PSI - Issue 78

Raffaele Laguardia et al. / Procedia Structural Integrity 78 (2026) 678–685

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of a Carbon Tax ( T C ) approach. The objective function can then be defined as follows:

LCCI = C I + C S + ( I I + I S ) × T c

(1)

The initial cost C I , derived from the parametric function in Braga et al. (2019), accounts for steel, dissipative devices, infills, and foundation works as a function of IVs. The environmental impact is assessed for the same items based on unitary data provided by the ICE database (Hammond G., 2019). As far as the damage due to seismic event is concerned, its economic and environmental impact is assessed in terms of Expected Annual Loss (EAL). EAL = L ( λ ) d λ (2) where L represents the loss value (i.e. economic or environmental) and λ the frequency of exceedance of that loss value. In this work, λ values are evaluated through a SAC-FEMA approach, by estimating the median demand via regression on a reduced sample of MRSA data at di ff erent hazard levels. This is done at each optimization step, and thus λ values are dependent on the IVs. Further information on median demand assessment can be found in (Laguardia and Franchin, 2022; Laguardia et al., 2023, 2025). The demand uncertainties adopted within the SAC FEMA framework (i.e., β L | IM ) have been evaluated following the procedure in Baker and Cornell (2008), with further details available in Laguardia et al. (2025). Once the EAL values are assessed, C S and I S values, needed for Eq. 1 can be obtained as follows: C S = EAL C × 1 − e − γ T o γ (3) I S = EAL I × 1 − e − γ T o γ (4) where EAL C and EAL I are the EAL assessed for economic and environmental parts, respectively, γ is the discount rate per year and T o is the observation period (i.e.,the nominal life of the building after intervention). As far as the constraint function is considered, its scope is to provide a design that satisfies safety requirements expressed in terms of Collapse Limit State (LS) MAF. This is done by using the following expression: λ LS = ∞ 0 P ( D ≥ ˆ C | IM = x ) d λ IM ( x ) (5) where λ LS is the MAF of exceedance of limit state LS, D is the demand, ˆ C is the median limit state capacity, λ IM is the MAF of exceedance of intensity measure IM derived from hazard and x is the intensity measure value. In this work the λ LS described in Eq.5 is assessed through a SAC-FEMA approach (Cornell et al., 2002) by the means of the

procedure exposed in Laguardia and Franchin (2022) and Laguardia et al. (2023). By the means of Eq.1 and Eq.5 the optimization problem can be formalized as follows:

find

min LCCI λ CP ≤ λ ∗ CP

(6)

subjected to