PSI - Issue 78

Parvane Rezaei Ranjbar et al. / Procedia Structural Integrity 78 (2026) 615–622

617

Incremental Dynamic Analysis (IDA) and Probabilistic Seismic Demand Models (PSDM), allow detailed modeling of nonlinear structural responses under seismic loads (Vamvatsikos and Cornell, 2002). This method has gained prominence due to its ability to model nonlinear structural behavior under seismic loading. Rossetto and Elnashai (2005) proposed a new procedure for the derivation of analytical displacement-based vulnerability curves for the seismic assessment of RC structures. Ranjbar and Naderpour (2020) use this method to reach the seismic resilience index of a reinforced concrete frame of a hospital. Numerous studies have also applied fragility curves to evaluate the performance of existing buildings in different regions (Kırçıl and Kocabey, 2019 ; Pohoryles et al., 2024). Yalcinkaya and Celik (2025) developed fragility curves for low-rise masonry buildings in low seismicity regions using IDA, emphasizing the role of induced seismicity and regional building typologies. Hybrid fragility curves were proposed in the literature (Kappos et al. , 2006) based on the combination of different methods for damage prediction. The aim is to compensate for the lack of observational data, the deficiencies of the structural models, and the subjectivity in expert opinion data. The probability of reaching or exceeding a specific damage state under earthquake excitation is defined as follows:   Y lim Fragility F P R r I = =  (1) where R is the response parameter (deformation, displacement, force, velocity, acceleration, etc.), r lim is the threshold of the response that is related to damage, and I is the severity of the event. In this research, a method based on coefficients (Cornell et al. , 2002; Konstantinidis and Makris, 2009; Su and Lee, 2013) is used to plot the fragility curves. In this method, the parameter D is assumed to have a log-normal distribution. In other words, the variable D , which has a log-normal distribution, is associated with the variable X , which has a normal distribution, by ln( D ). Therefore, parameter D is expressed by the following exponential model: ( ) b D a IM = (2) where a and b are unknown coefficients of regression that are obtained from the regression analysis of demand parameters through the IDA or the method based on coefficients. ( ) ( ) ( ) ln ln ln X D a b IM = = + (3) 2.2. Probabilistic approach

The mean and standard deviations are calculated as follows: ( ) ( ) ( ) ln b x m IM a IM =

(4)

2

  

   

  

  

1

i aIM 

n 

(5)

ln

= =

 

( ) D

x

1 2 −  i =

n

ln

b

i

where δ is a demand value. For the lognormally distributed random variable D , the fragility function ( P f ), which provides the probability that the demand D will exceed a certain threshold or capacity, C , conditional on a given IM , can be represented as ( ) ( ) ln 1 1 1 ln x f b x x C m IM C P P D C IM aIM    −      = − = −         (6) ( ) 2 ln 1 exp 2 2 u x x x C m IM u du   −  −     = −          ; ( ) ( ) ln x x u C m IM  = − (7)