Creating Fragility Functions for Performance-Based Earthquake Engineering:

TL;DR: In this paper, a set of procedures for creating fragility functions from various kinds of data is introduced, including actual EDP at which each specimen failed, bounding EDP, where some specimens failed and one knows the EDP to which each specimens was subjected, capable EDP where specimen EDPs are known but no specimens failed, and derived, where fragility function are produced analytically; expert opinion; and updating, in which one improves an existing fragility model using new observations.

Abstract: The Applied Technology Council is adapting PEER's performance-based earthquake engineering methodology to professional practice. The methodology's damage-analysis stage uses fragility functions to calculate the probability of damage to facility components given the force, deformation, or other engineering demand parameter (EDP) to which each is subjected. This paper introduces a set of procedures for creating fragility functions from various kinds of data: (A) actual EDP at which each specimen failed; (B) bounding EDP, in which some specimens failed and one knows the EDP to which each specimen was subjected; (C) capable EDP, where specimen EDPs are known but no specimens failed; (D) derived, where fragility functions are produced analytically; (E) expert opinion; and (U) updating, in which one improves an existing fragility function using new observations. Methods C, E, and U are all introduced here for the first time. A companion document offers additional procedures and more examples.

Summary

BACKGROUND AND OBJECTIVES

• This paper summarizes such a standard developed for ATC-58.
• See Porter et al. (2006) for more detail, examples, commentary, and alternative approaches.
• Method A is not applicable when one knows the maximum EDP to which each specimen was subjected, but not the value of EDP at which specimens actually failed.
• The methods proposed here are no substitute for understanding the processes that lead to damage, but are intended to help practitioners and scholars create fragility functions from damage data.

DOCUMENTATION REQUIREMENTS

• Four requirements are proposed for documenting fragility functions: 1. Description of specimens.
• Indicate whether EDP is the value at which damage occurred (Method A data) or the maximum to which each specimen was subjected (Methods B, C, and U).
• Define damage measures (DMs) quantitatively in terms of repairs required.

DAMAGE STATE PROBABILITY

• Fdm edp denotes the fragility function for damage state dm, defined as the probability that the component reaches or exceeds damage state dm, given a particular EDP value (Equation 1), and idealized by a lognormal distribution (Equation 2): Fdm edp P DM dm EDP = edp 1.
• Fdm edp = ln edp/xm 2 where denotes the standard normal cumulative distribution function (e.g., normsdist in Microsoft Excel), xm denotes the median value of the distribution, and denotes the logarithmic standard deviation.
• The authors use the lognormal because it fits a variety of structural component failure data well (e.g., Beck et al.
• Both xm and are established for each component type and damage state using methods presented later.
• The probability that the component is in damage state dm, given EDP=edp, is given by.

METHOD A, ACTUAL EDP: ALL SPECIMENS FAILED AT OBSERVED EDP

• These are the most informative data for creating fragility functions.
• They are most common where DM can be associated with a point on the observed force-deformation behavior of a component, such as a yield point.
• Alternatively, specimens are subjected to increasing levels of EDP.
• The test is interrupted after each level of EDP is imposed, and the specimen examined for damage.

M number of specimens tested to failure i index of specimens, i 1,2 , . . .M ri EDP at which damage was observed to occur in specimen i.

• One tests the resulting fragility function using the Lilliefors goodness-of-fit test (presented below).
• If it passes at the 5% significance level, the fragility function is acceptable.
• Example 1. Aslani (2005) provides a table of peak transient drift ratios at which 43 specimens of pre-1976 reinforced concrete slab-column connections experienced cracking of no more than 0.3 mm width, repaired by applying a surface coating.
• The data are repeated in Table 2 with original specimen numbers.
• The lognormal distribution with these parameters passes the Lilliefors goodness-of-fit test at the 5% significance level.

METHOD B, BOUNDING EDP: SOME SPECIMENS FAILED, PEAK EDP KNOWN

• Here, the data include the maximum EDP to which each of M specimens was subjected, and knowledge of whether the specimen exceeded the damage state of interest.
• Data must not be biased by damage state, i.e., specimens must not be selected because they experienced damage.
• The data are grouped into bins by ranges of EDP, where each bin has approximately the same number of specimens in it.
• These serve as independent data points of failure probability and EDP.
• The following approach converts Equation 2 to a linear regression problem by taking the inverse Gaussian cumulative distribution function of each side and fitting a line ŷ=sx+c to the data (e.g., see “probability paper” in Ang and Tang 1975).

