PSI - Issue 78

Federico Gusella et al. / Procedia Structural Integrity 78 (2026) 113–119

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( ≥ | = ) ≈ ( ≥ | = ) ≈ ( ( / ) )

(4)

where: θ i is the median of the fragility function and β i is the standard deviation of the ln(IM) at which the damage dm i occurs. In Eq. 4), as usually happens in decision-making concerned with the total risk estimation, the engineering demand parameter ( EDP ), such as an inter-story drift, is assumed to correspond to the physical damage ( DM ). Through Eq. 2) and Eq. 4) the probability of exceeding a specified damage level can be expressed by (Jalayer and Cornell, 2003): ∫ ( ≥ | = )| ( > )| = − 1 22 2 (5) Through Eq. 5), the expected annual loss (Eq. 1) can be arranged in (Gusella and Bartoli, 2025): ( ) = ∑ ( | = )( − 1 22 2 − − + 1 1 2 2 2 +1 ) = 1 (6) 2. Target Price for D structures A closed-form equation is provided to determine the initial target price ( P D,TARGET ) for a D building. This price enables the economic evaluation of the dissipative design concept compared to designing the same building as LD. In the proposed framework, the price of the D structure ( P D =d i P LD ) is expressed as a discount factor ( d i ) applied to the price of the LD structure ( P LD ). The target price ( P D,TARGET = d i,TARGET P LD ) represents the required price of the D structure for it to be competitive in the market while accounting for the expected seismic losses. As commonly assumed in practice, structures are designed to remain elastic under earthquakes corresponding to service limit states, and two discrete damage states ( d mi ), with i = 1, 2, are considered. For the LD structure (with seismic behavior factor q LD ≤ 1.5), the damage state d m1,LD corresponds to the failure of the first structural element, associated with reaching the Life Safety Limit State (LSLS), while d m2,LD represents collapse due to potential overcapacity of the structure. For the D structure, d m1,D corresponds to the initial plastic phase of the fuses, which exhibit a ductile post-elastic response consistent with the seismic behavior factor q D adopted in the design process. Reaching the ductile capacity of the structure corresponds to the LSLS and is defined by damage state d m2,D . 2.1. Medians of fragility curves A LD structure is designed elastically at the Life Safety Limit State (LSLS), with θ 1,LD denoting the median of the fragility function corresponding to damage state d m1,LD . The median of the fragility curve at collapse (damage state d m2,LD is defined as θ 2,LD =q LD θ 1,LD , where q LD ≤1.5. A dissipative design concept, based on the capacity design approach, enables an elastic design of the structure by adopting reduced seismic forces through the seismic behavior factor q D . In this case, the median θ 1,D of the fragility curve corresponding to the onset of plastic response (damage state d m1,D ) is given by: θ 1,D =θ 2,D /q D , where θ 2,D is the median of the fragility curve corresponding to achievement of the ultimate ductility capacity (damage state d m2,D ), which defines the LSLS. The medians of the fragility curves for both design concepts and associated damage states are summarized in Table 1. Figure 1 illustrates the fragility curves obtained by Eq. (4), using the values listed in Table 1, with θ 1,LD =θ 2,D =2.22 [g], a standard deviation of ln(IM ) β = 0.6 [g] and seismic behavior factors q LD =1.5 and q D =3.9 for LD and D