, are also illustrated.
The EDP was defined as the demand-
capacity (D/C) ratio of the roof-top displacement
(i.e.,
1
f
EDP

) and the end of each backbone
corresponds to the failure ductility


,
f f y
  
 

which was defined on the basis of the
displacement that determines a 50% drop from the
maximum base-shear on the original structure’s
SPO curve. These parameters, including the
yielding spectral acceleration at the equivalent
vibration period


y
Sa T

, are summarized in
Table 1 and Table 2.
Finally, a moderately-pinching, peak-
oriented hysteretic behavior without any cyclic
stiffness/strength deterioration (e.g., Vamvatsikos
and Cornell, 2006), was applied to the ESDoF
system. For more detailed information on
structural features, see Suzuki et al. (2018).
3.2. Original fragilities
To derive the original fragilities, MSA was
performed for ten IM, that is


0.5
Sa T s
,
values (i.e., stripes). The considered values
correspond to exceedance return periods


R
T
ranging from 10 to 105 at the site, in line with the
framework of the aforementioned RINTC project.
For each stripe, 20 GM records were collected
based on the CS method (Iervolino et al., 2018)
then the EDP was measured for each record.
Figure 2 shows the mean spectra of the GM
records and the obtained D/C ratios for all stripes
Table 1: Dynamic parameters of the ESDoF system.
Dir.
T

m




y
Sa T



X
0.65
1401
1.26
0.27
5%
Y
0.57
1251
1.33
0.37
5%
Table 2: SPO parameters of the ESDoF system.
Dir.
y
F

y


h
a
c

c
a
p
r
f

X
3671
0.0
3
0.0
1
9.0
-
0.02
0.66
30.0
Y
4581
0.03
0.17
2.2
-
0.03
0.67
17.9
Figure 1: Case study RC building;
(a) 3D model; (b) SPO curves.
0.5
1.0
4000
2000
F [k N]
 [m]
X
Y
X
0
0

y
*

y
*
f
*

f

c
F
y
*a K*
h
a K*
c

c
*
K*=F
y y
*/ *

(a) (b)
Y
X
6000
r =F /F
p p y
*
linear-fit (ESDoF)
-scaled (3D)
Y

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
5
of
1
IM
, against the two IMs. The observed trends
are then characterized by performing regression
analysis via Eq. (4).
Given the response data from NLDA, the
multiple linear regression analyses were
performed with Eq. (4) varying the number of the
intensity measures involved in the computation.
According to the adopted GMPE (to follow), the
considered
Sa
-based IM is the maximum
horizontal acceleration response, and the
considered GM characteristics were surface wave
magnitude and Joyner-Boore distance (
jb
R
;
Joyner and Boore, 1981). Table 3 provides the
regression results corresponding to the three IM
cases. It is observed that the joint consideration of
all four variables resulted in the lowest


.
Nonetheless, all the cases showed a comparable
standard deviation of the residuals.
Table 3: Multiple linear regression analysis results.
IM
0

1

2

3

4



4
v
IM
-2.71 0.72 0.19 0.14 -0.11 0.41
2
v
IM
-2.12 0.88 0.17 - - 0.43
1
IM
-2.23 1.03 - - - 0.44
For each case, the related fragility function
(i.e., P F
 
 
IM
) was derived using Eqs. (4)-(5).
As an example, the computed fragility (surface) is
shown in Figure 3 for the case of
2
.
v
IM It can be
seen that the failure probability increases
principally with
1
IM
and mildly with
2
IM
for a
given
1
IM
, which reflects the regression results in
Table 3.
Figure 3: Fragility surface for the IM2v case.
3.3. Hazard
PSHA was performed to characterize the
conditional distribution 1 2
, ,
IM IM M R
f for the site of
interest. For the conditional PDF of
1
IM
given
2
IM
, this study employed the GMPE by
Ambraseys et al. (1996) with the correlation
coefficients proposed by Baker and Jayaram
(2008). The correlation coefficient between PGA
and


0.5
Sa s
was 1 2
,
0.68
T T


.
To perform hazard disaggregation for the
site, this study utilized REASSESS (Chioccarelli
et al., 2018) considering the Branch 921 of the
official Italian hazard model (Stucchi et al., 2011)
and the GMPE cited above.
IM = PGA [g]
2
IM = Sa(0.5s) [g]
1
P
[
F
|
I
M
,
I
M
]
1 2
1
0.8
0.6
0.4
0.2
0
8
6
4
2
00
2
4
6
8
Figure 2: MSA for original fragility; (a) mean spectra of GM records; (b),(c) D/C ratios for the two IMs.
IM = Sa(T )
1 1
IM =PGA
2
mean maximum horizontal component
14
Periods [s]
S
a
(
T
)
IM
1
=Sa(0.5s) [g]
Num. of failure
IM
2
=PGA [g]
5 2
(a)
(b) (c)
D/C
EDP=1
f
EDP=1
f

