PSI - Issue 44

Valentin Cima et al. / Procedia Structural Integrity 44 (2023) 211–218 Valentina Cima et al. / Structural Integrity Procedia 00 (2022) 000–000

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(z>0). In the first case the ground displacement spectra have been used, while in the second one floor displacement spectra have been considered. Once determined the seismic demand, the Damage Indices DI DSi for the two damage limit states DS i have been evaluated as the ratio between the maximum displacement demand, D DSi , and the damage threshold, C Dsi . Finally, for each category, the fragility curves corresponding to the two DS i have been derived, considering the Peak Ground Acceleration, PGA, as an indicative parameter of the intensity of the seismic input. In particular, assuming a log-normal distribution of the DI Dsi conditioned on a given value of PGA, the following equation has been used for deriving the fragility curves, as reported in the guidelines of the CNR-DT212 (2014): f "#$ = p'DI "#$ ≥ 1 |-./ 0 = 1 − Φ3 ln1 − µ 78"9 :;<|=>? σ 78"9 :;<|=>? A = Φ3 µ 78"9 :;<|=>? σ 78"9 :;<|=>? A (3) where: • Φ [•] is the lognormal standard distribution function; • µ lnDIiIPGA is the mean value of the logarithm of the variable DI DSi conditioned on the given value of PGA; • σ lnDIiIPGA is the dispersion of the natural logarithm of the variable DI DSi conditioned on the given value of PGA. The conditional mean values of the variable DI DSi have been defined for each value of the PGA through the following expression proposed by Nielson and DesRoches (2007): µ 78"9 :;<|=>? = A "#$ + B "#$ · ln (PGA) (4) where A DSi and B DSi are coefficients obtained by a linear regression of the logarithm of the DI DSi versus the logarithm of the PGA (see Fig. 3).

ln (DI DSi )

ln (DI)

ln (PGA) ln (P )

Fig. 3. Linear regression of the logarithm of the DI DSi versus the logarithm of the PGA.

The dispersion values have been evaluated through the following equation: σ 789" :;<|=>? = Kσ L $ M + σ NON PNON $ $ M where: • σ m i represents the uncertainty of the model, estimated applying the following equation:

(5)