PSI - Issue 78

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

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with equal exceedance probability over a reference period (EN 1998-1, 2004; NTC-18, 2018). Although UHA ensures compliance with standards, it can result in different probabilities of structural failure (Iervolino et al., 2018; Žižmond & Dolšek, 2019 ). To address this, the Risk Targeted Approach (RTA) has been introduced. It calibrates seismic actions to satisfy a target failure probability for a specific performance level, using fragility curves that represent conditional failure probabilities for given Limit States (LS) and Intensity Measures ( IM ). Cornell et al. (2002) and FEMA 350 (2000) provided a probabilistic framework, including closed-form solutions for failure probability based on simplified assumptions about seismic hazard, structural demand and capacity distributions. As seismic hazard law is adopted the power law of Sewell et al. (1991). However, more accurate hazard law have since been proposed, such as hyperbolic (Bradley et al., 2007) or polynomial (Vamvatsikos, 2013) formulations in log-log space, addressing limitations of earlier approximations. Fragility curves, typically modeled with log-normal distributions (Tatangelo et al., 2024a), are derived through methods like Incremental Dynamic Analysis (IDA), Multiple Stripe Analysis (MSA), or the Cloud method. To reduce computational effort, simplified tools like SPO2FRAG or the Incremental N2 method have been developed. Recent research (e.g., Žižmond & Dolšek, 2019 ) proposes direct and indirect methods to define reliability targeted design spectra, including a reliability factor based on target failure probabilities. Despite significant progress, limited guidance exists on selecting appropriate target reliability indexes for different limit states and consequence classes. ASCE 7-16 (2016) define target probabilities equal to 2× 10⁻⁴ annual exceedance for Near Collapse. Model Code 2020 and the second-generation Eurocode 8 also adopt these values. This paper presents a reliability-based seismic approach in a certain reference period. It enables adjusting design spectral acceleration based on site hazard characteristics to satisfy probabilistic targets continuing to adopt the partial factor method. In addition, a procedure is proposed to derive target reliability indexes in certain reference period for Italy. 2. Reliability-based formulation for seismic design In a probabilistic framework the structural problem of a costruction exposed to a seismic action taken into account several uncertainties that may be divide into epistemic uncertainties, resulting from model limitations, data variability, and incomplete knowledge, and aleatory uncertainties, such as record-to-record variability. The failure probability (i.e. the exceedance probability of a certain limit state) can be determined through the convolution integral of seismic hazard, expressed by means seismic hazard function , with fragility curves. Cornell et al. (2002) proposed an approximate closed-form solution of the integral, where are adopted simplified expression of the based on the annual rate of exceedance (as described by Sewell et al., 1991), and a approximate relationship between the and Engineering Demand Parameter ( ). As a result, the closed-form expression for the annual failure probability, 1 , is expressed as follows: 1 = 0 ⋅ ( ̂ ) − 1 ⋅ � 0.5 ∙ 1 2 ⋅ 2 � (1) where 0 and 1 are positive empirical constants representing the intercept and slope, respectively, of the seismic hazard curve. ̂ denotes the median intensity measure causing the exceedance of a specific limit state, while represents the dispersion parameter of the fragility curve. In order to targeted a specific performance in terms of failure probability, is introduced the safety factor , expressed as follows ( Žižmond & Dolšek, 2019 ) = ̂ ∙ [( � ) −1 ] (2) where � is provided through the traditional uniform seismic hazard maps, referred to specific value of . In (2) ̂ is unknown, therefore it should be turned in order to explicate the value of ̂ , and expressing � as a function of . In this way 1 may be obtained with respect to the seismic hazard. Therefore, (1) become 1 = � ∙ − 1 � ⋅ � 0.5 ∙ 1 2 ⋅ 2 � = ( − 1 ) (3)