PSI - Issue 78

Iunio Iervolino et al. / Procedia Structural Integrity 78 (2026) 1553–1560

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[ ( ) > ] = [ ( ) > ∩ ] + [ ( ) > ∩ ] = = ∫ ∫ ∫ [ ( ) > | , , ] ∙ [ | , ] ∙ | ( | ) ∙ | ( | ) ∙ ( ) ∙ ∙ ∙ + ∫ ∫ ∫ ∫ [ ( ) > | , , , ] ∙ [ | , ] ∙ | ( | ) ∙ | ( | ) ∙ | ( | ) ∙ ( ) ∙ ∙ ∙ ∙ 2.1 Models for dip-slip ruptures In Eq.(1), [ ( ) > | , , ] is the exceedance probability of given { , , } , when the ground motion is not exhibiting pulse-like features. GMMs usually model the logarithm of ( ) as a normal RV conditional on and a metric for source-to-site distance such as Joyner and Boore distance or (Joyner and Boore 1981), which is univocally defined given { , , } . Denoting the conditional mean and standard deviation of the log of ( ) as [ ( )] ( , ) and [ ( )] ( , ) , respectively, it follows that: [ ( ) > | , , ] = [ ( ) > | , ] = 1 − Φ { ( ) − [ ( )] ( , ) [ ( )] ( , ) }, (2) where Φ(∙) is the standard normal cumulative distribution function (in fact, [ ( ) > | , ] generally also depends on other covariates such as local site conditions, neglected herein for simplicity). [ ( ) > | , , , ] is the exceedance probability of in the case of pulse-like ground motion, which is also conditional on the pulse duration (period), . This would require a GMM capable of accounting for pulse occurrence, with period . In practice, a traditional GMM — in the sense that it does not account for pulse period — is modified by applying the correction factor, , only to [ ( )] ( , ) from Shahi and Baker (2011): = {1.131 ∙ −3.11∙[ ( )+0.127] 2 + 0.058 , if ≤ 0.88 ∙ 0.924 ∙ −2.11∙[ ( )+0.255] 2 + 0.255, if > 0.88 ∙ Ǥ (3) To apply this factor to the GMM leads to replace [ ( )] ( , ) with [ ( )] ( , )+ in Eq. (2). Although was originally calibrated against the GMM of Boore and Atkinson (2008), the model considered in this study is that of Lanzano et al. (2019), which is based on data from Italian earthquakes. The term | ( | ) is the probability density function (PDF) of conditional on and allows to account for the fact that pulse period generally increases with earthquake magnitude. The log-normal model from Baltzopoulos et al. (2016) was assumed in the analyses: ( ) = −5.92 + 1.03 ∙ + , (4) with being the model residual ; i.e., a gaussian RV with zero mean and standard deviation equal to 0.63. [ | , ] is the probability of pulse occurrence conditional on { , , ℎ, , ℎ } that was modelled according to Iervolino and Cornell (2008), and [ | , ] its complement to one. This model relies on covariates that are determined once the magnitude-dependent rupture geometry, its position on the fault, and the hypocenter location are specified. In the case of dip-slip rupture, these parameters (also shown in Fig. 2c) are the along-dip length that the rupture propagates towards the site ( ), the site-to-rupture distance ( ) and the angle between the rupture plane and the line that passes through hypocenter and site. Thus, the pulse occurrence probability is [ | , , ] is obtained as per Eq. (5): (1)