PSI - Issue 78

Andrea Nettis et al. / Procedia Structural Integrity 78 (2026) 1412–1419 = ∫ ( ) | ( )|

1414

(1)

( ) = ∑

=1 ( )

(2)

( ) = { ( ) − +1 ( ), k=1,…,N −1 ( ), k=N DS

(3)

2.2. Fragility analysis In this study, system-level fragility functions are developed for a set of structure-level DSs by using the cloud analysis and multi-stripe analysis methods. In the cloud analysis (Jalayer et al., 2017a), the fragility function ( ) defined by Equation (4), represents the conditional probability that a given DS is reached or exceeded, that is, the probability that a selected engineering demand parameter (EDP) surpasses its corresponding DS threshold, conditioned on a specified level of IM. These functions are derived from a dataset of [ , ] pairs obtained through nonlinear seismic response analyses conducted using a suite of natural, unscaled ground motion records. The resulting [ , ] data are processed to evaluate two key probabilities: the conditional probability of reaching a specific DS given that collapse does not occur, ( | , ) and the probability of collapse ( | ) . The collapse probability ( | ) is estimated by applying logistic regression to the seismic response dataset. In contrast, the computation of ( | , ) requires the definition of a probabilistic seismic demand model capturing the analytical relationships between EDP and IM in case of no collapse. The power-law model functional form ̂ = is used in this study. The parameters [ , ] are determined by applying a linear regression to the cloud dataset ting a linear model to the cloud data [ , ] , after transforming both variables into the natural logarithmic scale. The dispersion σ of the EDP about its median estimated with the power-law model is assumed to be constant with respect to IM and is given by Equation (5) where M is the number of ground motions. The conditional probability that the EDP threshold associated with a specific DS is exceeded, given that collapse does not occur, is computed using Equation (6), where ϕ(∙) denotes the standard normal cumulative distribution function. ( ) = P(DS|IM) = P(DS|IM, NoC)(1 − ( | )) + ( | )) (4) = √∑ (ln − ln ( ) ) = 1 −2 (5) ( | , ) = 1 − ϕ (ln EDP DS σ −lnEDP̂ ) (6) The second strategy adopted for fragility analysis is the multi-stripe method(Baker, 2015). This technique involves selecting a discrete set of IM levels, referred to as “stripes”, and generating ground motion records that are spectrally compatible with each selected IM level. Seismic response analyses are then performed for each stripe to obtain sets of EDP values conditioned on the corresponding IM level. From the resulting EDP data, the number of ground motions that cause the structure to reach or exceed the DS under investigation can be determined for each stripe. The fragility function is represented by the lognormal cumulative distribution function expressed by Equation (7) based on the parameters and where the former represents the IM associated with a 50% probability of reaching the DS (commonly referred to as median of the fragility curve). These parameters are estimated by fitting the lognormal function to the empirical probabilities obtained from the stripe data using the maximum likelihood estimation method. ( | ) = ϕ (ln( β / )) (7)