PSI - Issue 78

Matteo Tatangelo et al. / Procedia Structural Integrity 78 (2026) 73–80

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1 = � � , 475 � � , 2475 � � ⋅ � � � , 475 � � , 2475 � �� (7) where � , 2475 � and � , 475 � are the intensity measure, while � , 2475 � and � , 475 � the corresponding annual exceedance probabilities. The capacity fragility curve should be obtained starting from the demand fragility curve that have as median value of � related to , associated with ULS, including Life Satety Limit State (LSLS) and Collapse Prevention Limit State (CPLS), and CC considered. The E , t ref considered are reported in Table 1 (NTC-18, 2018) Table 1. E , t ref , , with nominal life of 50 years for ULSs E , t ref , , Consequence Class (CC) - Consequence of failure Limit State CC1 Small CC2 Some CC3 Moderate CC4 Great LSLS 20% 10% 3% 1% CPLS 12% 5% 1% 0.3% A value of the fragility curve logarithmic standard deviation has to be assumed. As known, it represents the form of the fragility curve, providing the dispersion around the median value of the considered. In particular, takes into account sources of aleatory and epistemic uncertainties, where: the former are related to the variability associated with frequencies content and other attributes of the ground motion, and refers to a given intensity; while the latter are linked to structural modeling and analysis. The measure of the dispersion of both aleatory and epistemic uncertainties are combined by means the Square Root of the Sum of the Squares ( SRSS ), with the following equation (Cornell et al., 2002): = �� , � 2 + � , � 2 (8) where , and , are the measures of the logarithm standard deviation for randomness ( ) and epistemic ( ) uncertainties, respectively. A similar approach is adopted in the methodology presented in FEMA P-695 (2009), where a detailed procedure to quantify the logarithm standard deviation of is provided. So, , may derived by adopting the formulation seen in FEMA P-695 (2009), from which one obtain a value of 0.4. As for , , starting from the values proposed in literature, a value of , equal to 0.45 for ULSs is assumed. Therefore, by applying (5), a =0.60 is obtained, corresponding to the same value assumed in other works. In order to obtain a failure probability with (4), the target reliability factor should be defined, which is used for modifying the median value � of the demand fragility curve in order to satisfy a given reliability index . On the contrary, may be used to calibrate target reliability indexes ̅ . The values that may be assumed to calibrate ̅ can be obtained from Model Code (2020), where they suggest that for new constructions a in the order of 2.5, while for existing constructions a =1 . However, another way to assume could be that of calibrate in relation to the failure annual probability of 2×10 −4 provides from the ASCE 7-16 (2016), corresponding to the mean annual exceedance probability for the Near Collapse (NC) limit state, for a 2% probability of exceedance in 50 years for the U. S.. In particular, known the failure annual probability 1 , , 2 =2×10 −4 , the seismic hazard , and the fragility curve (by means of and � ), the value unknown of , , 2 is obtained by numerically solving the convolution integral, by iterating on the value � . The value of � so that 1 , , 2 is equal to 2×10 −4 corresponds to ̂ , representing the median value of the target fragility curve that has the same . Average a value of ( 1 , , 2 )=2.11 is obtained, such that consent to derive capacity fragility curves for a given construction so that the 1 , , 2 is uniform all-over the Italian national territory, valid in this case for NC and CC2. In general, the same value is applied for LSLS and CPLS. As it will be discussed in the next sections, starting from the ̂ so-derived a calibration for the Italian territory of ̅ , for the design and assessment of new and existing constructions may be performed. −1 =1.65 ⋅ � � � , 475 � � , 2475 � �� −1