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Developing Seismic Fragility Function of Structures by Stochastic Approach

K. Nasserasadi, M. Ghafory-Ashtiany, S. Eshghi and M.R. Zolfaghari
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Fragility function of structures is the major requirement of seismic loss estimation which is widely used in the seismic risk management. In this study, a comprehensive and simplifies stochastic methods are presented and the effect of damage threshold uncertainty on fragility functions is estimated. The method is employed to estimate the fragility of low-rise steel moment framed structure. It is shown that the results of the method are almost comparable with the result of previous studies and the effect of uncertainty of damage state on the deviation of fragility function in the lower intensity of ground motion is high. The deviation then gradually decreases.

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K. Nasserasadi, M. Ghafory-Ashtiany, S. Eshghi and M.R. Zolfaghari, 2008. Developing Seismic Fragility Function of Structures by Stochastic Approach. Journal of Applied Sciences, 8: 975-983.

DOI: 10.3923/jas.2008.975.983



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 well-known 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 structure-specific 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 Monte-Carlo 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 straight-forward and rapid procedure of structure-specific 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 structure-specific fragility function derivation. Besides, some challenging issues such as effect of damage threshold uncertainty on fragility dispersion and fragility function of structures with non-structural governing damage cases which is used for industrial facilities are discussed.


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, Fi (im) is the probability of exceeding damage D from damage state di 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 non-damage state (i = 0) to the nth 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, dit 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 (fim (di) and Fim(dit)), Eq. 2 is evaluated from probabilistic theorem:


In this study, PDF of DI is evaluated by multi-stripe 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 multi-stripe 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.


Two groups of low-rise steel frame which are compatible with high-code and moderate-code structural classifications of HAZUS were selected (Hazus, 1999). According to HAZUS, high-code frame corresponds to ductile steel structures designed for 0.133 fraction of building weight and moderate-code frame corresponds to semi-ductile steel frames which is designed for 0.067 fraction of building weight. For each classification, two low-rise 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 non-structural 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 source-to-site 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 source-to-site 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. P-Δ effect was considered and FEMA-356 force-deformation relationship (FEMA-356, 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 log-normal 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 (IŜDsd and βsd) of ISD distributions of high-code frames are shown in Table 3.

Table 1: Inter-story drift ration at thresholds of different damages states (Hazus, 1999)

Table 2: Selected strong motion records for non-linear 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 log-normal 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 (Fi(sd)) is estimated by changing the notation of Eq. 3 and replacing the distribution of damage index (fim(di)) by lognormal distribution of ISD (f(isd) = φ[ln(IŜDsd), βsd]):


where, ISDi 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 log-normal cumulative distribution function:


where, SDi and βi are mean and deviation of the function, respectively. These parameters for the fragility functions are shown in Table 4.

Non-structural damage governing case: Most direct and indirect loses stem from the damage to non-structural 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 non-structural elements govern the damage states such as supporting structures in industrial facilities. Since the damage to most of the non-structural damages are defined by ISD (Porter, 2000), the relevant fragility function is estimated by replacing the corresponding ISD damage threshold (ISDi) 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 non-structural damages.


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, εt(i) is a log-normal distributed random variable with mean of unit and deviation of βt(i).

Fig. 9: Deviation of fragility function dispersion (βt) 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., ISDi). 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 Monte-Carlos simulation for βt(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 (βt(sd)) is estimated by fitting log-normal distribution which is more proper distribution to the fragility distribution at every sd. Figure 9 has shown variation of βt(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., βt(i=1.4) = 0.4) in the lower IMs and unlike current assumption in the literature, it gradually decreases.


The proposed method derives the damage index distribution through substantial number of non-linear 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 push-over (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 displacement-base share space to R-μ space by| dividing the horizontal axis of the diagram by Δy and the vertical axis by Vy
Generating the set of IDA curves in R-μ space using SPO2IDA software which is available online (Vamvatsikos and Cornell, 2005)
Converting IDA curves from R-μ space to SD-ISD space by multiplying the horizontal axis by θmax y and vertical axis by g.Say2

The simplified method was applied to 3-story 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 R-μ space estimated by SPO2IDA software and converted to SD-ISD are shown in Fig. 12. For comparison, the results of non-linear 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, ISDsd is directly estimated from the 50% IDA curve (x50%) and βSD is estimated from and 16 or 84% IDA curves at each IM (x16% and x84%): 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) Base-shear vs. displacement and (b) Over strength vs. ductility

Fig. 12: Converted IDA curve for high code and moderate code structures to SD-ISD 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 non-structural 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 low-rise steel frames developed by Bazzurro et al. (2004) and Cherng (2000) and semi-ductile low-rise 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.


Development of structure-specific 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 non-structural 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.

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