N number of EDP bins

• Porter et al. (2006) presents an alternative approach using a leastsquares fit to the binary failure data, i.e., to the pairs of EDP and a binary (0,1) failure indicator.
• The alternative approach avoids errors associated with bin-average EDPs.
• Consider the damage statistics in Figure 2, which depicts motor control centers (MCCs) observed after various earthquakes in 45 facilities.
• Crosshatched boxes represent MCCs that experienced noticeable earthquake effect such as shifting but that remained operable.
• For each bin, the values of xj− x̄ and yj− ȳ are calculated as shown.

METHOD C, CAPABLE EDP: NO SPECIMENS FAILED, EDPS ARE KNOWN

• It addresses the best case for this type of data, i.e., many specimens, none of which had apparent distress, and several of which were subjected to EDP near the maximum value.
• The specimens in this bin without apparent distress are assigned 0% subjective failure probability, 10% for specimens with distress not suggestive of imminent failure, and 50% for specimens with distress suggestive of imminent failure.
• ANCO Engineers, Inc. (1983) performed shake-table tests on ceiling systems with various lateral restraints.
• Peak diaphragm acceleration (PDA) from nine of these tests is recorded in Table 5.
• Failure required replacement of damaged grid and tiles.

METHOD D, DERIVED FRAGILITY FUNCTIONS

• The capacity of some components can be calculated by modeling the component as a structural system, and determining the EDP (e.g., acceleration or shear deformation) that would cause the system to reach dm.
• Other components may be amenable to fault tree analysis; e.g., see Vesely et al. (1981).
• Let r denote the calculated capacity of the component to resist damage state dm, including consideration of any anchorage or bracing.

METHOD E, EXPERT OPINION

• There are several methods for eliciting expert opinion, from ad hoc to structured processes involving multiple experts, self-judgment of expertise, and iteration to examine major discrepancies between experts.
• The method (introduced for the first time here) employs Spetzler and von Holstein (1972) for probability encoding and Dalkey et al. (1970) for expert qualification, with some useful simplifications.
• To use Method E, select experts with professional experience in the design or postearthquake observation of the component.
• Representative images should be offered to the experts and recorded.
• If an expert refuses to provide estimates or limits them to certain conditions, either narrow the component definition accordingly and iterate, or ignore that expert’s response and analyze the remaining ones.

N number of experts providing judgment about a value i index of experts, i 1,2 , . . .N

• If the results of the survey produce 0.4, and this low value of cannot be justified, use the judged xl to anchor the fragility function, apply =0.4, and calculate the resulting value of xm.
• Kennedy and Short (1994) show that by establishing the EDP at which the component has 10% failure probability, the overall reliability of the component is insensitive to , hence the value of directly encoding experts’ judgment of this value in particular.
• Stone cladding on the exterior of retail buildings may fall in earthquakes.
• Consider 2-in. x 6-in. x 1-3/16-in. stone veneer adhered to a concrete masonry unit substrate with thin-bed mortar (liquid latex mixed with Portland cement, 100% coverage).
• Create a fragility function for the probability that any given stone would fall from the building (posing a life-safety threat) and require replacement, as a function of the peak transient drift ratio of the story on which the stone is applied.

METHOD U, UPDATING A FRAGILITY FUNCTION WITH NEW DATA

• Here, the data are a pre-existing fragility function and M specimens with known damage state and maximum EDP.
• It is not necessary that any of the specimens failed.
• The method uses Bayes’ Theorem (e.g., Ang and Tang 1975) to revise xm and of an existing fragility function with new observations of M specimens whose EDP and damage state have been observed.
• For those familiar with Bayesian updating, the prior probability distribution of xm is taken as lognormal with median equal to the xm value in the pre-existing fragility function, and logarithmic standard deviation taken as 0.707 of the pre-existing fragility function, consistent with a compound lognormal fragility function and r= u=0.707 .
• Their joint distribution is approximated by five discrete points (xmj, j), each with probability-like weight wj (where j=1,2 , . . .5).

ASSESSING FRAGILITY FUNCTION QUALITY

• The previous section provided mathematical procedures for developing fragility functions.
• Issues associated with the quality of those fragility functions are now addressed, particularly the treatment of competing EDPs, goodness-of-fit testing, dealing with fragility functions that cross, and how to assign an overall quality level to a fragility function.

CONSIDERING COMPETING EDPS

• One may be uncertain which is the best EDP to use.
• In such a case, create fragility functions for each alternative and choose the fragility function with the lowest .
• See Porter et al. (2006) for choosing between EDPs with differing COV.

