PSI - Issue 44

Fabrizio Paolacci et al. / Procedia Structural Integrity 44 (2023) 307–314 Fabrizio Paolacci et al. / Structural Integrity Procedia 00 (2022) 000–000 3 conclusions. In addition, regardless of (scalar or vector-valued), the record-to-record variability represents an additional dispersion measure around the target spectrum, which is added to the dispersion derived from the GMPE, which is already included in the spectrum formulation, making the risk assessment too conservative. This paper proposes a different approach. The hazard curves are evaluated without considering the randomness of the GMPE, which is instead transferred to the fragility curves. For this purpose, the hazard curves are calculated assuming that the factor (see Eq. (2)), which takes into account the record-to-record variability, is deterministic. The fragility curves are then built using sets of unscaled or lightly scaled accelerograms, selected so that their spectra are consistent with median and 84% fractile USH spectra, which are obtained assuming = 0 and 1, respectively. This offers considerable advantages, as it is no longer necessary to refer to a specific and allows to select pairs of spectrum-compatible natural records, solving the problem of the seismic assessment of three-dimensional structures. In what follows the formulation of seismic hazard curves with deterministic GMPE is provided and the way to generate the corresponding UHSs is described. Finally, the methodology for the risk assessment is described. With the aim of evaluating the seismic risk of a structure in a reference time, we consider the entire set of seismic events that could affect the site. In this respect the PSHA is executed and the UHS are evaluated. The ground motion records are then selected accordingly. If we want to control the dispersion of the selected records, we can release the assumption that the fragility curve is a characteristic of the structure, including also the uncertainties of the seismic hazard due to the ground motion prediction equation (GMPE), besides the variability in phasing and spectral shape. The UHS can be evaluated by well-known methods, where the potential seismic sources must be known, together to the Gutenberg-Richter parameters of each source and the GMPE. The latter allows to evaluate the intensity measure at the site as a function of magnitude (m), distance (r), soil type (s) and other parameters (c), depending on fault mechanism. The GMPE is usually expressed as: logY= ψ ( m,r,s,c ) + σ ε ε (2) where ψ is a deterministic function while ε is a Standard Normal random variable, so that σ ε is the standard deviation of logY. Given , , , , it can be assumed that logY is a Normal distributed random variable, the function ( , , , ) is the mean value of logY , and, for a value = 7 , 9(:,;,<, )> A B is the value with probability of exceedance 1- Φ ( ε 0 ) . If is deterministic, the probability that Y ( m,r,s,c| ε ) ≥ y is 1 if ψ ( m,r,s,c ) + σ ε ε ≥ y , 0 otherwise. If ( ) is the Heaviside function: H ( x ) = E 0, x<0 1, x ≥ 0 (3) The mean annual rate of earthquakes producing at the site an intensity Y > y for a given e is: ν (y| ε ) = ∑ ν 0i i ∫ ∫ H [ ψ ( m,r,s ) + σ ε ε - log ( y )] f R i (r) f M i (m) dm dr M i R i (4) By using the equation (4), if the dispersion in the attenuation law is removed by assuming e =0, we get the median uniform hazard spectrum. This spectrum considers the randomness due to the seismic event (magnitude and distance) but not the dispersion of the attenuation law. In the same way, if, in the equation (4), we assume ε =1 we get the fractile 84%. If the hazard is expressed in terms of mean annual rate ν (inverse of the return period, ν =1ÚT R ), given N values of ν , 2 × N spectra can be evaluated: S a (T, ν _i, ε =0), S a (T, ν _i, ε =1). For each of these N pairs of spectra, n records can be selected whose mean (logarithmic) spectrum approximates the reference spectrum S a ( ε =0), while the mean + one standard deviation spectrum matches the reference spectrum S a ( ε =1). In this way, the hazard analysis considers only the uncertainties in the magnitude, location and fault mechanism, while the uncertainties in the local ground motion, expressed by the dispersion of the spectral ordinates, are included in the fragility functions. For each hazard level, non-linear time history analyses are executed by applying the relevant set of records and the probability of exceeding the considered limit state, conditioned to the hazard J , is evaluated. The probability of failure conditioned to the generic hazard level can be evaluated by interpolation. Finally the probability of failure can be evaluated by using Eq. (1). For the application of the proposed approach, a procedure for the selection of recorded ground motion (each with one or two components) consistent with the reference spectrum in mean and dispersion has been developed. The 309