PSI - Issue 44

Laura Gioiella et al. / Procedia Structural Integrity 44 (2023) 1808–1815 Laura Gioiella et al. / Structural Integrity Procedia 00 (2022) 000–000 For what concerns Marche region, the seismic crater encompasses 87 towns, this means that the pairs � , � refer to 315 buildings only, over the 1240 structures of the whole regional school heritage. Fig. 4a) reports the distribution of the considered pairs � , � , determined in accordance with the procedures previously described, underlining their locations into the four provinces of Ancona (AN), Ascoli Piceno (AP), Fermo (FM) and Macerata (MC) (Fig. 4b). It can be observed that most of the pairs � , � showing the highest values are relative, as expected, to buildings located in the MC and AP areas that are closer to the epicentres of the main shocks. 1813 6

a

b

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

MC AP FM AN

damage index [-]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

PGA [g]

Fig. 4. (a) Distribution of the observed pairs � , � ; (b) locations of both schools belonging to the seismic crater and mainshock epicentres. 3.3. Evaluation of the damage model parameters and of the damage trends for the schools of the seismic crater This section refers to the choice of the shape functions 0 ( ; 0 ) and | ∗ ( | ; ) and the results of the optimization procedure based on the Maximum Likelihood Estimation (MLE) method, Lallemant et al. (2015). The evaluation of the parameters, 0 , has been conducted for the considered subset of schools, that is those located into the crater. It is worth to observe that, since no collapses have been recorded among the schools, the function 1 ( ; ) is assumed to be 1 ( ; ) = 0 . The first function, 0 ( ; 0 ) , is selected within the exponential family and the probability 0 of observing no damage ( = 0) varies with the intensity , according to a log-linear approximation, where only one parameter, 0 = [ Θ 01 ] , is required to define the function 0 ( ; 0 ) = −Θ 01 . A truncated Normal Distribution Function ( ) is assumed for the function | ∗ ( | ; ) by considering two parameters for each value of . The expression of the conditional is: f D | I * (d|i; Θ d )= n ( d;i,Θ d ) N ( 1;i,Θ d )- N ( 0;i,Θ d ) (4) where ( ; , ) is the normal distribution, and ( ; , ) is the CDF of the normal distribution. In this case the mean of the distribution varies linearly with the intensity, ′ � ; � = Θ 0 + Θ 1 , while the variance is constant, ( ; ) = Θ 0 , and the parameters are collected into the vector = � , � . Table 1 reports the values of the parameters achieved through the estimation. Table 1. Values of the parameters for the schools of the seismic crater. Damage model parameters – schools of the seismic crater Θ 0 μ ( i ; Θ μ d ) σ ( i ; Θ σ d ) MLE Θ 0 1 Θ μ d0 Θ μ d1 Θ σ d0 12.269 -0.972 4.625 0.223 -342.1 Fig. 5a), shows the mean trend of the damage derived by the model, when accounting also for the undamaged schools, obtained by the distribution | ( | ) for different values of the intensity i spanning from 0.1 g to 0.6 g. Fig. 5b), instead, reports the conditional | ( | ) of the damage for five specific intensities i ( i =0.1 g, 0.2 g, 0.3 g, 0.4 g, 0.5 g). Regarding Fig. 5a), the data are assessed only in the range of intensity i =0.02 g-0.29 g and the mean