GOODNESS OF FIT

• A goodness-of-fit test checks that an assumed distribution adequately fits the data.
• It is a special case of the Kolmogorov-Smirnov (K-S) test, applicable when the parameters of the distribution are estimated from the same data as are being compared with the distribution, as is the case here.

FRAGILITY FUNCTIONS THAT CROSS

• Some components have two or more fragility functions.
• Any two lognormal fragility functions i and j with medians xmj xmi and logarithmic standard deviations i j cross if: edp exp jln xmi − iln xmj j − i : i j edp exp jln xmi − iln xmj j − i : i j 25.
• This produces a negative probability of being in damage state i under Equation 3b.
• Figure 6a illustrates the point: F2 has a higher than F1, and F3 has a lower than F2.
• Two methods are proposed to deal with the problem.

ASSIGNING A SINGLE QUALITY LEVEL TO A FRAGILITY FUNCTION

• Fragility functions come from data with varying quantity and quality.
• It is based solely on the authors’ judgment.
• The analyst should report the quality of fragility functions used with any loss estimate.

CONCLUSIONS

• Six methods for creating fragility functions were presented, including three new ones: one for dealing with cases where no failure has been observed, another for situations where one must rely on expert opinion, and a third for updating an existing fragility function with new damage observations.
• The procedures are under consideration as a standard for ATC-58, a technology-transfer project by the Applied Technology Council to bring PEER’s performance-based earthquake engineering methodology to practice.
• The procedures are intended for engineering professionals who will eventually use PBEE.
• Little unfamiliar math is involved, and no calculus.
• A larger document, Porter et al. (2006), presents these procedures with more commentary, some alternative approaches, and more sample problems.

Figures

Table 1. Analysis methods and data employed

Figure 5. Substituting a sample of five points (bars) for a continuous joint distribution (surface).

Figure 2. Example 2 damage data.

Table 4. Values of exp −z

Table 6. Fragility function quality level

Figure 3. Checking Example 2 results.

Table 3. Example 2 solution data

Figure 6. (a) Fragility functions that cross; (b) solution with Equation 26; and (c) Equation 27.

Figure 4. Form for soliciting expert judgment on component fragility.

Figure 1. Example 1 fragility function (smooth curve) and sample cumulative distribution (stepped curve).

Table 2. Example 1 slab-column connection damage data; s = specimen, r = peak transient drift ratio, %

Full text

EARTHQUAKE ENGINEERING PRACTICE
Creating Fragility Functions for
Performance-Based Earthquake
Engineering
Keith Porter,
a)
M.EERI, Robert Kennedy,
b)
M.EERI,
and Robert Bachman,
c)
M.EERI
The Applied Technology Council is adapting PEER’s performance-based
earthquake engineering methodology to professional practice. The
methodology’s damage-analysis stage uses fragility functions to calculate the
probability of damage to facility components given the force, deformation, or
other engineering demand parameter (EDP) to which each is subjected. This
paper introduces a set of procedures for creating fragility functions from
various kinds of data: (A) actual EDP at which each specimen failed; (B)
bounding EDP, in which some specimens failed and one knows the EDP to
which each specimen was subjected; (C) capable EDP, where specimen EDPs
are known but no specimens failed; (D) derived, where fragility functions are
produced analytically; (E) expert opinion; and (U) updating, in which one
improves an existing fragility function using new observations. Methods C, E,
and U are all introduced here for the first time. A companion document offers
additional procedures and more examples. 关DOI: 10.1193/1.2720892兴
INTRODUCTION
BACKGROUND AND OBJECTIVES
A second-generation performance-based earthquake engineering (PBEE-2) proce-
dure has been developed by the Pacific Earthquake Engineering Research (PEER) Cen-
ter and others that estimates the probabilistic future seismic performance of buildings
and bridges in terms of system-level decision variables (DVs), i.e., performance mea-
sures that are meaningful to the owner, such as repair cost, casualties, and loss of use
(dollars, deaths, and downtime). Under contract to the Federal Emergency Management
Agency, the Applied Technology Council has undertaken to transfer the PEER method-
ology to professional practice (ATC 2005). The methodology involves four stages: haz-
ard analysis, structural analysis, damage analysis, and loss analysis. This paper addresses
the damage analysis, whose input is the engineering demand parameters (EDP) calcu-
lated in the structural analysis, and whose output is the damage measure (DM) of each
a)
California Institute of Technology, Pasadena, CA
b)
RPK Structural Mechanics Consulting, Inc., Escondido, CA
c)
Consulting Structural Engineer, Laguna Niguel, CA
471
Earthquake Spectra, Volume 23, No. 2, pages 471–489, May 2007; © 2007, Earthquake Engineering Research Institute

