PSI - Issue 44

Lucia Minnucci et al. / Procedia Structural Integrity 44 (2023) 729–736 Lucia Minnucci/ Structural Integrity Procedia 00 (2022) 000–000

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parameters are associated. A strategic choice for the performance levels should consider the possibility to associate them not only a physical mechanism of damage, but also the quantification of consequences after the damage itself, that translates into repair or reconstruction costs, if direct consequences are evaluated, or traffic and environmental costs, if indirect consequences are considered. The adoption of such a strategy is already known in the literature and widely adopted in case of bridge structures (Padgett and DesRoches 2007), but in this framework an additional effort is made to explicitly associate such damage levels to specific decisional processes firstly at bridge level and subsequently on the infrastructural system, as described in the following Section 2.4. To build fragility curves, different methods can be used (Jalayer and Cornell 2009), the most used ones being Incremental Dynamic Analysis (IDA) (Vamvatsikos and Cornell 2002), Cloud Analysis (Mackie and Stojadinović 2005) and Multiple-Stripe Analysis (MSA) (Scozzese et al. 2020). Using results from the seismic analyses according to one of the previous methods, fragility curves can be built for each bridge component of interest. The fragility function | ( | ) denotes the probability of exceeding a given threshold value d of a demand parameter D , given a value im of the seismic intensity IM , so that, for example in case of MSA, at each IM level the fraction of ground motions satisfying the exceedance condition ≥ can be computed. An appropriate fragility fitting strategy for this type of data is maximum likelihood, according to which the fragility function parameters ( � , ̂ ) are obtained by maximizing the logarithm of the likelihood function (Baker 2015): � � , ̂ � = argmax , �� � � + Φ � ln ( / ) � + ( − ) � 1 − Φ � ln ( / ) ��� =1 (1) where is the number of IM levels, is the total number of ground motion samples adopted at each IM level, is the fraction of ground motion samples leading to ≥ and Φ ( ∙ ) denoting the standard normal cumulative distribution function. At this point, looking at the entire road network, two passages are required before the decision-making process: 1) the evaluation of the expected value of IM characterising the specific site of the bridge of interest within the road network; 2) the analysis of the probability of occurrence of different limit states for each bridge component at the estimated IM from their respective fragility curves. To address the first one, the ground motion prediction equation recently proposed by Lanzano et al. (2019) can be adopted to estimate the expected IM values along the infrastructural network for a certain seismic magnitude and at a certain distance from the seismic source. To satisfy the second, a comparison between the different probabilities of exceedance from the fragility curves related to the analysed limit states is carried out. The comparison aims to quantify the most probable damage condition on the structure at the occurrence of an input from the considered seismic scenario (e.g., the Life safety limit state in Fig. 2 can be computed as the difference between the ordinates of the Life safety and Collapse fragility curves).

Fig. 2. Example of graphical evaluation of occurrence probabilities for different limit states (piers).