PSI - Issue 78

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

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with sufficient strength to remain elastic and ensure that energy dissipation can be maintained. Earthquake-resistant structures, classified as Dissipative (D) structures, are capable of dissipating energy exploiting the ductile hysteretic behavior of these fuses. For the design of a D structure, a linear elastic analysis can be performed using a reduced seismic action using the behavior factor ( q ). For structures classified as Low-Dissipative (LD), no allowance is made for hysteretic energy dissipation, and the behavior factor must not exceed 1.5. While a plastic design philosophy allows structures to be damaged, a clear procedure for determining the cost effectiveness of selecting a D structure over an LD structure is still lacking. This paper addresses this gap by proposing a closed-form equation to analytically estimate the target price of a D structure that would make it the most economical solution, taking into account the economic impact of the expected annual seismic loss. The economic impacts resulting from damages and disruptions, such as those observed during the 1994 Northridge (U.S.) earthquake, have prompted a new conceptual approach to design aimed at mitigating the devastating effects of earthquakes on society ( O’Reilly and Calvi, 2019) . Seismic codes should not focus solely on life safety and collapse prevention, as required by Performance-Based Earthquake Engineering (Bertero and Bertero, 2002) but should also account for the expected losses caused by seismic events. The necessity to integrate these performance metrics within PBEE has led to the development of Loss-Based Earthquake Engineering (LBEE). Relevant applications that aim to incorporate risk-targeted methods can be found in Miranda and Aslani (2003). Major steps toward an integrated probabilistic design and assessment approach to LBEE have been advanced by the Pacific Earthquake Engineering Research (PEER) Center (Cornell and Krawinkler, 2000). An important metric linking the seismic response of structures to financial consequences is the expected annual loss (Eq. 1), which quantifies the anticipated amount of loss experienced per year. ( ) = ∫ ∑ ( | = ) ( = | = )| ( > )| = 1 (1) where: E(C) is the expected value of the consequence C (e.g. monetary loss); DM is a vector damage measure indicating the discrete damage states of each component in the building; dm i is the i -th damage state, increasing values of i indicate more severe damages; IM is a ground-motion intensity measure; dλ(IM>im) is the ground motion hazard curve derivative; P(A=a|B=b) is the conditional probability for a , given b ; n is the number of damage states, assumed to be mutually exclusive and collectively exhaustive, and E(C|DM=dm i ) is the mean loss conditional to the damage i -th. The ground-motion hazard curve provides the mean annual frequency of exceeding a particular spectral acceleration for a given period and damping ratio and can be approximated by the power-law relationship: ( > ) = − (2) where: k o and k are parameters defining the shape of the hazard curve. The probability of damage state occurrence (i.e., of being in a particular dm i ) can be computed by subtracting the exceedance probabilities for sequential damage states: ( = | = ) = ( ≥ | = ) − ( ≥ +1 | = ) (3) Moreover, the lognormal cumulative distribution function can be used to describe the fragility function (Iervolino et al., 2023; Gusella et al. 2025 and Aljawhari et al. 2022), which gives the probability of a structure of reaching that damage state dm i or worse. 1.1. Loss-Based Earthquake Engineering