damageable structural and nonstructural component in the facility. The analysis uses fra-
gility functions, which in this context give the probability of exceeding a damage state (a
value of DM) as a function of EDP. One such fragility function is required for each com-
ponent type and damage state. Many building-component fragility functions have been
created in the past, but no comprehensive set of procedures exists on how to create them.
This paper summarizes such a standard developed for ATC-58. See Porter et al. (2006)
for more detail, examples, commentary, and alternative approaches.
Damage data come in many forms, but generally comprise knowledge of specimen
damage and the EDP imposed. Table 1 lists methods for six situations. Each addresses
different data and thus they are not interchangeable. For example, Method A is not ap-
plicable when one knows the maximum EDP to which each specimen was subjected, but
not the value of EDP at which specimens actually failed. One cannot use Method C if
some specimens failed.
The methods proposed here are no substitute for understanding the processes that
lead to damage, but are intended to help practitioners and scholars create fragility func-
tions from damage data. No calculus is required, and the only possibly unfamiliar ex-
pression is the Gaussian distribution, typically available in spreadsheet software.
DOCUMENTATION REQUIREMENTS
Four requirements are proposed for documenting fragility functions:
1. Description of specimens. What is the component type or taxonomic group the
fragility function addresses? (See Porter 2005 for an ATC-58 component tax-
onomy.) Where and how many specimens were tested or observed, how are they
counted, and what were their materials, material properties, configuration, and
building code (if applicable)? Provide a bibliographic reference of any data
source.
2. Excitation and EDP. Detail the loading protocol or characteristics of earthquake
motion. Identify the EDP(s) examined that might be most closely related to fail-
ure probability and define how EDP is calculated or inferred from the loading
protocol or observed excitation. Indicate whether EDP is the value at which
damage occurred (Method A data) or the maximum to which each specimen was
subjected (Methods B, C, and U).
Table 1. Analysis methods and data employed
Method name Data used
A. Actual failure EDP All specimens failed at observed values of EDP
B. Bounding EDP Some specimens failed; maximum EDP for each is known
C. Capable EDP No specimens failed; maximum EDP for each is known
D. Derived fragility Fragility functions produced analytically
E. Expert opinion Expert judgment is used
U. Updating Enhance existing fragility functions with new method-B data
472 K. PORTER, R. KENNEDY, AND R. BACHMAN

3. Damage evidence and DM. What kinds of physical damage or force-
deformation were observed? Define damage measures (DMs) quantitatively in
terms of repairs required. Damage is assumed to have a repair cost, but note
threats to life-safety or potential for loss of use. Explain how DM is inferred
from damage or force-deformation evidence.
4. Observation summary, analysis method, and results. Present a tabular or graphi-
cal listing of specimens, EDP, and DM. Which method was used to derive the
fragility function (Table 1)? Present resulting fragility function parameters
x
m
and
␤
and results of tests to establish fragility function quality (discussed be-
low). Provide sample calculations.
DAMAGE STATE PROBABILITY
F
dm
共edp兲 denotes the fragility function for damage state dm, defined as the probabil-
ity that the component reaches or exceeds damage state dm, given a particular EDP
value (Equation 1), and idealized by a lognormal distribution (Equation 2):
F
dm
共edp兲⬅P关DM ⱖ dm兩EDP = edp兴共1兲
F
dm
共edp兲 = ⌽
冉
ln共edp/x
m
兲
␤
冊
共2兲
where ⌽ denotes the standard normal (Gaussian) cumulative distribution function (e.g.,
normsdist in Microsoft Excel),
x
m
denotes the median value of the distribution, and
␤
denotes the logarithmic standard deviation.
We use the lognormal because it fits a variety of structural component failure data
well (e.g., Beck et al. 2002, Aslani 2005, Pagni and Lowes 2006), as well as nonstruc-
tural failure data (Reed et al. 1991 [Appendix J], Porter and Kiremidjian 2001, Badillo-
Almaraz et al. 2006), and building collapse by IDA (e.g., Cornell et al. 2005). It has
strong precedent in seismic risk analysis (e.g., Kennedy and Short 1994, Kircher et al.
1997). Finally, there is a strong theoretical reason to use the lognormal: it has zero prob-
ability density at and below zero EDP, is fully defined by measures of the first and sec-
ond moments—
ln共x
m
兲 and
␤
—and imposes the minimum information given these con-
straints, in the information-theory sense (Goodman 1985).
Both
x
m
and
␤
are established for each component type and damage state using
methods presented later. The probability that the component is in damage state dm,given
EDP= edp,isgivenby
P关DM = dm兩EDP = edp兴 =1−F
1
共edp兲 dm =0
=F
dm
共edp兲 − F
dm+1
共edp兲 1 ⱕ dm ⬍ N
=F
dm
共edp兲 dm = N 共3兲
CREATING FRAGILITY FUNCTIONS 473

