PSI - Issue 44

Romina Sisti et al. / Procedia Structural Integrity 44 (2023) 1380–1387 Romina Sisti et al. / Structural Integrity Procedia 00 (2022) 000–000

1384

5

& | ) = - '% (* 0 +,/.) .

(

(2)

where Φ represents the standard normal cumulative distribution function (CDF), Ɵ is the median, i.e., the PGA value corresponding to 50% probability of attaining D i , and β is the standard deviation of ln (PGA), respectively. The values of Ɵ and β are obtained by fitting the lognormal distribution given by Eq. (2) on the empirical distribution of damage data through the maximization of the likelihood function expressed as: ℎ = ∏ - 1 1 . 213" 9 '% ( %& ( ' 0 ) : 4 ) ;1 − 9 '% ( %& ( ' 0 ) :> % ) 54 ) (3) where z j is the number of the cases with damage exceeding a certain level D i , n j is the total cases belonging to the j- th range of PGA and considering the m independent observations of surveys with the definition of D i . Finally, for an immediate estimation of the losses associated with a certain event, the damage levels previously defined (from D1 to D5) are summarized in three main conditions of damage related to usability outcomes, such as: a) Safe → Slight Damage, (D1); b) Unsafe →Moderate Damage, (D2-D3); c) Collapse → Severe Damage, (D4-D5). D0 D1 D2 D3 D4 D5

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

7 7 4

6 5

9 6

7

8

5

6

4

9

6

10 10

8

10

19%

11

13%

20%

21%

10

24%

16%

16%

14%

17%

17%

14%

36%

12

23%

22%

20%

21%

12

19%

21%

18%

21%

22%

21%

17%

26%

41%

9

28%

20%

15%

41%

15%

47%

16

22%

47%

30%

23%

24%

24%

40%

43%

43%

9

17%

15 4 11 13

38%

39%

34%

32%

12

60%

23%

49%

21%

21%

25%

42%

35%

33%

6 9

31%

28%

24%

23%

22%

21%

14

14

13

14%

10

14%

13%

PGA≤0.1g(829) 0.1g

PGA≤0.1g(1700) 0.1g

PGA≤0.1g(1718) 0.1g

PGA≤0.1g(1517) 0.1g

0.5g

0.5g

0.2g

0.3g

0.5g

0.5g

0.2g

0.3g

b) Principal nave 0.2g

0.3g

0.2g

0.3g

a) Façade

c) Apse

d) Bell tower

Fig. 3. DPMs for the most recurring macro-elements: a) façade; b) principal nave; c) apse; d) bell tower. The number of churches falling in each range of PGA is indicated in round brackets. 4. Comparison of fragility curves: global, macro-elements and single mechanisms of churches The comparisons between the fragility curves related to the global damage index (continuous lines) and those related to the damage index of the single macro-element (dashed lines) are shown in Fig. 4. The sample size varies according to the number of churches where the macro-element analyzed is present. The smallest differences between the two types of curves are observed for the façade (Fig. 4a) and the principal nave (Fig. 4b) macro-elements. In fact, in comparison with the global behavior, both façade and principal nave show a lower vulnerability for ‘slight damage’ and almost overlapping curves for ‘moderate damage’. For ‘severe damage’, the vulnerability of the principal nave is still lower than the global one, while the vulnerability of the façade becomes slightly higher. More significant differences with the global damage can be observed for the damage in the apse and bell tower macro-elements. In particular, the bell tower shows a very large reduction in vulnerability for ‘slight damage’, a comparable behavior for ‘moderate damage’ and a significant increase for ‘severe damage’. The apse shows a vulnerability significantly lower than the global one for ‘slight’ and ‘moderate’ damage, while it becomes slightly higher in case of ‘severe damage’.