INTRODUCTION
Evaluation of seismic fragility functions of structures which defines the probability of physical damage as a function of ground motion intensity parameter has gained importance recently due to its key role in seismic loss assessment and risk management. Although some wellknown fragility databases such as ATC (1985) and Hazus (1999) are available, these fragility functions are developed for general types of structures with substantial amount of assumptions and uncertainties. Due to wide usage of fragility functions in the next generation of seismic design codes (Porter et al., 2007), need for development of structurespecific fragility functions has increased. As an answer to such demand, in this study, a comprehensive stochastic method with a proposed simplified procedure is introduced. From the fragility`s uncertainty point of view, few studied has been conducted. In the recent one, Kwon and Elnashai (2006) studied the effect of uncertainty of material and ground motion on the fragility function and has shown that the ground motion randomness have more effect on the fragility function. Among other factors which affect the uncertainty of the fragility, the uncertainties of damage threshold which describe the physical state of structure have not been investigated. In this study, this effect has been studies by a proposed methodology.
Fragility and vulnerability functions are developed by three main ways: expert opinion, analytical methods and damage data of structures from past events (Porter et al., 2007). Evaluation of fragility curves using existing data of earthquake damage is perhaps the best way to estimate potential damage of future earthquake and has been used for fragility functions development in several studies such as the work which have done by O`Rourke and So (2000), Sabbetta et al. (1998), Shinozoka et al. (2000a) and Sarabandari et al. (2004). In the absence of past damage data, the fragility functions are developed based on the opinion of experts. ATC (1985) is a good example of such approach. Nevertheless, when appropriate analytical tools are available, the analytical method is the proper method for fragility curve development of engineering and special structures.
Two general approaches have been utilized for development of analytical fragility functions: comparing capacity and demand of structures (Dimova and Hirata, 2000; Shinozoka et al., 2000a, b) and employing damage index (Hwang and Huo, 1994; Karim and Yamazaki, 2001, 2003; Smyth et al., 2004). Results of the first approach are more suitable for design purposes (Bazzurro et al., 2004) while the results of the second methodology are more appropriate for the loss estimation purposes due to its ability to define damage states.
Stochastic methods by the means of MonteCarlo simulation and artificial earthquake records generation have been employed for fragility functions (Hwang and Huo, 1994; Karim and Yamazaki, 2001, 2003; Singhal and Kiremidjian, 1996). However, due to existing uncertainties in these methods, demand for a straightforward and rapid procedure of structurespecific fragility function calculation is still existed.
The main objective of this study is demonstrating the procedure of structural fragility function development in a clear and transparent manner and presenting a simple methodology for structurespecific fragility function derivation. Besides, some challenging issues such as effect of damage threshold uncertainty on fragility dispersion and fragility function of structures with nonstructural governing damage cases which is used for industrial facilities are discussed.
DEFINITION OF FRAGILITY FUNCTION AND ESTIMATION METHOD
Due to practical reasons, continuous damage in structures is divided into several
discrete damage states (Porter, 2000). Fragility function estimates the conditional
exceeding probability of damage from a damage state at given ground motion intensity:
where, F_{i} (im) is the probability of exceeding damage D from damage
state d_{i} at given ground motion IM = im. Ground motion intensity
parameter denotes the magnitude of ground motion which is measured by Peak Ground
Acceleration (PGA), Peak Ground Velocity (PGV) or Spectral Displacement (SD).
Damage states i are defined from the nondamage state (i = 0) to the n^{th}
damage state (i = n) by qualitative and analytical definitions (Porter, 2000).
Since damage in structures is measured by Damage Index (DI), Eq.
1 is changed to:
where, di_{t} is the damage index at the threshold of damage states.
Having the Probability Density Function of DI or its cumulative distribution
function at every im (f_{im} (di) and F_{im}(di_{t})),
Eq. 2 is evaluated from probabilistic theorem:
In this study, PDF of DI is evaluated by multistripe analysis used by Jalayer
(2003) and Aslani and Miranda (2004). In this method, structure is analyzed
subjected to several real ground motion records that are all scaled to specific
IM level and distribution of structural response in the particular IM is estimated
from the results of the nonlinear analysis set.