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
6
As an example, Figure 4a shows hazard
disaggregation for PGA equal to 0.9g at the site of
interest (i.e., L’Aquila). Figure 4b gives
1 2
, ,
IM IM M R
f for


6, 6.5 km
M and


0km,5km
jb
R, which is the scenario
dominating the hazard being disaggregated in
Figure 4a.
3.4. Reference target fragility
To evaluate the considered conversion cases,
the PGA-based (reference) fragility curve was
also computed by performing NLDA via MSA;
i.e., as in Section 3.2 but with respect to PGA
rather than


0.5
Sa s
. The record selection was
analogous to


0.5
Sa s
(i.e., 20 GM records
fitting the CS given PGA). Figure 5 provides the
mean spectra of GM records conditioned at PGA
and the MSA results. A smaller number of failure
cases was observed in this case (5 vs 14 failure
cases at the tenth IM level; see Figure 2). Hence,
the analysis was also performed at the two
additional IM levels corresponding to
6
10
R
T
and
7
10
R
T years so as to observe failure cases
in at least 50% of the records. The target
lognormal fragility was fitted through a maximum
likelihood estimation criterion (Baker, 2015).
3.5. Conversion results
The PGA-based fragility curves were derived
from the presented fragility and hazard models
through the conversion formulas of Eqs. (1)-(3).
The results of the three conversion strategies, each
involving
4
v
IM
,
2
v
IM
and
1
IM
, are presented in
Figure 6, together with the reference fragility
curve described in Section 3.4. As seen in the
figure, the
4
v
IM
case provided the curve closest
to the reference, while all cases provided
apparently comparable results, showing the
curves located slightly at the left side of the
reference case. In fact, the median PGA causing
failure


.50%
f
PGA and the standard deviation



, computed as the difference between the 16th
and 50th percentiles of each converted curve,
Figure 4: PSHA results;(a) hazard
disaggregation for PGA = 0.90g corresponding
to
5580
R
T yrs at L’Aquila; (b) example of
conditional PDF of


0.5
Sa s
given PGA.
Figure 5: MSA for reference target fragility;
(a) mean spectra of GM records;
(b)D/C ratios and number of failure cases.
T = 5580 years
R
L’Aquila Soil C
PGA = 0.90g
R [km]
jb
M
(a)
Sa(0.5s) [g] PGA [g]
M = 6-6.5
R = 0-5km
jb
f (x|w,z,y)
IM1|M ,R, IM 2
(b)
f
(
w
,
z
|
y
)
M,R|IM 2
IM
2
= PGA
IM
1 1
= Sa(T )
mean maximum horizontal component
Periods [s]
S
a
(
T
)
(a)
(b)
11Num. of failure 8 5
IM
2
=PGA [g]
D/C

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
Seoul, South Korea, May 26-30, 2019
7
resulted in less than 12% difference with respect
to the reference target in all cases (Table 4).
Figure 6: Converted and reference fragility curves.
Table 4: Comparisons of the conversion strategies.
Parameter\IM Approach
4
v
IM
2
v
IM
1
IM
Reference
.50%
f
PGA [g] 4.34 3.96 4.19 4.46
Difference -3% -11% -6% 0%

0.65 0.62 0.72 0.71
Difference -8% -12% -2% 0%
All the converted curves appear to be similar,
likely because of the fact that the two GM
characteristics considered herein are accounted
for through the hazard-consistent record selection
via the CS approach, as well as owing to the
relative-sufficiency of the original
1
IM
with
respect to structural response.
4. CONCLUSIONS
The presented study discussed equations for
converting IMs of fragility functions with the aid
of the state-of-the art methods within PBEE,
without any additional structural analyses.
On the premise that structural response given
an IM is available from a preliminary analysis,
three possible conversion cases with different
assumptions on the IMs involved, were discussed.
In all cases, the fragility in terms of the target IM
was computed based on the total probability
theorem.
The considered conversion cases were
illustrated through the application study using the
ESDoF system of an Italian code-conforming RC
building. The PGA fragility curves were obtained
from the structural response given


Sa T
and
then were compared with a reference fragility
directly developed in terms of the target IM.
As far as the case study is concerned, all
cases provided the parameters of the converted
curves within 12% difference from those of the
reference fragility (the four-variables conversion,
involving the original and target IMs, magnitude
and distance, resulted to be the closest as
expected). This is likely owing to the hazard-
consistent record selection according to the CS
approach and to the explanatory power of the
original IM; i.e.,


Sa T
for the structural response
analysis.
5. ACKNOWLEGEMENTS
The study presented in this article was
developed within the activities of the ReLUIS-
DPC 2014-2018 research program, funded by
Presidenza del Consiglio dei Ministri –
Dipartimento della Protezione Civil and the
Horizon 2020 MSCA-RISE-2015 project No.
691213 entitled "Experimental Computational
Hybrid Assessment of Natural Gas Pipelines
Exposed to Seismic Risk” (EXCHANGE-RISK).
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IM = PGA
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P
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13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13
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8
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