where N denotes the number of possible damage states for the component, in addition to
the undamaged state, and
dm=0 denotes the undamaged state. Where Nⱖ2 and
␤
i
⫽
␤
j
for two damage states i ⫽ j , Equation 3 can produce a meaningless negative prob-
ability at some levels of EDP. This situation is addressed later.
CREATING FRAGILITY FUNCTIONS
This section provides mathematical procedures for developing fragility functions.
METHOD A, ACTUAL EDP: ALL SPECIMENS FAILED AT OBSERVED EDP
These are the most informative data for creating fragility functions. They are most
common where DM can be associated with a point on the observed force-deformation
behavior of a component, such as a yield point. Alternatively, specimens are subjected to
increasing levels of EDP. The test is interrupted after each level of EDP is imposed, and
the specimen examined for damage. Let
M⫽number of specimens tested to failure
i⫽index of specimens, i 苸 兵1,2, ...M其
r
i
⫽EDP at which damage was observed to occur in specimen i.
From the basic definitions of x
m
and
␤
共e.g., Ang and Tang 1975兲,
x
m
= exp
冉
1
M
兺
i=1
M
ln r
i
冊
␤
=
冑
1
M −1
兺
i=1
M
共ln共r
i
/x
m
兲兲
2
共4兲
One tests the resulting fragility function using the Lilliefors goodness-of-fit test (pre-
sented below). If it passes at the 5% significance level, the fragility function is accept-
able.
Example 1. Aslani (2005) provides a table of peak transient drift ratios at which 43
specimens of pre-1976 reinforced concrete slab-column connections experienced crack-
ing of no more than
0.3 mm width, repaired by applying a surface coating. The data are
repeated in Table 2 with original specimen numbers. Calculate the fragility function and
test goodness of fit.
Solution. The data are sorted in order of increasing r, an index i is added, the sta-
tistics
ln共r
i
兲 and ln共r
i
/x
m
兲
2
calculated and summed. Using Equation 4, x
m
=0.38, and
␤
=0.39. The lognormal distribution with these parameters passes the Lilliefors
goodness-of-fit test at the 5% significance level. The math is omitted here, but the test is
illustrated in Figure 1.
METHOD B, BOUNDING EDP: SOME SPECIMENS FAILED, PEAK EDP KNOWN
Here, the data include the maximum EDP to which each of M specimens was sub-
jected, and knowledge of whether the specimen exceeded the damage state of interest.
474 K. PORTER, R. KENNEDY, AND R. BACHMAN

Some specimens must be damaged. The method works best where M ⱖ 25. Data must
not be biased by damage state, i.e., specimens must not be selected because they expe-
rienced damage. The data are grouped into bins by ranges of EDP, where each bin has
approximately the same number of specimens in it. For each bin, one calculates the frac-
tion of specimens that failed and the bin-average EDP. These serve as independent data
points of failure probability and EDP. The following approach converts Equation 2 to a
linear regression problem by taking the inverse Gaussian cumulative distribution func-
tion of each side and fitting a line
y
ˆ
=sx+c to the data (e.g., see “probability paper” in
Ang and Tang 1975). Let (cont.)
Table 2. Example 1 slab-column connection damage data; s = specimen, r = peak transient drift
ratio, %
s rsrsrsrsr s r
3 0.43 11 0.28 22 0.40 60 0.54 72 0.19 80 0.43
4 0.30 12 0.35 23 0.36 61 0.40 73 0.28 81 0.50
5 0.28 16 0.31 24 0.28 62 0.80 74 0.28 82 0.70
6 0.65 17 0.31 25 0.20 66 0.50 75 0.40
⌺ ln r −41.6
7 0.22 18 0.28 26 0.50 67 0.50 76 0.74 x
m
0.38
8 0.32 19 0.22 27 0.25 68 0.50 77 0.54
⌺ ln共r/ x
m
兲
2
6.40
9 0.43 20 0.22 28 0.50 69 0.50 78 0.43
␤
0.39
10 0.42 21 0.31 59 0.64 71 0.19 79 0.71
(Specimens without PTD data are omitted)
Figure 1. Example 1 fragility function (smooth curve) and sample cumulative distribution
(stepped curve).
CREATING FRAGILITY FUNCTIONS 475

Frequently asked questions

Q1. What have the authors contributed in "Creating fragility functions for performance-based earthquake engineering" ?