Fig. 1: 
Procedure of estimating analytical fragility function 
Based on these assumptions, procedure of fragility curve development for real
structure(s) is summarized in five major steps shown in flowchart of methodology
given in Fig. 1:
• 
Selecting structure(s) with similar structural category and/or
behavior 
• 
Choosing a damage index and ground motion intensity measurement 
• 
Selecting group of time history records and scaling them to selected IMs
values. The selected records should represent the randomness of ground motion 
• 
Estimating the distribution of damage index at the selected IMs through
multistripe analyses and fitting proper distribution function to the results 
• 
Calculating fragility values using Eq. 3 and fitting
appropriate function to the results. 
Implementation of the above procedure is illustrated in the following example.
ILLUSTRATIVE EXAMPLE
Two groups of lowrise steel frame which are compatible with highcode and
moderatecode structural classifications of HAZUS were selected (Hazus, 1999).
According to HAZUS, highcode frame corresponds to ductile steel structures
designed for 0.133 fraction of building weight and moderatecode frame corresponds
to semiductile steel frames which is designed for 0.067 fraction of building
weight. For each classification, two lowrise steel moment resisting frames
(two bay; two and three story) which are selected from a middle frame in a hypothetical
building were designed. The elevation views of the frames are shown in Fig.
2.
Following assumptions are made for fragility curve development:
• 
Spectral Displacement (SD) is chosen as IM 
• 
ISD is selected for damage index, due to its good representation of damage
for structural and most of nonstructural elements (Porter, 2000) 
• 
Mean damage thresholds of HAZUS shown in Table 1 are
chosen as medium threshold of damage states (Hazus, 1999) 

Fig. 2: 
Elevation of frames selected for analysis; moderate code is
in parentheses (a) two story frames and (b) three story frames 
For the evaluation of ISD distribution in each ground motion value, a set of
20 records with a uniform distribution of sourcetosite at a minimum distance
of 10 km to reduce the near source effect were selected from the PEER (2007)
ground motion database (Table 2). The distribution of sourcetosite
distance with the magnitude of selected records is shown in Fig.
3. The records are scaled so that the 5% damped Spectral Displacement of
records at the period of structures (0.13 and 0.149 for moderate and high code)
equals to 2.5, 5, 10, 20, 30, 40, 50, 60, 70, 80 and 90 cm. Structures were
analyzed by OpenSees (2006) subjected to scaled records. PΔ effect was
considered and FEMA356 forcedeformation relationship (FEMA356, 2000) was
selected for the plastic hinge property of the structural elements.
The distribution of maximum ISD which is estimated from the analyses of moderate
and high code structures are shown in Fig. 4, 5
where the mean thresholds of damage states from Table 1 are
shown as well. To find appropriate distribution function for ISDs, normal and
lognormal distribution functions were tested. Even though lognormal distribution
rather inappropriate for higher level of SDs especially in the moderate code
frames, it is generally better fitted to ISD distributions in general. Mean
and lognormal deviation (IŜD_{sd} and β_{sd}) of ISD
distributions of highcode frames are shown in Table 3.
Table 1: 
Interstory drift ration at thresholds of different damages
states (Hazus, 1999) 

Table 2: 
Selected strong motion records for nonlinear dynamic analysis
(PEER, 2007) 


Fig. 3: 
Distribution of distance and magnitude of selected records 

Fig. 4: 
Maximum ISD of moderate code design frames; horizontal lines
are damage thresholds 
Table 3: 
Mean and deviation values of lognormal distribution of ISD
data in each SD for high code frame 

Table 4: 
Parameters of fragility functions estimated from comprehensive
procedure 


Fig. 5: 
Maximum ISD of high code design frames; horizontal lines are
damage thresholds 

Fig. 6: 
Fragility values and functions for moderated code design steel
moment resistant frame 

Fig. 7: 
Fragility values and functions for high code design steel
moment resistant frame 
The fragility value at each sd (F_{i}(sd)) is estimated by changing
the notation of Eq. 3 and replacing the distribution of damage index (f_{im}(di))
by lognormal distribution of ISD (f(isd) = φ[ln(IŜD_{sd}),
β_{sd}]):
where, ISD_{i} is the mean ISD threshold of damage states (Table
1). The results for the moderate and high code frames are shown in Fig.
6, 7 by dots. Fragility functions shown in the figures
are estimated by fitting a lognormal cumulative distribution function:
where, SD_{i} and β_{i} are mean and deviation of the
function, respectively. These parameters for the fragility functions are shown
in Table 4.
Nonstructural damage governing case: Most direct and indirect loses
stem from the damage to nonstructural elements and evaluation of damage probability
to these elements is important in the most of the cases. The proposed method
provides a useful tool for evaluation of fragility functions when nonstructural
elements govern the damage states such as supporting structures in industrial
facilities. Since the damage to most of the nonstructural damages are defined
by ISD (Porter, 2000), the relevant fragility function is estimated by replacing
the corresponding ISD damage threshold (ISD_{i}) in Eq.
4.

Fig. 8: 
Fragility function result for equipment governing damage state:
probability of leakage in the pipes and damage to equipment support (a)
High code and (b) Moderate code 
These thresholds can be estimated from experimental studies or working condition
of equipments.
For instance, in the illustrative example, two additional damage states are
defined to determine the pipe leakage and damage to equipment supports with
associated ISD of 0.0033 and 0.0083, respectively. The resulted fragility functions
which are shown in Fig. 8 can be used to evaluate the probability
of relative nonstructural damages.
EFFECT OF UNCERTAINTY OF DAMAGE THRESHOLDS
In the current literature, all kind of uncertainties including the uncertainty of damage thresholds are defined by a constant deviation. For example, Kinali and Ellingwood (2007) assumed 0.2 for that. In this stage, the contribution of damage threshold uncertainty on fragility uncertainty is investigated.
Basically, thresholds of damage states, like any other natural phenomena, have
statistic nature and are defined by a distribution. According to the existing
literature, the lognormal distribution can describe that well (Hazus, 1999;
Wen et al., 2003):
where, ε_{t(i)} is a lognormal distributed random variable with
mean of unit and deviation of β_{t(i)}.