This paper introduces a set of procedures for creating fragility functions from various kinds of data: ( A ) actual EDP at which each specimen failed ; ( B ) bounding EDP, in which some specimens failed and one knows the EDP to which each specimen was subjected ; ( C ) capable EDP, where specimen EDPs are known but no specimens failed ; ( D ) derived, where fragility functions are produced analytically ; ( E ) expert opinion ; and ( U ) updating, in which one improves an existing fragility function using new observations. 

Q2. what is the fragility function for a stone?

Create a fragility function for the probability that any given stone would fall from the building (posing a life-safety threat) and require replacement, as a function of the peak transient drift ratio of the story on which the stone is applied. 

Q3. What is the value of encoding experts’ judgment of the value of a component?

Kennedy and Short (1994) show that by establishing the EDP at which the component has 10% failure probability, the overall reliability of the component is insensitive to , hence the value of directly encoding experts’ judgment of this value in particular. 

Q4. how to create a fragility function from method c data?

To create a fragility function from Method-C data, letri EDP experienced by specimen i i=1,2 , . . .M rmax=maxi ri rd minimum EDP experienced by any specimen with distress ra the smaller of rd and 0.7rmax MA number of specimens without apparent distress and with ri ra MB number of specimens at any ri with distress not suggestive of imminent failure MC number of specimens at any ri with distress suggestive of imminent failure rm=rmax if MB+MC 0=0.5· rmax+ra otherwise S subjective failure probability at rmS = 0.5MC + 0.1MB / MA + MB + MC 14Use Table 4 to determine Fdm rm and Equation 15 to determine and xm.= 0.4z = −1 Fdm rmxm = rmexp − z 15Example. 

Q5. What is the prior probability distribution of xm?

For those familiar with Bayesian updating, the prior probability distribution of xm is taken as lognormal with median equal to the xm value in the pre-existing fragility function, and logarithmic standard deviation taken as 0.707 of the pre-existing fragility function, consistent with a compound lognormal fragility function and r= u=0.707 . 

Q6. What is the subjective failure probability for a bin of specimens without distress?

The specimens in this bin without apparent distress are assigned 0% subjective failure probability, 10% for specimens with distress not suggestive of imminent failure, and 50% for specimens with distress suggestive of imminent failure. 

Q7. What is the only possible expression for calculating fragility functions?

No calculus is required, and the only possibly unfamiliar expression is the Gaussian distribution, typically available in spreadsheet software. 

Q8. How many times does an expert say that the EDP is a given?

0.4xm = 1.67xl 18 Regarding Equation 18, it is common for experts to express overconfidence in an uncertain variable, such as the EDP at which damage will occur. 

Q9. What is the purpose of this paper?

This paper introduces a set of procedures for creating fragility functions from various kinds of data: (A) actual EDP at which each specimen failed; (B) bounding EDP, in which some specimens failed and one knows the EDP to which each specimen was subjected; (C) capable EDP, where specimen EDPs are known but no specimens failed; (D) derived, where fragility functions are produced analytically; (E) expert opinion; and (U) updating, in which one improves an existing fragility function using new observations. 

Q10. What is the probability that the component is in damage state?

The probability that the component is in damage state dm, given EDP=edp, is given by=Fdm edp − Fdm+1 edp 1 dm N=Fdm edp dm = N 3where N denotes the number of possible damage states for the component, in addition to the undamaged state, and dm=0 denotes the undamaged state. 

Q11. How do you calculate the fragility function of a ceiling?

The capacity of some components can be calculated by modeling the component as a structural system, and determining the EDP (e.g., acceleration or shear deformation) that would cause the system to reach dm. 

Q12. What is the simplest way to determine the fragility of a stone?

The method uses Bayes’ Theorem (e.g., Ang and Tang 1975) to revise xm and of an existing fragility function with new observations of M specimens whose EDP and damage state have been observed. 

Q13. What is the method for eliciting expert opinion on uncertain quantities?

To properly elicit expert opinion on uncertain quantities requires attention to clear definitions, biases, assumptions, and expert qualifications.