Fig. 9: 
Deviation of fragility function dispersion (β_{t})
as a function of SD (a) High code structure and (b) Moderate code structure 
The effect of this uncertainty on fragility functions can be quantified by
fragility dispersion. So far, the mean value of fragility function at every
ground motion is estimated by using the mean value of damage threshold in Eq.
5 (i.e., ISD_{i}). By the same token, if the dispersion of damage threshold
from Eq. 6 is utilized in Eq. 5, the distribution
of fragility value or fragility deviation at every ground motion can be calculated
as:
Equation 7 has been solved by MonteCarlos simulation for
β_{t(i=1.4)}, as suggested by Hazus (1999), for all fragility function
in the illustrative example and the distribution of fragility value is estimated
at every sd. Deviation of fragility function (β_{t}(sd)) is estimated
by fitting lognormal distribution which is more proper distribution to the
fragility distribution at every sd. Figure 9 has shown variation
of β_{t}(sd) vs. sd for the fragility functions shown in Fig.
6 and 7.
From Fig. 9 it can be observed that the deviation of fragility
function can reach to 3.5 times of damage threshold deviation (i.e., β_{t(i=1.4)}
= 0.4) in the lower IMs and unlike current assumption in the literature, it
gradually decreases.
SIMPLIFIED METHOD FOR DEVELOPMENT OF FRAGILITY FUNCTIONS
The proposed method derives the damage index distribution through substantial number of nonlinear dynamic analyses which is expensive and inconvenient for most of practical applications. A rapid method of fragility function development is introduced by taking advantage of recent development in estimation of structural capacity by converting pushover (SPO) curve of structures to distribution of incremental dynamic curves (Vamvatsikos and Cornell, 2005) which is very similar to ISD distribution.
The necessary steps for fast derivation of ISD distribution based on flowchart
of Fig. 10 are:
• 
Converting SPO curve of structure from top displacementbase
share space to Rμ space by dividing the horizontal axis of the
diagram by Δ_{y} and the vertical axis by V_{y} 
• 
Generating the set of IDA curves in Rμ space using SPO2IDA software
which is available online (Vamvatsikos and Cornell, 2005) 
• 
Converting IDA curves from Rμ space to SDISD space by multiplying
the horizontal axis by θ_{max y} and vertical axis by g.S_{ay}/ω^{2} 
The simplified method was applied to 3story structures of the examples shown
in Fig. 2. Pushover diagrams of these structures and its converted
diagram are shown in Fig. 11. The IDA curve in Rμ space
estimated by SPO2IDA software and converted to SDISD are shown in Fig.
12. For comparison, the results of nonlinear dynamic analyses are shown
in the figure as well.
Fragility value at each IM is estimated by applying Eq. 4
to new ISD distribution at each SD. In the Equation, ISD_{sd} is directly
estimated from the 50% IDA curve (x_{50%}) and β_{SD} is
estimated from and 16 or 84% IDA curves at each IM (x_{16%} and x_{84%}):
Fragility value and functions are shown in Fig. 13. The parameters
of fragility function are given in Table 5.

Fig. 10: 
Flowchart of simplified method of fragility function estimation 

Fig. 11: 
Pushover curves for high code and moderate code structures
(a) Baseshear vs. displacement and (b) Over strength vs. ductility 

Fig. 12: 
Converted IDA curve for high code and moderate code structures
to SDISD space (a) Moderate code IDA and (b) High code IDA 

Fig. 13: 
Fragility function result of simplified method (a) High code
and (b) Moderate code 

Fig. 14: 
Comparison of fragility functions for moderate code steel
frame structure (a) Slight damage state, (b) Moderate damage state, (c)
Extensive damage state and (d) Complete damage state 
Table 5: 
Parameters of fragility functions estimated from simplified
procedure 


Fig. 15: 
Comparison of fragility functions for high code steel frame
structure (a) Slight damage state, (b) Moderate damage state, (c) Extensive
damage state and (d) Complete damage state 
Validation: In this study, full and simplified methods for seismic fragility
function development of structures are introduced. In addition, application
of the method in development of nonstructural governing damages is shown and
effect of damage threshold uncertainty on the fragility dispersion is estimated.
The result of full and simplified method for high and medium code design frames
are compared with fragility of ductile lowrise steel frames developed by Bazzurro
et al. (2004) and Cherng (2000) and semiductile lowrise steel frames
developed by Smyth et al. (2004) shown in Fig. 14 and
15. It can be observed that in the first place, the results
of full and simplified method are very close in the case of extensive and complete
damage states and in the second place, the results of the method are almost
comparable with the results of previous studies.
CONCLUSION
Development of structurespecific fragility function is the key elements of accurate risk assessment. In this paper, full and simplified methods for seismic fragility function development of structures are introduced. In addition, application of the method in development of nonstructural governing damages is shown and effect of damage threshold uncertainty on fragility dispersion is estimated. It is observed that, the results of the proposed methods are almost comparable with the previous studies and the despite of current assumptions, effect of uncertainty of damage state on the deviation of fragility function in the lower intensity of ground motion is high which gradually decreases. This finding shows that current assumptions about constant uncertainty of fragility function might be correct and future investigation in this field is required.